MIXED BOUNDARY CONDITIONS IN EUCLIDEAN QUANTUM GRAVITY
Full-wave invisibility of active devices at all frequencies
a rX iv:mat h /611185v3[mat h.AP]16Mar27Full-wave invisibility of active devices at all frequencies Allan Greenleaf ∗Yaroslav Kurylev †Matti Lassas,‡Gunther Uhlmann §Final revision Abstract There has recently been considerable interest in the possibility,both theoretical and practical,of invisibility (or “cloaking”)from ob-servation by electromagnetic (EM)waves.Here,we prove invisibility with respect to solutions of the Helmholtz and Maxwell’s equations,for several constructions of cloaking devices.The basic idea,as in the papers [GLU2,GLU3,Le,PSS1],is to use a singular transformation that pushes isotropic electromagnetic parameters forward into singu-lar,anisotropic ones.We define the notion of finite energy solutions of the Helmholtz and Maxwell’s equations for such singular electro-magnetic parameters,and study the behavior of the solutions on the entire domain,including the cloaked region and its boundary.Weshow that,neglecting dispersion,the construction of [GLU3,PSS1]cloaks passive objects,i.e.,those without internal currents,at all fre-quencies k.Due to the singularity of the metric,one needs to workwith weak solutions.Analyzing the behavior of such solutions insidethe cloaked region,we show that,depending on the chosen construc-tion,there appear new“hidden”boundary conditions at the surfaceseparating the cloaked and uncloaked regions.We also consider theeffect on invisibility of active devices inside the cloaked region,inter-preted as collections of sources and sinks or internal currents.Whenthese conditions are overdetermined,as happens for Maxwell’s equa-tions,generic internal currents prevent the existence offinite energysolutions and invisibility is compromised.We give two basic constructions for cloaking a region D contained in a domainΩ⊂R n,n≥3,from detection by measurements madeat∂Ωof Cauchy data of waves onΩ.These constructions,the singleand double coatings,correspond to surrounding either just the outerboundary∂D+of the cloaked region,or both∂D+and∂D−,withmetamaterials whose EM material parameters(index of refraction orelectric permittivity and magnetic permeability)are conformal to asingular Riemannian metric onΩ.For the single coating construction,invisibility holds for the Helmholtz equation,but fails for Maxwell’sequations with generic internal currents.However,invisibility can berestored by modifying the single coating construction,by either in-serting a physical surface at∂D−or using the double coating.Whencloaking an infinite cylinder,invisibility results for Maxwell’s equa-tions are valid if the coating material is lined on∂D−with a surfacesatisfying the soft and hard surface(SHS)boundary condition,but ingeneral not without such a lining,even for passive objects.1IntroductionThere has recently been considerable interest[AE,MN,Le,PSS1,MBW] in the possibility,both theoretical and practical,of a region or object be-ing shielded(or“cloaked”)from detection via electromagnetic(EM)waves. [GLU1,§4]established a non-tunnelling result for time-independent Schr¨o dinger operators with highly singular potentials.This can be interpreted as cloaking, at any frequency,with respect to solutions of the Helmholtz equation using a layer of isotropic,negative index of refraction material.[GLU2,GLU3]raised the possibility of passive objects being undetectable,in the context of electri-2cal impedance tomography(EIT).Motivated by consideration of certain de-generating families of Riemannian metrics,families of singular conductivities, i.e.,not bounded below or above,were given and rigorous results obtained for the conductivity equation of electrostatics,i.e.,for waves of frequency zero.A related example of a complete but noncompact two-dimensional Rieman-nian manifold with boundary having the same Dirichlet-to-Neumann map as a compact one was given in[LTU].More recently,there has been exciting work based on the availability of meta-materials which allow fairly arbitrary behavior of EM material parameters. The constructions in[Le]are based on conformal mapping in two dimensions and are justified via change of variables on the exterior of the cloaked region. [PSS1]also proposes a cloaking construction for Maxwell’s equations based on a singular transformation of the original space,again observing that,outside the cloaked region,the solutions of the homogeneous Maxwell equations in the original space become solutions of the transformed equations.The trans-formation used is the same as used previously in[GLU2,GLU3]in the con-text of Calder´o n’s inverse conductivity problem.The paper[PSS2]contains analysis of cloaking on the level of ray-tracing,while full wave numerical sim-ulations are discussed in[CPSSP].Striking positive experimental evidence for cloaking from microwaves has recently been reported in[SMJCPSS]. Since the metamaterials proposed to physically implement these construc-tions need to be fabricated with a given wavelength,or narrow range of wavelengths,in mind,it is natural to consider this problem in the frequency domain.(As in the earlier works,dispersion,i.e.,dependance of the EM material parameters on k,which is certainly present for metamaterials,is neglected.)The question we wish to consider is then whether,at some(or all)frequencies k,these constructions allow cloaking with respect to solutions of the Helm-holtz equation or time-harmonic solutions of Maxwell’s equations.We prove that this indeed is the case for the constructions of[GLU3,PSS1],as long as the object being cloaked is passive;in fact,for the Helmholtz equation, the object can be an active device in the sense of having sources and sinks. On the other hand,for Maxwell’s equations with generic internal currents, invisibility in a physically realistic sense seems highly problematic.We give several ways of augmenting or modifying the original construction so as to obtain invisibility for all internal currents and at all frequencies.3Due to the degeneracy of the equations at the surface of the cloaked re-gion,is important to rigorously consider weak solutions to the Helmholtz and Maxwell’s equations on all of the domain,not just the exterior of the cloaked region.We analyze various constructions for cloaking from obser-vation on the level of physically meaningful EM waves,i.e.,finite energy distributional solutions of the equations,showing that careful formulation of the problem is necessary both mathematically and for correct understanding of the physical phenomena.It turns out that the cloaking structure imposes hidden boundary conditions on such waves at the surface of the cloak.When these conditions are overdetermined,finite energy solutions typically do not exist.The time-domain physical interpretation of this is not entirely clear, but it seems to indicate an accumulation of energy or blow-up of thefields which would compromise the desired cloaking effect.As mentioned earlier,the example in[PSS1]turns out to be a special case of one of the constructions from[GLU2,GLU3],which gave,in dimensions n≥3,counterexamples to uniqueness for Calder´o n’s inverse problem[C]for the conductivity equation.(Such counterexamples have now also been given for n=2[V,KSVW].)Thus,since the equations of electromagnetism(EM) reduce at frequency k=0to the conductivity equation with conductivity parameterσ(x),namely∇·(σ∇u)=0,for the electrical potential u,the invisibility question has already been answered affirmatively in the case of electrostatics.The present work addresses the invisibility problem at all frequencies k=0. We wish to cloak not just a passive object,but rather an active“device”, interpreted as a collection of sources and sinks,or an internal current,within D.Since the boundary value problems in question may fail to have unique solutions(e.g.,when−k2is a Dirichlet eigenvalue on D),it is natural,as in [GLU1],to use the set of Cauchy data at∂Ωof all of the solutions,rather than the Dirichlet-to-Neumann operator on∂Ω,which may not be well-defined. The basic idea of[GLU3,Le,PSS1]is to form new EM material parameters by pushing forward old ones via a singular mapping F.Solutions of the rel-evant equations,Helmholtz or Maxwell,with respect to the old parameters then push forward to solutions of the modified equations with respect to the new parameters outside the cloaked region.However,when dealing with a singular mapping F,the converse is not in general true.This means that out-side D,depending upon the class of solutions considered,there are solutions to the equations with respect to the new parameters which are not the push4forwards of solutions to the equations with the old parameters.Furthermore, it is crucial that the solutions be dealt with on all ofΩ,and not only outside D.Especially when dealing with the cloaking of active devices,this gives rise to the question of what are the proper transmission conditions on∂D,which allow arbitrary internal sources to be made invisible to an external observer. To address these issues rigorously,one needs to make a suitable choice of the class of weak solutions(on all ofΩ)to the singular equations being consid-ered.For both mathematical and,even more so,physical reasons,the weak solutions that are appropriate to consider seem to be the locallyfinite energy solutions;these belong to the Sobolev space H1with respect to the singular volume form| g|1/2dx onΩfor Helmholtz,and L2(Ω,| g|1/2dx)for Maxwell. These considerations are absent from[Le,PSS1,PSS2],where the cloaking is justified by appealing to both the transformation of solutions on the exterior of D under smooth mappings F(essentially the chain rule)and the fact, in the high frequency limit,that rays originating inΩ\D avoid∂D and do not enter D.As we will show,analysis of the transmission conditions at∂D shows that the constructions of[PSS1,PSS2],although adequate for cloaking active devices in the absence of polarization,i.e.,for Helmholtz,and cloaking passive devices in the presence of polarization,i.e.,for Maxwell,fail to admitfinite energy solutions to Maxwell when generic active devices are present.Furthermore,analysis of cloaking of an infinite cylinder,which was numerically explored in[CPSSP]and provides a model of the experimental verification of cloaking in[SMJCPSS],shows that even cloaking a passive object may be problematic.Fortunately,it is possible to remedy the situation by augmenting or modifying this construction.We now describe the results of this paper.For what we call the single coat-ing,which is the construction of[GLU3],and apparently that intended in [PSS1],we establish invisibility with respect to the Helmholtz equation at all frequencies k=0.In fact,one can not only cloak a passive object in a region D⊂⊂Ω,containing material with nonsingular index of refraction n(x),from all measurements made at the boundary∂Ω,but also an active device,interpreted as a collection of sources and sinks within D.Among the phenomena described here is thatfinite energy solutions to the single coating construction must satisfy certain“hidden”boundary condi-tions at∂D.For the Helmholtz equation,this is the Neumann boundary condition at∂D−,and it follows that waves which propagate inside D and are incident to∂D−behave as if the boundary were perfectly reflecting.Thus,5the cloaking structure on the exterior of D produces a“virtual surface”at ∂D−.However,for Maxwell’s equations with electric permittivityε(x)and magnetic permeabilityµ(x),the situation is more complicated.The hidden boundary condition forces the tangential components of both the electric field E and magneticfield H to vanish on∂D−.For cloaking passive objects, for which the internal current J=0,this condition can be satisfied,but for generic J,finite energy time-harmonic solutions fail to exist,and thus the single coating construction is insufficient for invisibility.In practice,even for cloaking passive objects,this may degrade the effective invisibility.Wefind two ways of dealing with this difficulty.One is to simply augment the single coating construction around a ball by adding a perfect electrical conductor(PEC)lining at∂D,in order to make the object inside the coating material to appear like a passive object.(Such a lining was apparently incor-porated,although claimed to be unnecessary,into the code used in[CPSSP] in an effort to stabilize the numerics.)However,for the sake of brevity,the necessary weak formulation of the boundary value problem for this setup will not be considered in this paper.Alternatively,one can introduce a more elaborate construction,which we refer to as the double coating.Mathematically,this corresponds to a singular Riemannian metric which degenerates in the same way as one approaches∂D from both sides;physically it would correspond to surrounding both the inner and outer surfaces of D with appropriately matched metamaterials.We show that,for the double coating,no lining is necessary and full invisibility holds for arbitrary active devices,at all nonzero frequencies,for both Helmholtz and Maxwell.It is even possible for thefield to be identically zero outside of D while nonzero within D,and vice versa.Finally,we also analyze cloaking within an infinitely long cylinder,D⊂R3. In the main result of§7and§8,we show that the cylinder D becomes invisible at all frequencies if we use a double coating together with the so-called soft and hard(SHS)boundary condition on∂D.For the origin and properties of the SHS condition and a description of how the SHS condition can be physically implemented,see[HLS,Ki1,Ki2,Li].We point out that there is some confusion in the physical literature concern-ing the theoretical possibility of invisibility.By this we mean uniqueness theorems for the inverse problem of recovering the electromagnetic param-eters from boundary information(nearfield)or scattering(farfield)at a6single frequency,or for all frequencies.There is a vast literature on this subject.We only mention here mathematical results directly related to the one mentioned in[Le,SMJCPSS].The Helmholtz operator at non-zero en-ergy for isotropic media is given by∆+k2n(x),where n(x)is the index of refraction and k=0.Unique determination of n(x)from boundary data for a single frequency k,under suitable regularity assumptions on n(x)and in dimension n≥3,was proved in[SyU],with a similar result for the acoustic wave equation in[N].