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a r X i v :h e p -t h /9806171v 1 19 J u n 1998UCB-PTH-98/35,LBNL-41948,NSF-ITP-98-070hep-th/9806171Glueballs and Their Kaluza-Klein CousinsHirosi Ooguri,Harlan Robins and Jonathan Tannenhauser Department of Physics,University of California at Berkeley,Berkeley,CA 94720Theoretical Physics Group,Mail Stop 50A-5101,Lawrence Berkeley National Laboratory,Berkeley,CA 94720Institute for Theoretical Physics,University of California,Santa Barbara,CA 93106Abstract Spectra of glueball masses in non-supersymmetric Yang-Mills theory in three and four dimensions have recently been computed using the conjectured duality between superstring theory and large N gauge theory.The Kaluza-Klein states of supergravity do not correspond to any states in the Yang-Mills theory and therefore should decouple in the continuum limit.On the otherhand,in the supergravity limit g 2Y M N →∞,we find that the masses of theKaluza-Klein states are comparable to those of the glueballs.We also showthat the leading (g 2Y M N )−1corrections do not make these states heavier than the glueballs.Therefore,the decoupling of the Kaluza-Klein states is not evident to this order.1IntroductionSpectra of glueball masses in non-supersymmetric Yang-Mills theory in three and four dimensions have recently been calculated[1]using the conjectured duality between string theory and large N gauge theory[2–5].The results are apparently in good numerical agreement with available lattice gauge theory data,although a direct comparison may be somewhat subtle,since the supergravity computation is expected to be valid for large ultraviolet couplingλ=g2Y M N,whereas we expect that QCD in the continuum limit is realized forλ→0[5,6].As explained in[6,1],the supergravity computation atλ≫1 gives the glueball masses in units of thefixed ultraviolet cutoffΛUV.Forfiniteλ,the glueball mass M is expected to be a function of the formM2=F(λ)Λ2UV.(1.1)In the continuum limitΛUV→∞,M should remainfinite and of orderΛQCD.This would require F(λ)→0asλ→0.In[1],the leading string theory corrections to the masses were computed and shown to be negative and of orderλ−3/2,in accordance with expectation.Witten has proposed[5]that three-dimensional pure QCD is dual to type IIB string theory on the product of an AdS5black hole and S5.This proposal requires that certain states in string theory decouple in the continuum limitλ→0.One class of such states are Kaluza-Klein excitations on S5.The supergravityfields on the AdS5black hole×S5 can be classified by decomposing them into spherical harmonics(the Kaluza-Klein modes) on S5[7,8].They fall into irreducible representations of the isometry group SO(6)of S5, which is the R-symmetry of the four-dimensional N=4supersymmetric gauge theory from which QCD3is obtained by compactification on a circle.Consequently,only SO(6) singlet states should correspond to physical states in QCD3in the continuum limit.These are the glueball states studied in[1].However,wefind that,in the supergravity limit, masses of the SO(6)non-singlet states are of the same order as the SO(6)singlet states. Since these states should decouple in the limitλ→0,it was speculated in[1]that the string theory corrections should make the non-singlet states heavier than the singlet states.The purpose of this paper is to test this idea.We compute the masses of the SO(6)non-singlet states coming from the Kaluza-Klein excitations of the dilaton in ten dimensions. Wefind the masses in the supergravity limit to be of the same order as those of the SO(6) singlet states.We then calculate the leading string theory corrections to the masses.We find that the leading corrections do not make the Kaluza-Klein states heavier than theglueballs.Therefore,the decoupling of the Kaluza-Klein states is not evident to this order.This suggests that the quantitative agreement between the glueball masses from supergravity and the lattice gauge theory data should be taken with a grain of salt.2The Supergravity LimitWe calculate the masses of the Kaluza-Klein states following the analysis of[1].Ac-cording to[5],QCD3is dual to type IIB superstring theory on the AdS5black hole×S5 geometry given bydx24πg2Y M N =dρ2ρ2 +ρ2−b4dρ (ρ4−1)ρd f0this shooting method can be used to compute k 20and the wavefunction f 0(ρ)to arbitrarilyhigh precision.The results of the numerical work are listed in Table 1.As expected,the masses are all of the order of the ultraviolet cutoffΛUV =b .l135711.5929.2654.9388.6034.5363.60100.6145.668.98109.5157.9214.3l 2sρ2−b 4ρ2 dτ2+ρ23 i =1dx 2i +d Ω25,(3.1)where δ1=+15γ 5b 4ρ8−19b 12ρ4+5b 8ρ12 ,(3.2)and γ=18γ b 42ρ8+b 1216πG 10 d 10x √2g µν∂µΦ∂νΦ+γe −3periodicity2πR ofτis also modified toR= 1−152b.(3.5) It is the inverse radius R−1that serves as the ultraviolet cutoffof QCD3.To solve the dilaton wave equation in theα′-corrected geometry(3.1),we writeΦ=Φ0+f(ρ)e ikx Y l(Ω5),(3.6) whereΦ0is the dilaton background given by(3.3),and expand f(ρ)and k2inγasf(ρ)=f0(ρ)+γh(ρ),k2=k20+γδk2.(3.7) Here f0(ρ)obeys the lowest order equation(2.3)and is a numerically given function,and k20is likewise determined from(2.3).The second-order differential equation obtained from the action(3.4)in the background metric(3.1)and dilatonfield(3.3)is,in units in which b=1,ρ−1ddρ −(k20+l(l+4)ρ2)h==(75−240ρ−8+165ρ−12)d2f0dρ+(δk2−120(k20+l(l+4)ρ2)ρ−12−405ρ−14)f0(ρ).(3.8) With f0(ρ)and k20given,one may regard this as an inhomogeneous version of the equation (2.3).We solve this equation for h(ρ)andδk2.We are now ready to present our results.Let us denote the lowest mass of the l-th Kaluza-Klein state by M l.In units of the ultraviolet cutoffΛUV=(2R)−1,with R given by(3.5),wefindM20=11.59×(1−2.78ζ(3)α′3+···)Λ2UVM21=19.43×(1−2.66ζ(3)α′3+···)Λ2UVM22=29.26×(1−2.62ζ(3)α′3+···)Λ2UVM23=41.10×(1−2.61ζ(3)α′3+···)Λ2UVM24=54.93×(1−2.63ζ(3)α′3+···)Λ2UVM25=70.76×(1−2.66ζ(3)α′3+···)Λ2UVM26=88.60×(1−2.69ζ(3)α′3+···)Λ2UVM27=108.4×(1−2.72ζ(3)α′3+···)Λ2UV.(3.9)Similar behavior is observed for the excited levels of each Kaluza-Klein state.Thus the corrections do not make the Kaluza-Klein states heavier than the glueballs, and the decoupling of the Kaluza-Klein states is not evident to this order.According to Maldacena’s duality,theλ−1/2expansion of the gauge theory corresponds to theα′-expansion of the two-dimensional sigma model with the AdS5black hole×S5as its target space.It is possible that the decoupling of the Kaluza-Klein states takes place only non-perturbatively in the sigma model.AcknowledgmentsWe thank Csaba Cs´a ki,Aki Hashimoto,Yaron Oz,John Terning,and especially David Gross for useful discussions.We thank the Institute for Theoretical Physics at Santa Barbara for its hospitality.This work was supported in part by the NSF grant PHY-95-14797and the DOE grant DE-AC03-76SF00098,and in part by the NSF grant PHY-94-07194through ITP.H.R. and J.T.gratefully acknowledge the support of the A.Carl Helmholz Fellowship in the Department of Physics at the University of California,Berkeley.Appendix:The Boundary Condition at the HorizonIn this appendix,we show that the boundary condition at the horizonρ=b used in the shooting method[1]is consistent,and that the eigenvalue k2and the wavefunction f(ρ)can be evaluated to an arbitrarily high precision using this method.In the neighborhood ofρ=b,the dilaton wave equation takes the form∂ρ(ρ−b)∂ρf(ρ)+···=0.(3.10) Its general solution is of the formf(ρ)=c1[1+α(ρ−b)+···]+c2[log(ρ−b)+···](3.11)with arbitrary coefficients c1,2(the constantαis determined by the wave equation and is in general non-zero).The regularity of the dilatonfield requires c2=0.In the shooting method,we numerically integrate the differential equation starting from a sufficiently large value ofρdown to the horizon.For generic k2,the function thus obtained,when expanded as in(3.11),would have c2=0.The task is to adjust k2so that c2=0.Since f(ρ)is divergent atρ=b for generic k2,it is numerically difficult to impose the boundary condition directly atρ=b.Instead,in[1]and in this paper,we required f′=0 atρ=b+ǫfor a given smallǫ(for example,ǫ=0.0000001b in this paper).By(3.11), this condition impliesc2=−c1αǫ+···.(3.12)Therefore,c2can be made arbitrarily small by adjustingǫ.This justifies the numerical method used in[1]and in this paper.We thank Aki Hashimoto for discussions on the numerical method.References[1]C.Cs´a ki,H.Ooguri,Y.Oz and J.Terning,“Glueball Mass Spectrum from Supergravity,”hep-th/9806021.[2]J.M.Maldacena,“The Large N Limit of Superconformal Field Theories and Supergravity,”hep-th/9711200.[3]S.S.Gubser,I.R.Klebanov and A.M.Polyakov,“Gauge Theory Correlators from Non-Critical String Theory,”hep-th/9802109.[4]E.Witten,“Anti-de Sitter Space and Holography,”hep-th/9802150.[5]E.Witten,“Anti-de Sitter Space,Thermal Phase Transition,And Confinement in GaugeTheories,”hep-th/9803131.[6]D.J.Gross and H.Ooguri,“Aspects of Large N Gauge Theory Dynamics as Seen by StringTheory,”hep-th/9805129.[7]H.J.Kim,L.J.Romans and P.Van Nieuwenhuizen,“Mass Spectrum of Chiral Ten-Dimensional N=2Supergravity on S5,”Phys.Rev.D32(1985)389.[8]M.G¨u naydin and N.Marcus,“The Spectrum of The S5Compactification of Chiral N=2,D=10Supergravity and The Unitary Supermultiplets of U(2,2/4),”Class.Quant.Grav.2(1985)L11.[9]S.S.Gubser,I.R.Klebanov and A.A.Tseytlin,“Coupling Constant Dependence in theThermodynamics of N=4Supersymmetric Yang-Mills Theory,”hep-th/9805156.[10]M.T.Grisaru,A.E.M.van de Ven and D.Zanon,“Four-Loop Beta Function for theN=1and N=2Supersymmetric Nonlinear Sigma-Model in Two Dimensions,”Phys.Lett.B173(1986)423;M.T.Grisaru and D.Zanon,“Sigma-Model Superstring Corrections to the Einstein-Hilbert Action,”Phys.Lett.B177(1986)347.[11] D.J.Gross and E.Witten,“Superstring Modifications of Einstein Equation,”Nucl.Phys.B277(1986)1.。
a rXiv:solv-int/9593v312Se p1995Proofs of Two Conjectures Related to the Thermodynamic Bethe Ansatz Craig A.Tracy Department of Mathematics and Institute of Theoretical Dynamics University of California,Davis,CA 95616,USA E-mail address:tracy@ Harold Widom Department of Mathematics University of California,Santa Cruz,CA 95064,USA E-mail address:widom@ Abstract We prove that the solution to a pair of nonlinear integral equations arising in the thermodynamic Bethe Ansatz can be expressed in terms of the resolvent kernel of the linear integral operator with kernel e −(u (θ)+u (θ′))2.I.Introduction Thermodynamic Bethe Ansatz techniques were introduced in the pioneering analysis of Yang and Yang [11]of the thermodyamics of a nonrelativistic,one-dimensional Bose gas with delta function ter this method was extended to a relativistic system with a factorizable S -matrix to give an exact expression for the ground state energy of this systemon a cylindrical space of circumference R [5,12].This was done by relating the ground state energy to the free energy of the same system on an infinite line at temperature T =1/R .In all cases one expresses the various quantities of interest in terms of “excitation energies”εa (θ)which are solutions of nonlinear integral equations of the formεa (θ)=u a (θ)− b φab (θ−θ′)log1+z b e −εb (θ′) dθ′matter.The methods used are either numerical or perturbative and there are,as far as the authors are aware,no known explicit solutions to the TBA equations.It thus came as a surprisewhen Cecottiet al.[2](see also [3]),in their analysis of certain N =2supersymmetric theories [1],discovered that a certain quantity (the “supersymmetric index”),expressible in terms of the solution of the pair of “TBA-like”integral equationsε(θ)=t cosh θ−1cosh(θ−θ′)dθ′η(θ)=2λ ∞−∞e −ε(θ′)2(cosh θ+cosh θ′)2.Zamolodchikov [13]then conjectured that the system of nonlinear equations could actually be solved in terms of this resolvent kernel.More precisely,if we denote the operator by K ,the kernel of the operator K (I −λ2K 2)−1by R +(θ,θ′),and set R +(θ):=R +(θ,θ)then the system should be satisfied ife −ε(θ)=R +(θ)and ηis defined by the second equation.In fact he conjectured that this should hold for operators with kernels of the more general forme −(u (θ)+u (θ′))2(1.1)if the first equation is replaced byε(θ)=2u (θ)−1cosh(θ−θ′)dθ′.In addition he conjectured that,with the same function η,R −(θ)=1cosh 2(θ−θ′)dθ′,where R −(θ,θ′)is the kernel of K 2(I −λ2K 2)−1and R −(θ):=R −(θ,θ).(We state everything in terms of kernels here;in the cited work the functions R ±were given by infinite series.2That they are the same follows from the Neumann series representation for the kernel of (I−λ2K2)−1.)We prove these conjectures here.One of the main ingredients is the fact that the equa-tions are in a sense equivalent to relations among the analytic continuations of the functions R±(θ)into a strip.(See formula(6.8)of[13].)Another is a particularly convenient represen-tations for these functions in terms of other functions,which we call Q(θ)and P(θ).(That these latter functions are fundamental is known from earlier work[4,8,9,10].)These rep-resentations are stated in Lemma1,and in Lemmas2and3we state precise versions of the equivalence alluded to before.In Lemmas4and5we derive general properties of functions in the range of the operator K in order to derive,as Lemma6,some basic properties of the functions Q(θ)and P(θ).If we try to prove the desired relation among the analytic continuations of R±(θ)wefind that we have to prove a certain crucial identity involving Q(θ)and P(θ)which is by no means obvious.But once conjectured it is not hard to prove,given the previous preparatory work, and that is stated as the Proposition which follows the lemmas.We show that a certain combination of these functions,which is clearly analytic in the strip,extends by periodicity to an entire bining this fact with the use of Liouville’s theorem,we deduce the identity.It should be mentioned that the main part of the argument may only be used if u belongs to a restricted class,but the result for general u follows by an approximation argument.This will be presented in the Appendix,as will proofs of some of the lemmas and some facts about Fourier transforms we shall use.II.PreliminariesWe shall assume throughout that u is continuous and bounded from below and that0<λ<e2min u/2π.