The Norma Cluster (ACO 3627) I. A Dynamical Analysis of the Most Massive Cluster in the Gre
The entropy formula for the Ricci flow and its geometric applications
a rX iv:mat h /211159v1[ma t h.DG]11Nov22The entropy formula for the Ricci flow and its geometric applications Grisha Perelman ∗November 20,2007Introduction 1.The Ricci flow equation,introduced by Richard Hamilton [H 1],is the evolution equation d ∗St.Petersburg branch of Steklov Mathematical Institute,Fontanka 27,St.Petersburg191011,Russia.Email:perelman@pdmi.ras.ru or perelman@ ;I was partially supported by personal savings accumulated during my visits to the Courant Institute in the Fall of 1992,to the SUNY at Stony Brook in the Spring of 1993,and to the UC at Berkeley as a Miller Fellow in 1993-95.I’d like to thank everyone who worked to make those opportunities available to me.1in dimension four converge,modulo scaling,to metrics of constant positivecurvature.Without assumptions on curvature the long time behavior of the metricevolving by Ricciflow may be more complicated.In particular,as t ap-proaches somefinite time T,the curvatures may become arbitrarily large in some region while staying bounded in its complement.In such a case,it isuseful to look at the blow up of the solution for t close to T at a point where curvature is large(the time is scaled with the same factor as the metric ten-sor).Hamilton[H9]proved a convergence theorem,which implies that asubsequence of such scalings smoothly converges(modulo diffeomorphisms) to a complete solution to the Ricciflow whenever the curvatures of the scaledmetrics are uniformly bounded(on some time interval),and their injectivity radii at the origin are bounded away from zero;moreover,if the size of thescaled time interval goes to infinity,then the limit solution is ancient,thatis defined on a time interval of the form(−∞,T).In general it may be hard to analyze an arbitrary ancient solution.However,Ivey[I]and Hamilton[H4]proved that in dimension three,at the points where scalar curvatureis large,the negative part of the curvature tensor is small compared to the scalar curvature,and therefore the blow-up limits have necessarily nonneg-ative sectional curvature.On the other hand,Hamilton[H3]discovered a remarkable property of solutions with nonnegative curvature operator in ar-bitrary dimension,called a differential Harnack inequality,which allows,inparticular,to compare the curvatures of the solution at different points and different times.These results lead Hamilton to certain conjectures on thestructure of the blow-up limits in dimension three,see[H4,§26];the presentwork confirms them.The most natural way of forming a singularity infinite time is by pinchingan(almost)round cylindrical neck.In this case it is natural to make a surgery by cutting open the neck and gluing small caps to each of the boundaries,andthen to continue running the Ricciflow.The exact procedure was describedby Hamilton[H5]in the case of four-manifolds,satisfying certain curvature assumptions.He also expressed the hope that a similar procedure wouldwork in the three dimensional case,without any a priory assumptions,and that afterfinite number of surgeries,the Ricciflow would exist for all timet→∞,and be nonsingular,in the sense that the normalized curvatures ˜Rm(x,t)=tRm(x,t)would stay bounded.The topology of such nonsingular solutions was described by Hamilton[H6]to the extent sufficient to makesure that no counterexample to the Thurston geometrization conjecture can2occur among them.Thus,the implementation of Hamilton program would imply the geometrization conjecture for closed three-manifolds.In this paper we carry out some details of Hamilton program.The more technically complicated arguments,related to the surgery,will be discussed elsewhere.We have not been able to confirm Hamilton’s hope that the so-lution that exists for all time t→∞necessarily has bounded normalized curvature;still we are able to show that the region where this does not hold is locally collapsed with curvature bounded below;by our earlier(partly unpublished)work this is enough for topological conclusions.Our present work has also some applications to the Hamilton-Tian con-jecture concerning K¨a hler-Ricciflow on K¨a hler manifolds with positivefirst Chern class;these will be discussed in a separate paper.2.The Ricciflow has also been discussed in quantumfield theory,as an ap-proximation to the renormalization group(RG)flow for the two-dimensional nonlinearσ-model,see[Gaw,§3]and references therein.While my back-ground in quantum physics is insufficient to discuss this on a technical level, I would like to speculate on the Wilsonian picture of the RGflow.In this picture,t corresponds to the scale parameter;the larger is t,the larger is the distance scale and the smaller is the energy scale;to compute something on a lower energy scale one has to average the contributions of the degrees of freedom,corresponding to the higher energy scale.In other words,decreasing of t should correspond to looking at our Space through a microscope with higher resolution,where Space is now described not by some(riemannian or any other)metric,but by an hierarchy of riemannian metrics,connected by the Ricciflow equation.Note that we have a paradox here:the regions that appear to be far from each other at larger distance scale may become close at smaller distance scale;moreover,if we allow Ricci flow through singularities,the regions that are in different connected compo-nents at larger distance scale may become neighboring when viewed through microscope.Anyway,this connection between the Ricciflow and the RGflow sug-gests that Ricciflow must be gradient-like;the present work confirms this expectation.3.The paper is organized as follows.In§1we explain why Ricciflow can be regarded as a gradientflow.In§2,3we prove that Ricciflow,considered as a dynamical system on the space of riemannian metrics modulo diffeomor-phisms and scaling,has no nontrivial periodic orbits.The easy(and known)3case of metrics with negative minimum of scalar curvature is treated in§2; the other case is dealt with in§3,using our main monotonicity formula(3.4) and the Gaussian logarithmic Sobolev inequality,due to L.Gross.In§4we apply our monotonicity formula to prove that for a smooth solution on a finite time interval,the injectivity radius at each point is controlled by the curvatures at nearby points.This result removes the major stumbling block in Hamilton’s approach to geometrization.In§5we give an interpretation of our monotonicity formula in terms of the entropy for certain canonical ensemble.In§6we try to interpret the formal expressions,arising in the study of the Ricciflow,as the natural geometric quantities for a certain Riemannian manifold of potentially infinite dimension.The Bishop-Gromov relative volume comparison theorem for this particular manifold can in turn be interpreted as another monotonicity formula for the Ricciflow.This for-mula is rigorously proved in§7;it may be more useful than thefirst one in local considerations.In§8it is applied to obtain the injectivity radius control under somewhat different assumptions than in§4.In§9we consider one more way to localize the original monotonicity formula,this time using the differential Harnack inequality for the solutions of the conjugate heat equation,in the spirit of Li-Yau and Hamilton.The technique of§9and the logarithmic Sobolev inequality are then used in§10to show that Ricciflow can not quickly turn an almost euclidean region into a very curved one,no matter what happens far away.The results of sections1through10require no dimensional or curvature restrictions,and are not immediately related to Hamilton program for geometrization of three manifolds.The work on details of this program starts in§11,where we describe the ancient solutions with nonnegative curvature that may occur as blow-up limits offinite time singularities(they must satisfy a certain noncollaps-ing assumption,which,in the interpretation of§5,corresponds to having bounded entropy).Then in§12we describe the regions of high curvature under the assumption of almost nonnegative curvature,which is guaranteed to hold by the Hamilton and Ivey result,mentioned above.We also prove, under the same assumption,some results on the control of the curvatures forward and backward in time in terms of the curvature and volume at a given time in a given ball.Finally,in§13we give a brief sketch of the proof of geometrization conjecture.The subsections marked by*contain historical remarks and references. See also[Cao-C]for a relatively recent survey on the Ricciflow.41Ricciflow as a gradientflow1.1.Consider the functional F= M(R+|∇f|2)e−f dV for a riemannian metric g ij and a function f on a closed manifold M.Itsfirst variation can be expressed as follows:δF(v ij,h)= M e−f[−△v+∇i∇j v ij−R ij v ij−v ij∇i f∇j f+2<∇f,∇h>+(R+|∇f|2)(v/2−h)]= M e−f[−v ij(R ij+∇i∇j f)+(v/2−h)(2△f−|∇f|2+R)], whereδg ij=v ij,δf=h,v=g ij v ij.Notice that v/2−h vanishes identically iffthe measure dm=e−f dV is keptfixed.Therefore,the symmetric tensor −(R ij+∇i∇j f)is the L2gradient of the functional F m= M(R+|∇f|2)dm, where now f denotes log(dV/dm).Thus given a measure m,we may consider the gradientflow(g ij)t=−2(R ij+∇i∇j f)for F m.For general m thisflow may not exist even for short time;however,when it exists,it is just the Ricciflow,modified by a diffeomorphism.The remarkable fact here is that different choices of m lead to the sameflow,up to a diffeomorphism;that is, the choice of m is analogous to the choice of gauge.1.2Proposition.Suppose that the gradientflow for F m exists for t∈[0,T]. Then at t=0we have F m≤nNow we computeF t≥2n( (R+△f)e−f dV)2=2t1and t2,are called Ricci solitons.(Thus,if one considers Ricciflow as a dy-namical system on the space of riemannian metrics modulo diffeomorphism and scaling,then breathers and solitons correspond to periodic orbits and fixed points respectively).At each time the Ricci soliton metric satisfies an equation of the form R ij+cg ij+∇i b j+∇j b i=0,where c is a number and b i is a one-form;in particular,when b i=1log V=1dtV(2−n)/nλ −RdV≥n2V2/n[ |R ij+∇i∇j f−1( (R+△f)2e−f dV−( (R+△f)e−f dV)2)]≥0,nwhere f is the minimizer for F.72.4.The arguments above also show that there are no nontrivial(that is with non-constant Ricci curvature)steady or expanding Ricci solitons(on closed M).Indeed,the equality case in the chain of inequalities above requires that R+△f be constant on M;on the other hand,the Euler-Lagrange equation for the minimizer f is2△f−|∇f|2+R=const.Thus,△f−|∇f|2=const=0, because (△f−|∇f|2)e−f dV=0.Therefore,f is constant by the maximum principle.2.5*.A similar,but simpler proof of the results in this section,follows im-mediately from[H6,§2],where Hamilton checks that the minimum of RV22e−f dV,(3.1)restricted to f satisfying(4πτ)−nM,τt=−1(3.3)2τThe evolution equation for f can also be written as follows:2∗u=0,where u=(4πτ)−ng ij|2(4πτ)−n2τalong the Ricciflow.It is not hard to show that in the definition ofµthere always exists a smooth minimizer f(on a closed M).It is also clear that limτ→∞µ(g ij,τ)=+∞whenever thefirst eigenvalue of−4△+R is positive. Thus,our statement that there is no shrinking breathers other than gradient solitons,is implied by the followingClaim For an arbitrary metric g ij on a closed manifold M,the function µ(g ij,τ)is negative for smallτ>0and tends to zero asτtends to zero.Proof of the Claim.(sketch)Assume that¯τ>0is so small that Ricci flow starting from g ij exists on[0,¯τ].Let u=(4πτ)−n2τ−1g ij,fτ,12τ−1g ij”converge”to the euclidean metric,and if we couldextract a converging subsequence from fτ,we would get a function f on R n, such that R n(2π)−n2|∇f|2+f−n](2π)−n2−t)=µ(g ij(0),12)satisfiesR ij+∇i∇j f−g ij=0.Of course,this argument requires the existence of minimizer,and justification of the integration by parts;this is easy if M is closed,but can also be done with more efforts on some complete M,for instance when M is the Gaussian soliton.93.3*The no breathers theorem in dimension three was proved by Ivey[I]; in fact,he also ruled out nontrivial Ricci solitons;his proof uses the almost nonnegative curvature estimate,mentioned in the introduction.Logarithmic Sobolev inequalities is a vast area of research;see[G]for a survey and bibliography up to the year1992;the influence of the curvature was discussed by Bakry-Emery[B-Em].In the context of geometric evolution equations,the logarithmic Sobolev inequality occurs in Ecker[E1].4No local collapsing theorem IIn this section we present an application of the monotonicity formula(3.4) to the analysis of singularities of the Ricciflow.4.1.Let g ij(t)be a smooth solution to the Ricciflow(g ij)t=−2R ij on[0,T). We say that g ij(t)is locally collapsing at T,if there is a sequence of times t k→T and a sequence of metric balls B k=B(p k,r k)at times t k,such that r2k/t k is bounded,|Rm|(g ij(t k))≤r−2k in B k and r−n k V ol(B k)→0.Theorem.If M is closed and T<∞,then g ij(t)is not locally collapsing at T.Proof.Assume that there is a sequence of collapsing balls B k=B(p k,r k) at times t k→T.Then we claim thatµ(g ij(t k),r2k)→−∞.Indeed one(x,p k)r−1k)+c k,whereφis a function of one can take f k(x)=−logφ(dist tkvariable,equal1on[0,1/2],decreasing on[1/2,1],and very close to0on [1,∞),and c k is a constant;clearly c k→−∞as r−n k V ol(B k)→0.Therefore, applying the monotonicity formula(3.4),we getµ(g ij(0),t k+r2k)→−∞. However this is impossible,since t k+r2k is bounded.4.2.Definition We say that a metric g ij isκ-noncollapsed on the scaleρ,if every metric ball B of radius r<ρ,which satisfies|Rm|(x)≤r−2for every x∈B,has volume at leastκr n.It is clear that a limit ofκ-noncollapsed metrics on the scaleρis also κ-noncollapsed on the scaleρ;it is also clear thatα2g ij isκ-noncollapsed on the scaleαρwhenever g ij isκ-noncollapsed on the scaleρ.The theorem above essentially says that given a metric g ij on a closed manifold M and T<∞,one canfindκ=κ(g ij,T)>0,such that the solution g ij(t)to the Ricciflow starting at g ij isκ-noncollapsed on the scale T1/2for all t∈[0,T), provided it exists on this interval.Therefore,using the convergence theorem of Hamilton,we obtain the following10Corollary.Let g ij (t ),t ∈[0,T )be a solution to the Ricci flow on a closed manifold M,T <∞.Assume that for some sequences t k →T,p k ∈M and some constant C we have Q k =|Rm |(p k ,t k )→∞and |Rm |(x,t )≤CQ k ,whenever t <t k .Then (a subsequence of)the scalings of g ij (t k )at p k with factors Q k converges to a complete ancient solution to the Ricci flow,which is κ-noncollapsed on all scales for some κ>0.5A statistical analogyIn this section we show that the functional W ,introduced in section 3,is in a sense analogous to minus entropy.5.1Recall that the partition function for the canonical ensemble at tem-perature β−1is given by Z = exp (−βE )dω(E ),where ω(E )is a ”density of states”measure,which does not depend on β.Then one computes the average energy <E >=−∂(∂β)2log Z.Now fix a closed manifold M with a probability measure m ,and suppose that our system is described by a metric g ij (τ),which depends on the temper-ature τaccording to equation (g ij )τ=2(R ij +∇i ∇j f ),where dm =udV,u =(4πτ)−n 2)dm.