(See[U]for a survey of related results).The article [N]was referred to in[Le,SMJCPSS]as showing that perfect invisibility was not possible.However,the results of[SyU,N]for the Helmholtz equation are valid only under the assumption that the medium is isotropic and that the index of refraction is bounded.This does not contradict the possibility of invisibility for an anisotropic index of refraction,nor for an unbounded isotropic index of refraction.The constructions of[GLU3,Le,PSS1]and the present paper violate both of these conditions.We also point out that the counterexamples given in[GLU1,Sec.4]yield invisibility for the Helm-holtz equation,in dimension n≥3,for certain isotropic negative indices of refraction which are highly singular(and negative)onΩ\D.We also note here that,atfixed energy,the Cauchy data is equivalent to the inverse scattering data.The connection between thefixed energy inverse scattering data,the Dirichlet-to-Neumann map and the Cauchy data is dis-cussed,for instance,in[N]for the Schr¨o dinger equation and in[U]for the Helmholtz equation in anisotropic media.The scattering operator is well defined for the degenerate metrics defined here;see,e.g.,[M].There is a large literature(see[U])on uniqueness in the Calder´o n problem for isotropic conductivities under the assumption of positive upper and lower bounds forσ.It was noted by Luc Tartar(see[KV]for an account)that uniqueness fails badly if anisotropic tensors are allowed,since if F:Ωis a smooth diffeomorphism with F|∂Ω=id,then F∗σandσhave the same Dirichlet-to-Neumann map(and Cauchy data.)Note that sinceεandµtransform in the same way,this already constitutes a form of invisibility, i.e.,from the Cauchy data one cannot distinguish between the EM material parameter pairsε,µand ε=F∗ε, µ=F∗µ.Thus,uniqueness for anisotropic media,in the mathematical literature,has come to mean uniqueness up to a pushforward by a(sufficiently regular) map F.Such uniqueness in the Calder´o n problem is known under various regularity assumptions on the anisotropic conductivity in two dimensions7[S,N1,SuU,ALP]and in three dimensions or higher[LaU,LeU,LTU], but for all of these results it is assumed that the eigenvalues ofσ(x)are bounded below and above by positive constants.Related to the Calder´o n problem is the Gel’fand problem,which uses Cauchy data at all frequencies, rather than at afixed one;for this problem,uniqueness results are also available,e.g.,[BeK,KK],with a detailed exposition in[KKL].For example, in the anisotropic inverse conductivity problem as above,Cauchy data at all frequencies determines the tensor up to a diffeomorphism F:Ω. Thus,a key point in the current works on invisibility that allows one to avoid the known uniqueness theorems for the Calder´o n problem is the lack of positive lower and upper bounds on the eigenvalues of these symmetric tensor fields.In this paper,as in[GLU3,Le,PSS1],the lower bound condition is violated near∂D,and there fails to be a global diffeomorphism F relating the pairs of material parameters having the same Cauchy data.For Maxwell’s equations,all of our constructions are made within the context of the permittivity and permeability tensorsεandµbeing conformal to each other,i.e.,multiples of each other by a positive scalar function;this condition has been studied in detail in[KLS].For Maxwell’s equations in the time domain,this condition corresponds to polarization-independent wave velocity.In particular,all isotropic media are included in this category.This seemingly special condition arises naturally from our construction,since the pushforward( ε, µ)of an isotropic pair(ε,µ)by a diffeomorphism need not be isotropic but does satisfy this conformality.For both mathematical and practical reasons,it would be very interesting to understand cloaking for general anisotropic materials in the absence of this assumption.We believe that our results suggest improvements which can be made in phys-ical implementations of cloaking.In the very recent experiment[SMJCPSS], the configuration corresponds to a thin slice of of an infinite cylinder,inside of which a homogeneous,highly conducting disk was placed in order to be cloaked.This corresponds to the single coating with the metric g2(see§2) on D being a constant multiple of the Euclidian metric.The analysis here suggests that lining the inside surface∂D−of the coating with a material implementing the SHS boundary condition[HLS,Ki1,Ki2,Li]should result in less observable scattering than occurs without the SHS lining,improving the partial invisibility already observed.The paper is organized as follows.In§2we describe the single and double8coating constructions.We then establish cloaking for the Helmholtz equation at all frequencies in§3.The notion of afinite energy solution for the single coating is defined in§§3.2and then the key step for showing invisibility is Proposition3.5.We discuss the Helmholtz equation for the double coating In§§3.3;there we define the notion of a weak solution and the Neumann boundary condition at the inner surface of the cloaked region.The key step for invisibility for Helmholtz at all frequencies in the presence of the double coating is Proposition3.11.In§4we study invisibility at all frequencies for Maxwell’s equations.We define the notion offinite energy solutions for the single and double coat-ings.In§5we demonstrate invisibility for Maxwell’s at all frequencies for the double coating;see Proposition5.1.In§6we show that,for the single coating construction,the Cauchy data for Maxwell’s equations must vanish on the surface of the cloaked region,showing that genericallyfinite energy solutions for Maxwell’s equations in the cloaked region do not exist.In§7we consider an infinite cylindrical domain and show invisibility at all frequencies for Maxwell’s equations for the double coating;the key result is Proposition 7.1.In§8,we consider how to cloak the cylinder,treating its surface as an obstacle with the SHS boundary condition.Finally,in§9,we briefly indicate how general the constructions can be made.In particular,we note that a modification the double coating allows one to change the topology of the domain and yet maintain invisibility.We would like to thank Bob Kohn for bringing the papers[Le,PSS1]to our attention,and Ismo Lindell for discussions concerning the SHS boundary condition.2Geometry and basic constructionsThe material parameters of electromagnetism,namely the conductivity,σ(x); electrical permittivity,ε(x);and magnetic permeability,µ(x),all transform as a product of a contravariant symmetric2-tensor and a(+1)−density.That is,if F:Ω1−→Ω2,y=F(x),is a diffeomorphism between domains in R n,thenσ(x)=(σjk(x))onΩ1pushes forward to(F∗σ)(y)onΩ2,given by9(F∗σ)jk(y)=1∂x k (x)]np,q=1∂F j∂x q(x)σpq(x) x=F−1(y),(1)with the same transformation rule for the other material parameters.It was observed by Luc Tartar(se[KV])that it follows that if F is a diffeomorphism of a domainΩfixing∂Ω,thenσand σ:=F∗σhave the same Dirichlet-to-Neumann map,producing infinite-dimensional families of indistinguishable conductivities.On the other hand,a Riemannian metric g=(g jk(x))is a covariant sym-metric two-tensor.Remarkably,in dimension three or higher,a material pa-rameter tensor and a Riemannian metric can be associated with each other byσjk=|g|1/2g jk,or g jk=|σ|2/(n−2)σjk,(2) where(g jk)=(g jk)−1and|g|=det(g).Using this correspondence,examples of singular anisotropic conductivities in R n,n≥3,that are indistinguishable from a constant isotropic conductivity,in that they have the same Dirichlet-to-Neumann map,were given in[GLU3].The two constructions there are based on two different types degenerations of Riemannian metrics,whose singular limits can be considered as coming from singular changes of vari-ables.The singular conductivities arising from these metrics via the above correspondence are then indistinguishable from a constant isotropicσ.In the current paper,we will continue to examine one of these constructions, correpsonnding to pinching offa neck of a Riemannian manifold;we refer to it as the single coating.We also introduce another construction,referred to as the double coating.We start by giving basic examples of each of these. For both examples,letΩ=B(0,2)⊂R3,the ball of radius2and center 0,be the domain at the boundary of which we make our observations;D= B(0,1)⊂Ωthe region to be cloaked;andΣ=∂D=S2the boundary of the cloaked region.Single coating construction:We begin by recalling an example from [GLU3,PSS1];the two dimensional examples in[Le,V]are either essentially the same or closely related in structure.10For the single coating,we blow up0using the mapF1:Ω\2+1)xB(0,2),let(g e)ij=δij be the Euclidian metric,corresponding to constant isotropic material parameters;via the map F1,g e pushes forward,i.e.,pulls back by F−1,to a metric on D,g1=(F1)∗g e:=(F−11)∗(g e).Introducing the boundary normal coordinates(ω,τ)in the annulusΩ\Ω\∂r =x j∂x jand two vectorfields v,w.Then,g1(∂r,∂r)=4, g1(∂r,v)= g1(∂r,w)=0, g1(w,v)=0,(5)g1(v,v)(r−1)2∈[c1,c2],where c1,c2>0.Thus, g1has one eigenvalue bounded from below(with eigenvector corresponding to the radial direction)and two eigenvalues that are of order(r−1)2(with eigenspace span{v,w}).In Euclidean coordinates, we have that,for| g1|=det( g1),| g1(x)|1/2∼C1(r−1)2,(6)| g ij1νi|≤C2,νi=2x(ν1,ν2,ν3)denotes the unit co-normal vectors of the surfaces{x∈Ω\Ω,g= g1,x∈Ω\D;in the sequel, we will identify the metric g and the corresponding pair( g1, g2).To unify notation for the two basic constructions,we will denote in the single coating case M1=Ω,M2=D and let M be the disjoint union M=M1∪M2. Also,for notational unity with the double coating,we letγ1={0}⊂M1,γ2=∅⊂M2,andγ=γ1∪γ2.Moreover,we denote N1=Ω\D=B(0,2)\The standard metric g on S31/πin these coordinates takes the formg2=sin2(πr)∂r =x j∂x jat x∈N2and two vectorsv,w such that in Euclidean metric(∂r,v,w)is a local orthonormal frame. Then,as follows from(7),at x∈N2with,say,1/2<r<1,g2(∂r,∂r)=1, g2(∂r,v)= g2(∂r,w)=0,g2(w,v)=0, g2(v,v)(1−r)2∈[c1,c2],where c1,c2>0.Thus, g2has one eigenvalue bounded from below(with eigenvector corresponding to the radial direction)and two eigenvalues that are of order(1−r)2,and with respect to the Euclidean coordinates on N2,| g2(x)|1/2≤C1(1−r)2,| g ij2νi|≤C2,νi=−x i2<r<1.(8)Set g1=(F1)∗g e on N1,where F1is defined as for the single coating example. Together,these define a singular metric g=( g1, g2)on the entire ball N= N1∪N2∪Σ=B(0,2).Comparing(4)and(7),we see that,in the Fermi coordinates1associated toΣ,| g|1/2 g ij is Lipschitz continuous on N;note also that| g|1/2 g ij is not invertible at∂B(0,1).Although they are distinct,each of these constructions may be summarized as follows.The domainΩ,which we will refer to as N,decomposes as N=N1∪Σ∪N2,where N1=Ω\1Recall that the Fermi coordinates associated toΣare(ω,τ),whereω=(ω1,ω2)are local coordinates onΣandτ=τ(x)is the distance from x toΣwith respect to the metric g,multiplied by+1in N1and−1in N2.13g=( g1, g2),arising as the pushforward of a(nonsingular)Riemannian metric g=(g1,g2)on a manifold with two components,M=M1∪M2,via a map F:M\γ−→N,F(x)= F1(x),x∈M1\γ,F2(x),x∈M2\γ.Here,M1and M2are disjoint,with N;γ1=γ∩M1is either a point(the point being blown up)for the single and double coatings, or a line(for the cloaking of an infinite cylinder in§7,8);andγ2=γ∩M2is either empty(for the single coating)or a point(for the double coating)or a line(for the cylinder.)In§9,we will show that such constructions exist in great generality,and for this reason the proofs will be expressed in terms of analysis on M and N.In this generality,we say that(M,N,F,γ,Σ,g)is a coating construction if (M,g)is a(nonsingular)Riemannian manifold,γ⊂M andΣ⊂N are as above,and F:M\γ→N\Σis diffeomorphism of either type.This then defines a singular Riemannian metric g everywhere on N\Σ=N1∪N2,byg= g1:=F1∗g1,x∈N1,g2:=F2∗g2,x∈N2.If we introduce Fermi coordinates(ω,τ)nearΣas above,the g satisfies (5),(6)or(8),with r−1replaced byτ,for the single and double coatings, resp.From these,one sees that| g|1/2g jk has a jump discontinuity acrossΣfor the single coating and is Lipschitz for the double coating.Note that in both examples,N=Ω=B(0,2),so that N and M1have the same topology. However,in a direct extension of the double coating construction,described in§9,the domain N containing the cloaked region N2need not even be diffeomorphic to M1.3The Helmholtz equationWe are interested in invisibility of a cloaked region with respect to the Cauchy data of solutions of the Helmholtz equation,(∆g+k2)u=f inΩ,(9)14where f represents a collection of sources and sinks.The Cauchy data C k g,f consists of the set of pairs of boundary measurements(u|∂Ω,∂νu|∂Ω)where u ranges over solutions to(9)in some function or distribution space(discussed below).Let(M,N,F,γ,Σ,g)be a single coating construction as in§2.For the moment,as in the Introduction,we continue to refer to N asΩ,N2as D andΣ+as∂D+;we may assume that M1=N,M2=D and F2=id, so that g2=g2is a(nonsingular)Riemannian metric on D.Thus, g is a Riemannian metric onΩ,singular onΩ\D,resulting from blowing up the metric g1onΩwith respect to a point O and inserting the(D,g2)into the resulting“hole”.We wish to show that C ke g,ef =C k g,0for all frequencies0<k<∞,if supp( f)⊂D and k is not a Neumann eigenvalue of(D,g2).Due to the singularity of g,it is necessary to consider nonclassical solutions to(9),and we will see that the exact notion of weak solution is crucial.Furthermore,a hidden Neumann boundary condition on∂D−is required for the existence offinite energy solutions.Physically,this means that the coating onΩ\。