(2.1) This assures that the series defining R±(θ)converge uniformly and that the operatorλK, acting on any of the usual function spaces,has norm less than1,so that I−λ2K2is invertible. (This follows from(4.5)below.)Since the parameterλmay be incorporated into u we may assume that in factλ=1.If we set√E(θ):=.(2.3)eθ+eθ′Our functions Q and P are defined byQ:=(I−K2)−1E,P:=(I−K2)−1KE.3Lemma1.We have the representationsR+(θ)=Q(θ)2−P(θ)2eθ.(2.4)Proof.We use the notations[A,B]:=AB−BA,{A,B}:=AB+BA and write X⊗Y for the operator with kernel X(θ)Y(θ′)and M for multiplication by eθ.Then we have immediately{M,K}=E⊗Efrom which it follows that also that[M,K2]=E⊗KE−KE⊗Eand then that[M,(I−K2)−1]=Q⊗P−P⊗Q.SinceK2(I−K2)−1=(I−K2)−1−I,(2.5) we deduce that the kernel of this operator is given by the formulaR−(θ,θ′)=Q(θ)P(θ′)−P(θ)Q(θ′)eθ+eθ′,and thefirst part of(2.4)follows.4Recall that a function f defined on R is said to belong to the Wiener space W if itsFourier transformˆf belongs to L1.Such a function is necessarily continuous and vanishes at ±∞.A sufficient condition that f∈W is that f and f′belong to L2.(See the Appendix.) We use the notation S a to denote the strip|ℑθ|<a in the complexθ-plane,and A(S a)to denote those functions g which are bounded and analytic in the strip,continuous on its closure,and for which g(θ+iy)→0asθ→+∞through real values when y∈R isfixed and satisfies|y|<a.The proofs of the next three lemmas will be found in the Appendix.Lemma2.Assume f∈W.Ifg(θ)=1cosh(θ−θ′)dθ′(2.6)then g∈A(Sπ/2)and its boundary functions satisfyg(θ+iπ/2)+g(θ−iπ/2)=f(θ)(2.7) for realθ.Conversely,if g∈A(Sπ/2)and if(2.7)holds then so does(2.6).Lemma3.Assume f,f′∈W.Ifg(θ)=1cosh2(θ−θ′)dθ′then g∈A(Sπ/2)and its boundary functions satisfyg(θ+iπ/2)−g(θ−iπ/2)=if′(θ)for realθ.Conversely,if g∈A(Sπ/2)and the second relation holds then so does thefirst. Lemma4.Assume f∈L1and set h(x)equal to f(log x)/x for x>0and equal to0for x≤0.If h∈W andg(θ)= ∞−∞f(θ′)this that if f∈L2then Kf and its derivative are exponentially small at±∞.In particular,any function in the range of K satisfies the hypothesis of the next lemma.Lemma5.If f is a bounded function on R with bounded derivative then Kf(θ)/E(θ) extends to a function in A(Sπ).The boundary functions of Kf satisfyKf(θ+iπ)+Kf(θ−iπ)=4πv(θ)f(θ)for realθ,wherev(θ):=e−(u(θ)+u(θ+iπ)).Proof.If we look at the the expression(2.3)for the kernel of K we see that Kf(θ)/E(θ) is of the form of the function g of Lemma4if f(θ)there is replaced by our E(θ)f(θ).It iseasy to see that if our f satisfies the stated conditions then the function h in the statement of Lemma4belongs to L2and has an L2derivative,so h∈W and the conclusion ofthe Lemma holds.Thus Kf/E∈A(Sπ).For the boundary function identity we use theexpression(1.1)for the kernel of K.If we make the substitutionsθ→2θ,θ′→2θ′we see that e u(2θ)(Kf)(2θ)=2eθ(Kf)(2θ)/E(2θ)is exactly of the form of the function g in thestatement of Lemma2if the function f(θ)there is replaced by our present4πe−u(2θ)f(2θ). Applying the identity stated there and using the periodicity of u give the identity stated here.We apply this to the functions Q(θ)and P(θ).Lemma6.The functionsQ(θ)(2.9)E(θ)belong to A(Sπ)and the boundary functions of Q and P satisfy the identities Q(θ+iπ)+Q(θ−iπ)=4πv(θ)P(θ),P(θ+iπ)+P(θ−iπ)=4πv(θ)Q(θ)(2.10) for realθ.Proof.Because of the relations Q=E+K2(I−K2)−1E,P=KQ all statements of thelemma except for thefirst part of(2.10)follow from Lemma5and the remark preceding it. Since Q−E=KP Lemma5gives the identity(Q−E)(θ+iπ)+(Q−E)(θ−iπ)=4πv(θ)P(θ).But it follows from the definition(2.2)of E and the fact that u has period2πi thatE(θ+iπ)+E(θ−iπ)=0.Thus we obtain the desired identity for Q.Using Lemma6and the fact that v is an entire function we can conclude that Q and P have analytic continuations to entire functions ofθ;each use of the pair of identities allows us6to widen byπthe strip of analyticity.Here is the crucial identity relating these continuations from which our results will follow.Proposition.We haveQ(θ+iπ/2)Q(θ−iπ/2)−P(θ+iπ/2)P(θ−iπ/2)=E(θ+iπ/2)E(θ−iπ/2).Proof.SetS(θ):=Q(θ+iπ/2)Q(θ−iπ/2)−P(θ+iπ/2)P(θ−iπ/2).ThenS(θ+iπ/2)=Q(θ+iπ)Q(θ)−P(θ+iπ)P(θ),S(θ−iπ/2)=Q(θ−iπ)Q(θ)−P(θ−iπ)P(θ),and so by(2.10),S(θ+iπ/2)+S(θ−iπ/2)=4πv(θ)[P(θ)Q(θ)−Q(θ)P(θ)]=0.It follows that S(θ)extends to an entire function of period2πi whose values atθ±iπ/2are negatives of each other.Therefore1−S(θ)/E(θ+iπ/2)E(θ−iπ/2)extends to an entire function of periodπi whose values atθ±iπ/2are equal.(We used again the fact that E(θ−iπ)=−E(θ+iπ).)To show that this is0(this is equivalent to the claimed identity)it suffices,by Liouville’s theorem,to show that it is bounded and that it tends to0asθ→+∞through real values.For aπi-periodic function it suffices to show that these properties hold in the strip Sπ/2.They do hold there because forθin this stripθ±iπ/2lie in the strip Sπ, for which we have the conclusions of Lemma6.III.Proof of the ConjecturesWe have to show that ifε:=−log R+and ifηis defined byη(θ)=2 ∞−∞e−ε(θ′)2π ∞−∞log(1+η2(θ′))πR+(θ) ∞−∞arctanη(θ′)We shall assume that we have a functionηsatisfying(3.1)–(3.3),and formally apply the first parts of Lemmas2and3to obtain the three identities(3.4)–(3.6)below.From these we shall see thatηmust have a certain representation in terms of Q and P.Then,using the identities of the Proposition,we shall show that(3.4)–(3.6)in fact hold ifηis defined this way.Finally,using the second parts of Lemmas2and3,we show that(3.1)–(3.3)hold.(Of course thefirst step is unnecessary for the proof,but it provides motivation for the eventual definition ofη.)Applying Lemmas2and3to(3.1)–(3.3)give4πR+(θ)=η(θ+iπ/2)+η(θ−iπ/2),(3.4) log(1+η2(θ))=2u(θ+iπ/2)−ε(θ+iπ/2)+2u(θ−iπ/2)−ε(θ−iπ/2),(3.5)2i η′(θ)R+(θ+iπ/2)−R−(θ−iπ/2)E(θ+iπ/2)2E(θ−iπ/2)2.(3.8)Lemma1shows that(3.6)may be writteni η′(θ)Q2−P2(θ+iπ/2)−Q′P−P′Q1+η2(θ)=Q′Q−P′PQ2−P2(θ−iπ/2)−E′E(θ−iπ/2).We have a choice now of either adding or subtracting the last two displayed formulas. Choosing the former,we obtainη′(θ)Q−P (θ+iπ/2)+Q′+P′E(θ+iπ/2)−E′η(θ)−i =ddθlog(Q+P)(θ−iπ/2)−dIntegrating and exponentiating gives the desired formula(Q−P)(θ+iπ/2)·(Q+P)(θ−iπ/2)η(θ)−i=−i.(3.10)E(θ+iπ/2)E(θ−iπ/2)We shall now show that ifηis defined by(3.9)then(3.10)also holds,as do relations (3.4)–(3.6).First,the statement that the right side of(3.10)minus the right side of(3.9)is equal to 2i follows from the Proposition.Thus(3.9)and(3.10)are completely equivalent.Second,taking the product of(3.9)and(3.10)gives(3.8)and hence(3.5).Third,reversing the argument that showed(3.8)and(3.6)imply(3.9)we see that(3.9) and(3.8),which we now have,imply(3.6).Finally,to obtain(3.4)we use(3.10)to expressη(θ+iπ/2)and(3.9)to express η(θ−iπ/2)andfind that their sum equals(Q−P)(θ)[(Q+P)(θ+iπ)+(Q+P)(θ−iπ)]i[Q(θ)2−P(θ)2]E(θ)E(θ+iπ)and by(2.4)and the definitions of E(θ)and v(θ)this equals4πR+(θ).So(3.4)–(3.6)are established.Now we show that they imply(3.1)–(3.3).By Lemmas2 and3this will be true if the functionsR+,log(1+η2),arctanη,η′/(1+η2)(3.11) belong to W and the functionsη,2u−ε,R−/R+(3.12) belong to A(Sπ/2).Our assumption has been thatλ=1satisfies inequality(2.1).This identity is still satisfied if u is increased or,equivalently,if E is decreased.It follows that we could,in our representation(2.3)of the kernel of K,have replaced E byδE for anyδ∈[0,1].It is clear that all quantites in(3.1)–(3.3)would then be real-analytic functions ofδforδ∈[0,1].So9if the relations hold for sufficiently smallδthey hold for all,includingδ=1.If we retrace the steps leading to bounds for the functions(2.9)wefind that they tend to0asδ→0. (In fact,they are O(δ2).)Hence we may assume that the bounds for these functions are as small as we like.In fact we assume that|Q(θ)4and|P(θ)4(θ∈Sπ).(3.13) We take in succession the items we have to verify.Recall that a function belongs to W if it and its derivative belong to L2.This is what we shall show for the functions(3.11).First, from(2.2)and(2.4)we see haveR+(θ)=Q(θ)2−P(θ)2E (θ+iπ/2)−Kfe2θ+e2θ′dθ′=O(sechθ)if,say,f∈L∞.This holds for|ℑθ|strictly less thanπ/4.Applying this to f=Q and f=P we deduce thatPE (θ−iπ/2)=O(sechθ),QE(θ−iπ/2)=O(sechθ)as long as|ℑθ|<π/8,for example.Now adding(3.9)and(3.10)gives the representation η(θ)=iQ(θ−iπ/2)P(θ+iπ/2)−Q(θ+iπ/2)P(θ−iπ/2)E(θ+iπ/2)+O(sechθ) P E(θ+iπ/2) PNow we show that the functions(3.11)all belong to A(Sπ/2).First,by(2.4),R−Q2−P2=2Q′P−P′Q1−(P/Q)2.By(3.13)P/Q is bounded in the larger strip Sπby1/3and so it follows that the last factor above is bounded.It also follows that P/Q∈A(Sπ)from which it follows that (P/Q)′∈A(Sπ/2).Hence R−/R+∈A(Sπ/2).Next,we use(2.4)again to write2u−ε=2u+log Q2−P2E2.By(3.13)we have|1−(Q±P)/E|<1/2in Sπ/2and we deduce as above that2u−ε∈A(Sπ/2).Finally since1−Q/E and P/E belong to A(Sπ),the functions1−Q(θ±iπ/2)E(θ±iπ/2)belong to A(Sπ/2).It follows from this and the representation(3.14)thatη∈A(Sπ/2).Thus we have proved the conjectures in the case where u(θ)has the special form(2.8). For a general u,more precisely for any u which is continuous and bounded below,we canfind a sequence of u n of the special type such that e−u n converge boundedly and locally uniformly to e−u.(This will be demonstrated in the Appendix.)This is enough to deduce the result for u from the results for the u n.IV.AppendixWe give details here of certain matters postponed from the previous sections.First we recall some facts about the Fourier transform,which we denote,as usual,by a circumflex:ˆf(ξ)= ∞−∞e−iξθf(θ)dθ.Ifˆf∈L1,in other words,if f∈W,we have the Fourier inversion formulaf(θ)=12π||ˆf||1.(4.1)Parseval’s identity reads||ˆf||2=12π||f||2,11and we have the general formula f′(ξ)=iξˆf(ξ).Proof that f,f′∈L2implies f∈W.It suffices to show that if we writeˆf(ξ)as [ˆf(ξ)(ξ+i)](ξ+i)−1then both factors on the right belong to L2.The second factor surely does,and the square of the absolute value of thefirst factor equals|ˆf(ξ)|2(ξ2+1)=| f′(ξ)|2+|ˆf(ξ)|2.The right side belongs to L1by Parseval’s identity and the assumption f,f′∈L2.Proof of Lemma2.It is clear that g,when defined by(2.6),is analytic in the strip.If we write g y(θ):=g(θ+iy)for realθtheng y(ξ)=ˆf(ξ)e−yξ2sinhπξ/2=ˆf(ξ)(ξ+i)ξe−yξand so the difference of the limiting values of g has Fourier transform−ˆf(ξ)ξ= if′.This establishes thefirst part of the lemma and the second follows just as before.Proof of Lemma4.Forfixedθ′the factor(eθ+eθ′)−1in the integral defining g(θ)is analytic inθ.Forθin any subset of Sπof the formℜθ≥θ0,|ℑθ|≤π−δ(δ>0)this factor is bounded uniformly inθ′and tends to0pointwise asℜθ→+∞.Since f∈L1 this is enough to conclude that g is analytic in Sπand tends to0in this strip asℜθ→+∞andℑθisfixed.It remains to prove boundedness of g and continuity near the boundary of Sπ,and for this we use the function h.In the lower part of the strip,−π<y<0,we set z=−eθ+iy withθreal,so thatℑz>0,and we can writeg(θ+iy)= ∞0h(x)x−z dx.Using the Fourier inversion formula and interchanging the order of integration,wefind the representationg(θ+iy)=i ∞0e izξˆh(ξ)dξ.Sinceˆh∈L1the integral is bounded uniformly forℑz>0and we deduce as in the Lemmas 2and3that it extends continuously toℑz=0.Thus g(θ+iy)is bounded for0≤y<πand extends continuously to y=π.A similar argument holds for the upper half of Sπ,and so g∈A(Sπ).Extension to general u.We begin with an approximation fact,reminiscent of the Weier-strass approximation theorem.Recall the notation C0(R)for the space of continuous func-tions f on R satisfying f(±∞)=0,which is a Banach space under the norm||f||:=sup{|f(x)|:x∈R}.The fact is that for eachδ>0thefinite linear combinations of the functionssinh kθe−δsinh2θ(k=0,1,···)are dense in C0(R);in other words,for any f∈C0(R)and anyδ′>0there exists afinitelinear combinationp(θ)=Nk=0a k sinh kθsuch that|p(θ)e−δsinh2θ−f(θ)|<δ′(4.3)13for allθ.This is true because the change of variable t=sinhθconverts it to the statementthat thefinite linear combinations of the functions t k e−δt2are dense in C0(R).And this in turn is true because if it weren’t then the Hahn-Banach theorem and Riesz representationtheorem([7],Thms.5.19and6.19)would imply that there there is a function of boundedvariation(=signed measure)µon R,not identically zero,such that ∞−∞t k e−δt2dµ(t)=0for all k≥0.But then the entire function F(z):= ∞−∞e izt e−δt2dµ(t)would satisfy F(k)(0)=0 for k≥0and so F≡0.This in turn implies e−δt2dµ(t)≡0and soµ(t)≡0,a contradiction.Here is how to construct the sequance u n described at the end of Section III.We may clearly assume that u is uniformly positive,i.e.that for someα>0we have u(θ)≥αfor allθ.Let n be given and definew:=min(u,n).Thenfind p(θ),a linear combination of the powers of sinhθ,such that(4.3)holds withf(θ)=w(θ)|<n−1e n−1sinh2θ.It is an easy exercise to deduce from this that for sufficiently large n|p(θ)2−u(θ)|<6n−1/2if u(θ)<n and sinh2θ<n.(We use here the facts that u is uniformly positive and that e<3.)The function p(θ)2is our u n(θ).We now deduce the identities(3.1)–(3.3)for general u.Denote by R n±the R±functionsassociated with the functions u n.If we can show that R n±(θ)→R±(θ)boundedly and pointwise then(3.1)–(3.3)for u will follow from the corresponding identities for the u n,since by the dominated convergence theorem we could take the limits as n→∞under the integral signs.The function R+(θ)is given by the series∞λ2m ∞−∞··· ∞−∞K(θ,θ1)···K(θ2m,θ)dθ1···dθ2m(4.4) m=0where K(θ,θ′)is given by(1.1).It follows from the fact∞dθ−∞u n≥0fo all n,and thatλ<1/2π.Denote by K n(θ,θ′)the kernel corresponding to u n so that R n+(θ)is given by the series∞λ2m ∞−∞··· ∞−∞K n(θ,θ1)···K n(θ2m,θ)dθ1···dθ2m.