(We do not discuss here what assumptions on g ij guarantee that the corre-sponding ”density of states”measure can be found)Then we compute<E >=−τ2 M(R +|∇f |2−n 2τg ij |2dmAlternatively,we could prescribe the evolution equations by replacing the t -derivatives by minus τ-derivatives in (3.3),and get the same formulas for Z,<E >,S,σ,with dm replaced by udV.Clearly,σis nonnegative;it vanishes only on a gradient shrinking soliton.<E >is nonnegative as well,whenever the flow exists for all sufficiently small τ>0(by proposition 1.2).Furthermore,if (a)u tends to a δ-function as τ→0,or (b)u is a limit of a sequence of functions u i ,such that each u i11tends to aδ-function asτ→τi>0,andτi→0,then S is also nonnegative. In case(a)all the quantities<E>,S,σtend to zero asτ→0,while in case (b),which may be interesting if g ij(τ)goes singular atτ=0,the entropy S may tend to a positive limit.If theflow is defined for all sufficiently largeτ(that is,we have an ancient solution to the Ricciflow,in Hamilton’s terminology),we may be interested in the behavior of the entropy S asτ→∞.A natural question is whether we have a gradient shrinking soliton whenever S stays bounded.5.2Remark.Heuristically,this statistical analogy is related to the de-scription of the renormalization groupflow,mentioned in the introduction: in the latter one obtains various quantities by averaging over higher energy states,whereas in the former those states are suppressed by the exponential factor.5.3*An entropy formula for the Ricciflow in dimension two was found by Chow[C];there seems to be no relation between his formula and ours.The interplay of statistical physics and(pseudo)-riemannian geometry occurs in the subject of Black Hole Thermodynamics,developed by Hawking et al.Unfortunately,this subject is beyond my understanding at the moment.6Riemannian formalism in potentially infi-nite dimensionsWhen one is talking of the canonical ensemble,one is usually considering an embedding of the system of interest into a much larger standard system of fixed temperature(thermostat).In this section we attempt to describe such an embedding using the formalism of Rimannian geometry.6.1Consider the manifold˜M=M×S N×R+with the following metric:˜g ij=g ij,˜gαβ=τgαβ,˜g00=N2N .It turns out that the components of the curvaturetensor of this metric coincide(modulo N−1)with the components of the matrix Harnack expression(and its traces),discovered by Hamilton[H3]. One can also compute that all the components of the Ricci tensor are equal12to zero(mod N−1).The heat equation and the conjugate heat equation on M can be interpreted via Laplace equation on˜M for functions and volume forms respectively:u satisfies the heat equation on M iff˜u(the extension of u to˜M constant along the S Nfibres)satisfies˜△˜u=0mod N−1;similarly,u satisfies the conjugate heat equation on M iff˜u∗=τ−N−12e−f dV).To achieve this,first apply to˜g a(small)diffeomor-phism,mapping each point(x i,yα,τ)into(x i,yα,τ(1−2fN)˜gαβ,˜g m00=˜g00−2fτ−fN)˜gαβ,g m00=˜g m00−|∇f|2=12−[τ(2△f−|∇f|2+R)+f−n]),g m i0=g mα0=g m iα=0Note that the hypersurfaceτ=const in the metric g m has the volume form τN/2e−f times the canonical form on M and S N,and the scalar curvatureof this hypersurface is12+τ(2△f−|∇f|2+R)+f)mod N−1.Thus theentropy S multiplied by the inverse temperatureβis essentially minus the total scalar curvature of this hypersurface.6.3Now we return to the metric˜g and try to use its Ricci-flatness by interpreting the Bishop-Gromov relative volume comparison theorem.Con-sider a metric ball in(˜M,˜g)centered at some point p whereτ=0.Then clearly the shortest geodesic between p and an arbitrary point q is always orthogonal to the S Nfibre.The length of such curveγ(τ)can be computedas τ(q)2τ+R+|˙γM(τ)|2dτ= √τ(R+|˙γM(τ)|2)dτ+O(N−3Thus a shortest geodesic should minimize L(γ)= τ(q)0√2Nτ(q) centered at p is O(N−1)-close to the hypersurfaceτ=τ(q),and its volume can be computed as V(S N) M( 2N L(x)+O(N−2))N dx,so the ratio of this volume to 2timesτ(q)−nMτ(R(γ(τ))+|˙γ(τ)|2)dτ(of course,R(γ(τ))and|˙γ(τ)|2are computed using g ij(τ))Let X(τ)=˙γ(τ),and let Y(τ)be any vectorfield alongγ(τ).Then the first variation formula can be derived as follows:δY(L)=14τ2τ1√τ(<Y,∇R >+2<∇X Y,X >)dτ= τ2τ1√dτ<Y,X >−2<Y,∇X X >−4Ric(Y,X ))dτ=2√τ<Y,∇R −2∇X X −4Ric(X,·)−12∇R +1τX (τ)has a limit as τ→0.From now on we fix p and τ1=0and denote by L (q,¯τ)the L -length of the L -shortest curve γ(τ),0≤τ≤¯τ,connecting p and q.In the computations below we pretend that shortest L -geodesics between p and q are unique for all pairs (q,¯τ);if this is not the case,the inequalities that we obtain are still valid when understood in the barrier sense,or in the sense of distributions.The first variation formula (7.1)implies that ∇L (q,¯τ)=2√¯τ(R +|X |2)−<X,∇L >=2√¯τ(R +|X |2)To evaluate R +|X |2we compute (using (7.2))dτR +2<∇R,X >−2Ric(X,X )−1τ(R +|X |2),(7.3)where H (X )is the Hamilton’s expression for the trace Harnack inequality (with t =−τ).Hence,¯τ32L (q,¯τ),(7.4)15where K =K (γ,¯τ)denotes the integral ¯τ0τ3¯τR −1¯τK (7.5)|∇L |2=−4¯τR +2¯τL −4¯τK (7.6)Finally we need to estimate the second variation of L.We computeδ2Y (L )=¯τ0√τ(Y ·Y ·R +2<∇X ∇Y Y,X >+2<R (Y,X ),Y,X >+2|∇X Y |2)dτNowd¯τ+¯τ0√2τY (7.8)We computed τ<Y,Y >,16so |Y (τ)|2=ττ(∇Y ∇Y R +2<R (Y,X ),Y,X >+2∇X Ric(Y,Y )−4∇Y Ric(Y,X )+2|Ric(Y,·)|2−22τ¯τ)dτTo put this in a more convenient form,observe thatdτRic(Y,Y )−2|Ric(Y,·)|2,so Hess L (Y,Y )≤1¯τ−2√τH (X,Y )dτ,(7.9)whereH (X,Y )=−∇Y ∇Y R −2<R (Y,X )Y,X >−4(∇X Ric(Y,Y )−∇Y Ric(Y,X ))−2Ric τ(Y,Y )+2|Ric(Y,·)|2−1τR +n τ−1dτ|Y |2=2Ric(Y,Y )+2<∇X Y,Y >=2Ric(Y,Y )+2<∇Y X,Y >=2Ric(Y,Y )+1¯τHess L (Y,Y )≤1√2H (X,˜Y )dτ,(7.11)where ˜Y is obtained by solving ODE (7.8)with initial data ˜Y (¯τ)=Y (¯τ).Moreover,the equality in (7.11)holds only if ˜Y is L -Jacobi and hence d √¯τ.17Now we can deduce an estimate for the jacobian J of the L-exponential map,given by L exp X(¯τ)=γ(¯τ),whereγ(τ)is the L-geodesic,starting at p and having X as the limit of√dτlog J(τ)≤n2¯τ−3√¯τg.Let l(q,τ)=1τL(q,τ)be thereduced distance.Then along an L-geodesicγ(τ)we have(by(7.4))d2¯τl+12¯τ−32exp(−l(τ))J(τ)is nonincreasing inτalongγ, and monotonicity is strict unless we are on a gradient shrinking soliton. Integrating over M,we get monotonicity of the reduced volume function ˜V(τ)= Mτ−n2¯τ≥0,(7.13) which follows immediately from(7.5),(7.6)and(7.10).Note also a useful inequality2△l−|∇l|2+R+l−nτL(q,τ),then from(7.5), (7.10)we obtain¯L¯τ+△¯L≤2n(7.15) Therefore,the minimum of¯L(·,¯τ)−2n¯τis nonincreasing,so in particular, the minimum of l(·,¯τ)does not exceed n2(τ0−τ),whenever theflow exists forτ∈[0,τ0].)7.2If the metrics g ij(τ)have nonnegative curvature operator,then Hamil-ton’s differential Harnack inequalities hold,and one can say more about the behavior of l.Indeed,in this case,if the solution is defined forτ∈[0,τ0],then H(X,Y)≥−Ric(Y,Y)(1τ0−τ)≥−R(1τ0−τ)|Y|2and18H(X)≥−R(1τ0−τ).Therefore,wheneverτis bounded away fromτ0(say,τ≤(1−c)τ0,c>0),we get(using(7.6),(7.11))|∇l|2+R≤Cldτlog|Y|2≤1n.We claim that˜V k(ǫk r2k)<3ǫn2ǫ−12k;on the otherhand,the contribution of the longer vectors does not exceed exp(−12k)by the jacobian comparison theorem.However,˜V k(t k)(that is,at t=0)stays bounded away from zero.Indeed,since min l k(·,t k−12,we can pick a point q k,where it is attained,and obtain a universal upper bound on l k(·,t k)by considering only curvesγwithγ(t k−12T].Sincethe monotonicity of the reduced volume requires˜V k(t k)≤˜V k(ǫk r2k),this is a contradiction.A similar argument shows that the statement of the corollary in4.2can be strengthened by adding another property of the ancient solution,obtained as a blow-up ly,we may claim that if,say,this solution is defined for t∈(−∞,0),then for any point p and any t0>0,the reduced volume function˜V(τ),constructed using p andτ(t)=t0−t,is bounded below byκ.7.4*The computations in this section are just natural modifications of those in the classical variational theory of geodesics that can be found in any textbook on Riemannian geometry;an even closer reference is[L-Y],where they use”length”,associated to a linear parabolic equation,which is pretty much the same as in our case.198No local collapsing theorem II8.1Let usfirst formalize the notion of local collapsing,that was used in7.3.Definition.A solution to the Ricciflow(g ij)t=−2R ij is said to be κ-collapsed at(x0,t0)on the scale r>0if|Rm|(x,t)≤r−2for all(x,t) satisfying dist t(x,x0)<r and t0−r2≤t≤t0,and the volume of the metric ball B(x0,r2)at time t0is less thanκr n.8.2Theorem.For any A>0there existsκ=κ(A)>0with the fol-lowing property.If g ij(t)is a smooth solution to the Ricciflow(g ij)t=−2R ij,0≤t≤r20,which has|Rm|(x,t)≤r−20for all(x,t),satisfying dist0(x,x0)<r0,and the volume of the metric ball B(x0,r0)at time zero is at least A−1r n0,then g ij(t)can not beκ-collapsed on the scales less than r0at a point(x,r20)with dist r20(x,x0)≤Ar0.Proof.By scaling we may assume r0=1;we may also assume dist1(x,x0)= A.Let us apply the constructions of7.1choosing p=x,τ(t)=1−t.Arguing as in7.3,we see that if our solution is collapsed at x on the scale r≤1,then the reduced volume˜V(r2)must be very small;on the other hand,˜V(1)can not be small unless min l(x,12(x,x0)≤13Kr0+r−10)(the inequality must be understood in the barrier sense,when necessary)(b)(cf.[H4,§17])Suppose Ric(x,t0)≤(n−1)K when dist t(x,x0)<r0, or dist t(x,x1)<r0.Thend3Kr0+r−10)at t=t0 Proof of Lemma.(a)Clearly,d t(x)= γ−Ric(X,X),whereγis the shortest geodesic between x and x0and X is its unit tangent vector,On the other hand,△d≤ n−1k=1s′′Y k(γ),where Y k are vectorfields alongγ,vanishing at20x0and forming an orthonormal basis at x when complemented by X,ands′′Yk (γ)denotes the second variation along Y k of the length ofγ.Take Y k to beparallel between x and x1,and linear between x1and x0,where d(x1,t0)=r0. Then△d≤n−1k=1s′′Y k(γ)= d(x,t0)r0−Ric(X,X)ds+ r00(s2r20)ds= γ−Ric(X,X)+ r00(Ric(X,X)(1−s2r20)ds≤d t+(n−1)(220),andrapidly increasing to infinity on(110),in such a way that2(φ′)2/φ−φ′′≥(2A+100n)φ′−C(A)φ,(8.1) for some constant C(A)<∞.Note that¯L+2n+1≥1for t≥12)is achieved for some y satisfying d(y,110.Now we compute2h=(¯L+2n+1)(−φ′′+(d t−△d−2A)φ′)−2<∇φ∇¯L>+(¯L t−△¯L)φ(8.2)∇h=(¯L+2n+1)∇φ+φ∇¯L(8.3) At a minimum point of h we have∇h=0,so(8.2)becomes2h=(¯L+2n+1)(−φ′′+(d t−△d−2A)φ′+2(φ′)2/φ)+(¯L t−△¯L)φ(8.4)Now since d(y,t)≥120),we can apply our lemma(a)to get d t−△d≥−100(n−1)on the set where φ′=0.Thus,using(8.1)and(7.15),we get2h≥−(¯L+2n+1)C(A)φ−2nφ≥−(2n+C(A))hThis implies that min h can not decrease too fast,and we get the required estimate.219Differential Harnack inequality for solutions of the conjugate heat equation9.1Proposition.Let g ij(t)be a solution to the Ricciflow(g ij)t=−2R ij,0≤t≤T,and let u=(4π(T−t))−ng ij|2(9.1)2(T−t)Proof.Routine computation.Clearly,this proposition immediately implies the monotonicity formula (3.4);its advantage over(3.4)shows up when one has to work locally.9.2Corollary.Under the same assumptions,on a closed manifold M,or whenever the application of the maximum principle can be justified,min v/u is nondecreasing in t.9.3Corollary.Under the same assumptions,if u tends to aδ-function as t→T,then v≤0for all t<T.Proof.If h satisfies the ordinary heat equation h t=△h with respect to the evolving metric g ij(t),then we have ddt hv≥0.Thus we only need to check that for everywhere positive h the limit of hv as t→T is nonpositive.But it is easy to see,that this limit is in fact zero.9.4Corollary.Under assumptions of the previous corollary,for any smooth curveγ(t)in M holds−d2(R(γ(t),t)+|˙γ(t)|2)−1and2(T−t)v≤0we get f t+12|∇f|2−f dt f(γ(t),t)=−f t−<∇f,˙γ(t)>≤−f t+12|˙γ|2.Summing these two inequalities, we get(9.2).9.5Corollary.If under assumptions of the previous corollary,p is the point where the limitδ-function is concentrated,then f(q,t)≤l(q,T−t),where l is the reduced distance,defined in7.1,using p andτ(t)=T−t.22。
A Tutorial on Spectral Clustering
Ulrike von Luxburg Max Planck Institute for Biological Cybernetics Spemannstr. 38, 72076 T¨ ubingen, Germany ulrike.luxburg@tuebingen.mpg.de
2
Similarity graphs
Given a set of data points x1 , . . . xn and some notion of similarity sij ≥ 0 between all pairs of data points xi and xj , the intuitive goal of clustering is to divide the data points into several groups such that points in the same group are similar and points in different groups are dissimilar to each other. If we do not have more information than similarities between data points, a nice way of representing the data is in form of the similarity graph G = (V, E ). Each vertex vi in this graph represents a data point xi . Two vertices are connected if the similarity sij between the corresponding data points xi and xj is positive or larger than a certain threshold, and the edge is weighted by sij . The problem of clustering can now be reformulated using the similarity graph: we want to find a partition of the graph such that the edges between different groups have very low weights (which means that points in different clusters are dissimilar from each other) and the edges within a group have high weights (which means that points within the same cluster are similar to each other). To be able to formalize this intuition we first want to introduce some basic graph notation and briefly discuss the kind of graphs we are going to study.