英语数学词汇E
数学专业词汇对照以字母 E 开头eccentric angle 离心角eccentric angle of an ellipse 椭圆的离心角eccentric anomaly 离心角eccentricity 离心率eccentricity of a hyperbola 双曲线的离心率echelon matrix 梯阵econometrics 计量经济学eddy 涡流旋涡edge 边edge connectivity 边连通度edge homomorphism 边缘同态edge of a solid 立体棱edge of regression 回归边缘edge of the wedge theorem 楔的边定理editcyclic markov chain 循环马尔可夫链effective area 有效面积effective convergence 有效收敛effective cross section 有效截面effective differential cross section 有效微分截面effective divisor 非负除数effective interest rate 有效利率effective number of replications 有效重复数effective variance of error 有效误差方差effectively computable function 能行可计算函数efficiency 效率efficiency factor 效率因子efficient estimator 有效估计量efficient point 有效点egyptian numerals 埃及数字eigenelement 特摘素eigenfunction 特寨数eigenspace 本照间eigenvalue 矩阵的特盏eigenvalue problem 特盏问题eigenvector 特镇量eigenvector of linear operator 线性算子的特镇量einstein equation 爱因斯坦方程einstein metric 爱因斯坦度量elastic coefficient 弹性常数elastic constant 弹性常数elastic deformation 弹性变形elastic limit 弹性限度elastic modulus 弹性模数elastic scattering 弹性散射elasticity 弹性elastodynamics 弹性动力学electrodynamics 电动力学electromagnetism 电磁electronic computer 电子计算机electronic data processing machine 电子数据处理机electronics 电子学electrostatics 静电学element 元件element of area 面积元素element of best approximation 最佳逼近元素element of finite order 有限阶元素element of surface 面元素elementary 基本的elementary chain 初等链elementary circuit 基本回路elementary conjunction 基本合取式elementary divisor 初等因子elementary divisor theorem 初等因子定理elementary event 简单事件elementary formula 原始公式elementary function 初等函数elementary geometry 初等几何elementary matrix 初等阵elementary number theory 初等数论elementary operation 初等运算elementary path 有向通路elementary set 初等集elementary subdivision 初等重分elementary symmetric function 初等对称函数elementary theory of numbers 初等数沦elevation 正视图eliminant 结式eliminate 消去elimination 消去elimination by substitution 代入消元法elimination method 消元法elimination of unknowns 未知数消去elimination theorem 消去定理ellipse 椭圆ellipse of deformation 变形椭圆ellipsograph 椭圆规ellipsoid 椭面ellipsoid of inertia 惯性椭球ellipsoid of revolution 回转椭面ellipsoid of rotation 回转椭面ellipsoidal 椭面的ellipsoidal coordinates 椭球面]坐标ellipsoidal harmonics 椭球低函数elliptic catenary 椭圆悬链线elliptic coordinates 椭圆坐标elliptic curve 椭圆曲线elliptic cylinder 椭圆柱elliptic cylinder function 椭圆柱函数elliptic differential operator 椭圆型微分算子elliptic equation 椭圆型微分方程elliptic function 椭圆函数elliptic function of the second kind 第二类椭圆函数elliptic function of the third kind 第三类椭圆函数elliptic geometry 椭圆几何elliptic integral 椭圆积分elliptic irrational function 椭圆无理函数elliptic modular function 椭圆模函数elliptic modular group 椭圆模群elliptic motion 椭圆运动elliptic orbit 椭圆轨道elliptic paraboloid 椭圆抛物面elliptic point 椭圆点elliptic quartic curve 椭圆四次曲线elliptic space 椭圆空间elliptic surface 椭圆曲面elliptic system 椭圆型方程组elliptic type 椭圆型ellipticity 椭圆率elongation 伸长embedding 嵌入embedding operator 嵌入算子embedding theorem 嵌入定理empirical curve 经验曲线empirical distribution curve 经验分布曲线empirical distribution function 经验分布函数empirical formula 经验公式empty mapping 空映射empty relation 零关系empty set 空集end 端end around carry 循环进位end device 输出设备endless 无穷的endomorphism 自同态endomorphism group 自同态群endomorphism ring 自同态环endpoint 端点energetic inequality 能量不等式energy 能量energy barrier 能量障碍energy distribution 能量分布energy integral 能量积分energy level 能级energy method 能量法energy momentum tensor 能量动量张量energy norm 能量范数energy operator 能量算子energy principle 能量原理energy space 能量空间energy surface 能面enlarge 扩大enneagon 九边形enriques surface 讹凯斯面ensemble 总体entire 整个的entire function 整函数entire modular form 整模形式entire rational function 整有理函数entire series 整级数entire transcendental function 整超越函数entrance angle 入射角entropy 熵enumerability 可数性enumerable 可数的enumerable set 可数集enumerate 列举enumeration 列举enumeration data 计数数据enumeration problem 列举问题enumerative geometry 枚举几何envelope 包络线envelope of holomorphy 正则包enveloping algebra 包络代数enveloping ring 包络环enveloping surface 包络面epicycle 周转圆epicycloid 外摆线epicycloidal 圆外旋轮线的epimorphic 满射的epimorphic image 满射像epimorphism 满射epitrochoid 长短辐圆外旋轮线epitrochoidal curve 圆外旋轮曲线epsilon chain 链epsilon function 函数epsilon map 映射epsilon neighborhood 邻域epsilon net 网epsilonnumber 数equal 相等的equal set 相等集equal sign 等号equality 等式equality constraint 等式约束equalization 平衡化;同等化equally possible event 相等可能事件equate 使...相等equation 方程equation of a circle 圆方程equation of a curve 曲线方程equation of continuity 连续方程equation of heat conduction 热传导方程equation of higher order 高阶方程式equation of jacobi 雅可比方程equation of mixed type 混合型方程equation of motion 运动方程equation of state 状态方程equation of the straight line 直线方程equation root 方程的根equation with integral coefficients 整系数方程equatorial coordinates 赤道座标equatorial radius 赤道半径equi asymptotic stability 等度渐近稳定性equi luminosity curve 均匀光度曲线equiangular 等角的equiangular spiral 对数螺线equiareal 保积的equiconjugate diameter 等共轭直径equicontinuity 同等连续性equicontinuous 等度连续的equicontinuous functions 等度连续函数equicontinuous set 等度连续集equiconvergence 同等收敛性equidimensional ideal 纯理想equidistant 等距的equidistant curve 等距曲线equilateral 等边的equilateral cone 等边锥面equilateral hyperbola 等轴双曲线equilateral triangle 等边三角形equilibrium 平衡equilibrium concentration 平衡浓度equilibrium conditions 平衡条件equilibrium constant 平衡常数equilibrium diagram 平衡图equilibrium point 平衡点equilibrium principle 平衡原理equilibrium state 平衡状态equipartition 匀分equipotent 等势的;对等的equipotent set 等势集equipotential 等势的equipotential line 等位线equipotential surface 等位面equivalence 等价equivalence class 等价类equivalence problem 等价问题equivalence relation 等价关系equivalent 等价的equivalent equation 等价方程equivalent fiber bundle in g g 等价纤维丛equivalent form 等价形式equivalent functions 等价函数equivalent knot 等价纽结equivalent mapping 保面积映射equivalent matrix 等价阵equivalent metric 等价度量equivalent neighborhood system 等价邻域系equivalent norm 等价范数equivalent point 等价点equivalent proposition 等值命题equivalent states 等价状态equivalent stochastic process 等价随机过程equivalent terms 等价项equivalent transformation 初等运算equivariant map 等变化映射erasing 擦除ergodic chain 遍历马尔可夫链ergodic hypothesis 遍历假说ergodic markov chain 遍历马尔可夫链ergodic property 遍历性ergodic state 遍历态ergodic theorem 遍历定理ergodic theorem in the mean 平均遍历定理ergodic theory 遍历理论ergodic transformation 遍历变换ergodicity 遍历性error 误差error analysis 误差分析error band 误差范围error coefficient 误差系数error component 误差分量error curve 误差曲线error equation 误差方程error estimation 误差估计error function 误差函数error in the input data 输入数据误差error law 误差律error limit 误差界限error mean square 误差方差error model 误差模型error of estimation 估计误差error of first kind 第一类误差error of measurement 测量误差error of observation 观测误差error of reading 读数误差error of second kind 第二类误差error of the third kind 第三类误差error of truncation 舍位误差error originated from input 输入误差error probability 误差概率error sum of squares 误差平方和error variance 误差方差escribe 旁切escribed 旁切的escribed circle 旁切圆essential 本性的essential boundary condition 本质边界条件essential convergence 本质收敛essential epimorphism 本质满射essential extension 本质开拓essential homomorphism 本质同态essential inferior limit 本质下极限essential infimum 本性下确界essential parameter 本质参数essential point 本质点essential singular kernel 本性奇核essential singularity 本性奇点essential spectrum 本质谱essential strategy 本质策略essential superior limit 本质上极限essential supremum 本性上确界essential undecidability 本质不可判定性essentially bounded 本质有界的essentially convergent sequence 本质收敛序列essentially self adjoint operator 本质自伴算子estimable function 可估计函数estimable hypothesis 可估计假设estimate 估计estimation 估计estimation of error 误差估计estimation of parameter 参数的估计estimation region 估计区域estimation theory 估计论estimator 估计量etale neighborhood 层邻域etale space 层空间etale topology 层拓扑etalon 标准euclid factorization theorem for rational integers 因子分解定理euclid lemma 欧几里得引理euclid parallel postulate 欧几里得平行公设euclidean algorithm 欧几里得算法euclidean domain 欧几里得整环euclidean geometry 欧几里得几何euclidean metric 欧几里得度量euclidean norm 欧几里得范数euclidean plane 欧几里得平面euclidean ring 欧几里得整环euclidean space 欧几里得空间euclidean vector space 欧几里得向量空间euler characteristic 欧拉示性数euler class 欧拉类euler constant 欧拉常数euler criterion 欧拉判别准则euler differential equation 欧拉微分方程euler formula 欧拉公式euler identity 欧拉恒等式euler number 欧拉数euler poincare formula 欧拉庞加莱公式euler poincare relation 欧拉庞加莱公式euler polyhedron theorem 欧拉多面体定理euler polynomial 欧拉多项式euler summation formula 欧拉总和公式eulerian angle 欧拉角evaluate 求...的值evaluation 计算evaluation of functions 函数值计算evaluation of integrals 积分计算even 偶数的even function 偶函数even number 偶数even parity 偶数同位even permutation 偶置换evenness 偶数性event 事件everywhere convergent sequence 处处收敛序列evidence 迷evident 迷的evolute 缩闭线evolute surface 渐屈面evolution 开方evolution equation 发展方程evolvent 渐伸线exact cohomology sequence 正合上同凋列exact differential equation 全微分方程exact division 正合除法exact homotopy sequence 正合同伦序列exact solution 精确解exact square 正合平方exactitude 精确度exactness axiom 正合性公理example 例exceed 超过excenter 外心exceptional curve 例外曲线exceptional jordan algebra 例外约当代数exceptional point 例外点exceptional value 例外值excess 超过excess function 超过函数excess of nine 舍九法exchange 交换exchange integral 交换积分exchange lattice 交换格exchange theorem 交换定理excircle 旁切圆excision 切除excision isomorphism 切除同构exclude 排除exclusion 排除exclusive disjunction 不可兼析取exclusive events 互斥事件exclusive or 不可兼的或executive program 执行程序exist 存在existence 存在existence conditions 存在条件existence of extremum 极值的存在existence theorem 存在定理existence theorem for roots 根的存在性定理existence theorem of implicit function 隐函数的存在性定理existential quantifier 存在量词exogenous variable 局外变量exotic space 异种空间expactation vector 期望值向量expand 展开expansion 展开expansion coefficient 展开系数expansion in series 级数展开expansion in terms of eigenfunction 本寨数展开expansion of a determinant 行列式的展开expansion theorem 展开定理expectation 期望值expected gain 期望增益expected payoff 期望增益expected value 期望值expected value vector 期望值向量experiment 实验experimental 实验的experimental error 实验误差explicit difference scheme 显式差分格式explicit differential equation 显式微分方程explicit function 显函数exponent 指数exponent notation 指数记法exponent of convergence 收敛指数exponential 指数函数exponential curve 指数曲线exponential distribution 指数分布exponential equation 指数方程exponential family 指数族exponential form of complex number 复数的指数形式exponential fourier transformation 指数型傅里叶变换exponential function 指数函数exponential integral 积分指数exponential law 指数定律exponential map 指数映射exponential p adic valuation 指数p 进赋值exponential process 指数过程exponential series 指数级数exponential sum 指数和exponential type 指数型exponential valuation 指数赋值exponentially asymptotic stability 指数式渐近稳定exportation 输出express 表示expression 式exradius 外圆半径extend 扩大extended commutator 广义换位子extended complex plane 扩张平面extended ideal 广义理想extended mean value theorem 广义均值定理extended plane 扩张平面extended point transformation 开拓的点变换extended predicate calculus 广义谓词演算extended riemann hypothesis 广义黎曼假设extended unitary group 广义酉群extension 扩张extension module 扩张模extension of a field 域的扩张extension of the residue field 剩余域的扩张extension principle of propositional logic 命题逻辑的外延性原理extension theorem 扩张定理extensionality 外延extensive quantity 外延量extent 范围exterior 外exterior algebra 外代数exterior angle 外角exterior approximation 外逼近exterior boundary problem 外边界问题exterior derivative 外导数exterior differential 外微分exterior differential form 外微分形式exterior differentiation 外微分法exterior domain 外域exterior interior angles 同位角exterior multiplication 外乘exterior normal 外法线exterior point 外点exterior power 外幂exterior problem 外边界问题exterior product 外积exterior product of tensors 张量的外积external 外部的external composition 外部合成external composition law 外部合成律external direct sum 外直和external division 外分external law of composition 外部合成律external memory 外存储器external program 外部程序external ratio 外分比external store 外存储器externally stable set 控制集externally tangent 外切的extract 开方extraction of a root 开方extraneous root 额外根extrapolate 外推extrapolation 外插extremal 极值曲线;极值的extremal element 极值元素extremal function 极值函数extremal length 极值长度extremal point 极值点extremal property 极值性质extremal surface 极值曲面extreme 外项extreme form 极型extreme point 极值点extreme term 外项extreme value 极值extreme value distribution 极值分布extreme value problem 极值问题extremity 端extremum 极值extremum conditions 极值条件extremum problem with subsidiary condition 附加条件极值问题extremum with a condition 条件极值extremum with a constraint 条件极值。
Black hole entropy in 3D gravity with torsion
Following the analogy with Poincar´ e gauge theory [11,12], Euclidean gravity with torsion in 3D can be formulated as a gauge theory of the Euclidean group E (3) = ISO (3) (EGT for short), the analytic continuation of the Poincar´ e group P (3) = ISO (1, 2). The underlying geometric structure is described by Riemann-Cartan space. EGT in brief. Basic gravitational variables in EGT are the triad field bi and the spin connection Aij = −Aji (1-forms). The corresponding field strengths are the torsion and the curvature: T i = dbi + Ai m ∧ bm , Rij = dAij + Ai m ∧ Amj (2-forms). Gauge symmetries of the theory are local translations and local rotations, parametrized by ξ µ and εij . In 3D, we can simplify the notation by introducing Aij = −εijk ωk , Rij = −εijk Rk , εij = −εijk θk .