(4.6) m=0It follows from(4.5)and the inequality e−u n≤1that the integral in the m th term of(4.6) is at most(2π)2m for all n and so,sinceλ<1/2π,the series converges uniformly in n. Thus we may take the limit as n→∞under the summation sign.Next,each integrand K n(θ,θ1)···K n(θ2m,θ)is uniformly bounded by K(θ,θ1)···K(θ2m,θ),which hasfinite integral over R2m,and so we may take each limit as n→∞under the integral sign(again by the dominated convergence theorem).The result is the series(4.4),and this giveslimR n+(θ)=R+(θ).n→∞Since0<R n+(θ)≤R+(θ)we have established that R n+(θ)→R+(θ)boundedly and pointwise.A similar argument applies to R n−(θ),and the proof is complete.AcknowledgementsThe authors wish to thank Paul Fendley for elucidating the conjectural status of the identities we prove here.This work was supported in part by the National Science Foundation through grants DMS–9303413and DMS–9424292.15References[1]Cecotti,S.,Vafa,C.:Topological–anti-topological fusion,Nucl.Phys.B367,359–461(1991);Ising model and N=2supersymmetric theories,Commun.Math.Phys.157, 139–178(1993)[2]Cecotti,S.,Fendley,P.,Intriligator,K.,Vafa,C.:A new supersymmetric index,Nucl.Phys.B386,405–452(1992)[3]Fendley,P.,Saleur,H.:N=2supersymmetry,Painlev´e III and exact scaling functionsin2D polymers,Nucl.Phys.B388,609–626(1992)[4]Its,A.R.,Izergin,A.G.,Korepin,V.E.,Slavnov,N.A.:Differential equations forquantum correlation functions,Int.J.Mod.Physics B4,1003–1037(1990)[5]Klassen,T.R.,Melzer,E.:The thermodynamics of purely elastic scattering theories andconformal perturbation theory,Nucl.Phys.B350,635–689(1991)[6]McCoy,B.M.,Tracy,C.A.,Wu,T.T.:Painlev´e functions of the third kind,J.Math.Phys.18,1058–1092(1977)[7]Rudin,W.:Real and Complex Analysis,3rd ed.New York:McGraw-Hill,1987[8]Tracy,C.A.,Widom,H.:Fredholm determinants,differential equations and matrixmun.Math.Phys.163,33–72(1994)[9]Tracy,C.A.,Widom,H.:Systems of partial differential equations for a class of operatordeterminants.Operator Theory:Adv.and Appls.78,381–388(1995)[10]Tracy,C.A.,Widom,H.:Fredholm determinants and the mKdV/sinh-Gordon hierar-chies,to appear in Commun.Math.Phys.,solv-int/9506006[11]Yang,C.N.,Yang,C.P.:Thermodynamics of a one-dimensional system of bosons withrepulsive delta-function interaction,J.Math.Phys.10,1115–1122(1969)[12]Zamolodchikov,Al.B.:Thermodynamic Bethe Ansatz in relativistic models:Scaling3-state Potts and Lee-Yang models,Nucl.Phys.B342,695–720(1990)[13]Zamolodchikov,Al.B.:Painlev´e III and2D polymers.Nucl.Phys.B432[FS],427–456(1994)16。
Integral Equation Formulation for ScatterDensity ProblemJ.H¨a m¨a l¨a inen,S.Savolainen,R.Wichman,K.Ruotsalainen and J.YlitaloAbstractIntegral equation formulation for the problem offinding a circularly symmetric scatter density(SD)around the mobile is deduced,and the SD is computed assuming that the distribution of angle-of-arrival(AoA)in base station is known.The corresponding integral equation is solved by using the spline collocation method.Thus,instead offitting a priori selected SD to estimated AoA distribution,we solve the SD based on AoA characteristics.Introduction:Recently,geometrical-based single bounce channel models have been proposed by different authors [1],[2],[3],where the distribution of the scatterers defines the model through the selected simple geometry.In[1], it was assumed that scatterers are uniformly spread over a disc with radius R and vanishes outside of this region. Similar analysis was carried out in[2]where the scatterers are uniformly distributed inside the ellipse with foci at Base Station(BS)and Mobile Station(MS).In[3],it was shown that angle-of-arrival(AoA)in BS,computed assuming Gaussian scatter density(SD),provides a betterfit to measurements of[4]when compared to model applying uniform distribution of scatterers.In[1],[2,[3],the form of SD isfirst selected,after that the distribution of AoA is computed,andfinally the result is compared to existing measurements.However,in order to maximise thefit between the AoA provided by the model and measurements the direction in the selection process concerning to SD should be reversed.We propose a novel approach where AoA in BS isfirst selected based on measurements and SD is then computed from the integral equation that defines the relation between SD and AoA seen by BS.The resulting SD can be used when building up a channel simulator modelling a certain environment.System Model and Notations:The notation used in the sequel is introduced in Fig.1,where r refers to the distance between MS and scatterer,d is the distance between scatterer and BS,D is the distance between MS and BS,andαandβdefine the departing(arriving)and arriving(departing)angles of the signal path(dashed lines)respectively.We assume that SD in polar coordinates(fα,r)depends only on r.Hence,AoA seen by MS is uniformly distributed. Furthermore,we assume that D≥R,i.e.,there are no local scatterers around the BS.This assumption is valid in macro-cell environments where BS antenna is placed well over the rooftops and angular spread(AS)is small.Due to the symmetry about the line joining MS and BS we can concentrate on the positive values ofβ.Integral Equation Formulation:We compute AoA in BS(fβ)by using the joint density function fβ,r that is obtained from fα,r after a standard change of variables.For that purpose we need the derivative ofβwith respect toα.From Fig.1β=arcsin(r/d·sinα),D+r cosα=d cosβ,(1) and the derivative ofβis computed from thefirst equation in(1),∂αβ=r(d2cosα+rD sin2α)d2d2−r2sin2α=d−D cosβd,(2)where the latter equality is obtained with the help of equations in(1).Furthermore,from Fig.1wefind thatd=D cosβ±r2−D2sin2β.(3)Here the sign changes when r=D sinβ.By(2)and(3)the joint distribution ofβand r attains the formfβ,r(β,r)=D cosβ±r2−D2sin2βr2−D2sin2βf r(r)2π,(4)where f r/2π=fα,r.The pdf of AoA in BS is obtained by integrating(4)over r.By taking into account the change of the sign in(4)we obtainfβ(β)=DπRD|sinβ|cosβf r(r)drr2−D2sin2β,|β|≤sin−1RD.(5)Numerical Solution of the Integral Equation:The numerical solution of(5)is obtained by using spline collocation method that is known to be fast and simple[5].For that purpose we define collocation pointsβn whereβ0=0,βn<βn+1andβN+1=sin−1(R/D).Furthermore,we define the mesh points r n by r n=D sinβn.In spline collocation method,f r in(5)is expressed in spline space,spanned by standard spline base functionsψn.Here we apply piecewise linear splines,i.e.ψn is a normalised hat function centred at r n and vanishing outside the interval (r n−1,r n+1).The coordinates of f r in spline space are denoted by u n.Since SD vanishes outside the circle with radius R,we have f r(r N+1)=0,and thusfβ(βn)=Nk=nu kπRr nD2−r2nr2−r2nψk(r)dr.(6)If fβis known,the coordinates u n can be solved from(6)in a recursive manner and they provide an approximation to values f r(r n).On the other hand,if f r is known,the point values of fβcan be computed directly from(6).We note that with piecewise linear splines,integral in(6)admits a closed-form expression in terms of elementary transcendental functions.The numerical accuracy of the method was studied in the case where the closed-form expression for fβwas deduced from(5)assuming that f r is Rayleigh,and the corresponding spline approximation˜f r was computed from(5)using the collocation method.Results show that the rms error between original f r and approximation˜f r decays proportionally to N−2.Absolute rms error was of the order of4·10−4when N=256and AS22=12◦.Fig.2depicts one-dimensional SDs for two example AoAs:Gaussian AoA[4]and the AoA following Student’s t-distribution[6].In both cases we take R=D=1and truncate distributions such that AoA vanishes if|β|>90◦and AS=12◦.Results show that the form of SD greatly depends on the selected distribution of AoA.The difference between SDs in Fig.2reflects the fact that tails of the Gaussian AoA decay rapidly while Student’s t-distributed AoA has a larger peak atβ=0.Furthermore,although the AS is the same for both AoAs,it is noticed that the expected scatterer distance(¯r)may vary depending on the AoA distribution.Finally,we note that relative scale in Fig.2can be easily replaced by absolute values having a physical meaning,Conclusions:Integral equation formulation was deduced tofind the SD provided that AoA in BS is known.The formulation provides effective means to compute SD numerically when AoA in BS is estimated from measurements. Resulting SD can then be used when building channel simulators for various types of environments.References[1]Petrus,P.,Reed J.H.,and Rappaport,T.S.:‘Geometrical-Based Statistical Macrocell Channel Model for Mobile Environments’,IEEE Trans.on Comm.,V ol.50,No.3,March2002.[2]Ertel,R.B.,and Reed,J.H.:‘Angle and Time Arrival Statistics for Circular and Elliptical Scattering Models’, IEEE J.on Selected Areas in Comm.,V ol.17,No.11,November1999.[3]Janaswamy,R.:‘Angle and Time of Arrival Statistics for the Gaussian Scatter Density Model’,IEEE Trans.on Wireless Comm.,V ol.1,No.3,July2002.[4]Pedersen,K.I.,Mogensen,P.M.,and B.H.Fleury,B.H.:‘A Stochastic Model of the Temporal and Azimuthal Dispersion Seen at the Base Station in Outdoor Propagation Environments’,IEEE Trans.Veh.Technol.,V ol.49,March2000.[5]Saranen,J.,and Vainikko,G.:‘Periodic Integral and Pseudodifferential Equations with Numerical Approximation’, Springer Verlag,Berlin,2002.[6]Andersen,J.B.,and Pedersen,K.I.:‘Angle-of-Arrival Statistics for Low Resolution Antennas’,IEEE Trans.on Antennas and Propagation,V ol.50,No.3,March2002.Author’s affiliations:J.H¨a m¨a l¨a inen and R.Wichman(Helsinki University of Technology,P.O.Box3000,FIN–02015HUT,Finland)S.Savolainen(Department of Mathematical Sciences,University of Oulu,FIN–90401Oulu,Finland)K.Ruotsalainen(Mathematics Division,University of Oulu,FIN–90401,Finland)J.Ylitalo(4G Lab/CWC,University of Oulu,P.O.Box4500,FIN–90014,Finland)Corresponding Author:R.Wichman,wichman@wooster.hut.fiFigure captions:Fig.1Problem geometry and notationsFig.2One-dimensional SD as a function of r when R=D=1and AoA follow Gaussian distribution(o)and Student’s t-distribution(x)with AS=12◦.Dashed lines refer to the expected scatterer distances。
Maxwell’s Equations in Medium *Zhang TaoInstitute of Low Energy Nuclear Physics, Beijing Normal University, Beijing Radiation Center, Beijing100875, Chinataozhang@AbstractReactions (magnetization, polarization, induced magnetization) of medium toelectromagnetic wave and propagation of electromagnetic wave in medium wereinvestigated. Faraday’s law in medium was presented, which is correct not only invacuum but also in medium. The Maxwell’s Equations in medium was modified.Keywords: Maxwell’s Equations, Faraday’s Law of induction, medium, electromagneticwave, total electric field, total magnetic induction1. IntroductionInteraction between electromagnetic wave and medium has been a very active research field. Maxwell’s Equations in vacuum have been verified by countless experiments. This theory resulted in the discovery of electromagnetic wave and has greatly changed our understanding of the world. The interaction between medium and electromagnetic wave is a very important subject and about it many problems have yet to be studied deeply. In this paper the behavior of electromagnetic wave in medium is analyzed and the Maxwell’s Equations in medium are reconsidered. For simplicity, the medium in this paper is an infinite, homogeneous and isolating medium, and there is no combining induced current in the medium [1].2. Problem with the Maxwell’s Equations in the mediumAs is well known, the existing theory of the Maxwell’s equations in the medium is0=⋅∇E , (1)t∂∂−=×∇B E , (2) 0=⋅∇B , (3)* Project supported by Beijing Science Technology New Star Program (Grant No. 952870400), the Beijing Municipal Commission of Education, Key Lab of Beam Technology and Material Modification of Ministry of Education in Beijing Normal University, and the Excellent Young Teachers Program of Ministry of Education, P. R. China.tµµt µ∂∂ε+×∇+∂∂=×∇E M P B 0000, (4) where E is electric field intensity, B magnetic induction, P polarization and M magnetization in the medium. Equation (2) is Faraday’s Law of induction and can be expressed as∫∫∫⋅∂∂−=⋅S B l E d d t. (5) Suppose that there is a varying magnetic field B through a toroid of a kind of the medium in vacuum (B is in vacuum). The symmetry axis of the magnetic field coincides with that of the toroid, as shown in Fig.1. Thus the induced electric field caused by ∫∫⋅∂∂−S B d t coincides with the center line l of the toroid, and the absolute value of E is the same one everywhere on the center line l . The induced electric field caused by ∫∫⋅∂∂−S B d t makes the medium (the toroid) polarized. Suppose the polarization is Pon the center line l . P is in the same direction as the induced electric field. Let t∂∂B keep unchanged during a period of time, then ∫∫⋅∂∂−S B d tin the area circled by the center line l keeps unchanged also. So the induced electric field keeps unchanged, and so does P . Therefore polarization current in the medium is 0 during this period of time, i.e., polarization of the medium does not influence the magnetic induction B through the toroid. P is different for different medium, and the macro electric field E and ∫⋅l l E d on the center line l are different for different medium, too. This means that ∫⋅ll E d is not equal to ∫∫⋅∂∂−S B d t . Hence, Eqs.