cfa三级笔记
cfa三级笔记Monte Carlo simulation（专题）定义：Monte Carlo simulation allows asset manager to model the uncertainty of several key variables. Generates random outcomes according to assumed probability distribution for these key variables. It is flexible approach for exploring different market or investment scenario. 蒙特卡洛模拟是将变量（事先定义好分布）的值随机发射，生成了结果，可灵活的探索不同市场、投资环境下的状态。
较MVO的优势：1, Rebalancing and taxes, Monte carlo simulation allow to analyze different rebalancing policies and their cost over time（in multi-period situation）. 蒙特卡洛模拟可以用于分析执行不同的再平衡策略、税收时的影响。
2, Path dependent. As there are cash out flow each year, terminal wealth（the portfolio’s value at a given point）will be path-dependent because of the interaction of cash flows and returns. 如果每年都有资金流出，指定时间的组合价值会受这些资金流出和收益的影响Cash flows in and out of a portfolio and the sequence of returns will have a material effect on terminal wealth, this is termed path-dependent.3, Monte Carlo can incorporate statistical properties outside the normal distribution, such as skewness and excess kurtosis.蒙特卡洛模拟可用于建模非正态分布。
The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time seri
904N.E.Huang and others10.Discussion98711.Conclusions991References993 A new method for analysing has been devel-oped.The key part of the methodany complicated data set can be decomposed intoof‘intrinsic mode functions’Hilbert trans-This decomposition method is adaptive,and,highly efficient.Sinceapplicable to nonlinear and non-stationary processes.With the Hilbert transform,Examplesthe classical nonlinear equation systems and dataare given to demonstrate the power new method.data are especially interesting,for serve to illustrate the roles thenonlinear and non-stationary effects in the energy–frequency–time distribution.Keywords:non-stationary time series;nonlinear differential equations;frequency–time spectrum;Hilbert spectral analysis;intrinsic time scale;empirical mode decomposition1.Introductionsensed by us;data analysis serves two purposes:determine the parameters needed to construct the necessary model,and to confirm the model we constructed to represent the phe-nomenon.Unfortunately,the data,whether from physical measurements or numerical modelling,most likely will have one or more of the following problems:(a)the total data span is too short;(b)the data are non-stationary;and(c)the data represent nonlinear processes.Although each of the above problems can be real by itself,the first two are related,for a data section shorter than the longest time scale of a sta-tionary process can appear to be non-stationary.Facing such data,we have limited options to use in the analysis.Historically,Fourier spectral analysis has provided a general method for examin-the data analysis has been applied to all kinds of data.Although the Fourier transform is valid under extremely general conditions(see,for example,Titchmarsh1948),there are some crucial restrictions of Proc.R.Soc.Lond.A(1998)Nonlinear and non-stationary time series analysis905the Fourier spectral analysis:the system must be linear;and the data must be strict-ly periodic or stationary;otherwise,the resulting spectrum will make little physicalsense.to the Fourier spectral analysis methods.Therefore,behoves us review the definitions of stationarity here.According to the traditional definition,a time series,X (t ),is stationary in the wide sense,if,for all t ,E (|X (t )2|)<∞,E (X (t))=m,C (X (t 1),X (t 2))=C (X (t 1+τ),X (t 2+τ))=C (t 1−t 2),(1.1)in whichE (·)is the expected value defined as the ensemble average of the quantity,and C (·)is the covariance function.Stationarity in the wide sense is also known as weak stationarity,covariance stationarity or second-order stationarity (see,forexample,Brockwell &Davis 1991).A time series,X (t ),is strictly stationary,if the joint distribution of [X (t 1),X (t 2),...,X (t n )]and [X (t 1+τ),X (t 2+τ),...,X (t n +τ)](1.2)are the same for all t i and τ.Thus,a strictly stationaryprocess with finite second moments is alsoweakly stationary,but the inverse is not true.Both definitions arerigorous but idealized.Other less rigorous definitions have also beenused;for example,that is stationary within a limited timespan,asymptotically stationary is for any random variableis stationary when τin equations (1.1)or (1.2)approaches infinity.In practice,we can only have data for finite time spans;these defini-tions,we haveto makeapproximations.Few of the data sets,from either natural phenomena or artificial sources,can satisfy these definitions.It may be argued thatthe difficulty of invoking stationarity as well as ergodicity is not on principlebut on practicality:we just cannot have enough data to cover all possible points in thephase plane;therefore,most of the cases facing us are transient in nature.This is the reality;we are forced to face it.Fourier spectral analysis also requires linearity.can be approximated by linear systems,the tendency tobe nonlinear whenever their variations become finite Compounding these complications is the imperfection of or numerical schemes;theinteractionsof the imperfect probes even with a perfect linear systemcan make the final data nonlinear.For the above the available data are ally of finite duration,non-stationary and from systems that are frequently nonlinear,either intrinsicallyor through interactions with the imperfect probes or numerical schemes.Under these conditions,Fourier spectral analysis is of limited use.For lack of alternatives,however,Fourier spectral analysis is still used to process such data.The uncritical use of Fourier spectral analysis the insouciant adoption of the stationary and linear assumptions may give cy range.a delta function will giveProc.R.Soc.Lond.A (1998)906N.E.Huang and othersa phase-locked wide white Fourier spectrum.Here,added to the data in the time domain,Constrained bythese spurious harmonics the wide frequency spectrum cannot faithfully represent the true energy density in the frequency space.More seri-ously,the Fourier representation also requires the existence of negative light intensity so that the components can cancel out one another to give thefinal delta function. Thus,the Fourier components might make mathematical sense,but do not really make physical sense at all.Although no physical process can be represented exactly by a delta function,some data such as the near-field strong earthquake records areFourier spectra.Second,tions;wave-profiles.Such deformations,later,are the direct consequence of nonlinear effects.Whenever the form of the data deviates from a pure sine or cosine function,the Fourier spectrum will contain harmonics.As explained above, both non-stationarity and nonlinearity can induce spurious harmonic components that cause energy spreading.The consequence is the misleading energy–frequency distribution forIn this paper,modemode functions The decomposition is based on the direct extraction of theevent on the time the frequency The decomposition be viewed as an expansion of the data in terms of the IMFs.Then,based on and derived from the data,can serve as the basis of that expansion linear or nonlinear as dictated by the data,Most important of all,it is adaptive.As will locality and adaptivity are the necessary conditions for the basis for expanding nonlinear and non-stationary time orthogonality is not a necessary criterionselection for a nonlinearon the physical time scaleslocal energy and the instantaneous frequencyHilbert transform can give us a full energy–frequency–time distribution of the data. Such a representation is designated as the Hilbert spectrum;it would be ideal for nonlinear and non-stationary data analysis.We have obtained good results and new insights by applying the combination of the EMD and Hilbert spectral analysis methods to various data:from the numerical results of the classical nonlinear equation systems to data representing natural phe-nomena.The classical nonlinear systems serve to illustrate the roles played by the nonlinear effects in the energy–frequency–time distribution.With the low degrees of freedom,they can train our eyes for more complicated cases.Some limitations of this method will also be discussed and the conclusions presented.Before introducing the new method,we willfirst review the present available data analysis methods for non-stationary processes.Proc.R.Soc.Lond.A(1998)Nonlinear and non-stationary time series analysis9072.Review of non-stationary data processing methodsWe willfirstgivea brief survey of themethodsstationary data.are limited to linear systems any method is almost strictly determined according to the special field in which the application is made.The available methods are reviewed as follows.(a )The spectrogramnothing but a limited time window-width Fourier spectral analysis.the a distribution.Since it relies on the tradition-al Fourier spectral analysis,one has to assume the data to be piecewise stationary.This assumption is not always justified in non-stationary data.Even if the data are piecewise stationary how can we guarantee that the window size adopted always coincides with the stationary time scales?What can we learn about the variations longer than the local stationary time scale?Will the collection of the locally station-ary pieces constitute some longer period phenomena?Furthermore,there are also practical difficulties in applying the method:in order to localize an event in time,the window width must be narrow,but,on the other hand,the frequency resolu-tion requires longer time series.These conflicting requirements render this method of limited usage.It is,however,extremely easy to implement with the fast Fourier transform;thus,ithas attracted a wide following.Most applications of this methodare for qualitative display of speech pattern analysis (see,for example,Oppenheim &Schafer 1989).(b )The wavelet analysisThe wavelet approach is essentially an adjustable window Fourier spectral analysiswith the following general definition:W (a,b ;X,ψ)=|a |−1/2∞−∞X (t )ψ∗ t −b ad t,(2.1)in whichψ∗(·)is the basic wavelet function that satisfies certain very general condi-tions,a is the dilation factor and b is the translationof theorigin.Although time andfrequency do not appear explicitly in the transformed result,the variable 1/a givesthe frequency scale and b ,the temporal location of an event.An intuitive physical explanation of equation (2.1)is very simple:W (a,b ;X,ψ)is the ‘energy’of X ofscale a at t =b .Because of this basic form of at +b involvedin thetransformation,it is also knownas affinewavelet analysis.For specific applications,the basic wavelet function,ψ∗(·),can be modified according to special needs,but the form has to be given before the analysis.In most common applications,however,the Morlet wavelet is defined as Gaussian enveloped sine and cosine wave groups with 5.5waves (see,for example,Chan 1995).Generally,ψ∗(·)is not orthogonalfordifferent a for continuous wavelets.Although one can make the wavelet orthogonal by selecting a discrete set of a ,thisdiscrete wavelet analysis will miss physical signals having scale different from theselected discrete set of a .Continuous or discrete,the wavelet analysis is basically a linear analysis.A very appealing feature of the wavelet analysis is that it provides aProc.R.Soc.Lond.A (1998)908N.E.Huang and othersuniform resolution for all the scales.Limited by the size of thebasic wavelet function,the downside of the uniform resolution is uniformly poor resolution.Although wavelet analysis has been available only in the last ten years or so,it hasbecome extremelypopular.Indeed,it is very useful in analysing data with gradualfrequency changes.Since it has an analytic form for the result,it has attracted extensive attention of the applied mathematicians.Most of its applications have been in edge detection and image compression.Limited applications have also been made to the time–frequency distribution in time series (see,for example,Farge 1992;Long et al .1993)andtwo-dimensionalimages (Spedding et al .1993).Versatile as the wavelet analysis is,the problem with the most commonly usedMorlet wavelet is its leakage generated by the limited length of the basic wavelet function,whichmakesthe quantitativedefinitionof the energy–frequency–time dis-tribution difficult.Sometimes,the interpretation of the wavelet can also be counter-intuitive.For example,to define a change occurring locally,one must look for theresult in the high-frequencyrange,for the higher the frequency the more localized thebasic wavelet will be.If a local event occurs only in the low-frequency range,one willstill be forced to look for its effects inthe high-frequencyrange.Such interpretationwill be difficultif it is possible at all (see,for example,Huang et al .1996).Another difficulty of the wavelet analysis is its non-adaptive nature.Once the basic waveletis selected,one will have to use it to analyse all the data.Since the most commonlyused Morlet wavelet is Fourier based,it also suffers the many shortcomings of Fouri-er spectral analysis:it can only give a physically meaningful interpretation to linear phenomena;it can resolve the interwave frequency modulation provided the frequen-cy variationis gradual,but it cannot resolve the intrawave frequency modulation because the basic wavelet has a length of 5.5waves.Inspite of all these problems,wavelet analysisisstillthe bestavailable non-stationary data analysis method so far;therefore,we will use it in this paper as a reference to establish the validity and thecalibration of the Hilbert spectrum.