多复变典型域上Poisson-华积分与Cauchy积分的边界性质
北京师范大学博士学位论文论文题目：多复变典型域上Poisson-华积分与Cauchy积分的边界性质作者：***导师：郑学安教授系别、年级：数学系97博学科、专业：基础数学多复变函数论完成日期：2000年5月北京师范大学研究生院多复变典型域上Poisson-华积分与Cauchy积分的边界性质目 录致谢 1 摘要..........................................................................................（2） Abstract .......................................................................................（3） 总论 (5)第一章 有界对称域上Poisson-华积分与Cauchy 积分...........................（8） §1.0 引言.................................................................................（8） §1.1 Poisson-华积分与Cauchy 积分 (13)第二章 复超球上Poisson-华积分与Cauchy 积分的边界性质 (16)第三章 第一类典型域上Poisson-华积分与Cauchy 积分的边界性质......（24） §3.0 引言..............................................................................（24） §3.1 R Ⅰ（m,n）及L Ⅰ（m,n）的分解 (26)§3.2 R Ⅰ（m,n）的不变Riemann 度量与体积元的计算 (29)§3.3 核函数的估计及Poisson-华积分与Cauchy 积分的边界性质......（41） §3.4 R Ⅰ上的Dirichlet 问题 (52)第四章 第二类典型域上Poisson-华积分与Cauchy 积分的边界性质......（54） §4.0 前言..............................................................................（54） §4.1 第二类典型域Harish--Chandra 模型及其Silov 边界的体积元...（56） §4.2 Poisson-华积分与Cauchy 积分的边界性质 (59)第五章 第三类典型域上Poisson-华积分与Cauchy 积分的边界性质......（65） §5.0 前言..............................................................................（65） §5.1 第三类典型域上Harish--Chandra 模型及其Silov 边界的体积元...（67） §5.2 Poisson-华积分与Cauchy 积分的边界性质 (71)参考文献 (78)本文是在导师郑学安教授悉心指导下完成的，作者对他多年来的关心、鼓励和耐心教导深表感谢，是他才使我走到今天。
A Beginner’s Course in Boundary Element Methods
Chapters 1 and 5 in“A Beginner’s Course in Boundary Element Methods”The materials in this document are taken from an earlier manuscript of the book “A Beginner’s Course in Boundary Element Methods”. The page numbers and the table of contents here do not correspond exactly to those in the published book.Details of the published book are as follows:WT Ang, A Beginner’s Course in Boundary Element Methods, Universal Publishers, Boca Raton, USA, 2007 (253 pages)./book.php?method=ISBN&book=1581129742____________________________________________________________________© 2007 WT AngComments on this document may be sent to the author at wtang@.sg.No part of this document may be reproduced and distributed in any form or used in any way for material profits without prior written permission of the author.The author does not bear any responsibility for injuries and/or damages to persons or properties that arise from the use of ideas or methods contained in this document.Contents1Two—dimensional Laplace’s Equation11.1Introduction (1)1.2Fundamental Solution (3)1.3Reciprocal Relation (4)1.4Boundary Integral Solution (5)1.5Boundary Element Solution with Constant Elements (10)1.6Formulae for Integrals of Constant Elements (12)1.7Implementation on Computer (15)1.8Numerical Examples (24)1.9Summary and Discussion (33)1.10Exercises (34)2Discontinuous Linear Elements372.1Introduction (37)2.2Boundary Element Solution with Discontinuous Linear Elements (38)2.3Formulae for Integrals of Discontinuous Linear Elements (41)2.4Implementation on Computer (44)2.5Numerical Examples (48)2.6Summary and Discussion (54)2.7Exercises (54)3Two-dimensional Helmholtz Type Equation573.1Introduction (57)3.2Homogeneous Helmholtz Equation (58)3.2.1Fundamental Solution (58)3.2.2Boundary Integral Solution (59)3.2.3Numerical Procedure (60)3.2.4Implementation on Computer (62)3.3Helmholtz Type Equation with Variable Coefficients (71)3.3.1Integral Formulation (71)3.3.2Approximation of Domain Integral (73)3.3.3Dual-reciprocity Boundary Element Method (75)2CONTENTS3.3.4Implementation on Computer (77)3.4Summary and Discussion (87)3.5Exercises (88)4Two-dimensional Diffusion Equation914.1Introduction (91)4.2Dual-reciprocity Boundary Element Solution (93)4.3Time-stepping Approach (96)4.4Implementation on Computer (100)4.5Numerical Examples (106)4.6Summary and Discussion (111)4.7Exercises (113)5Green’s Functions for Potential Problems1155.1Introduction (115)5.2Half Plane (116)5.2.1Two Special Green’s Functions (116)5.2.2Applications (118)5.3Infinitely Long Strip (132)5.3.1Derivation of Green’s Functions by Conformal Mapping (132)5.3.2Applications (136)5.4Exterior Region of a Circle (140)5.4.1Two Special Green’s Functions (140)5.4.2Applications (142)5.5Summary and Discussion (145)5.6Exercises (145)6Three-dimensional Problems1476.1Introduction (147)6.2Laplace’s Equation (147)6.2.1Boundary Value Problem (147)6.2.2Fundamental Solution (147)6.2.3Reciprocal Relation (148)6.2.4Boundary Integral Equation (148)6.2.5Boundary Element Method (151)6.2.6Computation of D(k)1(ξ,η,ζ)and D(k)2(ξ,η,ζ) (152)6.2.7Implementation on Computer (155)6.3Homogeneous Helmholtz Equation (167)6.4Helmholtz Type Equation with Variable Coefficients (173)6.4.1Dual-reciprocity Boundary Element Method (173)6.4.2Implementation on Computer (176)6.5Summary and Discussion (184)6.6Exercises (184)Chapter1TWO—DIMENSIONAL LAPLACE’S EQUATION1.1IntroductionPerhaps a good starting point for introducing boundary element methods is through solving boundary value problems governed by the two-dimensional Laplace’s equation∂2φ∂x2+∂2φ∂y2=0.(1.1)The Laplace’s equation occurs in the formulation of problems in many diverse fields of studies in engineering and physical sciences,such as thermostatics,elasto-statics,electrostatics,magnetostatics,idealfluidflow andflow in porous media.An interior boundary value problem which is of practical interest requires solving Eq.(1.1)in the two-dimensional region R(on the Oxy plane)bounded by a simple closed curve C subject to the boundary conditionsφ=f1(x,y)for(x,y)∈C1,∂φ=f2(x,y)for(x,y)∈C2,(1.2)where f1and f2are suitably prescribed functions and C1and C2are non-intersecting curves such that C1∪C2=C.Refer to Figure1.1for a geometrical sketch of the problem.The normal derivative∂φ/∂n in Eq.(1.2)is defined by∂φ=n x ∂φ+n y∂φ,(1.3)where n x and n y are respectively the x and y components of a unit normal vector to the curve C.Here the unit normal vector[n x,n y]on C is taken to be pointing away from the region R.Note that the normal vector may vary from point to point on C. Thus,[n x,n y]is a function of x and y.The boundary conditions given in Eq.(1.2)are assumed to be properly posed so that the boundary value problem has a unique solution,that is,it is assumed that one can alwaysfind a functionφ(x,y)satisfying Eqs.(1.1)-(1.2)and that there is only one such function.2Two—dimensional Laplace’s EquationFigure1.1For a particular example of practical situations involving the boundary value problem above,one may mention the classical heat conduction problem whereφdenotes the steady-state temperature in an isotropic solid.Eq.(1.1)is then the temperature governing equation derived,under certain assumptions,from the law of conservation of heat energy together with the Fourier’s heatflux model.The heat flux out of the region R across the boundary C is given by−κ∂φ/∂n,whereκis the thermal heat conductivity of the solid.Thus,the boundary conditions in Eq.(1.2) imply that at each and every given point on C either the temperature or the heat flux(but not both)is known.To determine the temperaturefield in the solid,one has to solve Eq.(1.1)in R tofind the solution that satisfies the prescribed boundary conditions on C.In general,it is difficult(if not impossible)to solve exactly the boundary value problem defined by Eqs.(1.1)-(1.2).The mathematical complexity involved depends on the geometrical shape of the region R and the boundary conditions given in Eq.(1.2).Exact solutions can only be found for relatively simple geometries of R(such as a square region)together with particular boundary conditions.For more complicated geometries or general boundary conditions,one may have to resort to numerical(approximate)techniques for solving Eqs.(1.1)-(1.2).This chapter introduces a boundary element method for the numerical solution of the interior boundary value problem defined by Eqs.(1.1)-(1.2).We show how a boundary integral solution can be derived for Eq.(1.1)and applied to obtain a simple boundary element procedure for approximately solving the boundary value problem under consideration.The implementation of the numerical procedure on the computer,achieved through coding in FORTRAN77,is discussed in detail.Fundamental Solution31.2Fundamental SolutionIf we use polar coordinates r andθcentered about(0,0),as defined by x=r cosθand y=r sinθ,and introduceψ(r,θ)=φ(r cosθ,r sinθ),we can rewrite Eq.(1.1)as1 r ∂∂r(r∂ψ∂r)+1r2∂2ψ∂θ2=0.(1.4)For the case in whichψis independent ofθ,that is,ifψis a function of r alone,Eq.(1.4)reduces to the ordinary differential equationd (r d[ψ(r)])=0for r=0.(1.5)The ordinary differential equation in Eq.(1.5)can be easily integrated twice to yield the general solutionψ(r)=A ln(r)+B,(1.6) where A and B are arbitrary constants.From(1.6),it is obvious that the two-dimensional Laplace’s equation in Eq.(1.1)admits a class of particular solutions given byφ(x,y)=A ln p x2+y2+B for(x,y)=(0,0).(1.7)If we choose the constants A and B in(1.7)to be1/(2π)and0respectively and shift the center of the polar coordinates from(0,0)to the general point(ξ,η),a particular solution of Eq.(1.1)isφ(x,y)=12πln p for(x,y)=(ξ,η).(1.8)As we shall see,the particular solution in Eq.(1.8)plays an important role in the development of boundary element methods for the numerical solution of the interior boundary value problem defined by Eqs.(1.1)-(1.2).We specially denote this particular solution using the symbolΦ(x,y;ξ,η),that is,we writeΦ(x,y;ξ,η)=1ln[(x−ξ)2+(y−η)2].(1.9)We refer toΦ(x,y;ξ,η)in Eq.(1.9)as the fundamental solution of the two-dimensional Laplace’s equation.Note thatΦ(x,y;ξ,η)satisfies Eq.(1.1)everywhere except at(ξ,η)where it is not well defined.4Two—dimensional Laplace’s Equation1.3Reciprocal RelationIfφ1andφ2are any two solutions of Eq.(1.1)in the region R bounded by the simple closed curve C then it can be shown thatZ C(φ2∂φ1−φ1∂φ2)ds(x,y)=0.(1.10)Eq.(1.10)provides a reciprocal relation between any two solutions of the Laplace’s equation in the region R bounded by the curve C.It may be derived from the two-dimensional version of the Gauss-Ostrogradskii(divergence)theorem as explained below.According to the divergence theorem,if F=u(x,y)i+v(x,y)j is a well defined vector function such that∇·F=∂u/∂x+∂v/∂y exists in the region R bounded by the simple closed curve C thenZ C F·n ds(x,y)=ZZ R∇·F dxdy,that is,Z C[un x+vn y]ds(x,y)=ZZ R[∂u+∂v]dxdy,where n=[n x,n y]is the unit normal vector to the curve C,pointing away from the region R.Sinceφ1andφ2are solutions of Eq.(1.1),we may write∂2φ1∂x2+∂2φ1∂y2=0,∂2φ2∂x2+∂2φ2∂y2=0.If we multiply thefirst equation byφ2and the second one byφ1and take the difference of the resulting equations,we obtain∂∂x (φ2∂φ1∂x−φ1∂φ2∂x)+∂∂y(φ2∂φ1∂y−φ1∂φ2∂y)=0,which can be integrated over R to giveZZ R[∂(φ2∂φ1−φ1∂φ2)+∂(φ2∂φ1−φ1∂φ2)]dxdy=0.Boundary Integral Solution5Application of the divergence theorem to convert the double integral over R into a line integral over C yieldsZ C[(φ2∂φ1−φ1∂φ2)n x+(φ2∂φ1−φ1∂φ2)n y]ds(x,y)=0which is essentially Eq.(1.10).Together with the fundamental solution given by Eq.(1.9),the reciprocal relation in Eq.(1.10)can be used to derive a useful boundary integral solution for the two-dimensional Laplace’s equation.1.4Boundary Integral SolutionLet us takeφ1=Φ(x,y;ξ,η)(the fundamental solution as defined in Eq.(1.9)) andφ2=φ,whereφis the required solution of the interior boundary value problem defined by Eqs.(1.1)-(1.2).SinceΦ(x,y;ξ,η)is not well defined at the point(ξ,η),the reciprocal relation in Eq.(1.10)is valid forφ1=Φ(x,y;ξ,η)andφ2=φonly if(ξ,η)does not lie in the region R∪C.Thus,Z C[φ(x,y)∂(Φ(x,y;ξ,η))−Φ(x,y;ξ,η)∂(φ(x,y))]ds(x,y)=0for(ξ,η)/∈R∪C.(1.11)A more interesting and useful integral equation than Eq.(1.11)can be derived from Eq.(1.10)if we take the point(ξ,η)to lie in the region R∪C.For the case in which(ξ,η)lies in the interior of R,Eq.(1.10)is valid if we replace C by C∪Cε,where Cεis a circle of center(ξ,η)and radiusεas shown in Figure1.2∗.This is becauseΦ(x,y;ξ,η)and itsfirst order partial derivatives(with respect to x or y)are well defined in the region between C and Cε.Thus,for C and Cεin Figure1.2,we can writeZ C∪Cε[φ(x,y)∂∂n(Φ(x,y;ξ,η))−Φ(x,y;ξ,η)∂∂n(φ(x,y))]ds(x,y)=0,that is,Z C[φ(x,y)∂∂n(Φ(x,y;ξ,η))−Φ(x,y;ξ,η)∂∂n(φ(x,y))]ds(x,y)=−Z Cε[φ(x,y)∂∂n(Φ(x,y;ξ,η))−Φ(x,y;ξ,η)∂∂n(φ(x,y))]ds(x,y).