(2) and (5) are not correct in the medium. Equations (2) and (5)are correct only in vacuum.3. Modified Maxwell’s Equations in MediumWhen propagating in the medium, electric field intensity E and magnetic induction B of electromagnetic wave cause polarization P and magnetization M in the medium [2]. We only consider refracted part and do not consider absorbed and scattered parts of electromagnetic wave in the medium. E , P , B and M are in the same phase. As E varies from its amplitude to 0, P varies synchronously. We separate this process into two steps to investigate the effects of E and P , respectively. Step 1: E varies from its amplitude to 0. Step 2: P varies from its amplitude to 0. Step 1 certainly causes a magnetic induction. Step 2 results in a varying electric field that causes another magnetic induction. So both E and P contribute to the magnetic induction of electromagnetic wave. Therefore P correlates an electric field, which should be E P =P /ε0. Since E P has the same phase and contributes to the magnetic induction of electromagnetic wave as E does, E P is part of the electric field of electromagnetic wave as E is. The same relation exists between B and M . The magnetic induction correlated to M is B M =−µ0M . B M has the same phase and contributes to the magnetic induction of electromagnetic wave as B does. B M is part of the magnetic induction of electromagnetic wave as B is. Maxwell’s theory believes that varying magnetic field produces electric field, and varying electric field produces magnetic field. Hence, propagation of electromagnetic wave in the medium should be the process that E +E P and B +B M translate into each other. That E , P , B and M are in the same phase meets the requirement of this process.For easy narrating, we call E T =E +E P =E +P /ε0 the total electric field, and B T =B +B M =B −µ0M the total magnetic induction. So Maxwell’s equations in the medium can be obtained by substituting electric field E and magnetic induction B of Maxwell’s equations in vacuum with total electric field E T and total magnetic induction B T , respectively, i.e.0T /ε=⋅∇ρE , (6)t∂∂−=×∇ΤB E T ， (7) 0T =⋅∇B , (8)tεµ∂∂=×∇T 00T E B . (9) Inserting E T =E +E P =E + P /ε0 and B T =B +B M = B −µ0M into Eqs. (6)-(9) yields00/)(ε=+⋅∇ρεP/E , (10)tε∂−∂−=+×∇)()(00M B P/E µ ， (11) 0)(0=−⋅∇M B µ, (12)tεεµµ∂+∂=−×∇)/()(0000P E M B . (13) Since E , B and M are curl vectors and there is no net charge in the medium, ∇⋅E =∇⋅P =∇⋅B =∇⋅M =0. Thus Eqs.(10)-(13) become0=⋅∇E , (14)00εtt P/M B E ×∇−∂∂+∂∂−=×∇µ, (15) 0=⋅∇B , (16)M P E B ×∇+∂∂+∂∂ε=×∇0000µtµt µ . (17) Except Eq.(15) which is derived from Eq.(11), Eqs. (14), (16) and (17) are consistent with Eqs (1), (3) and (4) of the existing theory, respectively. Equation (11) means that the change rate of B and the change rate of −µ0M in medium jointly induce an electric field, and this electric field becomes E and P /ε0 in the medium. Equations (11) and (15) are correct not only in vacuum, but also in the medium. They should be more universal forms of Faraday’s Law. The integral form of Eq. (11) is∫∫∫⋅∂−∂−=⋅ε+S M B l P/E d )(d )(00tμ (18) Equations (11) and (18) do not contradict existing experiments. For example, in the imagined experiment of Fig.1, since E +P /ε0 on the center line l is equal to the electric field E V in vacuum, i.e. the electric field without the toroid, and B −µ0M is equal to the magnetic induction B V in vacuum ( M =0 in vacuum), Eq. (18) becomes ∫∫∫⋅∂∂−=⋅S B l E d d V V tl . Obviously, this equation is correct. The reasonableness of Eq. (11) can be proved further by the symmetry property of electric field and magnetic field of electromagnetic wave. Equation (4) or (17) of the existing theory can be changed into Eq. (13). Equation (13) expresses the relation between curl of B +B M =B −µ0M and change rate of E +E P =E +P /ε0. According to the symmetry property, Faraday’s Law should be the relationship about curl of E +E P =E +P /ε0 and change rate of B +B M =B −µ0M , i.e., Eq. (11).Maxwell’s equations in the medium, i.e. Eqs. (6)-(9), have the same form as that in vacuum. This indicates that electromagnetic wave has the same propagation speed in the medium as in vacuum when there is no combining induced current. The refractive index (speed ratio) of electromagnetic wave is not directly related to polarization P and magnetization M of the medium.4. Conclusions Faraday’s law of induction of the Maxwell’s equations of the existing theory, i.e. ∫∫∫⋅∂∂−=⋅S B l E d d t, is correct only in vacuum. It is not correct in medium. When there is no combining induced current, the Faraday’s law of induction in medium should be expressed with total electric field and total magnetic induction, i.e. ∫∫∫⋅∂−∂−=⋅ε+S M B l P/E d )(d )(00tμ. Propagation speed of electromagnetic wave is not directly related to polarization P and magnetization M of medium, or relative dielectric constant and relative permeability.References[1] Zhang, T. (2004) Effect of magnetic field of light on refractive index, Chinese Physics 13 1358[2] Cai, S. S., Zhu, Y . and Xu, J. J. (2002) Electrodynamics , Beijing, Higher Education Press.注：本文中文（介质中的麦克斯韦方程组）已在中国科技论文在线发表，编号200603-519。
a r X i v :g r -q c /0607020v 2 7 D e c 2006Time (in)dependence in general relativityS.Deser ∗Department of Physics,Brandeis University,Waltham,Massachusetts 02454andLauritsen Laboratory,California Institute of Technology,Pasadena,California 91125J.Franklin †Department of Physics,Reed College,Portland,Oregon 97202Abstract We clarify the conditions for Birkhoff’s theorem,that is,time-independence in general relativity.We work primarily at the linearized level where guidance from electrodynamics is particularly useful.As a bonus,we also derive the equivalence principle.The basic time-independent solutions due to Schwarzschild and Kerr provide concrete illustrations of the theorem.Only familiarity with Maxwell’s equations and tensor analysis is required.I.INTRODUCTIONA major obstacle to teaching general relativity is the initially confusing mathematics underlying useful,physical simplifications.We focus in this paper on the conditions that lead to the simplest regime,time-independence.Because general relativity is coordinate-invariant,what does it mean to speak of a particular coordinate’s independence?The answer is illuminating.Loosely,we expect that there exists a choice of coordinate frame in which the gravitationalfield does not depend on t.But is this a meaningful,that is,invariant criterion?The answer is yes:it means that the spacetime geometry allows the existence of a Killing vectorfield fµ(x)that obeys the tensor equationDνfµ+Dµfν≡∂νfµ+∂µfν−gσρ(∂νgµρ+∂µgνρ−∂ρgµν)fσ=0,(1)where gµνis the metric and Dµis the covariant derivative with respect to it,as defined in Eq.(1).We use the signature(−+++)and units such that c=1.If fµis also timelike (f2<0),then the solution in the frame where fµ=g0µ(more manifestly,the contravariant form fµof the vector is fµ=δµ0)implies that∂0gµν=0,(2)and there is no time dependence.(A special property of time-independent geometries is that in(and only in)them,matter systems such as particles retain a conserved energy,just as in flat space.)Our main point is that we have re-expressed the issue of when a given geometry is time-independent,that is,when there exists a frame where Eq.(2)holds,as a covariant (coordinate-independent)criterion:the existence of solutions to Eq.(1).All this transcrip-tion makes no reference tofield equations.There exist many frames where t-dependence is present,but that is not the point.It is not true false that every geometry has a static frame –the Killing equation is a strong requirement.II.MAXWELLWe begin with electrodynamics whosefield equations outside sources,unlike general relativity,can be written entirely in terms of gauge invariantfield strengths,∇·E=0(3a)∇·B=0(3b)˙E=−∇×B(3c)˙B=∇×E.(3d)The˙E equation’s longitudinal part(see the following)implies that˙E L=0,which exhibits the fact that the“Coulomb”part of E is always time-independent,whatever the behavior of the interior charges.The remaining,dynamical transverse part E T and its partner B (transverse by definition)cannot depend on time if they vanish identically,which is the case for spherically symmetric configurations:any E(r)is necessarily of the form∇S(r)and is purely longitudinal.There is no monopole radiation;it is also the only guaranteed static case,as dipole and higher configurations define transverse vectors.Equation(3)does not therefore require time-dependence,or electro/magneto-statics would not exist.For future use we recall that the transverse/longitudinal division of any vectorfield V is a decomposition of unity,V i= (δij−ˆk iˆk j)+ˆk iˆk j V j,(4) along some arbitrary unit vector directionˆk.Its more familiar Fourier transform isV=V T+V L,(5)where∇·V T=∇×V L=0.Our discussion has been couched in terms of the gauge invariantfield strengths E and B, whose time(in-)dependence is unaffected by the choice of gauge.The underlying potentials (A0,A)are another story:even if(E,B)are static,there exist gauge choices for which the potentials do depend on t by adding gauge terms∂µΛ(r,t)that do not affect Fµν=∂µAν−∂νAµ.In any case the transverse vector potentials are unaffected,being gauge invariant.Only(A0,A L)can be altered,keeping E L unchanged.It is instructive to analyze the equations in terms of the Aµin parallel with the general relativity discussion in Sec.III where potentials are unavoidable.III.GENERAL RELATIVITYFor our purposes the gravitationalfield is a glorified tensor version of the vector Maxwell field Aµ,and we expect similar properties of the results there to apply.At the linearized level,the Einstein equations outside sources are2Gµν≡ hµν− ∂µ∂αhαν+∂ν∂αhαµ +∂µ∂νh−ηµν h−∂α∂βhαβ =0(6) for thefield hµνwith h≡hαα;all indices are moved by the Minkowski metricηµν.As for Maxwell’s equations,we decompose Eq.(6)into space and time components,with the simplifying notation h0i≡N i and h00≡N.The theory is invariant under linearized gauge/coordinate transformations hµν→hµν+∂µξν+∂νξµ,that is,Gµν(∂µξν+∂νξµ)=0, an invariance that is useful to exploit.The component form(the linearized version of a decomposition used long ago to analyze the full theory1)of Eq.(6)is2G00=∇2˜h−∂i∂j h ij(7a)2G0i=∇2N i−∂j˙h ji−∂i∂j N j+∂i˙˜h(7b)2G ij= h ij+∂i˙N j+∂j˙N i−(∂i∂k h kj+∂j∂k h ki)+ δij∇2−∂i∂j (N−˜h)+¨˜hδij+δij ∂m∂n h mn−2∂k˙N k ,(7c) with˜h≡h i i the trace of the spatial part of thefield.This slightly complicated set of equations simplifies when we decompose the spatial tensors h ij and the vectors N i,the latter into transverse/longitudinal parts via Eq.(4),the former by the following partition of unity:h ij=h T T ij+h T ij+∂i h j+∂j h i,(8a)∂i h T T ij=∂i h T ij=0=h T T ii(8b)1h T ij=be set to zero,leaving the gauge invariant set(h T T ij,h T,N T i,N)once we use the available gauge invariance.Now Eq.(7)reduces to2G00=∇2h T=0(9a)2G0i=∇2N T i+∂i˙h T=0(9b)2G ij= h T T ij+ ∂i˙N T j+∂j˙N T i + δij∇2−∂i∂j N−1(δij+∇−2∂i∂j)¨h T=0.(9c)2The time-independence of h T follows from the longitudinal part of Eq.(9b),and the relation N=1Unlike Maxwell,there is another category offields lacking a TT part,namely those with dipole character.As we saw there,dipoles permit a transverse vector,but their single direction is not generic enough to construct a TT tensor.Axial symmetry does permit TT, for example via the tensor harmonic P2(cosθ).To summarize at this point,both Maxwell and linearized general relativity gaugefields only allow time-dependence of their true dynamical excitations,and only when those modes can be present,which always excludes spherical symmetry and also dipole symmetry for the general relativity case.IV.KERR AND SCHW ARZSCHILDIt is instructive,at the linearized level,to relate the exterior solution properties to explicit matter sources.In electrodynamics the current consists of two parts:the charge densityρand the longitudinal current j L,which obey the continuity equation˙ρ+∇·j L=0,and the transverse current j T.The(ρ,j L)subset couples only to the longitudinal electricfield, which is equivalent to it,and as we saw,is time-independent away from sources.The transverse electric and magneticfields are generated by the transverse current and can be time dependent if j T is.Similar reasoning applies to general relativity:the source here is the tensor Tµν,whose(T00,T L0i)components are like(ρ,j L).They obey the same continuity equation and excite only the metric component h T,which is also t-independent outside of source distributions.Because general relativity is a tensor theory,there is another“charge”associated with momentum like T00was with energy,namely(T0i,T L ij),which also obeys continuity and is coupled to N T i.The remaining source part,T T T ij,which may,but need not, depend on time,excites the dynamical h T T ijfields.An important example of time-independence is furnished by the Kerr solution3,4of full general relativity,which we will reproduce in the following.In our linearized context,the static metric is generated by a time-independent spinning point mass withT00=mδ3(r),T0i=amǫijk s j∂kδ3(r),(11)where s j denotes the(constant)unit spin vector.As explained in Ref.5the space integral of T00is the total mass m,and that of T0i vanishes because there is no momentum.Itsfirst moment,the angular momentum J,is given by J=am s.The notation choice that expresses J∼am is historical,but has the virtue that m=0is actually justflat space(also in fullgeneral relativity)and the parameter a reduces to that defining ellipsoidal coordinates in ordinary euclidean 3-space.The opposite limit,a =0,defines the spherically symmetric static Schwarzschild solution.We will not discuss in detail the full general relativity extensions of our linear results.Consider,without deriving it (there is no simple way to do so)the full Kerr intervalds 2=−g tt dt 2+g rr dr 2+g θθdθ2+g φφdφ2+2g tφdtdφ.(12)There are five functions of (r,θ)which are (in units of c =1=16πG ),g tt =−(1−2Mr/ρ2)(13a)g rr =ρ2/∆(13b)g θθ=ρ2(13c)g φφ=sin 2θ r 2+a 2)+2a 2Mr sin 2θ/ρ2 (13d)g tφ=−2aMr sin 2θ/ρ2,(13e)with ρ2≡r 2+a 2cos 2θand ∆≡a 2−2Mr +r 2.The linearized limit of Eqs.(12)and (13),or equivalently its asymptotic form,is a superposition of the (linearized)Schwarzschild solution and a spin term h 0φcorresponding to the source (11)h 00=2mr(14b)h ij =2m r 2.