(c )The Wigner–Ville distributionThe Wigner–Ville distribution is sometimes alsoreferred toas the Heisenberg wavelet.By definition,it is the Fourier transform of the central covariance function.For any time series,X (t ),we can define the central variance as C c (τ,t )=X (t −12τ)X ∗(t +12τ).(2.2)Then the Wigner–Ville distribution is V (ω,t )=∞−∞C c (τ,t )e −i ωτd τ.(2.3)This transform has been treated extensively by Claasen &Mecklenbr¨a uker (1980a ,b,c )and by Cohen (1995).It has been extremely popular with the electrical engi-neering community.The difficulty with this method is the severe cross terms as indicated by the exis-tence of negativepowerfor some frequency ranges.Although this shortcoming canbe eliminated by using the Kernel method (see,for example,Cohen 1995),the resultis,then,basically that of a windowed Fourier analysis;therefore,itsuffers all thelim-itations of the Fourier analysis.An extension of this method has been made by Yen(1994),who used the Wigner–Ville distribution to define wave packets that reduce Proc.R.Soc.Lond.A (1998)Nonlinear and non-stationary time series analysis909 a complicated data set to afinite number of simple components.This extension is very powerful and can be applied to a variety of problems.The applications to complicated data,however,require a great amount of judgement.(d)Evolutionary spectrumThe evolutionary spectrum wasfirst proposed by Priestley(1965).The basic idea is to extend the classic Fourier spectral analysis to a more generalized basis:from sine or cosine to a family of orthogonal functions{φ(ω,t)}indexed by time,t,and defined for all realω,the frequency.Then,any real random variable,X(t),can beexpressed asX(t)= ∞−∞φ(ω,t)d A(ω,t),(2.4)in which d A(ω,t),the Stieltjes function for the amplitude,is related to the spectrum asE(|d A(ω,t)|2)=dµ(ω,t)=S(ω,t)dω,(2.5) whereµ(ω,t)is the spectrum,and S(ω,t)is the spectral density at a specific time t,also designated as the evolutionary spectrum.If for eachfixedω,φ(ω,t)has a Fourier transformφ(ω,t)=a(ω,t)e iΩ(ω)t,(2.6) then the function of a(ω,t)is the envelope ofφ(ω,t),andΩ(ω)is the frequency.If, further,we can treatΩ(ω)as a single valued function ofω,thenφ(ω,t)=α(ω,t)e iωt.(2.7) Thus,the original data can be expanded in a family of amplitude modulated trigono-metric functions.The evolutionary spectral analysis is very popular in the earthquake communi-ty(see,for example,Liu1970,1971,1973;Lin&Cai1995).The difficulty of its application is tofind a method to define the basis,{φ(ω,t)}.In principle,for this method to work,the basis has to be defined a posteriori.So far,no systematic way has been offered;therefore,constructing an evolutionary spectrum from the given data is impossible.As a result,in the earthquake community,the applications of this method have changed the problem from data analysis to data simulation:an evo-lutionary spectrum will be assumed,then the signal will be reconstituted based on the assumed spectrum.Although there is some general resemblance to the simulated earthquake signal with the real data,it is not the data that generated the spectrum. Consequently,evolutionary spectrum analysis has never been very useful.As will be shown,the EMD can replace the evolutionary spectrum with a truly adaptive representation for the non-stationary processes.(e)The empirical orthogonal function expansion(EOF)The empirical orthogonal function expansion(EOF)is also known as the principal component analysis,or singular value decomposition method.The essence of EOF is briefly summarized as follows:for any real z(x,t),the EOF will reduce it toz(x,t)=n1a k(t)f k(x),(2.8)Proc.R.Soc.Lond.A(1998)910N.E.Huang and othersin whichf j·f k=δjk.(2.9)The orthonormal basis,{f k},is the collection of the empirical eigenfunctions defined byC·f k=λk f k,(2.10)where C is the sum of the inner products of the variable.EOF represents a radical departure from all the above methods,for the expansion basis is derived from the data;therefore,it is a posteriori,and highly efficient.The criticalflaw of EOF is that it only gives a distribution of the variance in the modes defined by{f k},but this distribution by itself does not suggest scales or frequency content of the signal.Although it is tempting to interpret each mode as indepen-dent variations,this interpretation should be viewed with great care,for the EOF decomposition is not unique.A single component out of a non-unique decomposition, even if the basis is orthogonal,does not usually contain physical meaning.Recently, Vautard&Ghil(1989)proposed the singular spectral analysis method,which is the Fourier transform of the EOF.Here again,we have to be sure that each EOF com-ponent is stationary,otherwise the Fourier spectral analysis will make little sense on the EOF components.Unfortunately,there is no guarantee that EOF compo-nents from a nonlinear and non-stationary data set will all be linear and stationary. Consequently,singular spectral analysis is not a real improvement.Because of its adaptive nature,however,the EOF method has been very popular,especially in the oceanography and meteorology communities(see,for example,Simpson1991).(f)Other miscellaneous methodsOther than the above methods,there are also some miscellaneous methods such as least square estimation of the trend,smoothing by moving averaging,and differencing to generate stationary data.Methods like these,though useful,are too specialized to be of general use.They will not be discussed any further here.Additional details can be found in many standard data processing books(see,for example,Brockwell &Davis1991).All the above methods are designed to modify the global representation of the Fourier analysis,but they all failed in one way or the other.Having reviewed the methods,we can summarize the necessary conditions for the basis to represent a nonlinear and non-stationary time series:(a)complete;(b)orthogonal;(c)local;and (d)adaptive.Thefirst condition guarantees the degree of precision of the expansion;the second condition guarantees positivity of energy and avoids leakage.They are the standard requirements for all the linear expansion methods.For nonlinear expansions,the orthogonality condition needs to be modified.The details will be discussed later.But even these basic conditions are not satisfied by some of the above mentioned meth-ods.The additional conditions are particular to the nonlinear and non-stationary data.The requirement for locality is the most crucial for non-stationarity,for in such data there is no time scale;therefore,all events have to be identified by the time of their occurences.Consequently,we require both the amplitude(or energy) and the frequency to be functions of time.The requirement for adaptivity is also crucial for both nonlinear and non-stationary data,for only by adapting to the local variations of the data can the decomposition fully account for the underlying physics Proc.R.Soc.Lond.A(1998)Nonlinear and non-stationary time series analysis911of the processes and not just to fulfil the mathematical requirements for fitting the data.This is especially important for the nonlinear phenomena,for a manifestation of nonlinearity is the ‘harmonic distortion’in the Fourier analysis.The degree of distortion depends on the severity of nonlinearity;therefore,one cannot expect a predetermined basis to fit all the phenomena.An easy way to generate the necessary adaptive basis is to derive the basis from the data.In this paper,we will introduce a general method which requires two steps in analysing the data as follows.The first step is to preprocess the data by the empirical mode decomposition method,with which the data are decomposed into a number of intrinsic mode function components.Thus,we will expand the data in a basis derived from the data.The second step is to apply the Hilbert transform to the decomposed IMFs and construct the energy–frequency–time distribution,designated as the Hilbert spectrum,from which the time localities of events will be preserved.In other words,weneed the instantaneous frequency and energy rather than the global frequency and energy defined by the Fourier spectral analysis.Therefore,before goingany further,we have to clarify the definition of the instantaneous frequency.3.Instantaneous frequencyis to accepting it only for special ‘monocomponent’signals 1992;Cohen 1995).Thereare two basicdifficulties with accepting the idea of an instantaneous fre-quency as follows.The first one arises from the influence of theFourier spectral analysis.In the traditional Fourier analysis,the frequency is defined for thesineor cosine function spanning the whole data length with constant ampli-tude.As an extension of this definition,the instantaneous frequencies also have torelate to either a sine or a cosine function.Thus,we need at least one full oscillationof a sineor a cosine wave to define the local frequency value.According to this logic,nothing full wave will do.Such a definition would not make sense forThe secondarises from the non-unique way in defining the instantaneousfrequency.Nevertheless,this difficulty is no longer serious since the introduction ofthe meanstomakethedata analyticalthrough the Hilbert transform.Difficulties,however,still exist as ‘paradoxes’discussed by Cohen (1995).For an arbitrary timeseries,X (t ),we can always have its Hilbert Transform,Y (t ),as Y (t )=1πP∞−∞X (t )t −t d t,(3.1)where P indicates the Cauchy principal value.This transformexists forallfunctionsof class L p(see,for example,Titchmarsh 1948).With this definition,X (t )and Y (t )form the complex conjugate pair,so we can have an analytic signal,Z (t ),as Z (t )=X (t )+i Y (t )=a (t )e i θ(t ),(3.2)in which a (t )=[X 2(t )+Y 2(t )]1/2,θ(t )=arctanY (t )X (t ).(3.3)Proc.R.Soc.Lond.A (1998)912N.E.Huang andothers Theoretically,there are infinitely many ways of defining the imaginary part,but the Hilbert transform provides a unique way of defining the imaginary part so that the result is ananalyticfunction.A brief tutorial on the Hilbert transform with theemphasis on its physical interpretation can be found in Bendat &Piersol is the bestlocal fitan amplitude and phase varying trigonometric function to X (t ).Even with the Hilbert transform,there is still controversy in defining the instantaneous frequency as ω=d θ(t )d t .(3.4)This leads Cohen (1995)to introduce the term,‘monocomponent function’.In prin-ciple,some limitations on the data are necessary,forthe instantaneous frequencygiven in equation (3.4)is a single value function of time.At any given time,thereis only one frequency value;therefore,it can only represent one component,hence ‘monocomponent’.Unfortunately,no cleardefinition of the ‘monocomponent’signalwas given to judge whether a function is or is not ‘monocomponent’.For lack ofa precise definition,‘narrow band’was adopted a on the data for the instantaneous frequency to make sense (Schwartz et al .1966).There are two definitions for bandwidth.The first one is used in the study of the probability properties of the signalsand waves,wherethe processes are assumed tobe stationary and Gaussian.Then,the bandwidth can be defined in spectral moments The expected number of zero crossings per unit time is given byN 0=1π m 2m 0 1/2,(3.5)while the expected number of extrema per unit time is given byN 1=1π m 4m 2 1/2,(3.6)in which m i is the i th moment of the spectrum.Therefore,the parameter,ν,definedas N 21−N 20=1π2m 4m 0−m 22m 2m 0=1π2ν2,(3.7)offers a standard bandwidth measure (see,for example,Rice 1944a,b ,1945a,b ;Longuet-Higgins 1957).For a narrow band signal ν=0,the expected numbers extrema and zero crossings have to equal.the spectrum,but in a different way.coordinates as z (t )=a (t )e i θ(t ),(3.8)with both a (t )and θ(t )being functions of time.If this function has a spectrum,S (ω),then the mean frequency is given byω = ω|S (ω)|2d ω,(3.9)Proc.R.Soc.Lond.A (1998)Nonlinear and non-stationary time series analysis913which can be expressed in another way asω =z ∗(t )1i dd tz (t )d t=˙θ(t )−i ˙a (t )a (t )a 2(t )d t =˙θ(t )a 2(t )d t.(3.10)Based on this expression,Cohen (1995)suggested that ˙θbe treated as the instanta-neous frequency.With these notations,the bandwidth can be defined asν2=(ω− ω )2 ω 2=1 ω 2(ω− ω )2|S (ω)|2d ω=1 ω 2z ∗(t ) 1i d d t− ω 2z (t )d t =1 ω 2 ˙a 2(t )d t +(˙θ(t )− ω )2a 2(t )d t .(3.11)For a narrow band signal,this value has to be small,then both a and θhave to begradually varying functions.Unfortunately,both equations (3.7)and (3.11)defined the bandwidth in the global sense;they are both overly restrictive and lack preci-sion at the same time.Consequently,the bandwidth limitation on the Hilbert trans-form to give a meaningful instantaneous frequency has never been firmly established.For example,Melville (1983)had faithfully filtered the data within the bandwidth requirement,but he still obtained many non-physical negative frequency values.It should be mentioned here that using filtering to obtain a narrow band signal is unsat-isfactory for another reason:the filtered data have already been contaminated by the spurious harmonics caused by the nonlinearity and non-stationarity as discussed in the introduction.In order to obtain meaningful instantaneous frequency,restrictive conditions have to be imposed on the data as discussed by Gabor (1946),Bedrosian (1963)and,more recently,Boashash (1992):for any function to have a meaningful instantaneous frequency,the real part of its Fourier transform has to have only positive frequency.This restriction can be proven mathematically as shown in Titchmarsh (1948)but it is still global.For data analysis,we have to translate this requirement into physically implementable steps to develop a simple method for applications.For this purpose,we have to modify the restriction condition from a global one to a local one,and the basis has to satisfy the necessary conditions listed in the last section.Let us consider some simple examples to illustrate these restrictions physically,by examining the function,x (t )=sin t.(3.12)Its Hilbert transform is simply cos t .The phase plot of x –y is a simple circle of unit radius as in figure 1a .The phase function is a straight line as shown in figure 1b and the instantaneous frequency,shown in figure 1c ,is a constant as expected.If we move the mean offby an amount α,say,then,x (t )=α+sin t.(3.13)Proc.R.Soc.Lond.A (1998)。