(1.12)∗The divergence theorem is not only applicable for simply connected regions but also for multiply connected ones such as the one shown in Figure1.2.For the region in Figure1.2,the unit normal vector to Cε(the inner boundary)points towards the center of the circle.6Two—dimensional Laplace’sEquationFigure1.2Eq.(1.12)holds for any radiusε>0,so long as the circle Cε(in Figure1.2) lies completely inside the region bounded by C.Thus,we may letε→0+in Eq.(1.12).This givesZ C[φ(x,y)∂(Φ(x,y;ξ,η))−Φ(x,y;ξ,η)∂(φ(x,y))]ds(x,y) =−limε→0+Z Cε[φ(x,y)∂(Φ(x,y;ξ,η))−Φ(x,y;ξ,η)∂(φ(x,y))]ds(x,y).(1.13)Using polar coordinates r andθcentered about(ξ,η)as defined by x−ξ= r cosθand y−η=r sinθ,we may writeΦ(x,y;ξ,η)=1ln(r),∂[Φ(x,y;ξ,η)]=n x ∂[Φ(x,y;ξ,η)]+n y∂[Φ(x,y;ξ,η)]=n x cosθ+n y sinθ.(1.14)The Taylor’s series ofφ(x,y)about the point(ξ,η)is given byφ(x,y)=∞X m=0m X k=0(∂mφ∂x k∂y m−k)¯¯¯¯(x,y)=(ξ,η)(x−ξ)k(y−η)m−kk!(m−k)!.On the circle Cε,r=ε.Thus,φ(x,y)=∞X m=0m X k=0(∂m[φ(x,y)])¯¯¯¯(x,y)=(ξ,η)εm cos kθsin m−kθfor(x,y)∈Cε.(1.15)Boundary Integral Solution 7Similarly,we may write∂∂n [φ(x,y )]=∞X m =0m X k =0(∂m ∂x k ∂y m −k {∂∂n [φ(x,y )]})¯¯¯¯(x,y )=(ξ,η)×εm cos k θsin m −k θfor (x,y )∈C ε.(1.16)Using Eqs.(1.14),(1.15)and (1.16)and writing ds (x,y )=εd θwith θranging from 0to 2π,we may now attempt to evaluate the limit on the right hand side of Eq.(1.13).On C ε,the normal vector [n x ,n y ]is given by [−cos θ,−sin θ].Thus,ZC εφ(x,y )∂[Φ(x,y ;ξ,η]ds (x,y )=−1φ(ξ,η)2πZ 0d θ−12π∞X m =1m X k =0εm k !(m −k )!(∂m φ∂x k ∂y m −k )¯¯¯¯(x,y )=(ξ,η)2πZ 0cos k θsin m −kθd θ→−φ(ξ,η)as ε→0+,(1.17)andZC εΦ(x,y ;ξ,η)∂[φ(x,y )]ds (x,y )=1∞X m =0m X k =0(∂m (∂[φ(x,y )]))¯¯¯¯(x,y )=(ξ,η)×εm +1ln(ε)k !(m −k )!2πZ 0cos k θsin m −k θd θ→0as ε→0+,(1.18)since εm +1ln(ε)→0as ε→0+for m =0,1,2,···.Consequently,as ε→0+,Eq.(1.13)yieldsφ(ξ,η)=ZC[φ(x,y )∂(Φ(x,y ;ξ,η))−Φ(x,y ;ξ,η)∂(φ(x,y ))]ds (x,y )for (ξ,η)∈R.(1.19)Together with Eq.(1.9),Eq.(1.19)provides us with a boundary integralsolution for the two-dimensional Laplace’s equation.If both φand ∂φ/∂n are known8Two—dimensional Laplace’s Equation at all points on C,the line integral in Eq.(1.19)can be evaluated(at least in theory) to calculateφat any point(ξ,η)in the interior of R.From the boundary conditions (1.2),at any given point on C,eitherφor∂φ/∂n,not both,is known,however.To solve the interior boundary value problem,we mustfind the unknownφand∂φ/∂n on C2and C1respectively.As we shall see later on,this may be done through manipulation of data on the boundary C only,if we can derive a boundary integral formula forφ(ξ,η),similar to the one in Eq.(1.19),for a general point(ξ,η) that lies on C.For the case in which the point(ξ,η)lies on C,Eq.(1.10)holds if we replace the curve C by D∪Dε,where the curves D and Dεare as shown in Figure1.3.(If Cεis the circle of center(ξ,η)and radiusε,then D is the part of C that lies outside Cεand Dεis the part of Cεthat is inside R.)Thus,Z D[φ(x,y)∂∂n(Φ(x,y;ξ,η))−Φ(x,y;ξ,η)∂∂n(φ(x,y))]ds(x,y)=−Z Dε[φ(x,y)∂∂n(Φ(x,y;ξ,η))−Φ(x,y;ξ,η)∂∂n(φ(x,y))]ds(x,y).(1.20)Let us examine what happens to Eq.(1.20)when we letε→0+.Asε→0+,the curve D tends to C.Thus,we may writeZ C[φ(x,y)∂∂n(Φ(x,y;ξ,η))−Φ(x,y;ξ,η)∂∂n(φ(x,y))]ds(x,y)=−limε→0+Z Dε[φ(x,y)∂(Φ(x,y;ξ,η))−Φ(x,y;ξ,η)∂(φ(x,y))]ds(x,y).(1.21)Boundary Integral Solution 9Note that,unlike in Eq.(1.13),the line integral over C in Eq.(1.21)is improper as its integrand is not well de fined at (ξ,η)which lies on C .Strictly speaking,the line integration should be over the curve C without an in finitesimal segment that contains the point (ξ,η),that is,the line integral over C in Eq.(1.21)has to be interpreted in the Cauchy principal sense if (ξ,η)lies on C .To evaluate the limit on the right hand side of Eq.(1.21),we need to know what happens to D εwhen we let ε→0+.Now if (ξ,η)lies on a smooth part of C (not at where the gradient of the curve changes abruptly,that is,not at a corner point,if there is any),one can intuitively see that the part of C inside C εapproaches an in finitesimal straight line as ε→0+.Thus,we expect D εto tend to a semi-circle as ε→0+,if (ξ,η)lies on a smooth part of C .It follows that in attempting to evaluate the limit on the right hand side of Eq.(1.21)we have to integrate over only half a circle (instead of a full circle as in the case of Eq.(1.13)).Modifying Eqs.(1.17)and (1.18),we obtain lim ε→0+Z D εφ(x,y )∂∂n [Φ(x,y ;ξ,η)]ds (x,y )=−12φ(ξ,η),lim ε→0+Z D εΦ(x,y ;ξ,η)∂[φ(x,y )]ds (x,y )=0.Hence Eq.(1.21)gives 1φ(ξ,η)=Z C[φ(x,y )∂(Φ(x,y ;ξ,η))−Φ(x,y ;ξ,η)∂(φ(x,y ))]ds (x,y )for (ξ,η)lying on a smooth part of C.(1.22)Together with the boundary conditions in Eq.(1.2),Eq.(1.22)may be applied to obtain a numerical procedure for determining the unknown φand/or ∂φ/∂n on the boundary C .Once φand ∂φ/∂n are known at all points on C ,the solution of the interior boundary value problem de fined by Eqs.(1.1)-(1.2)is given by Eq.(1.19)at any point (ξ,η)inside R .More details are given in Section 1.5below.For convenience,we may write Eqs.(1.11),(1.19)and (1.22)as a single equation given by λ(ξ,η)φ(ξ,η)=Z C[φ(x,y )∂(Φ(x,y ;ξ,η))−Φ(x,y ;ξ,η)∂(φ(x,y ))]ds (x,y ),(1.23)if we de fine λ(ξ,η)=⎧⎨⎩0if (ξ,η)/∈R ∪C,1/2if (ξ,η)lies on a smooth part of C,1if (ξ,η)∈R.(1.24)10Two—dimensional Laplace’s Equation1.5Boundary Element Solution with Constant ElementsWe now show how Eq.(1.23)may be applied to obtain a simple boundary element procedure for solving numerically the interior boundary value problem defined by Eqs.(1.1)-(1.2).The boundary C is approximated as an N-sided polygon with sides C(1),C(2),···,C(N−1)and C(N),that is,C'C(1)∪C(2)∪···∪C(N−1)∪C(N).(1.25) The sides or the boundary elements C(1),C(2),···,C(N−1)and C(N)are constructed as follows.We put N well spaced out points(x(1),y(1)),(x(2),y(2)),···,(x(N−1),y(N−1))and(x(N),y(N))on C,in the order given,following the counter clockwise direction.Defining(x(N+1),y(N+1))=(x(1),y(1)),we take C(k)to be theboundary element from(x(k),y(k))to(x(k+1),y(k+1))for k=1,2,···,N.As an example,in Figure1.4,the boundary C=C1∪C2in Figure1.1is approximated using5boundary elements denoted by C(1),C(2),C(3),C(4)and C(5).For a simple approximation ofφand∂φ/∂n on the boundary C,we assume that these functions are constants over each of the boundary elements.Specifically, we make the approximation:φ'φ(k)and ∂φ∂n=p(k)for(x,y)∈C(k)(k=1,2,···,N),(1.26)whereφk)and p(k)are respectively the values ofφand∂φ/∂n at the midpoint of C(k).Figure1.4Boundary Element Solution with Constant Elements11 With Eqs.(1.25)and(1.26),wefind that Eq.(1.23)can be approximately written asλ(ξ,η)φ(ξ,η)=NX k=1{φ(k)F(k)2(ξ,η)−p k)F(k)1(ξ,η)},(1.27)whereF(k)1(ξ,η)=Z C(k)Φ(x,y;ξ,η)ds(x,y),F(k)2(ξ,η)=Z C(k)∂[Φ(x,y;ξ,η)]ds(x,y).(1.28)For a given k,eitherφ(k)or p(k)(not both)is known from the boundary conditions in Eq.(1.2).Thus,there are N unknown constants on the right hand side of Eq.(1.27).To determine their values,we have to generate N equations containing the unknowns.If we let(ξ,η)in Eq.(1.27)be given in turn by the midpoints of C(1),C(2),···,C(N−1)and C(N),we obtain1φ(m)=NX k=1{φ(k)F(k)2(x(m),y(m))−p(k)F(k)1(x(m),y(m))}for m=1,2,···,N,(1.29)where(x m),y(m))is the midpoint of C(m).In the derivation of Eq.(1.29),we takeλ(x(m),y(m))=1/2,since(x(m),y(m)) being the midpoint of C(m)lies on a smooth part of the approximate boundary C(1)∪C(2)∪···∪C(N−1)∪C(N).Eq.(1.29)constitutes a system of N linear algebraic equations containing the N unknowns on the right hand side of Eq.(1.27).We may rewrite it asNX k=1a(mk)z(k)=N X k=1b(mk)for m=1,2,···,N,(1.30)12Two—dimensional Laplace’s Equation where a(mk),b(mk)and z(k)are defined bya(mk)=(−F(k)1(x(m),y(m))ifφis specified over C(k),F(k)2(x(m),y(m))−1δ(mk)if∂φ/∂n is specified over C(k),b(mk)=(φ(k)(−F(k)2(x(m),y(m))+1δ(mk))ifφis specified over C(k),p(k)F(k)1(x(m),y(m))if∂φ/∂n is specified over C(k),δ(mk)=½0if m=k,1if m=k,z(k)=(p(k)ifφis specified over C(k),φ(k)if∂φ/∂n is specified over C(k).(1.31)Note that z(1),z(2),···,z(N−1)and z(N)are the N unknown constants on the right hand side of Eq.(1.27),while a(mk)and b(mk)are known coefficients.Once Eq.(1.30)is solved for the unknowns z(1),z(2),···,z(N−1)and z(N),the values ofφand∂φ/∂n over the element C(k),as given byφ(k)and p(k)respectively, are known for k=1,2,···,N.Eq.(1.27)withλ(ξ,η)=1then provides us with an explicit formula for computingφin the interior of R,that is,φ(ξ,η)'NX k=1{φ(k)F(k)2(ξ,η)−p(k)F(k)1(ξ,η)}for(ξ,η)∈R.(1.32)To summarize,a boundary element solution of the interior boundary value problem defined by Eqs.(1.1)-(1.2)is given by Eq.(1.32)together with Eqs.(1.28), (1.30)and(1.31).Because of the approximation in Eqs.(1.25)and(1.26),the solution is said to be obtained using constant elements.Analytical formulae for calculating F(k)1(ξ,η)and F(k)2(ξ,η)in Eq.(1.28)are given in Eqs.(1.37),(1.38), (1.40)and(1.41)(together with Eq.(1.35))in the section below.1.6Formulae for Integrals of Constant ElementsThe boundary element solution above requires the evaluation of F(k)1(ξ,η)and F(k)2(ξ,η). These functions are defined in terms of line integrals over C(k)as given in Eq.(1.28). The line integrals can be worked out analytically as follows.Points on the element C(k)may be described using the parametric equationsx=x(k)−t`(k)n(k)yy=y(k)+t`(k)n(k)x)from t=0to t=1,(1.33)where`(k)is the length of C(k)and[n(k)x,n(k)y]=[y(k+1)−y(k),x(k)−x(k+1)]/`(k)is the unit normal vector to C(k)pointing away from R.Formulae for Integrals of Constant Elements13 For(x,y)∈C(k),wefind that ds(x,y)=p=`(k)dt and(x−ξ)2+(y−η)2=A(k)t2+B(k)(ξ,η)t+E(k)(ξ,η),(1.34) whereA(k)=[`(k)]2,B(k)(ξ,η)=[−n(k)y(x(k)−ξ)+(y(k)−η)n(k)x](2`(k)),E(k)(ξ,η)=(x(k)−ξ)2+(y(k)−η)2.(1.35) The parameters in Eq.(1.35)satisfy4A(k)E(k)(ξ,η)−[B(k)(ξ,η)]2≥0for any point(ξ,η).To see why this is true,consider the straight line defined by the parametric equations x=x(k)−t`(k)n(k)y and y=y(k)+t`(k)n(k)x for−∞<t<∞.Note that C(k)is a subset of this straight line(given by the parametric equations from t=0to t=1).Eq.(1.34)also holds for any point(x,y)lying on the extended line.If(ξ,η) does not lie on the line then A(k)t2+B(k)(ξ,η)t+E(k)(ξ,η)>0for all real values of t (that is,for all points(x,y)on the line)and hence4A(k)E(k)(ξ,η)−[B(k)(ξ,η)]2>0. On the other hand,if(ξ,η)is on the line,we canfind exactly one point(x,y)such that A(k)t2+B(k)(ξ,η)t+E(k)(ξ,η)=0.As each point(x,y)on the line is given by a unique value of t,we conclude that4A(k)E(k)(ξ,η)−[B(k)(ξ,η)]2=0for(ξ,η)lying on the line.From Eqs.(1.28),(1.33)and(1.34),F(k)1(ξ,η)and F(k)2(ξ,η)may be written asF(k)1(ξ,η)=`(k)1Z0ln[A(k)t2+B(k)(ξ,η)t+E(k)(ξ,η)]dt,F(k)2(ξ,η)=`(k)2π1Z0n(k)x(x(k)−ξ)+n(k)y(y(k)−η)A(k)t2+B(k)(ξ,η)t+E(k)(ξ,η)dt.(1.36)The second integral in Eq.(1.36)is the easiest one to work out for the case in which4A(k)E(k)(ξ,η)−[B(k)(ξ,η)]2=0.For this case,the point(ξ,η)lies on the straight line of which the element C(k)is a subset.Thus,the vector[x(k)−ξ,y(k)−η] is perpendicular to[n(k)x,n(k)y],that is,n(k)x(x(k)−ξ)+n(k)y(y(k)−η)=0,and we obtainF(k)2(ξ,η)=0for4A(k)E(k)(ξ,η)−[B(k)(ξ,η)]2=0.(1.37) From the integration formulaZ dt at2+bt+c=√4ac−b2√4ac−b2+constantfor real constants a,b and c such that4ac−b2>0,14Two—dimensional Laplace’s Equation wefind thatF(k)2(ξ,η)=(k)(k)(k)−ξ)+n(k)(k)−η)]p×2A(k)+B(k)(ξ,η)p4A(k)E(k)(ξ,η)−[B(k)(ξ,η)]2−B(k)(ξ,η)p4A(k)E(k)(ξ,η)−[B(k)(ξ,η)]2for4A(k)E(k)(ξ,η)−[B(k)(ξ,η)]2>0.(1.38) If4A(k)E(k)(ξ,η)−[B(k)(ξ,η)]2=0,we may writeA(k)t2+B(k)(ξ,η)t+E(k)(ξ,η)=A(k)(t+B(k)(ξ,η)2A(k))2.Thus,F(k)1(ξ,η)=`(k)1Z0ln[A(k)(t+B(k)(ξ,η))2]dtfor4A(k)E(k)(ξ,η)−[B(k)(ξ,η)]2=0.(1.39) Now if(ξ,η)lies on a smooth part of C(k),the integral in Eq.(1.39)is im-proper,as its integrand is not well defined at the point t=t0≡−B(k)(ξ,η)/(2A(k))∈(0,1).Strictly speaking,the integral should then be interpreted in the Cauchy princi-pal sense,that is,to evaluate it,we have to integrate over[0,t0−ε]∪[t0+ε,1]instead of[0,1]and then letε→0to obtain its value.However,in this case,it turns out that the limits of integration t=t0−εand t=t0+εeventually do not contribute anything to the integral.Thus,for4A(k)E(k)(ξ,η)−[B(k)(ξ,η)]2=0,thefinal ana-lytical formula for F(k)1(ξ,η)is the same irrespective of whether(ξ,η)lies on C(k)or not.If(ξ,η)lies on C(k),we may ignore the singular behaviour of the integrand and apply the fundamental theorem of integral calculus as usual to evaluate the definite integral in Eq.