(14c)We emphasize that the time-independence here is derivable directly from the exterior equa-tions,apart from details of the interior source,as we would expect for a spinning spherical ball of charge in E&M,its natural analogue.6V.CONCLUSIONSBy working primarily in the linearized limit,we have provided,using the Maxwell tem-plate,a framework for understanding the basis of time-independence in general relativity in terms of the underlying physics and source geometry.Our main conclusion is that thetime-dependence of solutions of gauge theories such as Maxwell’s or general relativity is a property of their radiation modes.If these are forbidden due to spherical(dipole)symmetry, then time-independence is guaranteed.In particular,the Kerr and Schwarzschild solutions illustrate the absence of dipole and monopole excitations.Although the full general relativ-ity is unavoidably more complicated(and involves global issues we have bypassed here),our results capture at least its long distance properties.AcknowledgmentsWe are grateful to Prof.J.Hartle for stimulating criticism that led to this(we hope) improved version of our earlier paper.This work was supported by NSF grant PHY-04-01667.∗Electronic address:deser@†Electronic address:jfrankli@1R.Arnowitt,S.Deser,and C.W.Misner,“The dynamics of general relativity,”in Gravitation: An Introduction to Current Research,edited by L.Witten(John Wiley&Sons,New York, 1962).Reprinted as gr-qc/0405109.2For the history and a modern derivation,see S.Deser and J.Franklin,“Schwarzschild and Birkhoffa la Weyl,”Am.J.Phys.73(3),261–264(2005),gr-qc/0408067.3Roy P.Kerr,“Gravitationalfield of a spinning mass as an example of algebraically special metrics,”Phys.Rev.Lett.11(5),237–238(1963).4Robert H.Boyer and Richard W.Lindquist,“Maximal analytic extension of the Kerr metric,”J.Math.Phys.8(2),265–281(1967).5Charles W.Misner,Kip S.Thorne,and John Archibald Wheeler,Gravitation(W.H.Freeman, New York,1973).6J.Franklin and P.T.Baker,“Linearized Kerr and spinning massive bodies:An electrodynamics analogy,”Am.J.Phys.,to appear.。
CHAPTER IGLOBAL WEAK FORMS,WEIGHTED RESIDUALS,FINITE ELEMENTS,BOUNDARY ELEMENTS,&LOCAL WEAK FORMS1.1IntroductionDifferential equations(“ordinary differential equations”in one dimensional, and“partial differential equations”in two or three dimensional physical do-mains)and boundary conditions are abstract,and often highly approximate, characterizations of physical processes in engineering.In physical processes that evolve with time,derivatives with respect to time,and appropriate initial conditions,may also appear in these equations.Exact mathematical solutions are often possible only for problems defined in the simplest of geometrical domains,and only mostly for linear problems[i.e.,for differential equations and boundary/initial conditions wherein the unknown variable(s)and its(their) derivatives appear only in linear combinations.]However,most engineering processes occur mostly in complex geometrical domains,and most often,in-volve nonlinearities of the unknown variable(s)and its(their)derivatives.To solve such practical problems,to aid in engineering analysis,design,and syn-thesis,(digital)computer-based modeling&simulation techniques are neces-sary.We label this as“Computer-Modeling&Simulation-Based Engineering”. The aim is to use the power of the modern digital computer,high-speed sci-entific computing,visualization techniques,virtual reality,and the attendant information technologies,to achieve a near-real-time simulation of complex en-gineering phenomena,in order to facilitate integrated product&process design for real-life engineering systems.These include aircraft,spacecraft,missiles, automobiles,ships,power plants,bridges,biological systems,micro-electronic devices,computer chips,micro-electro-mechanical systems,systems involving heat and mass transfer,compressors,turbines,systems involvingfluidflow in closed and open domains,atmospheric polution,sustainable ecological tech-2Global Weak Forms,Weighted Residuals,Finite Elements,Boundary Elements,&Local Weak Forms nologies,nano,micro,meso and macro mechanics of engineered materials,etc. In as much as there is a great deal of commonality in the differential equations governing these processes,with appropriately defined unknown variable(s)gov-erning each process,one can envision a common“tool kit”that will enable com-puter modeling and simulation of a diverse variety of engineering processes.Engineering computations have made a startling progress over the past three decades.The most successful methods among them are thefinite element method and the boundary element method.However,as compared to its sound theoretical foundation,thefinite element method suffers from drawbacks such as locking(shear,membrane,incompressibility,etc),and tedious meshing or re-meshing in problems of crack propagation,shear-band formation,large de-formations,etc.Due to these reasons,in recent years,meshless approaches in solving the boundary value problems have received a considerable attention as new ways to alleviate some of these drawbacks.Meshless methods,as alternative numerical approaches to eliminate the well-known drawbacks in thefinite element and boundary element methods, have attracted much attention in recent decades,due to theirflexibility,and, most importantly,due to their potential in negating the need for the human-labor intensive process of constructing geometric meshes in a domain.Such meshless methods are especially useful in those problems with discontinuities or moving boundaries.The main objective of the meshless methods is to get rid of,or at least alleviate the difficulty of,meshing and remeshing the entire structure,by only adding or deleting nodes in the entire structure,instead.Meshless meth-ods may also alleviate some other problems associated with thefinite element method,such as locking,element distortion,and others.The initial idea of meshless methods dates back to the smooth particle hy-drodynamics(SPH)method for modeling astrophysical phenomena(Gingold and Monaghan,1977).The research into meshless methods has become very active,only after the publication of the Diffuse Element Method by Nayroles, Touzot&Villon(1992).Several so-called meshless methods[Element Free Galerkin(EFG)]by Belytschko,Lu&Gu(1994);Reproducing Kernel Particle Method(RKPM)by Liu,Chen,Uras&Chang(1996);the Partition of Unity Finite Element Method(PUFEM)by Babuska and Melenk(1997);hp-cloud method by Duarte and Oden1996;Natural Element Method(NEM)by Suku-mar,Moran&Belytschko(1998);Meshless Galerkin methods using Radial Basis Functions(RBF)by Wendland(1999);have also been reported in litera-1.1:Introduction3 ture since then.A detailed bibliography on meshless methods is provided at the end of this monograph.The major differences in these meshless methods come only from the techniques used for interpolating the trial function.Even though no mesh is required in these methods for the interpolation of the trial and test functions for the solution variables,the use of shadow elements is inevitable in these methods,for the integration of the weak-form,or of the energy.Therefore, these methods are not truly meshless.Recently,two truly meshless methods,the meshless local boundary inte-gral equation(LBIE)method,and the meshless local Petrov-Galerkin(MLPG) method,have been developed in Zhu,Zhang,&Atluri(1998a,b),Atluri&Zhu (1998a,b)and Atluri,Kim&Cho(1999),for solving linear and non-linear boundary problems.Both these methods are truly meshless,as no domain/or boundary meshes are required in these two approaches,either for the purposes of interpolation of the trial and test functions for the solution variables,or for the purposes of integration of the weak-form.All pertinent integrals can be easily evaluated over over-lapping,regularly shaped,domains(in general,spheres in three-dimensional problems)and their boundaries.In fact,The LBIE approach can be treated simply as a special case of the MLPG approach(Atluri,Kim, &Cho,1999).Zhang and Yao(2001)present a new regular hybrid boundary node method by combining the advantages of both LBIE and the boundary node method.Remarkable successes of the MLPG method have been reported in solving the convection-diffusion problems[Lin&Atluri(2000)];fracture me-chanics problems[Kim&Atluri(2000),Ching&Batra(2001)];Navier-Stokes flows[Lin&Atluri(2001)];Shear-deformable beams[Cho&Atluri(2001)]; and plate bending problems[Gu&Liu(2001),Long&Atluri(2002)],etc.In summary,the MLPG is a truly meshless method,which involves not only a meshless interpolation for the trial functions(such as MLS,PU,Shepard func-tion or RBF),but also a meshless integration of the weak-form(i.e.all integra-tions are always performed over regularly shaped sub-domains such as spheres, parallelopipeds,and ellipsoids in3-D).The aim of this monograph is an exposition of the MLPG method,in all its generality,and in all its variations.It is shown that some of the variations of the MLPG method not only eliminate the human-labor cost of meshing a domain altogether,but also involve less computational cost as compared to thefinite element and boundary element methods.4Global Weak Forms,Weighted Residuals,Finite Elements,Boundary Elements,&Local Weak Forms1.2Global weak forms and the weighted residual method(WRM) To set the stage for a discourse on the MLPG method,we start off by discussing a variety of modeling strategies,using Global Weak Forms,through an example problem.It can be shown that all mathematical formulations governing a variety of linear boundary value problems in engineering may be written in the form: L mΨf(1.1) where L m is a linear differential operator of order m.L m can be an“ordinary differential operator”in a one-dimensional problem,or a“partial differential operator”in2and3-dimensional physical domains.In a general3-D domain, imagine a line,going through the domain,and connecting two different points on the boundary.Along this line,one can imagine an ordinary differential equa-tion of order“m”.Then,at each point on the boundary,we may have m2 number of boundary conditions,each involving a differential operator of order m1.For example,we consider the linear Poisson’s equation in2-dimmensional space,defined by the Cartesian Coordinates x i.Consider the domainΩto be as sketched in Fig.1.1.The boundary ofΩ,denoted byΓ[or,sometimes, by∂Ω],is defined by the curve A-B-C-D-E-F-A,and the curve GH.Along the segment AB,the boundary conditionφφis prescribed.Like wise,the boundary conditions along the various segments are considered to be:∂φ∂n q along BC;φφand∂φ∂n q along parts of CD;∂φ∂n0along the slit (or“crack”)D-E-F;φφand∂φ∂n q along parts of FA;and∂φ∂n0 along the“hole”GH.The boundary value problem defined in Fig.1.1is a fairly realistic engineering problem.Hence,AB belongs to the essential boundaryΓu, and EF,ED and GH belong to theflux boundaryΓq.The Poisson equation can be can be written as:∇2φx p x xΩ(1.2)where x x i e i is the vector of position of a point insideΩ,p is a given source function,and the domainΩis enclosed byΓΓuΓq,with boundary condi-tions:φφonΓu(1.3a)∂φq q onΓq(1.3b)∂n1.2:Global weak forms and the weighted residual method(WRM)5Figure1.1:A realistic boundary value problem,∇2φx p x.The condition(1.3a)is often referred to as the Dirichlet boundary condition; and(1.3b)as the Neumann boundary condition.The Poisson equation arises in very many branches of engineering:the torsion of a solid rod of an arbitrary cross-section,undergoing an infinitesimal twist;heat transfer in homogeneous media;seepageflow;fluidfilm lubrication;eletro-staticfield;magneto-static field,etc.Let u,hereafter called the trial function,be an approximate solution to the problem(1.2).In as much as u may not be an exact solution,substituting u in the differential equation and the boundary conditions Eqs.(1.2-1.3),we obtain the following relations for the“error residuals”.Interior Error:R I x∇2u x p x0(1.4)Boundary Error:R B1u x u0atΓuR B2∂u∂nq atΓq(1.5)6Global Weak Forms,Weighted Residuals,Finite Elements,Boundary Elements,&Local Weak Formswhere R I is the interior error;and R Bii12are the errors in the boundary conditions.The“best”approximation is the one that nullifies the errors R I and R Biin some fashion.In general,the trial functions must satisfy several minimum re-quirements to make the process of error nullification meaningful.The functions that satisfy such minimum requirements are called the admissible functions.For example,in Eq.(1.4),u must be2times differentiable and u i must be contin-uous(or u is C1continuous);if u is not C1continuous,u ii will be“infinite”where u i is discontinuous.In that case,R I u is infinite at these points;and the minimization of R I u no longer makes any sense.Consider a linear combination of N admissible functions u i(i12N), such that:uN∑i1a i u i x a i u i x(1.6)where a i(i12N)are undetermined coefficients.We use the convention that summation is implied over a repeated index.Here,choosing an admissible u x[i.e.,C1continuous u x in Eq.(1.4)]over a global domainΩ,of arbitrary shape is a non-trivial proposition.Much of Chapter2in this monograph is devoted to this topic.However,ifΩis a simple geometry,such as a circle, square,etc.,u x can be in the form of trigonometric or other functions.Figure1.2:Schematics of the point collocation methodWe now discuss several methods,to reduce the errors R I and R Bito be as1.2:Global weak forms and the weighted residual method(WRM)7 small as possible,and in the limit,make them uniformly zero over the globaldomainΩ.1.2.1Point collocation methodWe assume that u is C1continuous all overΩin Eq.(1.4).We consider making the error vanish at certain points x j j12M,in the domainΩ.This leads to the following equations for the undetermined coefficients.R I x j a i∇2u i x j p x j0j12M(1.7) andR Bix j0i12(1.8) Suppose that we use the conditions of vanishing of the two types of boundary residuals[Eq.