牛顿-拉夫逊潮流计算中检测雅可比矩阵奇异性和网络孤岛的新方法
由 （ 式可得：Ｉ 【 ０由于 Ｄ是对角矩 ３ ） ＝ 阵， ， 因此 至少有一对角元 素为 ０ 。 因为 Ｕ＝ ＵＬ Ｄ Ｄ ，ＶＬ 设该潮流计算 是 ｎ 节点 系统 。 所以（ ） ２） ２ 或（ 工 ａ ｂ弋有一个成立 ， Ｕ 中有一 Ｈ子矩阵奇异 ，那 么 Ｈ矩阵各 个列向量线 性相 即 ｎ 一１ 零行 或 中有一零列 。 ｕ 中行为零 ， 是行相关 隋况 ；丰中列 为 关 ， ： 这 Ｌ 即 － ＝ （ 不全为 ０ ｑ ０ ） 零， 这是列相关 隋况。 其 中： 是 Ｈ矩 阵的列 向量 ，１是相关 系 ｃ Ｔ Ａ矩 阵奇异 ， 那么 Ａ矩 阵行 向量 、 向量线 列 数 。由潮流雅可 比矩阵元素计算可知 ： 性相关 ， 即： 对 同一节点 ， 素和 Ｊ 素的计 算具 有完 Ｈ元 元 全相似 的表达式 ，因此 ，矩 阵的各个列 向量也 Ｊ （ａ ４） 应满足（ ， 即：
中国新技术新产 品
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高 新 技 术
新型停 水 自动关 闭阀结构 、 点及操作要 点 特
张金龙 曹 艳
（ 西安航 空技 术高等专科学校机械 工程 系， 陕西 西安 ７ ０ ７ ） １０ ７
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Ｉ 言 。在 日常生 活 中 ， 前 由于停 水时 忘记 关 闭 阀门 ， 水 时 也没 能及 时 关 闭 阀门 ， 来 造成 水 资源 浪 费甚 至形 成安 全 隐 患 的情况 屡 见不 鲜 。 着全 民节 水 概念 不 断深入 人 心 ， 一 问 随 这 题 引起 各方 关 注 。 因此 急 需设 计 一 款可 以在 停 水 时 自动关 闭 的水 阀 ，它 能够 在停 水 后 即 使 人们 忘记 关 闭 水 龙 头 也 能实 现 自动 关 闭 ， 而再 次 来水 时 不 至于 出 现水 患 的情 况 ，能够 有 效 的节 约水 资源 。 要 实 现 自动 关 闭 功 能首 先 要 有 动 力 ， 这 方 面可 以借 助 磁性 元件 的磁 力 、弹性 元 件 的 弹力 、 力 等外 力 ， 时考 虑供 水 和停 水 时 的 重 同 水 压变 化 ， 通过 联 动机 构实 现 。 ２停 水 自动关 闭 阀 的结 构 及 特点 。利用 水 压 、 力 等 力 学 特 性 ， 过 一 系 列 的实 验 、 重 经 改 进 ， 发 出一 种 简单 、 行 的带 有 停水 自锁 研 可 机 构 的水 阀 。 款 水 阀为纯 机 械构 造 ， 阀体 这 以 为 主体 框 架 ， 有 阀 芯 、 封 圈 、 心 轮 以及 配 密 偏 手柄 ， 无弹 性元 件 ， 作状 况 不 受环 境 和时 间 工 的 限制 ， 构 简 单 ， 价 低 廉 并 方 便拆 换 ， 结 造 整 体 可靠 性 高 。 停 水 自动关 闭 阀结 构 原 理 如 图 １ 示 ， 所 实 物 如 图 ２所示 。序号 ｌ 水 阀 的偏 心轮 ， 为 ２ 为 ０ 型密 封 圈 ， 为 Ｖ型 密封 圈 ， 阀体 ， ３ ４为 ５ 为 阀芯 ， 销 轴 ， 手 柄 。 阀体 ４是 主 框 ６为 ７为 架 ， 来装 配其 它 元 件 ， 进 水 口和 出 水 口； 用 有 阀芯 ５的顶 端 与末 端分 别 装有 Ｖ 型密 封 圈 ３ 和 ０ 型 密 封 圈 ２ｖ 型 密 封 圈 ３利 用 其 锥 面 ， 与 阀体 ４内部 锥 面 配合 实 现 停 水 时 密 封 ， 而 ０型密 封 圈 ２与 阀体 ４内壁 的接 触 实 现来 水 时对 水 阀末 端 的密 封 ，在 阀 芯 ５的 中部 开两
Superfluid Friction and Late-time Thermal Evolution of Neutron Stars
Bennett Link1 blink@ Montana State University, Department of Physics, Bozeman MT 59717 ABSTRACT The recent temperature measurements of the two older isolated neutron stars PSR 1929+10 and PSR 0950+08 (ages of 3 × 106 and 2 × 107 yr, respectively) indicate that these objects are heated. A promising candidate heat source is friction between the neutron star crust and the superfluid it is thought to contain. We study the effects of superfluid friction on the long-term thermal and rotational evolution of a neutron star. Differential rotation velocities between the superfluid and the crust (averaged over the inner crust moment of inertia) of ω ¯ ∼ 0.6 rad s−1 for PSR 1929+10 and ∼ 0.02 rad s−1 for PSR 0950+08 would account for their observed temperatures. These differential velocities could be sustained by pinning of superfluid vortices to the inner crust lattice with strengths of ∼ 1 MeV per nucleus. Pinned vortices can creep outward through thermal fluctuations or quantum tunneling. For thermally-activated creep, the coupling between the superfluid and crust is highly sensitive to temperature. If pinning maintains large differential rotation (∼ 10 rad s−1 ), a feedback instability could occur in stars younger than ∼ 105 yr causing oscillations of the temperature and spin-down rate over a period of ∼ 0.3tage . For stars older than ∼ 106 yr, however, vortex creep occurs through quantum tunneling, and the creep velocity is too insensitive to temperature for a thermal-rotational instability to occur. These older stars could be heated through a steady process of superfluid friction. Subject headings: stars: interiors — stars: neutron — stars: evolution — stars: stability — superfluid — dense matter
XPSPEAK 说明书
Using XPSPEAK Version 4.1 November 2000Contents Page Number XPS Peak Fitting Program for WIN95/98 XPSPEAK Version 4.1 (1)Program Installation (1)Introduction (1)First Version (1)Version 2.0 (1)Version 3.0 (1)Version 3.1 (2)Version 4.0 (2)Version 4.1 (2)Future Versions (2)General Information (from R. Kwok) (3)Using XPS Peak (3)Overview of Processing (3)Appearance (4)Opening Files (4)Opening a Kratos (*.des) text file (4)Opening Multiple Kratos (*.des) text files (5)Saving Files (6)Region Parameters (6)Loading Region Parameters (6)Saving Parameters (6)Available Backgrounds (6)Averaging (7)Shirley + Linear Background (7)Tougaard (8)Adding/Adjusting the Background (8)Adding/Adjusting Peaks (9)Peak Types: p, d and f (10)Peak Constraints (11)Peak Parameters (11)Peak Function (12)Region Shift (13)Optimisation (14)Print/Export (15)Export (15)Program Options (15)Compatibility (16)File I/O (16)Limitations (17)Cautions for Peak Fitting (17)Sample Files: (17)gaas.xps (17)Cu2p_bg.xps (18)Kratos.des (18)ASCII.prn (18)Other Files (18)XPS Peak Fitting Program for WIN95/98 XPSPEAKVersion 4.1Program InstallationXPS Peak is freeware. Please ask RCSMS lab staff for a copy of the zipped 3.3MB file, if you would like your own copyUnzip the XPSPEA4.ZIP file and run Setup.exe in Win 95 or Win 98.Note: I haven’t successfully installed XPSPEAK on Win 95 machines unless they have been running Windows 95c – CMH.IntroductionRaymond Kwok, the author of XPSPEAK had spent >1000 hours on XPS peak fitting when he was a graduate student. During that time, he dreamed of many features in the XPS peak fitting software that could help obtain more information from the XPS peaks and reduce processing time.Most of the information in this users guide has come directly from the readme.doc file, automatically installed with XPSPEAK4.1First VersionIn 1994, Dr Kwok wrote a program that converted the Kratos XPS spectral files to ASCII data. Once this program was finished, he found that the program could be easily converted to a peak fitting program. Then he added the dreamed features into the program, e.g.∙ A better way to locate a point at a noise baseline for the Shirley background calculations∙Combine the two peaks of 2p3/2 and 2p1/2∙Fit different XPS regions at the same timeVersion 2.0After the first version and Version 2.0, many people emailed Dr Kwok and gave additional suggestions. He also found other features that could be put into the program.Version 3.0The major change in Version 3.0 is the addition of Newton’s Method for optimisation∙Newton’s method can greatly reduce the optimisation time for multiple region peak fitting.Version 3.11. Removed all the run-time errors that were reported2. A Shirley + Linear background was added3. The Export to Clipboard function was added as requested by a user∙Some other minor graphical features were addedVersion 4.0Added:1. The asymmetrical peak function. See note below2. Three additional file formats for importing data∙ A few minor adjustmentsThe addition of the Asymmetrical Peak Function required the peak function to be changed from the Gaussian-Lorentzian product function to the Gaussian-Lorentzian sum function. Calculation of the asymmetrical function using the Gaussian-Lorentzian product function was too difficult to implement. The software of some instruments uses the sum function, while others use the product function, so both functions are available in XPSPEAK.See Peak Function, (Page 12) for details of how to set this up.Note:If the selection is the sum function, when the user opens a *.xps file that was optimised using the Gaussian-Lorentzian product function, you have to re-optimise the spectra using the Gaussian-Lorentzian sum function with a different %Gaussian-Lorentzian value.Version 4.1Version 4.1 has only two changes.1. In version 4.0, the printed characters were inverted, a problem that wasdue to Visual Basic. After about half year, a patch was received from Microsoft, and the problem was solved by simply recompiling the program2. The import of multiple region VAMAS file format was addedFuture VersionsThe author believes the program has some weakness in the background subtraction routines. Extensive literature examination will be required in order to revise them. Dr Kwok intends to do that for the next version.General Information (from R. Kwok)This version of the program was written in Visual Basic 6.0 and uses 32 bit processes. This is freeware. You may ask for the source program if you really want to. I hope this program will be useful for people without modern XPS software. I also hope that the new features in this program can be adopted by the XPS manufacturers in the later versions of their software.If you have any questions/suggestions, please send an email to me.Raymund W.M. KwokDepartment of ChemistryThe Chinese University of Hong KongShatin, Hong KongTel: (852)-2609-6261Fax:(852)-2603-5057email: rmkwok@.hkI would like to thank the comments and suggestions from many people. For the completion of Version 4.0, I would like to think Dr. Bernard J. Flinn for the routine of reading Leybold ascii format, Prof. Igor Bello and Kelvin Dickinson for providing me the VAMAS files VG systems, and my graduate students for testing the program. I hope I will add other features into the program in the near future.R Kwok.Using XPS PeakOverview of Processing1. Open Required Files∙See Opening Files (Page 4)2. Make sure background is there/suitable∙See Adding/Adjusting the Background, (Page 8)3. Add/adjust peaks as necessary∙See Adding/Adjusting Peaks, (Page 9), and Peak Parameters, (Page 11)4. Save file∙See Saving Files, (Page 6)5. Export if necessary∙See Print/Export, (Page 15)AppearanceXPSPEAK opens with two windows, one above the other, which look like this:∙The top window opens and displays the active scan, adds or adjusts a background, adds peaks, and loads and saves parameters.∙The lower window allows peak processing and re-opening and saving dataOpening FilesOpening a Kratos (*.des) text file1. Make sure your data files have been converted to text files. See the backof the Vision Software manual for details of how to do this. Remember, from the original experiment files, each region of each file will now be a separate file.2. From the Data menu of the upper window, choose Import (Kratos)∙Choose directory∙Double click on the file of interest∙The spectra open with all previous processing INCLUDEDOpening Multiple Kratos (*.des) text files∙You can open up a maximum of 10 files together.1. Open the first file as above∙Opens in the first region (1)2. In the XPS Peak Processing (lower) window, left click on 2(secondregion), which makes this region active3. Open the second file as in Step2, Opening a Kratos (*.des) text file,(Page 4)∙Opens in the second region (2)∙You can only have one description for all the files that are open. Edit with a click in the Description box4. Open further files by clicking on the next available region number thenfollowing the above step.∙You can only have one description for all the files that are open. Edit with a click in the Description boxDescriptionBox 2∙To open a file that has already been processed and saved using XPSPEAK, click on the Open XPS button in the lower window. Choose directory and file as normal∙The program can store all the peak information into a *.XPS file for later use. See below.Saving Files1. To save a file click on the Save XPS button in the lower window2. Choose Directory3. Type in a suitable file name4. Click OK∙Everything that is open will be saved in this file∙The program can also store/read the peak parameter files (*.RPA)so that you do not need to re-type all the parameters again for a similar spectrum.Region ParametersRegion Parameters are the boundaries or limits you have used to set up the background and peaks for your files. These values can be saved as a file of the type *.rpa.Note that these Region Parameters are completely different from the mathematical parameters described in Peak Parameters, (Page 11) Loading Region Parameters1. From the Parameters menu in the upper window, click on Load RegionParameters2. Choose directory and file name3. Click on Open buttonSaving Parameters1. From the Parameters menu in the XPS Peak Fit (Upper) window, clickon Save Region Parameters2. Choose directory and file name3. Click on the Save buttonAvailable BackgroundsThis program provides the background choices of∙Shirley∙Linear∙TougaardAveraging∙ Averaging at the end points of the background can reduce the time tofind a point at the middle of a noisy baseline∙ The program includes the choices of None (1 point), 3, 5, 7, and 9point average∙ This will average the intensities around the binding energy youselect.Shirley + Linear Background1. The Shirley + Linear background has been added for slopingbackgrounds∙ The "Shirley + Linear" background is the Shirley background plus astraight line with starting point at the low BE end-point and with a slope value∙ If the slope value is zero , the original Shirley calculation is used∙ If the slope value is positive , the straight line has higher values atthe high BE side, which can be used for spectra with higher background intensities at the high BE side∙ Similarly, a negative slope value can be used for a spectrum withlower background intensities at the high BE side2. The Optimization button may be used when the Shirley background is higher at some point than the signal intensities∙ The program will increase the slope value until the Shirleybackground is below the signal intensities∙ Please see the example below - Cu2p_bg.xps - which showsbackground subtraction using the Shirley method (This spectrum was sent to Dr Kwok by Dr. Roland Schlesinger).