(1.39)directly over[0,1].The integration required in Eq.(1.39)can be easily done to giveF(k)1(ξ,η)=`(k){ln(`(k))+(1+B(k)(ξ,η))ln|1+B(k)(ξ,η)|−B(k)(ξ,η)ln|B(k)(ξ,η)|−1}for4A(k)E(k)(ξ,η)−[B(k)(ξ,η)]2=0.(1.40)Implementation on Computer 15UsingZ ln(at 2+bt +c )dt =t [ln(a )−2]+(t +b 2a )ln[t 2+b a t +c a]+1a √√+constant for real constants a,b and c such that 4ac −b 2>0,we obtainF (k )1(ξ,η)=`(k )4π{2[ln(`(k ))−1]−B (k )(ξ,η)2A (k )ln |E (k )(ξ,η)A (k )|+(1+B (k )(ξ,η)2A (k ))ln |1+B (k )(ξ,η)A (k )+E (k )(ξ,η)A (k )|+p ×2A (k )+B (k )(ξ,η)p 4A (k )E (k )(ξ,η)−[B (k )(ξ,η)]2−(k )p }for 4A (k )E (k )(ξ,η)−[B (k )(ξ,η)]2>0.(1.41)1.7Implementation on ComputerWe attempt now to develop double precision FORTRAN 77codes which can be used to implement the boundary element procedure described in Section 1.5on the computer.In our discussion here,syntaxes,variables and statements in FORTRAN 77are written in typewriter fonts,for example,xi ,eta and A=L**2d0.One of the tasks involved is the setting up of the system of linear algebraic equations given in Eqs.(1.30)and (1.31).To do this,the functions F (k )1(ξ,η)and F (k )2(ξ,η)have to be computed using the formulae in Section 1.6.We create a sub-routine called CPF which accepts the values of ξ,η,x (k ),y (k ),n (k )x ,n (k )y and `(k )(storedin the real variables xi ,eta ,xk ,yk ,nkx ,nky and L )in order to calculate and returnthe values of πF (k )1(ξ,η)and πF (k )2(ξ,η)(in the real variables PF1and PF2).The subroutine CPF is listed below.subroutine CPF(xi,eta,xk,yk,nkx,nky,L,PF1,PF2)double precision xi,eta,xk,yk,nkx,nky,L,PF1,PF2,&A,B,E,D,BA,EA16Two—dimensional Laplace’s Equation A=L**2d0B=2d0*L*(-nky*(xk-xi)+nkx*(yk-eta))E=(xk-xi)**2d0+(yk-eta)**2d0D=dsqrt(dabs(4d0*A*E-B**2d0))BA=B/AEA=E/Aif(D.lt.0.0000000001d0)thenPF1=0.5d0*L*(dlog(L)&+(1d0+0.5d0*BA)*dlog(dabs(1d0+0.5d0*BA))&-0.5d0*BA*dlog(dabs(0.5d0*BA))-1d0)PF2=0d0elsePF1=0.25d0*L*(2d0*(dlog(L)-1d0)-0.5d0*BA*dlog(dabs(EA))&+(1d0+0.5d0*BA)*dlog(dabs(1d0+BA+EA))&+(D/A)*(datan((2d0*A+B)/D)-datan(B/D)))PF2=L*(nkx*(xk-xi)+nky*(yk-eta))/D&*(datan((2d0*A+B)/D)-datan(B/D))endifreturnendCPF is repeatedly called in the subroutine CELAP1.CELAP1reads in the num-ber of boundary elements(N)in the real variable N,the midpoints(x(k),y(k))in the real arrays xm(1:N)and ym(1:N),the boundary points(x(k),y(k))in the real arrays xb(1:N+1)and yb(1:N+1),the normal vectors(n(k)x,n(k)y)in the real arrays nx(1:N)and ny(1:N),the lengths of the boundary elements in the real array lg(1:N) and the types of boundary conditions(on the boundary elements)in the integer ar-ray BCT(1:N)together with the corresponding boundary values in the real array BCV(1:N),set up and solve Eq.(1.30),and return all the values ofφ(k)and p k) in the arrays phi(1:N)and dphi(1:N)respectively.(More details on the arrays BCT(1:N)and BCV(1:N)will be given later on in Section1.8.)Thus,a large part of the boundary element procedure(with constant elements)for the numerical solution of the boundary value problem is executed in CELAP1.The subroutine CELAP1is listed as follows.subroutine CELAP1(N,xm,ym,xb,yb,nx,ny,lg,BCT,BCV,phi,dphi)integer m,k,N,BCT(1000)double precision xm(1000),ym(1000),xb(1000),yb(1000),。
丘成桐大学生数学竞赛数学专业大纲(英汉对照版)
dirichlet-to-nenumann算子的定义
dirichlet-to-nenumann算子的定义Dirichlet-Neumann Laplacian, also known as theDirichlet-to-Neumann operator, is a mathematical operator that arises in the study of partial differential equations. It plays a fundamental role in mathematical analysis and has applications in various fields such as physics, engineering, and signal processing. In this article, we will explore the definition of theDirichlet-Neumann operator and its properties step-by-step.1. Introduction to the Laplacian Operator:Before diving into the definition of the Dirichlet-Neumann operator, it is essential to understand the concept of the Laplacian operator. The Laplacian operator is a second-order partial differential operator commonly denoted by Δor ∇^2 and is defined as the divergence of the gradient of a function in Cartesian coordinates. In simple terms, it measures the local rate of change or curvature of a function.2. Introduction to Dirichlet and Neumann Boundary Conditions: When solving partial differential equations, boundary conditions are necessary to uniquely determine the solution. Dirichlet boundary conditions specify the value of the function on theboundary of the domain. Neumann boundary conditions, on the other hand, specify the normal derivative of the function on the boundary. These two types of boundary conditions are widely used in various physical and mathematical problems.3. Definition of the Dirichlet-Neumann Operator:The Dirichlet-Neumann operator is a mapping that relates the Dirichlet and Neumann boundary conditions for the Laplacian operator. Let's consider a bounded domain Ωin n-dimensional Euclidean space R^n and a function u defined on Ω. The Dirichlet-Neumann operator, denoted as D(u), is defined as the outer unit normal derivative of u on the boundary of Ω.Mathematically, the Dirichlet-Neumann operator can be expressed as:D(u) = ∂u/∂nwhere ∂u/∂n is the outward normal derivative of u on the boundary of Ω.4. Properties of the Dirichlet-Neumann Operator:The Dirichlet-Neumann operator possesses several importantproperties that make it a powerful tool in the study of partial differential equations. Here are some key properties:4.1. Linearity: The Dirichlet-Neumann operator is a linear operator, which means it satisfies the superposition principle. If u and v are functions defined on Ωand αand βare scalar constants, then D(αu + βv) = αD(u) + βD(v).4.2. Adjoints and Self-adjointness: The Dirichlet-Neumann operator has a formal adjoint defined as D*(u) = −∂u/∂n, where ∂u/∂n is the inward normal derivative. This adjoint allows us to establish duality between Dirichlet and Neumann boundary conditions. Moreover, under certain conditions, the Dirichlet-Neumann operator can be shown to be self-adjoint, which ensures important symmetry properties.4.3. Spectrum and Eigenvalues: The Dirichlet-Neumann operator has a spectrum consisting of eigenvalues and associated eigenvectors. These eigenvalues provide important information about the behavior and properties of the operator and the underlying partial differential equation.4.4. Applications: The Dirichlet-Neumann operator finds applications in various areas such as inverse problems, shape optimization, optimal control, and spectral theory. It is a crucial tool in the study of the behavior and solutions of partial differential equations with mixed boundary conditions.5. Conclusion:The Dirichlet-Neumann operator is a mathematical operator that relates the Dirichlet and Neumann boundary conditions for the Laplacian operator. It plays a fundamental role in the study of partial differential equations and finds applications in diverse fields. Understanding the definition and properties of theDirichlet-Neumann operator provides valuable insights into the behavior and solutions of partial differential equations and can lead to advancements in various scientific and engineering disciplines.。
Spectral Boundary Conditions in the Bag Model
a rXiv:h ep-th/04192v18O ct24Spectral Boundary Conditions in the Bag Model A.A.Abrikosov,jr.∗117259,Moscow,ITEP.Abstract We propose a reduced form of spectral boundary conditions for holding fermions in the bag in a chiral invariant way.Our boundary conditions do not depend on time and allow Hamiltonian treatment of the system.They are suited for studies of chiral phenomena both in Minkowski and Euclidean spaces.Introduction The two principal problems of QCD are confinement and spontaneous break-ing of chiral invariance.Both phenomena take place in the strongly interact-ing domain where the theory becomes nonperturbative.Probably they are interrelated.However,usually they were considered separately.Up to nowthe spontaneous chiral invariance breaking (SCIB)was discussed mostly in the infinite space.It would be interesting to address the features of SCIB that appear due to localization of quarks in finite volume.In order to do that one needs to hold quarks in a chiral invariant way.There exists an entire family of bag models.The famous MIT bag [1]successfully reproduced the spectrum and other features of hadrons.A gen-eralization of the MIT model are so-called chiral bags [2,3].An apparent drawback of all these models is that the boundary conditions are explicitly chiral noninvariant.Attempts to save the situation led to the cloudy bag[4]where the chiral symmetry was restored by pions emitted from the bag surface(the pion cloud).But this model is sensitive to details of the adopted scheme of quark-pion interaction.Thus neither of the models is suited to the discussion of SCIB infinite volume.The way to lock fermions in afinite volume without spoiling the chi-ral invariance is to impose the so-called spectral boundary conditions (SBC).They werefirst introduced by Atiah,Patodi and Singer(APS)who investigated anomalies on manifolds with boundaries[5].Later these bound-ary conditions were widely applied in studies of index theorems on various manifolds[6].In distinction from those mentioned before the APS conditions are non-local.They are defined on the boundary as a whole.This looks natural forfinite Euclidean manifolds but is inconvenient for physical models.In the process of evolution the spatial boundary of a static physical bag becomes an infinite space-time cylinder.Constrainingfields on the entire“world cylin-der”would violate causality and complicate continuation to Minkowski space.In this talk we shall demonstrate how to avoid this difficulty.One can consider a restricted spatial version of spectral boundary conditions.The modified conditions do not depend on time and are acceptable from the physical point of view.This makes possible the Hamiltonian description of the system.The paper has the following structure.We shall review the classical APS boundary conditions in Section1.In Section2we shall formulate the modified spectral conditions and discuss their properties.In conclusion we shall summarize the results and mention future prospects.1The physics of APS boundary conditions 1.1ConventionsWe shall start from the traditional form of SBC.First we will introduce coordinates,Dirac matrices and the gauge that allow to define the spectral boundary conditions in a convenient way.Let us consider massless fermions interacting with gaugefieldˆA in a closed 4-dimensional Euclidean domain B4.We choose a curvilinear coordinate frame such that in the vicinity of the boundary∂B4thefirst coordinateξ2points along the outward normal(ξ=0corresponds to∂B4)while the three others,q i,parametrize∂B4itself.For simplicity we shall assume that near the surface the metric gαβdepends only on q so thatds2=dξ2+g ik(q)dq i dq k.(1) Moreover we choose the gauge so that on the boundaryˆAξ=0.Now we mustfix the Dirac matrixγξ.Let I be the2×2unity matrix.Thenγξ= 0iI−iI0 ;γq= 0σqσq0 .(2)Matricesσq are the ordinary Pauliσ-matrices.With these definitions the Dirac operator of massless fermions on the surface takes the form,−i/∇|∂B4=−iγα∇α= 0ˆMˆM†0 = 0I∂ξ−iˆ∇−I∂ξ−iˆ∇0,(3)whereˆ∇=σq∇q is the convolution of covariant gradient along the boundary ∇q withσ-matrices.Note that Hermitian conjugated operatorsˆM andˆM†differ only by the sign of∂ξ-derivative.Further on we shall call the covariant derivative−iˆ∇on the boundary the boundary operator.It is a linear differential operator acting on2-spinors.It is Hermitian and includes tangential gaugefieldˆA q and the spin connection which arises from the curvature of∂B4.The massless Dirac operator anticommutes withγ5-matrix:−i/∇,γ5 =0,γ5= I00−I .(4)Thus the Lagrangian of Quantum Chromodynamics is chiral invariant.In order to exploit this one needs chiral invariant boundary conditions.1.2The APS boundary conditions1.2.1The definitionAtiah,Patodi and Singer investigated spectra of Dirac operator on manifolds with boundaries.If we separate upper and lower(left and right)components of4-spinors the corresponding eigenvalue equation for−i/∇will take the form −i/∇ψΛ=−i/∇ uΛvΛ =Λ uΛvΛ =ΛψΛ.