(1.8)]to generate N M equations;then,by using M collocation points in the interior of the domainΩ,we can generate N equations for the N undetermined coefficients a i.While a solution a j for such a system of equationswill make the errors R I and R Bigo to zero identically at afinite number of points,the residuals R I and R Bimay oscillate in between these points and may indeed be quite ing more collocation points than M will result in more equations than the unknowns.In this case,it is not possible,in general,tofind a set of coefficients a i that will drive the error at the collocation points to zero.Suppose that point collocation of Eqs.(1.7)at an arbitrarily large number of points,along with enforcing(1.8)at an arbitrary number of points,lead,in general,to the over-determined system of equations:A i j a j p i(1.9) where i12K;j12N and K N.We seek an approximate solu-tion a j for a j,such thatA i j a j p iεi0(1.10) and a j minimize the square error,εiεi.Thus,∂∂a j εiεi∂∂a j A ip a pp i A iq a q p i0(1.11)8Global Weak Forms,Weighted Residuals,Finite Elements,Boundary Elements,&Local Weak FormsorA i j A iq a q A i j p i(1.12) orB jq a q p i(1.13) where i12K;j12K and q12N.The above method is an algebraic least-squares procedure to approximately solve an over-determined system of equations,provided that B jq is positive def-inite.For an underdetermined system of equations,however,only some of the unknowns can be expressed in terms of the other unknowns.It is noted that Eq.(1.11)also amounts to minimizing the summed point-wise square error. 1.2.2Weighted integral square errorAs discussed above,in the point collocation method,we can use more colloca-tion points than the number of unknown coefficients in the trial function,using a least square error approach.In the limit,as the number of collocation points tends to∞,we may consider minimizing the integrated square error:εσΩ∇2u x p x2dΩβΓuu u2dΓγΓq∂u∂nq2dΓ(1.14)whereσ,β,γare weighting constants,which are also often referred to as “penalty-parameters”.By minimizing the aboveεwith respect to a i,one may obtain the desired system of N algebraic equations to determine the N unknownsa i.1.2.3Subdomain integral/average error methodAlternatively,we may consider making each of the averages of the error,over a set of subdomains,go to zero.For example,we may setΩk R i x dΩ0;∂ΩkR i x dΓ0(1.15)The number of subdomains is chosen such that we have a solvable set of equa-tions for a i,i12N[i.e.N,or more than N subdomains,to yield a1.2:Global weak forms and the weighted residual method(WRM)9 system of N equations,or of more than N equations].This is illustrated in Fig.1.3.Note that the subdomainsΩi i12k are continguous(i.e.,non-overlapping),and span the entire domainΩ.While most ofΩi may be of a regular[say triangular]shape,others may be approximately triangular.Figure1.3:Schematics of the subdomains1.2.4Finite volume methodThis is similar to the Subdomain Integral method,and has been popularized in the literature dealing with unsteady aerodynamics,and heat transfer.The prin-ciple of thefinite volume scheme is to integrate Eq.(1.2)on a control volume (i.e.subdomain)Ωk,which yields:∂Ωk u i n i dΓΩkpdΩ0(1.16)where∂Ωk is the boundary of the subdomainΩk,andΩkΩ;next,thefluxes need to be approximated on the boundary∂Ωk of the control volume.Hence, we shall approximate the integral of u i n i over each edge of the mesh.Note, once again,that the sub-domainsΩk are contiguous,as in Fig.1.3.10Global Weak Forms,Weighted Residuals,Finite Elements,Boundary Elements,&Local Weak Forms Another variant of the Finite V olume Method may be derived,by rewriting Eq.(1.2)in a“mixed”form:φiφi;φi i p(1.17) wherein,φandφi are treated as independent variables.If u and u i are the trial functions forφandφi,respectively,we may write the counterparts of(1.16)as:∂Ωk un i dΓΩku i dΩ0(1.18)and∂Ωk u i n i dΓΩkpdΩ0(1.19)Instead of the“global trial function”,u a k u k x;and u i a k u ki x,one may assume a linear interpolation of u and u i over each element,as shown in Fig.1.3.Note again,all the subdomainsΩk are contiguous in Fig. 1.3,and the method involves the generation of a“mesh”,for both the purpose of interpola-tion,as well as for purpose of integration,as in Eqs.(1.18)and(1.19)]1.2.5The weighted residual method and global weak formsIf the function F x satisfies the conditionΩF x g x dΩ0(1.20) for all g x onΩ,then,F x0onΩ.Conversely,therefore,we canfind the trial function u a i u i such that R I u0[where R I is defined in Eq.(1.7)] by enforcing the weak form equationΩR I x v x dΩ0(1.21) for all possible functions v x onΩ.We denote v x as“test”functions.In general,the trial function u x is chosen to be a linear combination of a set of basis functions,u i(i12N),with undetermined coefficients,a i,i.e. u a i u i x,xΩ.Likewise,the test function v x is chosen as v b j v j x, j12M;where v j are the basis functions and b j are the undetermined1.2:Global weak forms and the weighted residual method(WRM)11 coefficients.The“best”approximation is obtained by determining that combi-nation of u i that satisfies the weak form equation(1.21)for the chosen test func-tions v j.Different choices of the basis functions for the trial function and the test functions will lead to different approximation methods.For example,it reduces to the Point Collocation Method when Dirac Delta functions,v xδx x j, are used for the test functions.It reduces to the subdomain integral method, when the following piecewise constant functions[or Heaviside step functions] are used as test functions:v x1xΩk[see Fig.1.3]0elsewhere(1.22)When the test function v x is taken to be the same as the error function R I, Eq.(1.21)reduces to the least square error method.Now,consider the weighted interior residual equation for the Poisson’s prob-lem,governed by Eq.(1.2):Ω∇2u x p x v x dΩ0(1.23)which requires u to be twice differentiable and u i to be continuous[or u is C1 continuous].If not,u ii will be∞where u i is discontinuous and(1.23)makes no sense.On the other hand,there is no continuity requirement on the test function v in Eq.(1.23).Thus,the requirements on u and v are“unsymmetric”, in order to be admissible in Eq.(1.23).Hence,we denote Eq.(1.23)as a global unsymmetric weak form(GUSWF).To reduce this high-order differentiability requirement on u,we can inte-grate Eq.(1.23)by parts,by using∇2u v u ii v u i v i u i v i,to obtain the following global symmetric weak formulation(GSWF),Γu i n i vdΓΩu i v i pv dΩ0(1.24)which has the same differentiability requirements on both the trial function u and the test function v in the interior of the domainΩ,namely,both u and v are required to be once differentiable,and both u and v are required to be C0continuous.If they are not C0continuous,u i and v i may tend to∞when u and v are discontinuous and Eq.(1.24)is no longer meaningful.Thus,the12Global Weak Forms,Weighted Residuals,Finite Elements,Boundary Elements,&Local Weak Forms requirement on the test function is“increased”,while the requirement on the trial function is“decreased”.We may directly substitute the natural boundary condition Eq.(1.3b)into the symmetric weak form equation,i.e.Eq.(1.24),and noticing that u i n i∂u∂n q in Eq.(1.24),we obtainΓu u i n i vdΓΓqqvdΓΩu i v i pv dΩ0(1.25)When all the boundary conditions are natural,(or“Generalized Force”type), we may rewrite Eq.(1.25)asΓq qvdΓΩu i v i pv dΩ0(1.26)We can see that Eq.(1.26)is actually the combined weak form equation for both the differential equation(D.E.)and the boundary conditions(B.C.),if these B.C. are purely of the“natural”or“kinetic”,or“generalized-force”type.Let u be approximated using u i(i12N),i.e.u a i u i i12N(1.27)where a i are undermined coefficients.Note that,in order to be admissible,u i need only to be C0continuous in the domainΩ,,but they need not obey any boundary conditions a priori,in as much as the boundary condition for u,if they are of the“natural”or“kinetic”type,i.e.,Eq.(1.3b),is already embedded in the weak form Eq.(1.26).Let v be any function that can be represented asv b j v j j12M(1.28)where b i are arbitrary constants.Note again,that in as much as the prescribed boundary conditions for u are purely natural as in Eq.(1.3b),the test func-tion v need not obey any boundary conditions,as seen from Eq.(1.26).Using Eqs.(1.27)and(1.28)in Eq.(1.26),we obtainΓq qb j v j dΓΩa i u i kb j v j k pb j v j dΩ0(1.29)Since Eq.(1.29)is true for any b j,we have the following linear system of equa-tions for a i.A i j a i f j(1.30)1.2:Global weak forms and the weighted residual method(WRM)13 where i12N;j12M,andA i jΩu i k v j k dΩ(1.31)f jΩpv j dΩΓqqv j dΓ(1.32)If M N[i.e.,the number of basis functions in the trial function u,is the same as the number of basis functions in the test function v],and u j are identically the same as v j j12N,we have a symmetric A i j.When v j u j,the method is generally known as the Galerkin symmetric weak form approach.When the test functions v j are in general different from u j,the method is known as the Petrov-Galerkin approach.In this case,even when M N and even when a symmetric weak form is used,the coefficient matrix A i j is unsymmetric.When M N,the system of Eq.(1.30)is over-determined,and can be solved by the method of algebraic least squares.Also,when m n,and u j v j,the coefficient matrix A i j in Eq.(1.30)is symmetric and positive definite,as long as“rigid body modes”[i.e.u j is a constant or linear function],are removed.Suppose now that the B.C.are purely kinematic[i.e.,ΓΓu],i.e.essen-tial boundary condition Eq.(1.3a)exists all over the boundary.If we examine the GSWF(1.24),we notice that the above“lower-order”or purely“kinematic”B.C.(1.3a)cannot be directly substituted into the GSWF(1.24).Unless we choose to add additional terms to Eq.(1.24)to enforce essential boundary con-dition(1.3a)in a weak form,we must choose the trial function u to satisfy these conditions(1.3a)identically,a priori.In this case,we are left with the GSWF (1.24)with boundary terms as indicated in Eq.(1.24).We can eliminate these boundary terms onΓin Eq.(1.24),if we choose the test functions such that v x0,xΓ.If,on the otherhand,in the case when the boundary conditions are entirely kinematic,i.e.,only on u,and if,correspondingly,v does not vanish at the boundary,we have the discrete weak form(similar to1.29)as:Γa i u i k n k b j v j dΓΩa i u i kb j v j k pb j v j dΩ0(1.33)leading to the discrete equation A i j a j f i,whereA i jΩu i k v j k dΩΓu i k n k v j dΓ(1.34a)and f jΩpv j dΩ(1.34b)14Global Weak Forms,Weighted Residuals,Finite Elements,Boundary Elements,&Local Weak Forms It is seen that nothing can be said of the positive definiteness of A i j in (1.34a),which does not assure the solvability of the problem.On the other-hand,if v j vanish at the boundary,A i j reduces to that in Eq.(1.31),which is positive definite(when rigid modes are suppressed).Hence a solution always exists for a i.Thus,if the given problem has prescribed boundary conditions that are purely of the kinematic type,i.e.,as in Eq.(1.3a)[i.e.,ΓΓu],we choose u to obey these conditions a priori,and correspondingly choose v such that v x0,xΓ.In this case,the GSWF reduces to:Ωu i v i pv dΩ0(1.35)when all the B.C.are purely kinematic.If the trial function u,especially in2and3-dimensional problems,cannot be chosen so as to satisfy the essential boundary conditions a priori,one can enforce them by:(i)choosing a boundary weighting function that may depend on the higher-order derivatives of the test function v,such that the kinematic boundary conditions are enforced by the weak-form:Γuu u v i n i dΓ0(1.36)However,when the bilinear form arising out ofΓu uv i n i dΓis added to the A i jin Eq.(1.34b),the positive definiteness of the resulting coefficient matrix A i j cannot be guaranteed;(ii)by choosing an independent Lagrange multiplierfield at the boundary[this is also not desirable,since the solvability of the problem will involve certain stability conditions which must be satisfied a priori[Brezzi (1974)];(iii)by using boundary collocation for the essential B.C.at a desired number of points;(iv)by using the least-squares integral error condition for boundary error residuals;or(v)by using a“Penalty”approach.Methods(iii)to (v)are in general simpler than the others.More will be said of these essential boundary conditions,and their satisfaction,later in this book,in the context of meshless methods.In general,in the GSWF(1.24),one can directly substitute any higher-order B.C.(natural)into(1.24),and thus enforce them a posteriori;while all lower-order(essential)B.C.must be satisfied a priori,and the GSWF is then simpli-fied by enforcing v to satisfy the homogeneous condition atΓu.1.2:Global weak forms and the weighted residual method(WRM)15Now,we use the formalism that the test function v is considered to be a “variation”of the trial function u,in the traditional“calculus of variations”sense.For,example,in general,if L2m is a self-adjoint operator,i.e.L2m u vdΩL m v L m u dΩB(1.37) where B are boundary integral terms,then aπexist so thatδπ0leads to L2m u p,whereπ12L m u2pu dΩB(1.38) In a elastic system with a conservative(dead-load)forces,πis the“Total Po-tential Energy”,andδπ0is the condition of stationarity of the total potential energy.It may simply be viewed as a mathematical equivalent of the symmetric weak form,which,however,is more direct.In general,the weak form approach is,however,more general than the above“energy approach”,in as much as a weak form can be stated for non-self-adjoint operators as well.The Galerkin Finite Element Method(GFEM)is based on the Global Symmetric Weak Form (GSWF),Eq.(1.24).To reduce the C0continuity conditions as demanded by the Symmetric Weak form(1.24),we can integrate Eq.