∙ A shows the problematic background when the Shirley backgroundis higher than the signal intensities. In the Shirley calculation routine, some negative values were generated and resulted in a non-monotonic increase background∙ B shows a "Shirley + Linear" background. The slope value was inputby trial-and-error until the background was lower than the signal intensities∙ C was obtained using the optimisation routineA slope = 0B slope = 11C slope = 15.17Note: The background subtraction calculation cannot completely remove the background signals. For quantitative studies, the best procedure is "consistency". See Future Versions, (Page 2).TougaardFor a Tougaard background, the program can optimise the B1 parameter by minimising the "square of the difference" of the intensities of ten data points in the high binding energy side of the range with the intensities of the calculated background.Adding/Adjusting the BackgroundNote: The Background MUST be correct before Peaks can be added. As with all backgrounds, the range needs to include as much of your peak as possible and as little of anything else as possible.1. Make sure the file of interest is open and the appropriate region is active2. Click on Background in the upper window∙The Region 0 box comes up, which contains the information about the background3. Adjust the following as necessary. See Note.∙High BE (This value needs to be within the range of your data) ∙Low BE (This value needs to be within the range of your data) NOTE: High and Low BE are not automatically within the range of your data. CHECK CAREFULLY THAT BOTH ENDS OF THE BACKGROUND ARE INSIDE THE EDGE OF YOUR DATA. Nothing will happen otherwise.∙No. of Ave. Pts at end-points. See Averaging, (Page 7)∙Background Type∙Note for Shirley + Linear:To perform the Shirley + Linear Optimisation routine:a) Have the file of interest openb) From the upper window, click on Backgroundc) In the resulting box, change or optimise the Shirley + LinearSlope as desired∙Using Optimize in the Shirley + Linear window can cause problems. Adjust manually if necessary3. Click on Accept when satisfiedAdding/Adjusting PeaksNote: The Background MUST be correct before peaks can be added. Nothing will happen otherwise. See previous section.∙To add a peak, from the Region Window, click on Add Peak ∙The peak window appears∙This may be adjusted as below using the Peak Window which will have opened automaticallyIn the XPS Peak Processing (lower) window, there will be a list of Regions, which are all the open files, and beside each of these will be numbers representing the synthetic peaks included in that region.Regions(files)SyntheticPeaks1. Click on a region number to activate that region∙The active region will be displayed in the upper window2. Click on a peak number to start adjusting the parameters for that peak.∙The Processing window for that peak will open3. Click off Fix to adjust the following using the maximum/minimum arrowkeys provided:∙Peak Type. (i.e. orbital – s, p, d, f)∙S.O.S (Δ eV between the two halves of the peak)∙Position∙FWHM∙Area∙%Lorenzian-Gaussian∙See the notes for explanations of how Asymmetry works.4. Click on Accept when satisfiedPeak Types: p, d and f.1. Each of these peaks combines the two splitting peaks2. The FWHM is the same for both the splitting peaks, e.g. a p-type peakwith FWHM=0.7eV is the combination of a p3/2 with FWHM at 0.7eV anda p1/2 with FWHM at 0.7eV, and with an area ratio of 2 to 13. If the theoretical area ratio is not true for the split peaks, the old way ofsetting two s-type peaks and adding the constraints should be used.∙The S.O.S. stands for spin orbital splitting.Note: The FWHM of the p, d or f peaks are the FWHM of the p3/2,d5/2 or f7/2, respectively. The FWHM of the combined peaks (e.g. combination of p3/2and p1/2) is shown in the actual FWHM in the Peak Parameter Window.Peak Constraints1. Each parameter can be referenced to the same type of parameter inother peaks. For example, for four peaks (Peak #0, 1, 2 and 3) with known relative peak positions (0.5eV between adjacent peaks), the following can be used∙Position: Peak 1 = Peak 0 + 0.5eV∙Position: Peak 2 = Peak 1 + 0.5eV∙Position: Peak 3 = Peak 2 + 0.5eV2. You may reference to any peak except with looped references.3. The optimisation of the %GL value is allowed in this program.∙ A suggestion to use this feature is to find a nice peak for a certain setting of your instrument and optimise the %GL for this peak.∙Fix the %GL in the later peak fitting process when the same instrument settings were used.4. This version also includes the setting of the upper and lower bounds foreach parameter.Peak ParametersThis program uses the following asymmetric Gaussian-Lorentzian sumThe program also uses the following symmetrical Gaussian-Lorentzian product functionPeak FunctionNote:If the selection is the sum function, when the user opens a *.xps file that was optimised using the Gaussian-Lorentzian product function, you have to re-optimise the spectra using the Gaussian-Lorentzian sum function with a different %Gaussian-Lorentzian value.∙You can choose the function type you want1. From the lower window, click on the Options button∙The peak parameters box comes up∙Select GL sum for the Gaussian-Lorentzian sum function∙Select GL product for the Gaussian-Lorentzian product function. 2. For the Gaussian-Lorentzian sum function, each peak can have sixparameters∙Peak Position∙Area∙FWHM∙%Gaussian-Lorentzian∙TS∙TLIf anyone knows what TS or TL might be, please let me know. Thanks, CMH3. Each peak in the Gaussian-Lorentzian product function can have fourparameters∙Peak Position∙Area∙FWHM∙%Gaussian-LorentzianSince peak area relates to the atomic concentration directly, we use it as a peak parameter and the peak height will not be shown to the user.Note:For asymmetric peaks, the FWHM only refers to the half of the peak that is symmetrical. The actual FWHM of the peak is calculated numerically and is shown after the actual FWHM in the Peak Parameter Window. If the asymmetric peak is a doublet (p, d or f type peak), the actual FWHM is the FWHM of the doublet.Region ShiftA Region Shift parameter was added under the Parameters menu∙Use this parameter to compensate for the charging effect, the fermi level shift or any change in the system work function∙This value will be added to all the peak positions in the region for fitting purposes.An example:∙ A polymer surface is positively charged and all the peaks are shifted to the high binding energy by +0.5eV, e.g. aliphatic carbon at 285.0eV shifts to 285.5eV∙When the Region Shift parameter is set to +0.5eV, 0.5eV will be added to all the peak positions in the region during peak fitting, but the listed peak positions are not changed, e.g. 285.0eV for aliphatic carbon. Note: I have tried this without any actual shift taking place. If someone finds out how to perform this operation, please let me know. Thanks, CMH.In the meantime, I suggest you do the shift before converting your files from the Vision Software format.OptimisationYou can optimise:1. A single peak parameter∙Use the Optimize button beside the parameter in the Peak Fitting window2. The peak (the peak position, area, FWHM, and the %GL if the "fix" box isnot ticked)∙Use the Optimize Peak button at the base of the Peak Fitting window3. A single region (all the parameters of all the peaks in that region if the"fix" box is not ticked)∙Use the Optimize Region menu (button) in the upper window4. All the regions∙Use the Optimize All button in the lower window∙During any type of optimisation, you can press the "Stop Fitting" button and the program will stop the process in the next cycle.Print/ExportIn the XPS Peak Fit or Region window, From the Data menu, choose Export or Print options as desiredExport∙The program can export the ASCII file of spectrum (*.DAT) for making high quality figures using other software (e.g. SigmaPlot)∙It can export the parameters (*.PAR) for further calculations (e.g. use Excel for atomic ratio calculations)∙It can also copy the spectral image to the system clipboard so that the spectral image can be pasted into a document (e.g. MS WORD). Program Options1. The %tolerance allows the optimisation routine to stop if the change inthe difference after one loop is less that the %tolerance2. The default setting of the optimisation is Newton's method∙This method requires a delta value for the optimisation calculations ∙You may need to change the value in some cases, but the existing setting is enough for most data.3. For the binary search method, it searches the best fit for each parameterin up to four levels of value ranges∙For example, for a peak position, in first level, it calculates the chi^2 when the peak position is changed by +2eV, +1.5eV, +1eV, +0.5eV,-0.5eV, -1eV, -1.5eV, and -2eV (range 2eV, step 0.5eV) ∙Then, it selects the position value that gives the lowest chi^2∙In the second level, it searches the best values in the range +0.4eV, +0.3eV, +0.2eV, +0.1eV, -0.1eV, -0.2eV, -0.3eV, and -0.4eV (range0.4eV, step 0.1eV)∙In the third level, it selects the best value in +0.09eV, +0.08eV, ...+0.01eV, -0.01eV, ...-0.09eV∙This will give the best value with two digits after decimal∙Level 4 is not used in the default setting∙The range setting and the number of levels in the option window can be changed if needed.4. The Newton's Method or Binary Search Method can be selected byclicking the "use" selection box of that method.5. The selection of the peak function is also in the Options window.6. The user can save/read the option parameters with the file extension*.opa∙The program reads the default.opa file at start up. Therefore, the user can customize the program options by saving the selectionsinto the default.opa file.CompatibilityThe program can read:∙Kratos text (*.des) files together with the peak fitting parameters in the file∙The ASCII files exported from Phi's Multiplex software∙The ASCII files of Leybold's software∙The VAMAS file format∙For the Phi, Leybold and VAMAS formats, multiple regions can be read∙For the Phi format, if the description contains a comma ",", the program will give an error. (If you get the error, you may use any texteditor to remove the comma)The program can also import ASCII files in the following format:Binding Energy Value 1 Intensity Value 1Binding Energy Value 2 Intensity Value 2etc etc∙The B.E. list must be in ascending or descending order, and the separation of adjacent B.E.s must be the same∙The file cannot have other lines before and after the data∙Sometimes, TAB may cause a reading error.File I/OThe file format of XPSPEAK 4.1 is different from XPSPEAK 3.1, 3.0 and 2.0 ∙XPSPEAK 4.1 can read the file format of XPSPEAK 3.1, 3.0 and 2.0, but not the reverse∙File format of 4.1 is the same as that of 4.0.LimitationsThis program limits the:∙Maximum number of points for each spectrum to 5000∙Maximum of peaks for all the regions to 51∙For each region, the maximum number of peaks is 10. Cautions for Peak FittingSome graduate students believe that the fitting parameters for the best fitted spectrum is the "final answer". This is definitely not true. Adding enough peaks can always fit a spectrum∙Peak fitting only assists the verification of a model∙The user must have a model in mind before adding peaks to the spectrum!Sample Files:gaas.xpsThis file contains 10 spectra1. Use Open XPS to retrieve the file. It includes ten regions∙1-4 for Ga 3d∙5-8 for Ga 3d∙9-10 for S 2p2. For the Ga 3d and As 3d, the peaks are d-type with s.o.s. = 0.3 and 0.9respectively3. Regions 4 and 8 are the sample just after S-treatment4. Other regions are after annealing5. Peak width of Ga 3d and As 3d are constrained to those in regions 1 and56. The fermi level shift of each region was determined using the As 3d5/2peak and the value was put into the "Region Shift" of each region7. Since the region shift takes into account the Fermi level shift, the peakpositions can be easily referenced for the same chemical components in different regions, i.e.∙Peak#1, 3, 5 of Ga 3d are set equal to Peak#0∙Peak#8, 9, 10 of As 3d are set equal to Peak#78. Note that the %GL value of the peaks is 27% using the GL sum functionin Version 4.0, while it is 80% using the GL product function in previous versions.18 Cu2p_bg.xpsThis spectrum was sent to me by Dr. Roland Schlesinger. It shows a background subtraction using the Shirley + Linear method∙See Shirley + Linear Background, (Page 7)Kratos.des∙This file shows a Kratos *.des file∙This is the format your files should be in if they have come from the Kratos instrument∙Use import Kratos to retrieve the file. See Opening Files, (Page 4)∙Note that the four peaks are all s-type∙You may delete peak 2, 4 and change the peak 1,3 to d-type with s.o.s. = 0.7. You may also read in the parameter file: as3d.rpa. ASCII.prn∙This shows an ASCII file∙Use import ASCII to retrieve the file∙It is a As 3d spectrum of GaAs∙In order to fit the spectrum, you need to first add the background and then add two d-type peaks with s.o.s.=0.7∙You may also read in the parameter file: as3d.rpa.Other Files(We don’t have an instrument that produces these files at Auckland University., but you may wish to look at them anyway. See the readme.doc file for more info.)1. Phi.asc2. Leybold.asc3. VAMAS.txt4. VAMASmult.txtHave Fun! July 1, 1999.。
TW 142数学名词-中小学教科书名词中英对照术语
A New Equation of State for Carbon Dioxide Covering the Fluid Region
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3.3 3.4 3.5 3.6 3.7 3.8 4.