(5)3The next step is to Fourier expand u and v on the boundary.Let2-spinors eλ(q)be eigenfunctions of the boundary operator−iˆ∇:−iˆ∇eλ(q)=λeλ(q).(6) Note that the form of this equation and the eigenfunctions eλ(q)depend on gauge.It is here that the gauge conditionˆAξ(0,q)=0becomes important. The operator−iˆ∇is Hermitian soλare real.The functions eλform an orthogonal basis.In principle−iˆ∇may have zero-modes on∂B4but convexmanifolds are not the case.In the vicinity of the boundary spinors uΛand vΛmay be expanded in series in eλ:uΛ(ξ,q)= λfλΛ(ξ)eλ(q),fλΛ(ξ)= ∂B4e†λ(q)uΛ(ξ,q)√g d3q;(7b)where g=det||g ik||is the determinant of metric on the boundary.The spectral boundary condition states that on the boundary,i.e.at ξ=0fλΛ ∂B4=0forλ>0;(8a)gλΛ ∂B4=0forλ<0.(8b)Another way to say this is to introduce integral projectors P+and P−onto boundary modes with positive and negativeλ:P+(q,q′)= λ>0eλ(q)e†λ(q′);P−(q,q′)= λ<0eλ(q)e†λ(q′).(9)Let I be the unity operator on the function space spanned by eλ.Then, obviously,P++P−=I.(10) If we join two-dimensional projectors P+and P−into4×4matrix P the spectral boundary condition for4-spinorψwill look as follows:Pψ|∂B4= P+00P− u v ∂B4=0.(11)4The projector P commutes with matrixγ5:P,γ5 =0.(12)Therefore boundary condition(11)by construction preserves the chiral in-variance.1.2.2The physicsNow we shall prove that the spectral boundary conditions are acceptable and explain their physical ly,we shall show that SBC provide Hermicity of the Dirac operator and conservation of fermions in the bag. After that we will explain the origin of requirements(8).First let us prove that Dirac operator is Hermitian.As usually,we inte-grate by parts the expressionB4dV f†(−i/∇g)= B4dV(−i/∇f)†g+ ∂B4dS f†(−iγξ)g.(13)Now we need to show that if f and g satisfy(8)then the last term vanishes.Conditions(8)mean that on the boundary4-spinors f and g may be written as:f= f−f+ and g= g−g+ ,where f±and g±include only components with positive and negativeλrespectively,see(7).Rewriting the boundary term in(13)we get∂B4dS f†(−iγξ)g= ∂B4dS (f−)†g+−(f+)†g− =0,(14)due to the orthogonality of eigenfunctions of the boundary operator.Thus the APS boundary conditions indeed ensure the Hermicity of Dirac operator.In addition,relation(14)guarantees conservation of fermions in the bag. Indeed,for f=g the LHS is nothing but the net fermionic current through the boundary, ∂B4dS jξ=−i ∂B4dS f†γξf=0.(15)Therefore the number of fermions is conserved and particles in the spectral bag are confined.In order to understand the origin of SBC let us rewrite the eigenvalue condition(5)near the boundary in terms of components.(∂ξ+λ)gλΛ(ξ)=ΛfλΛ(ξ);(16a)−(∂ξ−λ)fλΛ(ξ)=ΛgλΛ(ξ).(16b)5Depending on the sign ofλthese relations reduce on the boundary either to∂ξgλΛfλΛ ξ=0=λ<0,gλΛ(0)=0atλ<0.(17b) Thus both components either vanish on the boundary or have a negative logarithmic derivative along the normal.This requirement has a simple physical interpretation.Suppose that out of the bag the metric and the gaugefield remain the same as on the boundary. Then we can continue the functions f and g toξ=∞.Outside the bag the functions will be square integrable falling exponents as if the particle was locked in a potential well.The only difference is that now the potential for every mode is adjusted specially.We may conclude that the spectral boundary conditions claim that wave functions must have square integrable continuation to the infinite space.2The SBC for physical bags2.1The truncated SBCNow let us turn to fermions in the infinite Euclidean cylinder B3⊗R.We shall call thefirst three coordinates“space”and the fourth one“time”.The boundary operator consists of spatial and temporal parts:−iˆ∇∂B3⊗R=−iˆ∇∂B3−iσz∂4.(18) We shall call the spatial part−iˆ∇∂B3the truncated boundary operator. Let its eigenfunctions be e±λ:−iˆ∇∂B3e±λ(q)=±λe±λ(q),λ>0.(19) Wave functions on the space-time boundary∂B3⊗R can be expanded in and longitudinal plane waves:e±λuΛ= λ>0 dke ikt g+λ,kΛe+λ+g−λ,kΛe−λ .(20b)2π6The truncated operator−iˆ∇∂B3anticommutes withσz.Thereforeσz changes the sign of e-eigenvalues.A possible choice of eigenvectors is(see [7,8]for the sphere)=±iσz e∓λ.(21)e±λThus the last term in(18)mixes positive and negative spatial harmonics.In classical approach this would mean that now SBC should be writ-ten in terms of k-dependent eigenfunctions of the full boundary operator (18).However physically these“future-sensitive”boundary conditions look strange.Therefore we propose to apply independent of k truncated APS constraints:f+λ,kΛ ∂B3=0;(22a)g−λ,kΛ ∂B3=0.(22b) These conditions do not depend on time and allow Hamiltonian treatment of the system.Moreover,they may be applied both in Euclidean and Minkowski spaces.Now let us show that they are acceptable.2.2ConsistencyWe are going to prove that the truncated form of SBC fulfills necessary ly that they are chiral invariant,that the Dirac operator is Hermitian and the fermionic current is conserved and after all that wave functions may be continued out of the bag.The proof of thefirst three points literally follows the4-dimensional case. Everything that concerns formulae(9–15)remains true for truncated(T) 3-dimensional SBC(22).One may define on∂B3projectors,P±T(q,q′)= λ>0e±λ(q) e±λ(q′) †.(23)Then the truncated boundary conditions may be written in the manifestly γ5-invariant form,P Tψ|∂B3= P+T00P−T u v ∂B3=0.(24)7Hermicity of the Dirac operator and conservation of fermions are proven in the same way as before,see(13–15).We don’t rewrite the formula.The last point is more delicate.We already mentioned that theσz term in (18)mixes positive and negative harmonics.Therefore they must be analysed together and instead of two eigenvalue equations(16)we get four(ξis the normal to the spatial boundary):(∂ξ+λ)g+λ,kΛ=Λf+λ,kΛ+ik g−λ,kΛ;(25a)−(∂ξ−λ)f+λ,kΛ=Λg+λ,kΛ+ik f−λ,kΛ:(25b)(∂ξ−λ)g−λ,kΛ=Λf−λ,kΛ−ik g+λ,kΛ;(25c)−(∂ξ+λ)f−λ,kΛ=Λg−λ,kΛ−ik f+λ,kΛ.(25d) The new feature with respect to(16)are ik addends that appear due to the mixing.However one may notice that the terms in the RHS of(25)come in pairs f+,g−and f−,g+.Therefore according to conditions(22)the RHS of equations(25a,25d)vanish on the boundary.Thus the behaviour of g+and f−on the boundary is governed by the homogeneous equations and∂ξf−λ,kΛg+λ,kΛξ=0=−λ<0.(26)Hence despite the presence of extra pieces the nonvanishing components g+and f−have negative logarithmic derivatives.This means that solutions of eigenvalue equations may be continued outwards of the“world cylinder”in an integrable way and the last of our requirements is fulfilled.This completes the proof of acceptability of the truncated SBC.ConclusionThe truncated version of APS boundary conditions offers a number of possi-bilities.It allows to formulate a chiral invariant bag model and to approach chiral properties of fermionicfield in the closed volume.The constraints are imposed on the spatial boundary of the bag so one may write down the Hamiltonian and study the energy spectrum of the system.Another advan-tage is that the modified SBC do not depend on time and may be used both in Euclidean and Minkowsky space.8A new feature that SBC may bring to the bag physics is their nonlocality. Other bag models[1,2,3]employed local boundary conditions which corre-spond to the thin wall approximation.The nonlocal spectral conditions refer to the boundary as a whole.Therefore in a sense hadrons are also treated as a whole.This looks promising by itself.To begin with it would be interesting to investigate hadronic spectra in chiral invariant bags.This could answer whether the model is realistic and indicate missing elements.Another question is more mathematical.Chiral symmetry is a specific of fermions in even dimensional spaces.Hence the spectral boundary conditions were also considered in even dimensions.The truncated SBC are formulated in the odd dimensional space that remains after discarding the time.This might have interesting consequences.For example,the boundary of odd-dimensional bag is an even-dimensional manyfold and one can introduce a sort of internal chirality for surface modes.The question is if there is a way for this hidden symmetry to reveal itself.In conclusion I would like to express my gratitude to Professor A.Wipf for elucidating discussions.I thank the Organizers forfinancial support.The work was supported by RFBR grant03–02–16209and Federal Program of the Russian Ministry of Industry,Science and Technology No40.052.1.1.1112. References[1]A.Chodos,R.L.Jaffe,C.B.Thorn,and V.F.Weisskopf,Phys.Rev.D9(1974)3471.[2]S.Duerr,A.Wipf,Nucl.Phys.B443(1995)201.[3]G.Esposito,K.Kirsten,Phys.Rev.D66(2002)085014.[4]S.Th´e berge,A.W.Thomas,ler,Phys.Rev.D22(1980)2838;Phys.Rev.D23(1981)2106(E).[5]M.F.Atiah,V.K.Patodi,I.M.Singer,Math.Proc.Camb.Phil.Soc.77(1975)43.[6]T.Eguchi,P.B.Gilkey,A.J.Hanson,Phys.Rep.C66(1980)213.[7]A.A.Abrikosov,Int.Journ.of Mod.Phys.A17(2002)885.[8]A.A.Abrikosov,hep-th/0212134.9。
丘成桐大学生数学竞赛试卷
S.-T.Yau College Student Mathematics Contests 2010Analysis and Differential EquationsTeam(Please select 5problems to solve)1.a)Let f (z )be holomorphic in D :|z |<1and |f (z )|≤1(z ∈D ).Prove that|f (0)|−|z |1+|f (0)||z |≤|f (z )|≤|f (0)|+|z |1−|f (0)||z |.(z ∈D )b)For any finite complex value a ,prove that 12π 2π0log |a −e iθ|dθ=max {log |a |,0}.2.Let f ∈C 1(R ),f (x +1)=f (x ),for all x ,then we have ||f ||∞≤ 10|f (t )|dt + 10|f (t )|dt.3.Consider the equation¨x +(1+f (t ))x =0.We assume that ∞|f (t )|dt <∞.Study the Lyapunov stability of the solution (x,˙x )=(0,0).4.Suppose f :[a,b ]→R be a L 1-integrable function.Extend f to be 0outside the interval [a,b ].Letφ(x )=12h x +h x −hf Show thatb a |φ|≤ b a |f |.5.Suppose f ∈L 1[0,2π],ˆf (n )=12π 2π0f (x )e −inx dx ,prove that 1)∞ |n |=0|ˆf(n )|2<∞implies f ∈L 2[0,2π],2)n |n ˆf (n )|<∞implies that f =f 0,a.e.,f 0∈C 1[0,2π],where C 1[0,2π]is the space of functions f over [0,1]such that both f and its derivative f are continuous functions.126.SupposeΩ⊂R3to be a simply connected domain andΩ1⊂Ωwith boundaryΓ.Let u be a harmonic function inΩand M0=(x0,y0,z0)∈Ω1.Calculate the integral:II=−Γu∂∂n(1r)−1r∂u∂ndS,where 1r=1(x−x0)2+(y−x0)2+(z−x0)2and∂∂ndenotes theout normal derivative with respect to boundaryΓof the domainΩ1.(Hint:use the formula∂v∂n dS=∂v∂xdy∧dz+∂v∂ydz∧dx+∂v∂zdx∧dy.)S.-T.Yau College Student Mathematics Contests 2010Applied Math.,Computational Math.,Probability and StatisticsTeam(Please select 5problems to solve)1.Let X 1,···,X n be independent and identically distributed random variables with continuous distribution functions F (x 1),···,F (x n ),re-spectively.Let Y 1<···<Y n be the order statistics of X 1,···,X n .Prove that Z j =F (Y j )has the beta (j,n −j +1)distribution (j =1,···,n ).2.Let X 1,···,X n be i.i.d.random variable with a continuous density f at point 0.LetY n,i =34b n (1−X 2i /b 2n )I (|X i |≤b n ).Show that n i =1(Y n,i −EY n,i )(b n n i =1Y n,i )1/2L −→N (0,3/5),provided b n →0and nb n →∞.3.Let X 1,···,X n be independently and indentically distributed ran-dom variables with X i ∼N (θ,1).Suppose that it is known that |θ|≤τ,where τis given.Showmin a 1,···,a n +1sup |θ|≤τE (n i =1a i X i +a n +1−θ)2=τ2n −1τ2+n −1.Hint:Carefully use the sufficiency principle.4.The rules for “1and 1”foul shooting in basketball are as follows.The shooter gets to try to make a basket from the foul line.If he succeeds,he gets another try.More precisely,he make 0baskets by missing the first time,1basket by making the first shot and xsmissing the second one,or 2baskets by making both shots.Let n be a fixed integer,and suppose a player gets n tries at “1and 1”shooting.Let N 0,N 1,and N 2be the random variables recording the number of times he makes 0,1,or 2baskets,respectively.Note that N 0+N 1+N 2=n .Suppose that shots are independent Bernoulli trails with probability p for making a basket.(a)Write down the likelihood for (N 0,N 1,N 2).12(b)Show that the maximum likelihood estimator of p is ˆp =N 1+2N 2N 0+2N 1+2N 2.(c)Is ˆp an unbiased estimator for p ?Prove or disprove.(Hint:E ˆp is a polynomial in p ,whose order is higher than 1for p ∈(0,1).)