(1.23)by parts twice,by using∇2u vu ii v u i v i uv i i uv ii,to obtain another global unsymmetric weak for-mulation(GUSWF)as follows,Γu i n i vdΓΓuv i n i dΓΩu∇2v pv dΩ0(1.39)which requires the test function v to be twice differentiable and v i to be contin-uous[or v is C1continuous].If not,v ii will be∞where v i is discontinuous and (1.39)makes no sense.On the other hand,there is no continuity requirement on the trial function u in Eq.(1.39).Thus,the requirements on u and v are also “unsymmetric”,in order to be admissible in Eq.(1.39).Hence,we also denote Eq.(1.39)as a global unsymmetric weak form(GUSWF2).In this case,we may directly substitute the natrural and essential bound-ary conditions Eqs.(1.3a)and(1.3b)into the global unsymmetric weak form equation,i.e.Eq.(1.39),we obtainΓu qvdΓΓqqvdΓΓuuv i n i dΓΓquv i n i dΓΩu∇2v pv dΩ0(1.40)16Global Weak Forms,Weighted Residuals,Finite Elements,Boundary Elements,&Local Weak Forms1.3The Galerkinfinite element methodIn the previous section,the basis functions u j and v j for the trial function u and the test function v,were at once assumed as“global”functions,and each of it was nonzero over the entire problem-domain,Ω.Figure1.4:Use of a patch work of generic triangular elements to approximate the given geometry in Fig.1.1In a one-dimensional problem,assuming component trial and test functions u j and v j over the entire problem-domain is relatively simple.On the other hand,if the problem is defined by a partial differential equation in2or3dimen-sions,and if the global domain is of an arbitrary shape,assuming the trial and test functions as simple polynomials over the entire global domain is inconve-nient at best.To overcome this curse of arbitrary shaped geometrical domains wherein the problem is defined,one may resort to assuming trial and test func-tions over a generic simpler-shaped subdomain(element)of the given problem, and create a non-overlapping patchwork of the subdomains(elements)to geo-metrically approximate the given domain(see Fig.1.4).We can assume the trial and test functions over each subdomain(element)as“local basis”or“element basis”functions.。
Allowable stress：许用应力Allowable load：许用荷载Buckling：失稳Flexural rigidity：弯曲刚度，bending 弯曲，normal stress in bending 弯曲正应力Shear stress in bending 弯曲切应力Bending moment ：弯曲Bending moment diagram：弯矩图Critical stress ：临界应力，critical load ：临界荷载Eccentric tension 偏心拉伸Limit load：极限荷载，ultimate stress 极限应力，ultimate stress in tension 抗拉强度Normal Principal stress 主应力，principal stress trajectory 主应力迹线Theorem of conjugate shearing stress切应力互等定律，reciprocal-displacement theorem位移互等定律Torsion ：扭转弹性力学elasticity弹性理论theory of elasticity均匀应力状态homogeneous state of stress应力不变量stress invariant应变不变量strain invariant应变椭球strain ellipsoid均匀应变状态homogeneous state of strain应变协调方程equation of strain compatibility 拉梅常量Lame constants各向同性弹性isotropic elasticity旋转圆盘rotating circular disk楔wedge开尔文问题Kelvin problem 布西内斯克问题Boussinesq problem艾里应力函数Airy stress function克罗索夫--穆斯赫利什维利法Kolosoff- Muskhelishvili method基尔霍夫假设Kirchhoff hypothesis板Plate矩形板Rectangular plate圆板Circular plate环板Annular plate波纹板Corrugated plate加劲板Stiffened plate,reinforced Plate中厚板Plate of moderate thickness弯[曲]应力函数Stress function of bending壳Shell扁壳Shallow shell旋转壳Revolutionary shell球壳Spherical shell[圆]柱壳Cylindrical shell锥壳Conical shell环壳Toroidal shell封闭壳Closed shell波纹壳Corrugated shell扭[转]应力函数Stress function of torsion翘曲函数Warping function半逆解法semi-inverse method瑞利--里茨法Rayleigh-Ritz method松弛法Relaxation method莱维法Levy method松弛Relaxation量纲分析Dimensional analysis自相似[性] self-similarity影响面Influence surface接触应力Contact stress赫兹理论Hertz theory协调接触Conforming contact滑动接触Sliding contact滚动接触Rolling contact压入Indentation各向异性弹性Anisotropic elasticity颗粒材料Granular material散体力学Mechanics of granular media 热弹性Thermoelasticity超弹性Hyperelasticity粘弹性Viscoelasticity对应原理Correspondence principle褶皱Wrinkle塑性全量理论Total theory of plasticity 滑动Sliding微滑Microslip粗糙度Roughness非线性弹性Nonlinear elasticity大挠度Large deflection突弹跳变snap-through有限变形Finite deformation格林应变Green strain阿尔曼西应变Almansi strain弹性动力学Dynamic elasticity运动方程Equation of motion准静态的Quasi-static气动弹性Aeroelasticity水弹性Hydroelasticity颤振Flutter弹性波Elastic wave简单波Simple wave 柱面波Cylindrical wave水平剪切波Horizontal shear wave竖直剪切波Vertical shear wave体波body wave无旋波Irrotational wave畸变波Distortion wave膨胀波Dilatation wave瑞利波Rayleigh wave等容波Equivoluminal wave勒夫波Love wave界面波Interfacial wave边缘效应edge effect塑性力学Plasticity可成形性Formability金属成形Metal forming耐撞性Crashworthiness结构抗撞毁性Structural crashworthiness拉拔Drawing破坏机构Collapse mechanism回弹Springback挤压Extrusion冲压Stamping穿透Perforation层裂Spalling塑性理论Theory of plasticity安定[性]理论Shake-down theory运动安定定理kinematic shake-down theorem 静力安定定理Static shake-down theorem率相关理论rate dependent theorem载荷因子load factor加载准则Loading criterion加载函数Loading function加载面Loading surface塑性加载Plastic loading塑性加载波Plastic loading wave简单加载Simple loading比例加载Proportional loading卸载Unloading卸载波Unloading wave冲击载荷Impulsive load阶跃载荷step load脉冲载荷pulse load极限载荷limit load中性变载nentral loading拉抻失稳instability in tension加速度波acceleration wave本构方程constitutive equation完全解complete solution名义应力nominal stress过应力over-stress真应力true stress等效应力equivalent stress流动应力flow stress应力间断stress discontinuity应力空间stress space主应力空间principal stress space静水应力状态hydrostatic state of stress 对数应变logarithmic strain工程应变engineering strain等效应变equivalent strain应变局部化strain localization应变率strain rate应变率敏感性strain rate sensitivity应变空间strain space 有限应变finite strain塑性应变增量plastic strain increment累积塑性应变accumulated plastic strain永久变形permanent deformation内变量internal variable应变软化strain-softening理想刚塑性材料rigid-perfectly plasticMaterial刚塑性材料rigid-plastic material理想塑性材料perfectl plastic material材料稳定性stability of material应变偏张量deviatoric tensor of strain应力偏张量deviatori tensor of stress应变球张量spherical tensor of strain应力球张量spherical tensor of stress路径相关性path-dependency线性强化linear strain-hardening应变强化strain-hardening随动强化kinematic hardening各向同性强化isotropic hardening强化模量strain-hardening modulus幂强化power hardening塑性极限弯矩plastic limit bending Moment塑性极限扭矩plastic limit torque弹塑性弯曲elastic-plastic bending弹塑性交界面elastic-plastic interface弹塑性扭转elastic-plastic torsion粘塑性Viscoplasticity非弹性Inelasticity理想弹塑性材料elastic-perfectly plastic Material极限分析limit analysis极限设计limit design极限面limit surface上限定理upper bound theorem上屈服点upper yield point下限定理lower bound theorem下屈服点lower yield point界限定理bound theorem初始屈服面initial yield surface后继屈服面subsequent yield surface屈服面[的]外凸性convexity of yield surface 截面形状因子shape factor of cross-section沙堆比拟sand heap analogy屈服Yield屈服条件yield condition屈服准则yield criterion屈服函数yield function屈服面yield surface塑性势plastic potential能量吸收装置energy absorbing device能量耗散率energy absorbing device塑性动力学dynamic plasticity塑性动力屈曲dynamic plastic buckling塑性动力响应dynamic plastic response塑性波plastic wave运动容许场kinematically admissible Field静力容许场statically admissible Field流动法则flow rule速度间断velocity discontinuity滑移线slip-lines滑移线场slip-lines field移行塑性铰travelling plastic hinge塑性增量理论incremental theory of Plasticity 米泽斯屈服准则Mises yield criterion普朗特--罗伊斯关系prandtl- Reuss relation 特雷斯卡屈服准则Tresca yield criterion洛德应力参数Lode stress parameter莱维--米泽斯关系Levy-Mises relation亨基应力方程Hencky stress equation赫艾--韦斯特加德应力空间Haigh-Westergaard stress space洛德应变参数Lode strain parameter德鲁克公设Drucker postulate盖林格速度方程Geiringer velocity Equation 结构力学structural mechanics结构分析structural analysis结构动力学structural dynamics拱Arch三铰拱three-hinged arch抛物线拱parabolic arch圆拱circular arch穹顶Dome空间结构space structure空间桁架space truss雪载[荷] snow load风载[荷] wind load土压力earth pressure地震载荷earthquake loading弹簧支座spring support支座位移support displacement支座沉降support settlement超静定次数degree of indeterminacy机动分析kinematic analysis结点法method of joints截面法method of sections结点力joint forces共轭位移conjugate displacement影响线influence line三弯矩方程three-moment equation单位虚力unit virtual force刚度系数stiffness coefficient柔度系数flexibility coefficient力矩分配moment distribution力矩分配法moment distribution method力矩再分配moment redistribution分配系数distribution factor矩阵位移法matri displacement method单元刚度矩阵element stiffness matrix单元应变矩阵element strain matrix总体坐标global coordinates贝蒂定理Betti theorem高斯--若尔当消去法Gauss-Jordan elimination Method屈曲模态buckling mode复合材料力学mechanics一.综合类1.geotechnical engineering岩土工程2.foundation engineering基础工程3.soil, earth土4.soil mechanics土力学cyclic loading周期荷载unloading卸载reloading再加载viscoelastic foundation粘弹性地基viscous damping粘滞阻尼shear modulus剪切模量5.soil dynamics土动力学6.stress path应力路径7.numerical geotechanics 数值岩土力学二. 土的分类1.residual soil残积土groundwater level地下水位2.groundwater 地下水groundwater table地下水位3.clay minerals粘土矿物4.secondary minerals次生矿物ndslides滑坡6.bore hole columnar section钻孔柱状图7.engineering geologic investigation工程地质勘察8.boulder漂石9.cobble卵石10.gravel砂石11.gravelly sand砾砂12.coarse sand粗砂13.medium sand中砂14.fine sand细砂15.silty sand粉土16.clayey soil粘性土17.clay粘土18.silty clay粉质粘土19.silt粉土20.sandy silt砂质粉土21.clayey silt粘质粉土22.saturated soil饱和土23.unsaturated soil非饱和土24.fill (soil)填土25.overconsolidated soil超固结土26.normally consolidated soil正常固结土27.underconsolidated soil欠固结土28.zonal soil区域性土29.soft clay软粘土30.expansive (swelling) soil膨胀土31.peat泥炭32.loess黄土33.frozen soil冻土三. 土的基本物理力学性质 compression index2.cu undrained shear strength3.cu/p0 ratio of undrained strength cu to effective overburden stress p0(cu/p0)NC ,(cu/p0)oc subscripts NC and OC designated normally consolidated and overconsolidated, respectively4.cvane cohesive strength from vane test5.e0 natural void ratio6.Ip plasticity index7.K0 coefficient of “at-rest ”pressure ,for total stressesσ1 andσ28.K0‟ domain fo r effective stressesσ1 … andσ2‟9.K0n K0 for normally consolidated state10.K0u K0 coefficient under rapid continuous loading ,simulating instantaneous loading or an undrained condition11.K0d K0 coefficient under cyclic loading(frequency less than 1Hz),as a pseudo- dynamic test for K0 coefficient12.kh ,kv permeability in horizontal and vertical directions, respectively13.N blow count, standard penetration test14.OCR over-consolidation ratio 15.pc preconsolidation pressure ,from oedemeter test16.p0 effective overburden pressure17.p s specific cone penetration resistance, from static cone test18.qu unconfined compressive strength19.U, Um degree of consolidation ,subscript m denotes mean value of a specimen20.u ,ub ,um pore (water) pressure, subscriptsb and m denote bottom of specimen and mean value, respectively21.w0 wL wp natural water content, liquid and plastic limits, respectively22.σ1,σ2principal stresses, σ1 … andσ2‟ denote effective principal stresses23.Atterberg limits阿太堡界限24.degree of saturation饱和度25.dry unit weight干重度26.moist unit weight湿重度27.saturated unit weight饱和重度28.effective unit weight有效重度29.density密度pactness密实度31.maximum dry density最大干密度32.optimum water content最优含水量33.three phase diagram三相图34.tri-phase soil三相土35.soil fraction粒组36.sieve analysis筛分37.hydrometer analysis比重计分析38.uniformity coefficient不均匀系数39.coefficient of gradation级配系数40.fine-grained soil(silty and clayey)细粒土41.coarse- grained soil(gravelly and sandy)粗粒土42.Unified soil classification system土的统一分类系统43.ASCE=American Society of Civil Engineer 美国土木工程师学会44.AASHTO= American Association State Highway Officials美国州公路官员协会45.ISSMGE=International Society for Soil Mechanics and Geotechnical Engineering 国际土力学与岩土工程学会四. 渗透性和渗流1.Darcy‟s law达西定律2.piping管涌3.flowing soil流土4.sand boiling砂沸5.flow net流网6.seepage渗透（流）7.leakage渗流8.seepage pressure渗透压力9.permeability渗透性10.seepage force渗透力11.hydraulic gradient水力梯度12.coefficient of permeability渗透系数五. 地基应力和变形1.soft soil软土2.(negative) skin friction of driven pile打入桩（负）摩阻力3.effective stress有效应力4.total stress总应力5.field vane shear strength十字板抗剪强度6.low activity低活性7.sensitivity灵敏度8.triaxial test三轴试验9.foundation design基础设计10.recompaction再压缩11.bearing capacity承载力12.soil mass土体13.contact stress (pressure)接触应力（压力）14.concentrated load集中荷载15.a semi-infinite elastic solid半无限弹性体16.homogeneous均质17.isotropic各向同性18.strip footing条基19.square spread footing方形独立基础20.underlying soil (stratum ,strata)下卧层（土）21.dead load =sustained load恒载持续荷载22.live load活载23.short。