Melting Pressure. . . . . . . . . . . . . . . . . . . . . . .. Sublimation Pressure. . . . . . . . . . . . . . . . . . . .. Vapor Pressure ....................... " Saturated Liquid Density. . . . . . . . . . . . . . . .. Saturated Vapor Density. . . . . . . . . . . . . . . . .. Caloric Data on the Liquid-Vapor Phase Boundary. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Experimental Basis of the New Equation of State. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4.1 Thermal Properties .................... " 4.2 Specific Isobaric Heat Capacity. . . . . . . . . . .. 4.2.1 Experimental Results for the Specific Isobaric Heat Capacity. . . . . . . . . . . . .. 4.2.2 Results for the Specific Isobaric Heat
The absorption spectrum of V838 Mon in 2002 February - March. I. Atmospheric parameters and
a r X i v :a s t r o -p h /0411032v 1 1 N o v 2004Mon.Not.R.Astron.Soc.000,1–7(2004)Printed 2February 2008(MN L A T E X style file v2.2)The absorption spectrum of V838Mon in 2002February -March.I.Atmospheric parameters and iron abundance.⋆Bogdan M.Kaminsky 1†,Yakiv V.Pavlenko 1‡1Main Astronomical Observatory of Ukrainian Academy of Sciences,Golosiiv woods,03680Kyiv-127,UkraineReceived ;acceptedABSTRACTWe present a determination of the effective temperatures,iron abundances,and mi-croturbulent velocities for the pseudophotosphere of V838Mon on 2002February 25,and March 2and 26.Physical parameters of the line forming region were obtained in the framework of a self-consistent approach,using fits of synthetic spectra to observed spectra in the wavelength range 5500-6700˚A .We obtained T eff=5330±300K,5540±270K and 4960±190K,for February 25,March 2,and March 26,respectively.The iron abundance log N (Fe)=−4.7does not appear to change in the atmosphere of V838Mon from February 25to March 26,2002.Key words:stars:atmospheres –stars:abundances –stars:individual:V838Mon1INTRODUCTIONThe peculiar variable star V838Mon was discovered during an outburst in the beginning of 2002January (Brown 2002).Two further outbursts were then observed in 2002February (Munari et al.2002a;Kimeswenger et al.2002;Crause et al.2003)and in general the optical brightness in V-band of the star increased by 9mag.Since 2002March,a gradual fall in V-magnitude began which,by 2003January,was re-duced by 8mag.The suspected progenitor of V838Mon was identified by Munari et al.(2002a)as a 15mag F-star on the main sequence.Possibly V838Mon might have a B3V companion (Desidera &Munari 2002),but it could be a background star.The discovery of a light echo (Henden et al.2002)allowed an estimate of the distance to V838Mon and,according to recent works based on HST data (Bond et al.2003;Tylenda 2004)its distance is 5-6kpc.If these estimations are correct,at the time of maximum brightness V838Mon was the most luminous star in our Galaxy.Details of the spectral evolution of the star are described in Kolev et al.2002;Wisniewski et al.2003;Osiwala et al.2002).During outbursts (except for the last)the spec-trum displayed numerous emission lines with P Cyg pro-files,formed in the expanding shell and around an F-or A-star (Kolev et al.2002).On the other hand,absorption spectra appropriate to a red giant or supergiant were ob-served in quiescent periods.Strong lines of hydrogen,D lines of sodium,triplets of calcium and other elements show P⋆Based in part on observations collected with the 1.83m tele-scope of the Astronomical Observatory in Asiago,Italy †E-mail:bogdan@mao.kiev.ua ‡E-mail:yp@mao.kiev.uaCyg profiles.They have similar profiles and velocities vary-ing from −500km s −1in late January to −280km s −1in late March (Munari et al.2002a).Since the middle of 2002March,the emissions are considerably weakened and the spectrum of V838Mon evolved to later spectral classes.In middle of 2002April,there were present some lines of TiO;in May the spectrum evolved to the“very cold”M-giant (Banerjee &Ashok 2002).In October Evans et al.(2003)characterized it as a L-supergiant.Recently Kipper et al.(2004)found for iron group ele-ments [m/H]=−0.4,while abundances of lithium and some s-process elements are clearly enhanced.This results was obtain using the static LTE model.These results are very dependent on the model atmo-sphere and spectrum synthesis assumptions.The nature of the outbursts remains a mystery.Possible explanations include various thermonuclear processes (very slow nova,flare post-AGB),and the collision of two stars (Soker &Tylenda 2003).Munari et al.(2002a)suggested that V838Mon is a new type of a variable star,because comparison with the closely analogous V4334Sgr and M31RV has shown significant enough differences in the observed parameters.In this paper we discuss the results of the determina-tion of iron abundance and atmospheric parameters of V838Mon.These we obtained from an analysis of absorption spec-tra of V838Mon on 2002February 25and March 2and 26.The complexity and uniqueness of the observed character-istics of V838Mon practically excluded a definition of the parameters of the atmosphere using conventional methods,based on calibration on photometric indices,ionization bal-ance,profiles of hydrogen lines.Indeed,the presence around the star of a dust shell,and the uncertain determination of2Bogdan M.Kaminsky,Yakiv V.Pavlenko interstellar reddening(from E B−V=−0.25to E B−V=−0.8 Munari et al.2002a),affects the U−B and B−V colours. Emission in the hydrogen lines provides severe problems for their application in the estimation of effective temperature. Moreover,both the macroturbulent motions and expansion of the pseudophotosphere merges the numerous lines in wide blends.As a result,a single unblended line in the spectrum of V838Mon cannot be found at all,and any analysis based on measurements of equivalent widths is completely excluded.The observational data used in this paper are described in section2.Section3explains some background to our work and some details of the procedure used.We attempt to de-termine T eff,the microturbulent velocity V t and the iron abundance log N(Fe)in the atmosphere of V838Mon in theframework of the self-consistent approach in section4.Some results are discussed in section5.2OBSER V ATIONSSpectra of V838Mon were obtained on2002February25 and March26with the Echelle+CCD spectrograph on the 1.82m telescope operated by Osservatorio Astronomico di Padova on Mount Ekar(Asiago),and freely available to the community from http://ulisse.pd.astro.it/V838Mon/.A 2arcsec slit was used withfixed E-W orientation,produc-ing a PSF with a FWHM of1.75pixels,corresponding to a resolving power close to20000.The detector was a UV coated Thompson CCD1024×1024pixel,19micron square size,covering in one exposure the wavelength range4500to 9480˚A(echelle orders#49to#24).The short wavelength limit is set by a2mm OG455long-passfilter,inserted in the optical train to cut the second order from the cross-disperser. The wavelength range is covered without gaps between ad-jacent echelle orders up to7300˚A.The spectra have been extracted and calibrated using IRAF software running un-der Linux operating system.The spectra are sky-subtracted andflat-fielded.The wavelength solution was derived simul-taneously for all26echelle orders,with an average r.m.s of 0.18km s−1.The8480-8750˚A wavelength range of these Asi-ago spectra has been described in Munari et al.(2002a,b).Another set of spectra(R∼32000)for March2was obtained with the echellefibre-fed spectrograph on the1.9-m SAAO telescope kindly provided for us by Dr.Lisa Crause (see Crause et al.2003for details).3PROCEDURETo carry out our analysis of V838Mon we used the spectral synthesis techniques.Our synthetic spectra were computed in the framework of the classical approach:LTE,plane-parallel media,no sinks and sources of energy inside the atmosphere,and transfer of energy provided by the radia-tionfield and by convection.Strictly speaking,none of these assumptions is100% valid in atmosphere of V838Mon.Clearly we have non-static atmosphere which may well have shock waves mov-ing trough it.Still we assumed that in any moment the structure of model atmosphere of V838Mon is similar to model atmospheres of supergiants.Indeed,temporal changes of the absorption spectra on the days were rather marginal.0.00050.0010.00150.0020.00250.0030.00350.004-200-150-100-50 0 50 100 150 200Velocity (km s-1)V exp=160 km s-1V*sin i=80 km s-1V macro=50 km s-1parison of expansion(V exp=160km s−1),rota-tional(v∗sin i=80km s−1)and macroturbulent(V macro=50 km s−1)profiles used in this paper to convolve synthetic spectra.Most probably,for this object,we see only a pseudophoto-sphere,which is the outermost part of an expanding enve-lope.Therefore,ourfirst goal was to determine whether it is possible tofit our synthetic spectra to the observed V838 Mon spectra.At the time of the observations the spectral class of V838Mon was determined as K-type(Kolev et al.2002). Absorption lines in spectrum of V838Mon form compara-tively broad blends.Generally speaking,there may be a number of broad-ening mechanisms:•Microturbulence,which is formed by small scale(i.e τ≪1)motions in the atmosphere.In the case of a super-giant,V t usualy does not exceed10km s−1.In our analysis we determined V t from a comparison of observed and com-puted spectra.•Stellar rotation.Our analysis shows that,in the case of V838,we should adopt v∗sin i=80km s−1tofit the observed spectra.This value is too high for the later stages of stellar evolution,for obvious reasons.In reality rotation cannot contribute much to the broadening of lines observed in spectra of most supergiants.•Expansion of the pseudophotosphere of the star.Asym-metrical profiles of expansion broadening can be described, to afirst approximation,by the formulaG(v,λ,∆λ)=const∗∆λThe absorption spectrum of V838Mon3−0.50.511.522.5566056705680569057005710N o r m a l i s e d F l u xWavelength (Å)February 3February 25March 2March 26Figure 2.Spectra of V838Mon observed on February 3,Febru-ary 25,March 2and March 262002emission:many lines are observed in emission.This demon-strates that effects of the radial expansion of the line-forming layers were not significant for the dates of our data and for-mally obtained value V exp =160km s −1is not real.•Macroturbulence.After the large increase of luminos-ity in 2002January-February,large scale (i.e.of magnitude τ>1)macroturbulent motions should be very common in the disturbed atmosphere of V838Mon.Our numerical ex-periments showed that,to get appropriate fits to the ob-served spectra taking into account only macroturbulent ve-locities,we should adopt V macro ∼50km s −1.In any case,for the times of our observations the spectra of V838Mon resemble the spectra of “conventional”super-giants.Our V838Mon spectra for February 25,March 2and 26agree,at least qualitatively,with the spectrum of Arcturus (K2III),convolved with macroturbulent velocity profile,given by a gaussian of half-width V macro =50km s −1(Fig.3).The observed emissions in the cores of the strongest lines are formed far outside,perhaps at the outer boundary of the expanding envelope,i.e.in the region which is heated by shock wave dissipation.As result of our first numerical experiments,we con-cluded that the spectra of V838Mon in 2002February -March were similar to the spectrum of a normal late (su-per)giant,broadened by strong macroturbulence motions and/or expansion of its pseudophotosphere.Unfortunately we cannot,from the observed spectra,distinguish between broadening due to the macroturbulence and expansion (see next section).It is worth noting that the observed spectra of V838Mon are formed in a medium with decreasing temperature to the outside,i.e.in the local co-moving system of co-ordinates the atmosphere,to a first approximation,can be described by a “normal”model,at least in the region of formation of weak or intermediate strength atomic lines.0.10.20.3 0.4 0.5 0.6 0.7 0.8 0.91 1.1 570057105720573057405750N o r m a l i s e d F l u xWavelength (Å)V 838 Mon ArcturusArcturus conv. V macro =50 km s −1Figure parison of the spectrum of V838Mon and that of Arcturus,convolved with macroturbulent profile V macro =50km s −13.1Fits to observed spectraWe computed a sample of LTE synthetic spectra for a grid of Kurucz (1993)model atmospheres with T eff=4000–6000K using the WITA612program (Pavlenko 1997).Synthetic spectra were computed with wavelength step 0.02˚A ,micro-turbulent velocities 2–18km s −1with a step 1km s −1,iron abundances log N (Fe)=−5.6→−3.6dex 1,with a step 0.1dex.Then,due to the high luminosity of the star,we formally adopt log g =0.Synthetic spectra were computed using the VALD (Kupka et al.1999)line list.For atomic lines the line broadening constants were taken from VALD or computed following Unsold (1955).For the dates of our observations lines of neutral iron dominate in the spectra.Fortunately,they show rather weak gravity/pressure dependence,therefore the uncertainty in the choice of log g will not be important in determining our main results;the dependence of the computed spectra on T effis more significant (see Fig.4).The computed syn-thetic spectra were convolved with different profiles,and then fitted to the observed spectra following the numeri-cal scheme described in Jones et al.(2002)and Pavlenko &Jones (2002).In order to determine the best fit parameters,we com-pared the observed residual fluxes r obsλwith computed values H theorλ+f s .We let H obs λ= F theor x −y ∗G (y )∗dy ,where F theor λis the theoretical flux and G (y )is the broadening profile.In our case G (y )may be wavelength dependent.To get the best fit we find the minima per point of the 2D functionS (f s ,f g )=Σ(1−H synt /H obs )2.We calculated these minimization parameters for our grid of synthetic spectra to determine a set of parameters f s (wavelength shift parameter)and f g (convolution parame-ter).The theoretical spectra were convolved with a gaussian profile.Our convolution profile is formed by both expan-sion and macroturbulent motions.We cannot distinguish between them in our spectra.To get a numerical estimate1in the paper we use the abundance scaleN i =14Bogdan M.Kaminsky,Yakiv V.Pavlenko0.60.650.7 0.75 0.8 0.85 0.90.95 1 6306 6308 6310 6312 6314 6316 6318 6320 6322 6324N o r m a l i s e d F l u xWavelength (Å)T eff =4000 KT eff =5000 K logg=0T eff =5000 K logg=1T eff =6000 KFigure 4.Dependence of computed spectra on T effand log gof the broadening processes in the pseudophotosphere,we use a formal parameter V g ,which describes the cumulative effect of broadening/expansion motions.The parameters f s and f g were determined by the min-imization procedure;the procedure was carried out for dif-ferent spectral regions.We selected for analysis 6spectral orders in the interval 5600-6700˚A .In the red,spectral lines are blended by telluric spectra,and are of lower S/N.In the blue the blending of the spectra are rather high.Our main intention was to obtain a self-consistent solution sep-arately for different echelle orders,and then compare them.If we could obtain similar parameters from different spec-tral regions it can be evidence of the reality of the obtained solution.4RESULTS 4.1The SunTo be confident in our procedure,we carried out a similar analysis for the Sun.For this case we know the solar abun-dances and other basic parameters,therefore our analysis provides an independent estimation of the quality of our procedure:•From the solar atlas of Kurucz et al (1984)we ex-tract spectral regions corresponding to our observed orders of V838Mon;•we convolve the solar spectra with a gaussian of V macro =50km s −1.•we carried out a spectral analysis of the spectral regions with our procedure;again,model atmospheres from Kurucz (1993)with a grid of different log g ,T eff,log N (Fe)were used.The results of our “re-determination”of parameters of the solar atmosphere are given in Table 1.The best fit to one spectral region is shown in Fig.5.From our analysis of the solar spectrum we obtained T eff=5625±125K,log N (Fe)=−4.48±0.15dex,V t =1.2±0.4km s −1.Here and below we used the standard deviation for error esti-mates.All these parameters are in good agreement with theTable 1.Parameters of the solar atmosphere116480–668555001-4.545.8126300–649057502-4.646.4136125–631557501-4.442.9145960–614557501-4.244.1165660–581055001-4.643.9175520–567055001-4.642.9Averaged56251.2-4.4844.3The absorption spectrum of V838Mon5–For February25we obtained T eff=5330±300K,log N(Fe)=−4.7±0.14dex and V t=13.