(d)Find the asymptotic distribution of ˆp as n tends to ∞.5.When considering finite difference schemes approximating partial differential equations (PDEs),for example,the scheme(1)u n +1j =u n j −λ(u n j −u n j −1)where λ=∆t ∆x ,approximating the PDE (2)u t +u x =0,we are often interested in stability,namely(3)||u n ||≤C ||u 0||,n ∆t ≤T for a constant C =C (T )independent of the time step ∆t and the spa-tial mesh size ∆x .Here ||·||is a given norm,for example the L 2norm orthe L ∞norm,of the numerical solution vector u n =(u n 1,u n 2,···,u n N ).The mesh points are x j =j ∆x ,t n =n ∆t ,and the numerical solutionu n j approximates the exact solution u (x j ,t n )of the PDE (2)with aperiodic boundary condition.(i)Prove that the scheme (1)is stable in the sense of (3)for boththe L 2norm and the L ∞norm under the time step restriction λ≤1.(ii)Since the numerical solution u n is in a finite dimensional space,Student A argues that the stability (3),once proved for a spe-cific norm ||·||a ,would also automatically hold for any other norm ||·||b .His argument is based on the equivalency of all norms in a finite dimensional space,namely for any two norms ||·||a and ||·||b on a finite dimensional space W ,there exists a constant δ>0such thatδ||u ||b ≤||u ||a ≤1δ||u ||b .Do you agree with his argument?If yes,please give a detailed proof of the following theorem:If a scheme is stable,namely (3)holds for one particular norm (e.g.the L 2norm),then it is also stable for any other norm.If not,please explain the mistake made by Student A.6.We have the following 3PDEs(4)u t +Au x =0,(5)u t +Bu x =0,3 (6)u t+Cu x=0,C=A+B.Here u is a vector of size m and A and B are m×m real matrices. We assume m≥2and both A and B are diagonalizable with only real eigenvalues.We also assume periodic initial condition for these PDEs.(i)Prove that(4)and(5)are both well-posed in the L2-norm.Recall that a PDE is well-posed if its solution satisfies||u(·,t)||≤C(T)||u(·,0)||,0≤t≤Tfor a constant C(T)which depends only on T.(ii)Is(6)guaranteed to be well-posed as well?If yes,give a proof;if not,give a counter example.(iii)Suppose we have afinite difference schemeu n+1=A h u nfor approximating(4)and another schemeu n+1=B h u nfor approximating(5).Suppose both schemes are stable in theL2-norm,namely(3)holds for both schemes.If we now formthe splitting schemeu n+1=B h A h u nwhich is a consistent scheme for solving(6),is this scheme guar-anteed to be L2stable as well?If yes,give a proof;if not,givea counter example.S.-T.Yau College Student Mathematics Contests2010Geometry and TopologyTeam(Please select5problems to solve)1.Let S n⊂R n+1be the unit sphere,and R n⊂R n+1the equator n-plane through the center of S n.Let N be the north pole of S n.Define a mappingπ:S n\{N}→R n called the stereographic projection that takes A∈S n\{N}into the intersection A ∈R n of the equator n-plane R n with the line which passes through A and N.Prove that the stereographic projection is a conformal change,and derive the standard metric of S n by the stereographic projection.2.Let M be a(connected)Riemannian manifold of dimension2.Let f be a smooth non-constant function on M such that f is bounded from above and∆f≥0everywhere on M.Show that there does not exist any point p∈M such that f(p)=sup{f(x):x∈M}.3.Let M be a compact smooth manifold of dimension d.Prove that there exists some n∈Z+such that M can be regularly embedded in the Euclidean space R n.4.Show that any C∞function f on a compact smooth manifold M (without boundary)must have at least two critical points.When M is the2-torus,show that f must have more than two critical points.5.Construct a space X with H0(X)=Z,H1(X)=Z2×Z3,H2(X)= Z,and all other homology groups of X vanishing.6.(a).Define the degree deg f of a C∞map f:S2−→S2and prove that deg f as you present it is well-defined and independent of any choices you need to make in your definition.(b).Prove in detail that for each integer k(possibly negative),there is a C∞map f:S2−→S2of degree k.1S.-T.Yau College Student Mathematics Contests 2010Algebra,Number Theory andCombinatoricsTeam(Please select 5problems to solve)1.For a real number r ,let [r ]denote the maximal integer less or equal than r .Let a and b be two positive irrational numbers such that 1a +1b = 1.Show that the two sequences of integers [ax ],[bx ]for x =1,2,3,···contain all natural numbers without repetition.2.Let n ≥2be an integer and consider the Fermat equationX n +Y n =Z n ,X,Y,Z ∈C [t ].Find all nontrivial solution (X,Y,Z )of the above equation in the sense that X,Y,Z have no common zero and are not all constant.3.Let p ≥7be an odd prime number.(a)Evaluate the rational number cos(π/7)·cos(2π/7)·cos(3π/7).(b)Show that (p −1)/2n =1cos(nπ/p )is a rational number and deter-mine its value.4.For a positive integer a ,consider the polynomialf a =x 6+3ax 4+3x 3+3ax 2+1.Show that it is irreducible.Let F be the splitting field of f a .Show that its Galois group is solvable.5.Prove that a group of order 150is not simple.6.Let V ∼=C 2be the standard representation of SL 2(C ).(a)Show that the n -th symmetric power V n =Sym n V is irre-ducible.(b)Which V n appear in the decomposition of the tensor productV 2⊗V 3into irreducible representations?1。
Chapter 1 Derivation of Reaction-Diffusion Equations
where J is the flux of P , d(x) is called diffusion coefficient at x, and ∇x is the gradient operator ∇x f (x) = (∂f /∂x1 , ∂f /∂x2 , · · · , ∂f /∂xn ). On the other hand, the number of particles at any point may change because of other reasons like birth, death, hunting, or chemical reactions. We assume that the rate of change of the density function due to these reasons is f (t, x, P ), which we usually call the reaction rate. Now we derive a differential equation using the balanced law. We choose any region O, then the total population in O is O P (t, x)dx, and the rate of change of the total population is d dt P (t, x)dx.
Chapter 1
Derivation of Reaction-Diffusion Equations
1.1 Fick’s Law
Diffusion mechanism models the movement of many individuals in an environment or media. The individuals can be very small such as basic particles in physics, bacteria, molecules, or cells, or very large objects such as animals, plants, or certain kind of events like epidemics, or rumors. The particles reside in a region, which we call Ω, and we assume that Ω is open subset of Rn (the n-th dimensional space with Cartesian coordinate system) with n ≥ 1. In particular, we are interested in the cases of n = 1, 2 and 3, but most material here are true regardless of the dimensions of the space (sometimes n = 1 and n ≥ 2 may be different as we will see.) The main mathematical variable we consider here is the density function of the particles: P (t, x), where t is the time, and x ∈ Ω is the location. The dimension of the population density usually is number of particles or organisms per unit area (if n = 2) or unit volume (if n = 3). For example, the human population density is often expressed in number of people per square kilometer. A list of world population and population density can be found at /wiki/List_of_countries_by_population_density. However in such data table, we can only find the population density for countries, and a country is not a “point” on our map. In reality, population density is always associated with a scale, like country, city, county, town, and street. But as in many other mathematical models, we will assume that the function P (t, x) has nicer properties, like continuity and differentiability, which is in fact reasonable, when a population with a large number of organisms is considered. Technically, we define the population density function P (t, x) as follows: let x be a point in the habitat Ω, and let {On }∞ n=1 be a sequence of spatial regions (which have the same dimension as Ω) surrounding x; here On is chosen in a way that the spatial measurement |On | of On (length, area, volume, or 1
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Mixed boundary conditions in Euclidean quantum gravity
= cij = 1 gij , where cij is the metric on a unit 3-sphere), one has
b ij h
1 hij + 1 'ijj + 2 'jji + Kij '0 2
(1:9)
the invariance of (1.1) under (1.9) is guaranteed providing (1.4)-(1.5) hold. Strictly, one has to require that 'ijj
June 1995, DSF preprint 95/23
MIXED BOUNDARY CONDITIONS IN EUCLIDEAN QUANTUM GRAVITY
Giampiero Esposito1;2 and Alexander Yu Kamenshchik3
PostScript® processed by the SLAC/DESY Libraries on 3 Jul 1995.
GR-QC-9506092
Mixed boundary conditions in Euclidean quantum gravity
1. Introduction
Since the main problem in quantum cosmology is to nd a suitable set of boundary conditions which completely determine the path integral for the quantum state of the universe 1-2], the search for such boundary conditions is of crucial importance in Euclidean quantum gravity 3]. For this purpose, it has been found in 4] that, following earlier work in 5], a set of gauge-invariant boundary conditions for Euclidean quantum gravity can be written as
h i
@M
= 0, but this is automatically satis ed by (1.5), at least
in the case of a 3-sphere boundary. The remaining boundary conditions (1.2)-(1.3) are obtained by setting to zero at the boundary the de Donder functional de ned in (1.7). Their invariance under (1.8) is guaranteed providing (1.4)-(1.5) hold. Note that the boundary conditions (1.1)-(1.5) are non-local in that they cannot be written in terms of complementary projection operators, and can instead be re-expressed as integral equations on metric perturbations at the boundary. The aim of our paper is to study in detail another relevant choice of mixed boundary conditions in Euclidean quantum gravity. The motivations of our analysis are as follows. (i) In classical general relativity one can also x at the boundary the trace of the extrinsiccurvature tensor (rather than the induced 3-metric) 8]. This choice of boundary conditions is also relevant for the theory of the quantum state of the universe 1,9]. (ii) In one-loop quantum cosmology, one can also consider Hawking's magnetic boundary conditions for quantum gravity, which set to zero at the boundary the linearized magnetic curvature, and hence the rst derivatives with respect to of the perturbed 3-metric (see section 7.3 of 2]). 4
1 Istituto
Nazionale di Fisica Nucleare, Sezione di Napoli
2 Dipartimento
Mostra d'Oltremare Padiglione 20, 80125 Napoli, Italy; di Scienze Fisiche Mostra d'Oltremare Padiglione 19, 80125 Napoli, Italy;
Z
M
Ig:a: = 32 G
where
1
p
det g d4x
(1:6)
is taken to be the de Donder functional (r being the 4-dimensional Levi-Civita
connection of the background) 4,6]
dD (h)onditions in Euclidean quantum gravity
2,8]. Since, denoting by Kij the unperturbed extrinsic-curvature tensor of the boundary (Kij
1 @gij 2 @
b h
1 h + 2r ' + 1r ' : 2 and
(1:8) 7]. This property
The boundary conditions for ' are the same as those for
should not be surprising, since already in the simpler case of Euclidean Maxwell theory the ghost and anti-ghost obey the same boundary conditions imposed on the scalar eld occurring in the gauge transformations of the potential (see Appendix A). It is therefore su cient to consider ' , and then multiply by ?2 the resulting contribution to the amplitudes, since the ghosts are anti-commuting and complex-valued. The boundary conditions (1.1) re ect a natural choice in the classical variational problem in general relativity, where one often xes at the boundary the intrinsic 3-geometry
quantum gravity. These involve, in particular, Robin boundary conditions on the perturbed 3-metric and hence lead, by gauge invariance, to Robin conditions on the whole ghost 1form. The corresponding trace anomaly is evaluated in the case of at Euclidean 4-space bounded by a 3-sphere. In general, this anomaly di ers from the ones resulting from other local or non-local boundary conditions studied in the recent literature. PACS: 0370, 0460, 9880 1
h
hij
i
@M
=0 =0
(1:1) (1:2) (1:3) (1:4) (1:5)
@h00 + 6 h ? @ (gij h ) + 2 h ji 00 @ ij 2 0i @ @h0i + 3 h ? 1 h 0i 2 00ji @
h i
@M
@M
=0
'0
@M
=0 = 0:
h i
'i
@M
With our notation 4], g is the at background 4-metric, h is its perturbation, is the Euclidean-time coordinate, which becomes a radial coordinate if at 4-space is bounded by a 3-sphere (see section 3). Moreover, ' is the ghost 1-form, with normal component '0 and tangential components 'i, and the stroke denotes covariant di erentiation tangentially with respect to the 3-dimensional Levi-Civita connection of the boundary. We are interested