a r X i v :m a t h /0112201v 1 [m a t h .D G ] 19 D e c 2001KILLING SPINOR EQUATIONS IN DIMENSION 7AND GEOMETRY OFINTEGRABLE G 2-MANIFOLDSTHOMAS FRIEDRICH AND STEF AN IVANOV Abstract.We compute the scalar curvature of 7-dimensional G 2-manifolds admitting a connection with totally skew-symmetric torsion.We prove the formula for the general so-lution of the Killing spinor equation and express the Riemannian scalar curvature of the solution in terms of the dilation function and the NS 3-form field.In dimension n =7the dilation function involved in the second fermionic string equation has an interpretation as a conformal change of the underlying integrable G 2-structure into a cocalibrated one of pure type W 3.Contents 1.Introduction 12.General properties of G 2-structures 33.Conformal transformations of G 2-structures 44.Connections with torsion,parallel spinors and Riemannian scalar curvature 55.Solutions to the Killing spinor equations in dimension 77References 81.Introduction Riemannian manifolds admitting parallel spinors with respect to a metric connection with totally skew-symmetric torsion became a subject of interest in theoretical and mathematical physics recently.One of the main reasons is that the number of preserved supersymmetries in string theory depends essentially on the number of parallel spinors.In 10-dimensional string theory,the Killing spinor equations with non-constant dilation Φand the 3-form field strength H can be written in the following way [39],(see [25,24,16])(∗)∇Ψ=0,(d Φ−1Received by the editors 1st February 2008.Key words and phrases.G 2-structures,string equations.Supported by the SFB 288”Differential geometry and quantum physics”of the DFG and the European Human Potential Program EDGE,Research Training Network HPRN-CT-2000-00101.S.Ivanov thanks ICTP for the support and excellent environment.12THOMAS FRIEDRICH AND STEFAN IV ANOVproblems in string theory[32,37,21].One possible generalization of Calabi-Yau manifolds, hyper-K¨a hler manifolds,parallel G2-manifolds and parallel Spin(7)-manifolds are manifolds equipped with linear metric connections having skew-symmetric torsion and holonomy con-tained in SU(n),Sp(n),G2,Spin(7).One remarkable fact is that the existence(in small dimen-sions)of a parallel spinor with respect to a metric connection∇with skew-symmetric torsion determines the connection in a unique way if its holonomy group is a subgroup of SU,Sp,G2, provided that some additional differential conditions on the structure are fulfilled[39,16],and always in dimension8for a subgroup of the group Spin(7)[23].The case of16-dimensional Riemannian manifolds with Spin(9)-structure was investigated in[13],homogeneous models are discussed in[2].The existence of∇-parallel spinors in the dimensions4,5,6,7,8is studied in [39,10,24,16,17,23].In dimension7,thefirst consequence is that the manifold should be a G2-manifold with an integrable G2-structure[16],i.e.,the structure group could be reduced to the group G2and the corresponding3-formω3should obey d∗ω3=θ∧∗ω3for some special 1-formθ.In this paper we study solutions to the Killing spinor equations(∗)in dimension 7and the geometry of integrable G2-manifolds.Wefind a formula for the Riemannian scalar curvature in terms of the fundamental3-form.Ourfirst main result is the following Theorem1.1.Let(M,g,ω3)be an integrable G2-manifold with the fundamental3-formω3. The Riemannian scalar curvature Scal g is given in terms of the fundamental3-formω3byScal g=112||T||2+3δθ,(1.1)whereθand T are the Lee form and the torsion of the unique G2-connection given by(1.2)T=−∗dω3+13∗(∗dω3∧ω3)=112·||T||2−6·△Φ,where△Φ=δdΦis the Laplacian.The solution has constant dilation if and only if the G2-structure is cocalibrated of pure type W3.Our proof relies on the existence theorem for a G2-connection with torsion,the Schr¨o dinger-Lichnerowicz formula for the connection with torsion(both established in[16])and the special properties of the Clifford action on the special parallel spinor.KILLING SPINOR EQUATIONS IN DIMENSION7AND GEOMETRY OF INTEGRABLE G2-MANIFOLDS32.General properties of G2-structuresLet us consider R7endowed with an orientation and its standard inner product.Denote an oriented orthonormal basis by e1,...,e7.We shall use the same notation for the dual basis. We denote the monomial e i∧e j∧e k by e ijk.Consider the3-formω3on R7given by(2.6)ω3=e127+e135−e146−e236−e245+e347+e567.The subgroup of SO(7)thatfixesω3is the exceptional Lie group G2.It is a compact,simply-connected,simple Lie group of dimension14[34].The3-formω3corresponds to a real spinor and,therefore,G2is the isotropy group of a non-trivial real spinor.A G2-structure on a7-manifold M7is a reduction of the structure group of the tangent bundle to the exceptional group G2.This can be described geometrically by a nowhere vanishing differential3-formω3 on M7,which can be locally written as(2.6).The3-formω3is called the fundamental form of the G2-manifold M7(see[3])and it determines the metric completely.The action of G2on the tangent space gives an action of G2on k-forms and we obtain the following splitting[11,6]:Λ1(M7)=Λ17,Λ2(M7)=Λ27⊕Λ214,Λ3(M7)=Λ31⊕Λ37⊕Λ327,whereΛ27={α∈Λ2(M7)|∗(α∧ω3)=2α},Λ214={α∈Λ2(M7) ∗(α∧ω3)=−α},Λ37={∗(β∧ω3)|β∈Λ1(M7)},Λ327={γ∈Λ3(M7)|γ∧ω3=0,γ∧∗ω3=0}. Following[8]we consider the1-formθdefined by(2.7)3θ=−∗(∗dω3∧ω3)=∗(δω3∧∗ω3).We shall call this1-form the Lee form associated with a given G2-structure.If the Lee form vanishes,then we shall call the G2-structure balanced.The classification of the different types of G2-structures was worked out by Fernandez-Gray[11],and Cabrera used the Lee form to characterize each of the16classes.An integrable G2-structure(or a structure of type W1⊕W3⊕W4)is characterized by the differential equationd∗ω3=θ∧∗ω3,and a cocalibrated G2-structure is defined by the conditiond∗ω3=0.A cocalibrated G2-structure of pure type W3is characterized by the two condtions d∗ω3= 0,dω3∧ω3=0.Then the following proposition follows immediately.Proposition2.1.If the Lee1-form is closed,then the G2-structure is locally conformal to a balanced G2-structure.We shall call locally conformally parallel G2-manifolds that are not globally conformally parallel strict locally conformally parallel.Example2.1.Any7-dimensional oriented spin Riemannian manifold admits a certain G2-structure,in general a non-parallel one(see for example[29]).Thefirst known examples of complete parallel G2-manifold were constructed by Bryant and Salamon[7],thefirst compact examples by Joyce[26,27,28].There are many known examples of compact nearly parallel G2-manifolds:S7[11],SO(5)/SO(3)[7,35],the Aloff-Wallach spaces N(g,l)=SU(3)/U(1)g,l [9],any Einstein-Sasakian and any3-Sasakian space in dimension7[14,15].There are also some non-regular3-Sasakian manifolds(see[4,5]).Moreover,compact nearly parallel G2-manifolds with large symmetry group are classified in[15].Compact integrable nilmanifolds are constructed and studied in[12].Any minimal hypersurface N in R8admits a cocalibrated G2-structure[11].Moreover,the structure is parallel,nearly parallel,cocalibrated of pure type if and only if the hypersurface N is totally geodesic,totally umbilic or minimal,respectively.4THOMAS FRIEDRICH AND STEFAN IV ANOV3.Conformal transformations of G2-structuresWe study the conformal transformation of G2-structures(see[11]).Proposition3.1.Let¯g=e2f·g,¯ω3=e3f·ω3be a conformal change of a G2-structure(g,ω3) and denote by¯θ,θthe corresponding Lee forms,respectively.Then(3.8)¯θ=θ+4d f.Proof.We have the relationsvol¯g=e7f·vol g,d¯ω3=e3f·(3d f∧ω3+dω3).We calculate¯∗d¯ω3=e4f(∗dω3+3∗(d f∧ω3)),¯∗d¯ω3∧¯ω3=e7f(∗dω3∧ω3−12∗d f),where we used the general identity∗(ω3∧γ)∧ω3=4∗γ,which is valid for any1-formγ. Consequently,we obtain¯θ=−13 ∗(∗dω3∧ω3)−12∗2d f =θ+4d f. Proposition3.1allows us tofind a distinguished G2-structure on a compact7-dimensional G2-manifold.Theorem3.1.Let(M7,g,ω3)be a compact7-dimensional G2-manifold.Then there exists a unique(up to homothety)conformal G2-structure g0=e2f·g,ω30=e3f·ω3such that the corresponding Lee form is coclosed,δ0θ0=0.Proof.We shall use the Gauduchon Theorem for the existence of a distinguished metric on a compact,hermitian or Weyl manifold[19,20].We shall use the expression of this theorem in terms of a Weyl structure(see[40],Appendix1).We consider the Weyl manifold(M7,g,θ,∇W) with the Weyl1-formθ,where∇W is a torsion-free linear connection on M7determined by the condition∇W g=θ⊗g.Applying the Gauduchon Theorem we canfind,in a unique way, a conformal metric g0such that the corresponding Weyl1-form is coclosed with respect to g0. The key point is that,by Proposition3.1,the Lee form transforms under conformal rescaling according to(3.8),which is exactly the transformation of the Weyl1-form under conformal rescaling of the metric¯g=e4f·g.Thus,there exists(up to homothety)a unique conformal G2-structure(g0,ω30)with coclosed Lee form. We shall call the G2-structure with coclosed Lee form the Gauduchon G2-structure. Corollary3.1.Let(M7,g,Φ)be a compact G2-manifold and(g,Φ)be the Gauduchon struc-ture.Then the following formula holds:∗ dδω3∧∗ω3 =||δω3||2.In particular,if the structure is integrable,then∗ dδω3∧∗ω3 =24||θ||2.ing(2.7),we calculate that0=3·δθ=∗d(δω3∧∗ω3)=∗ dδω3∧∗ω3−∗d∗ω3∧d∗ω3 =∗ dδω3∧∗ω3−||δω3||2·vol . If the structure is integrable,then||δω3||2=24||θ||2. Corollary3.2.On a compact G2-manifold with closed Lee form whose Gauduchon G2-structure is not balanced,thefirst Betti number satisfies b1(M)≥1.For integrable G2-manifolds one can define a suitable elliptic complex as well as cohomolgy groups˜H i(M7)(see[12]).Thefirst cohomolgy group is given by˜H1(M7)={α∈Λ1(M7):dα∧∗ω3=0,d∗α=0}.KILLING SPINOR EQUATIONS IN DIMENSION7AND GEOMETRY OF INTEGRABLE G2-MANIFOLDS5 Corollary3.3.On a compact integrable manifold which is not globally conformally balanced, one has˜b1≥1.Proof.By the condition of the theorem the Gauduchon structure has a non-identically zero Lee form.Then0=δω3=∗(dθ∧∗ω3),since the structure is integrable.Adding the condition δθ=0,we obtain˜b1≥1.4.Connections with torsion,parallel spinors and Riemannian scalar curvature The Ricci tensor of an integrable G2-manifold was expressed in principle by the structure form ω3in the paper[16].Here we intend tofind an explicit formula for the Riemannian scalar ing the unique connection with skew-symmetric torsion preserving the given integrable G2-structure found in[16],we apply the Schr¨o dinger-Lichnerowicz formula for the Dirac operator of a metric connection with totally skew-symmetric torsion(see[16])in order to derive the formula for the scalar curvature.First,let us summarize the mentioned results from[16].Theorem4.1.(see[16])Let(M7,g,ω3)be a G2-manifold.Then the following conditions are equivalent:(1)The G2-structure is integrable,i.e.,d∗ω3=θ∧∗ω3;(2)There exists a unique linear connection∇preserving the G2-structure with totally skew-symmetric torsion T given by(4.9)T=−∗dω3+17·(dω3,∗ω3)∗ω3,π47(dω3)=36·λ·Ψ0−θ·Ψ0,λ=−14||T||2.The4-formσT defined by the formulaσT=16THOMAS FRIEDRICH AND STEFAN IV ANOVTheorem 4.2.(see [16])Let Ψbeaparallelspinor with respect to a metric connection ∇with totallyskew-symmetric torsion T on a Riemannian spin manifold M n .Then the following formulas hold3·dT ·Ψ−2·σT ·Ψ+Scal ·Ψ=0,D(T ·Ψ)=dT ·Ψ+δT ·Ψ−2·σT ·Ψ.Proof of Theorem 1.1.Let Ψ0be the ∇-parallel spinor corresponding to the fundamental 3-form ω3.Then the Riemannian Dirac operator D g and the Levi-Civita connection ∇g act on Ψ0by the rule(4.12)∇g X Ψ0=−14·T ·Ψ0=−74·θ·Ψ0,where we used Theorem 4.1.We are going to apply the well known Schr¨o dinger-Lichnerowicz formula [31,38](D g )2=△g +18·D g λ·Ψ0 +364·λ2+94·δθ ·Ψ0−74·dθ·Ψ0+34·n i =1 ∇e i i e i T )·Ψ0−14·δT ·Ψ0−12·||T ||2·Ψ0.Substituting (4.13)and (4.14)into the SL-formula,multiplying the obtained result by Ψ0and taking the real part,we arrive at (4.15) 4916·||θ||2+3322+18·(σT ·Ψ0,Ψ0).On the other hand,using (4.10),we obtainD(T ·Ψ0)=D(76·λ·Ψ0−θ·Ψ0)=76dλ·Ψ0−d ∇θ·Ψ0−δθ·Ψ0=dT ·Ψ0−2σT ·Ψ0+δT ·Ψ0.Multiplying the latterequality by Ψ0and taking the real part,we obtain −δθ·||Ψ0||2=(dT ·Ψ0,Ψ0)−(2σT ·Ψ0,Ψ0).Consequently,Theorem (4.2)and (4.11)imply (4.16) −3·δθ−112·||dω3||2.KILLING SPINOR EQUATIONS IN DIMENSION7AND GEOMETRY OF INTEGRABLE G2-MANIFOLDS7 Proof.In the case of a cocalibrated G2-structure of pure type,the torsion3-form T=−∗dω3. The claim follows from Theorem1.1. Using the results in[11]we derive immediately the following formula,which is essentially the reformulated Gauss equation.Corollary4.4.Let M7be a hypersurface in R8the with second fundamental form S and mean curvature H.Then the Riemannian scalar curvature on M7is given by the formulaScal g=4912·||S0||2,(4.17)where S0is the image of the traceless part of the second fundamental form via the isomorphism S20(R7)→Λ327.In particular,if M is a minimal hypersurface,thenScal g=−12||θ||2since the structure is locally conformally parallel.Then,Theo-rem1.1leads to the formula(4.19)Scal g=156 M||θ||2d vol>0,since the structure is strictly locally conformally parallel.The second assertion is a consequence of Corollary3.2.5.Solutions to the Killing spinor equations in dimension7We consider the Killing spinor equations(∗)in dimension7.The existence of a∇-parallel spinor is equivalent to the existence of a∇-parallel integrable G2-structure and the3-formfield strength H=T is given by(4.9).We now investigate the second Killing spinor equation(∗). Proof of Theorem1.2.LetΨbe an arbitrary∇-parallel such that(dΦ−T)·Ψ=0.The spinor fieldΨdefines a second G2-structureω30such thatΨ=Ψ0is the canonical spinorfield.Since the connection preserves the spinorfieldΨ,it preserves the G2-structureω30,too.On the other hand,the connection preservingω30is unique.Consequently,the torsion T0coincides with the torsion form T and for the G2-structureω30we have∇Ψ0=0,(dΦ−T0)·Ψ0=0.The Clifford action T0·Ψ0depends only on the(Λ31⊕Λ37)-part of ing(4.9)and the algebraic formulas∗(γ∧ω30)·Ψ0=−γ(∗ω30)·Ψ0=−4·γ·Ψ0,ω30·Ψ0=−7·Ψ0we calculate(5.20)T0·Ψ0=−θ·Ψ0−18THOMAS FRIEDRICH AND STEFAN IV ANOVComparing with the second Killing spinor equation(∗)wefind2·dΦ=−β,(dω30,∗ω30)=0 which completes the proof. As a corollary we obtain the result from[21],which states that any solution to both equations (∗)has necessarily the NS three form H=T given by(1.4).A more precise analysis using Proposition3.1and Theorem1.1of the explicit solutions constructed in[21]shows that these solutions are conformally equivalent to a cocalibrated structure of pure type.In other words, the multiplication of the G2-structures(g±,ω3±)by(eΦ·g±,e(3/2)Φ·ω3±)is a new example of a cocalibrated G2-structure of pure type W3,and it is a solution to the Killing spinor equations with constant dilation.The same conclusions are valid for the solutions constructed in[1,37,32]. Theorem1.2allows us to construct a lot of compact solutions to the Killing spinor equations.If the dilation is a globally defined function,then any solution is globally conformally equivalent to a cocalibrated G2-structure of pure type.For example,any conformal transformation of a compact7-dimensional manifold with a Riemannian holonomy group G2constructed by Joyce [26,27]is a solution with non-constant dilation.Another source of solutions are conformal transformations of the cocalibrated G2-structures of pure type W3induced on any minimal hypersurface in R8.Summarizing,we obtain:Corollary5.1.Any solution(M7,g,ω3)to the Killing spinor equations(∗)in dimension7with non-constant globally defined dilation functionΦcomes from a solution with constant dilation by a conformal transformation(g=eΦ·g0,ω3=e(3/2)Φ·ω30),where(g0,ω30)is a cocalibrated G2-structure of pure type W3.References[1]B.Acharya,J.Gauntlett,N.Kim,Fivebranes wrapped on associative three-cycles,to appear in 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