±2.8km s−1.–For the March26data the mean values are T eff=4960±270K,log N(Fe)=−4.68±0.11dex,V t=12.5±1.7km s−1.–And for March2the mean values are T eff=5540±190K,log N(Fe)=−4.75±0.14dex,V t=13.3±3.2kms−1.–We obtained V g=54±3,47±3and42±5km s−1for February25,March2and March26,respectively.–The f s parameter provides the heliocentric velocity ofV838Mon.We obtained V radial=−76±3,−70±3and−65±3km s−1for February25,March2and March26,respectively.Most probably,we see some reduction in theexpansion velocity of the envelope.5DISCUSSIONFrom a comparison of our results for all three dates we see that:•The effective temperature for March26is somewhat lower then for the previous dates.This is an expected re-sult,in view of the gradual cooling of envelope.However, for March2we found a slightly higher value of temperature than for February25.A possible explanation is the heating of the pseudophotosphere as result of the third outburst.•The microturbulent velocities are very similar and ex-tremely high for all three dates.•Our analysis shows a lower value of V g for the later dates:the effects of expansion and macroturbulence were weakened at the later stages of evolution of the pseudopho-tosphere of V838Mon.•The iron abundances log N(Fe)=-4.7±0.14are similar for all dates.Our estimates of effective temperature are in a good agreement with Kipper et al.(2004),although we used dif-ferent procedures of analysis.The iron abundance([Fe/H]=−0.4)and microturbulent velocity(V t=12km s−1)found by Kipper et al.(2004)for March18are in agreement with our results.Our deduced“effective temperatures”as well as those in Kipper et al(2004)do not correspond with values ob-tained from photometry(T eff∼4200K).We assume that in our analysis we deal with temperatures in the line forming region,rather than with the temperatures at photospheric levels which determine the spectral energy distribution of V838Mon and the photometric indices.Indeed,the formally determined microturbulent velocity V t=13km s−1exceeds the sound velocity in the atmosphere(4-5km s−1).This means that the region of formation of atomic lines should be heated by dissipation of supersonic motions:the temper-ature there should be higher than that given in a plane-parallel atmosphere of T eff∼4200K.Certainly the effect cannot be explained by sphericity effects:the temperature gradients in the extended atmo-spheres should be steeper(see Mihalas1978),therefore tem-peratures in the line forming regions should be even lower, in contradiction with our results.Strong deviations from LTE are known to occur during the photospheric stages of the evolution of novae and super-0.40.50.60.70.80.911.16480 6500 6520 6540 6560 6580 6600 6620 6640 6660 6680 6700 NormalisedFluxWavelength (Å)V 838 MonT eff=5250 KT eff=4500 K0.40.50.60.70.80.911.16300 6320 6340 6360 6380 6400 6420 6440 6460 6480 6500 NormalisedFluxWavelength (Å)V 838 MonT eff=5000 KT eff=4500 K0.40.50.60.70.80.911.16120 6140 6160 6180 6200 6220 6240 6260 6280 6300 6320 NormalisedFluxWavelength (Å)V 838 MonT eff=5250 KT eff=4500 KFigure6.The bestfits of synthetic spectra to11-13orders of the observed spectrum of V838Mon on February25,found by the minimization procedure.novae.The main effect there should be caused by deviations from LTE in the ionization balance.However,in our case we used lines of the neutral iron,which dominate by number. We cannot expect a reduction in the density of Fe I atoms in the comparatively cool atmosphere of the star.Further-more,we exclude from our analysis strong lines with P Cyg profiles.Lines of interest in our study have normal profiles.6Bogdan M.Kaminsky,Yakiv V.PavlenkoTable2.Atmospheric parameters for V838MonAsiago spectraFebruary25116480–6685525015-4.753.2-79.6126300–6490500014-4.954.5-76.3136125–6315525010-4.756.0-82.7145960–6145575017-4.552.5-79.5165660–581050009-4.960.7-67.1175520–5670575014-4.751.1-73.6Averaged533013.2-4.7354.7-76.5March26116480–6685475012-4.843.7-67.3126300–6490475014-4.844.7-68.3136125–6315475010-4.546.0-74.3145960–6145500015-4.842.1-65.8165660–5810500011-4.738.8-52.2175520–5670550013-4.539.7-63.6Average496012.5-4.6842.5-65.2SAAO spectraMarch2116480–6685550012-4.645.3-80.6126300–6490525016-4.955.8-77.3136125–631552507-4.642.8-78.9145960–6145575014-4.849.0-80.6165660–5810550015-4.951.7-68.1175520–5670600016-4.742.1-80.0Average554013.3-4.7547.8-77.6The absorption spectrum of V838Mon70.40.50.6 0.7 0.8 0.9 1 1.1 5960 5980 6000 6020 6040 6060 6080 6100 6120 6140 6160N o r m a l i s e d F l u xWavelength (Å)V 838 Mon T eff =5750 K T eff =4500 K0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 5660 5680570057205740576057805800N o r m a l i s e d F l u xWavelength (Å)V 838 Mon T eff =5000 K T eff =4500 K0.40.50.6 0.7 0.8 0.9 1 1.1 5500 5520 5540 5560 5580 5600 5620 5640 5660 5680N o r m a l i s e d F l u xWavelength (Å)V 838 Mon T eff =5750 K T eff =4500 K Figure 7.The best fits of synthetic spectra to orders 14,16and 17of the observed spectrum of V838Mon on February 25,found by the minimization procedure.•most probably,the line-forming region is heated by su-personic motions –our spectroscopic temperatures exceed photometrically determined T effby ∼1000K;•we do not find any significant change in the iron abun-dance in atmosphere V838from February 25to March 26.•we derived a moderate deficit of iron log N (Fe)∼−4.7in the atmosphere of V838Mon.ACKNOWLEDGMENTSWe thank Drs.Ulisse Munari,Lisa Crause,Tonu Kipper and Valentina Klochkova for providing spectra and for discus-sions of our results.We thank Dr.Nye Evans for improving text of paper.We thank unknown referee for many helpful remarks.This work was partially supported by a PPARC visitors grants from PPARC and the Royal Society.YPs studies are partially supported by a Small Research Grant from American Astronomical Society.This research has made use of the SIMBAD database,operated at CDS,Strasbourg,France.REFERENCESAllen C.W.,1973,Astrophysical quantities,3rd edition,TheAthlone Press,LondonBanerjee D.P.K.,Ashok N.M.,2002,A&A,395,161Bond H.E.,et al.,2003,Natur,422,405Brown N.J.,2002,IAU Circ,7785,1Crause L.A.,Lawson W.A.,Kilkenny D.,van Wyk F.,MarangF.,Jones A.F.,2003,MNRAS,341,785Desidera,S.,Munari,U.,2002,IAU Circ,7982,1Evans A.,Geballe T.R.,Rushton M.T.,Smalley B.,van LoonJ.Th.,Eyres S.P.S.,Tyne V.H.,2003,MNRAS,343,1054Henden A.,Munari U.,Schwartz M.B.,2002,IAU Circ,7859Jones H.R.A.,Pavlenko Ya.,Viti S.,Tennyson J.,2002,MNRAS,330,675JKimeswenger S.,Ledercle C.,Schmeja S.,Armsdorfer B.,2002,MNRAS,336,L43Kipper T.,et al.,2004,A&A,416,1107Kolev D.,Mikolajewski M.,Tomow T.,Iliev I.,Osiwala J.,Nirski J.,Galan C.,2002,Collected Papers,Physics (Shu-men,Bulgaria:Shumen University Press),147Kupka F.,Piskunov N.,Ryabchikova T.A.,Stempels H.C.,WeissW.W.,1999,A&AS,138,119Kurucz R.L.,Furenlid I.,Brault J.,Testerman L.,1984,Nationalsolar obs.-Sunspot,New Mexico Kurucz R.L.,1993,CD-ROM 13Mihalas D.,1978,Stellar atmospheres,Freeman &Co.Munari U.,et al.,2002a,A&A,389,L51Munari U.,Henden A.,Corradi R.M.L,Zwitter T.,2002b,in”Classical Nova Explosions”,M.Hernanz and J.Jos´e eds.,AIP Conf.Ser.637,52Osiwala J.P.,Mikolajewski M.,Tomow T.,Galan C.,Nirski J.,2003,ASP Conf.Ser.,303,in pressPavlenko Y.V.,1997,Astron.Reps,41,537Pavlenko Ya.V.,Jones H.R.A.,2002.A&A,397,967Pavlenko Ya.V.,2003.Astron.Reps,47,59Soker N.,Tylenda R.,2003,ApJ,582,L105Tylenda R.,2004,A&A,414,223Unsold A.,1955Physik der Sternatmospheren,2nd ed.Springer.BerlinWisniewski J.P.,Morrison N.D.,Bjorkman K.S.,MiroshnichenkoA.S.,Gault A.C.,Hoffman J.L.,Meade M.R.,Nett J.M.,2003,ApJ,588,486This paper has been typeset from a T E X/L A T E X file preparedby the author.。
MODEL EVALUATION GUIDELINES FOR__ SYSTEMATIC
D. N. Moriasi, J. G. Arnold, M. W. Van Liew, R. L. Bingner, R. D. Harmel, T. L. Veith
Sensitivity analysis is the process of determining the rate of change in model output with respect to changes in model inputs (parameters). It is a necessary process to identify key parameters and parameter precision required for calibration (Ma et al., 2000). Model calibration is the process of estimating model parameters by comparing model predictions (output) for a given set of assumed conditions with observed data for the same conditions. Model validation involves running a model using input parameters measured or determined during the calibration process. According to Refsgaard (1997), model validation is the process of demonstrating that a given site-specific model is capable of making “sufficiently accurate” simulations, although “sufficiently accurate” can vary based on project goals. According to the U.S. EPA (2002), the process used to accept, reject, or qualify model results should be established and documented before beginning model evaluation. Although ASCE (1993) emphasized the need to clearly define model evaluation criteria, no commonly accepted guidance has been established, but specific statistics and performance ratings for their use have been developed and used for model evaluation (Donigian et al., 1983; Ramanarayanan et al., 1997; Gupta et al., 1999; Motovilov et al., 1999; Saleh et al., 2000; Santhi et al., 2001; Singh et al., 2004; Bracmort et al., 2006; Van Liew et al., 2007). However, these performance ratings are model and project specific. Standardized guidelines are needed to establish a common system for judging model performance and comparing various models (ASCE, 1993). Once established, these guidelines will assist modelers in preparing and reviewing quality assurance project plans for modeling (U.S. EPA, 2002) and will increase accountability and public a
2012_Nature Materials_Fano-resonant asymmetric metamaterials for ultrasensitive spectroscopy
surface-enhanced spectroscopies9,11,12 . Especially important for biosensing applications are infrared metamaterials, because biomolecules possess mid-infrared vibrational fingerprints that can be used for their identification, thereby improving the specificity of biodetection26 . Whereas the concept of tuning metamaterial resonances to specific vibrational lines of a biomolecule is straightforward13 , its actual implementation is complicated by dense spectral spacing of infrared-active vibrations. For example, selectively tuning resonant metamaterials to commonly used amide vibrations of the protein backbone (amide I at ωI = 1,620 − 1,680 cm−1 and amide II at ωII = 1,510 − 1,580 cm−1 ) requires the structures’ quality factors q ∼ 10. Even narrower resonances may be required to resolve the secondary protein structure. Such sharp resonant lines are difficult to obtain with most plasmonic metamaterials despite relatively low Ohmic losses at infrared frequencies11,13 . Narrow linewidth is also requisite for obtaining large field enhancements over the surface of a planar plasmonic metamolecule. For example, a half-wave antenna provides neither strong field enhancement nor spectrally narrow response. By arranging metamolecules into periodic arrays, it is possible to suppress radiative decay and boost field enhancement owing to the formation of collective high-quality resonances27 . The drawback of such coherent metamaterials11,28 is their reliance on long-range interactions between unit cells, making their performance sensitive to fabrication imperfections and angular spread of the infrared beam. Another recently emerged approach to achieving sharp spectral response and high field enhancement is based on Fano resonances29–37 . The Fano-resonance approach relies on local resonances of the metamolecules and therefore is free of the disadvantages of coherent metamaterials. Fano resonances
北太平洋海温Victoria模态与ENSO年际关系的非对称特征
北太平洋海温Victoria模态与ENSO年际关系的非对称特征作者：祁莉毛欣张文君来源：《大气科学学报》2022年第02期摘要利用美国NOAA海表温度资料，重点分析了北太平洋海温异常EOF第二模态Victoria模态（VM）与ENSO年际关系的非对称特征。
研究发现，VM和ENSO在年代际尺度上相关性较弱，而在年际尺度上有很好的相关关系，两者同期为负相关，VM超前1a为正相关。
然而，正负VM事件与ENSO冷暖位相在年际尺度上的联系存在着一定的非对称性。
正VM事件与同年冬季热带中东太平洋海温异常的联系较弱，但次年常有厄尔尼诺事件发生;相比较而言，负VM事件在同年一般都有厄尔尼诺事件伴随发生，而与次年冬季热带中东太平洋海温没有显著联系，且很少有ENSO事件发生。
由此可见，正VM事件对次年厄尔尼诺的发生发展似乎有促进作用，可作为ENSO前期预报因子之一，而负VM事件不能作为ENSO的前期预报因子。
关键词北太平洋海温;Victoria模态;ENSO;非对称性北太平洋是重要的海域之一，它对热带太平洋地区、东亚地区的气候都有着明显的影响（LatifandBarnett，1994;DeserandBlackmon，1995;Nakamuraetal.，2008;方陆俊等，2016;张文君等，2018;祁莉等，2019;陈海山等，2020）。
众所周知，北太平洋海表温度经验正交函数分析（EmpiricalOrthogonalFunction，EOF）的第一模态是太平洋年代际振荡（PacificDecadalOscillation，PDO）模態（Mantuaetal.，1997）。
它是北太平洋上显著的年代际信号，研究指出PDO会造成局部地区气候的年代际变化（魏凤英和宋巧云，2005），并且PDO与热带太平洋的ENSO事件紧密联系（Wang，1995），有些研究甚至通过PDO的位相预测ENSO的频率与振幅（Papineau，2001;杨修群等，2004），而不同PDO位相背景下ENSO 对东亚冬、夏季风和气候的影响也存在明显的不同（WuandWang，2002;朱益民和杨修群，2003;Wangetal.，2008;YoonandYeh，2010;Kimetal.，2014）。
A normal form for beam and non-local nonlinear schrodinger equation
Home Search Collections Journals About Contact us My IOPscienceA normal form for beam and non-local nonlinear Schrödinger equationsThis article has been downloaded from IOPscience. Please scroll down to see the full text article.2010 J. Phys. A: Math. Theor. 43 434028(/1751-8121/43/43/434028)View the table of contents for this issue, or go to the journal homepage for moreDownload details:IP Address: 218.49.116.75The article was downloaded on 01/04/2011 at 01:32Please note that terms and conditions apply.IOP P UBLISHING J OURNAL OF P HYSICS A:M ATHEMATICAL AND T HEORETICAL J.Phys.A:Math.Theor.43(2010)434028(13pp)doi:10.1088/1751-8113/43/43/434028A normal form for beam and non-local nonlinear Schr¨o dinger equationsM ProcesiUniversit`a di Napoli“Federico II”,Naples,I-80126ItalyE-mail:michela.procesi@unina.itReceived30March2010,infinal form1July2010Published12October2010Online at /JPhysA/43/434028AbstractWe discuss normal forms of the completely resonant nonlinear beam equationand nonlinear Schr¨o dinger equation.We work in n>1spatial dimensions andstudy both periodic and Dirichlet boundary conditions on cubes.We discussthe applications to the problem offinding quasi-periodic solutions.In the caseof periodic boundary and the dimension n=2,we apply KAM theory andprove the existence and stability of quasi-periodic solutions.PACS numbers:05.45.−a,02.30.JrMathematics Subject Classification:37K55,35Q551.IntroductionIn this paper we discuss an approximate normal form for the completely resonant nonlinear beam and nonlinear Schr¨o dinger equation(NLS for short):v tt+ 2v=κv3(1)i u t− u=κ| s(u)|2 s(u),(2) either for x∈T n(periodic boundary conditions)or with Dirichlet boundary conditions on the cube D=[0,π]n.In equation(2)the symbol s,s 0,is a linear smoothing operators e i k·x=|k|−s e i k·x,so equation(2)reduces to the standard local NLS for s=0.It is well known that equations(1)and(2)are infinite-dimensional Hamiltonian systems with an ellipticfixed point at v=0,so one wishes to apply in this infinite-dimensional setting the powerful tools of Birkhof normal form and KAM theory.One must note that both(1)and(2)are completely resonant,namely in the linear part the frequencies are rational and all the bounded solutions are periodic.One expects small 1751-8113/10/434028+13$30.00©2010IOP Publishing Ltd Printed in the UK&the USA1periodic,quasi-periodic(and hopefully almost periodic)solutions to occur due to the presence of the nonlinear term.To prove suitable‘non-degeneracy’assumptions one generally performsa step of resonant Birkhof normal form.There are essentially two problems.(1)Hamiltonians(1)and(2)after the normal form step appear to have a very intricate behavior.Already in the case offinite-dimensional systems this may pose serious problems in proving the existence of invariant tori.(2)Finding quasi-periodic solutions for nonlinear PDEs in n>1spatial dimension requiressolving rather delicate small divisor problems related to the high multiplicity of the eigenvalues of the Laplacian .In this paper we focus on problem1.As an application we prove the existence and stability of quasi-periodic solutions in the case of n=2and periodic boundary conditions.1.1.Some backgroundThere are two main methods for constructing quasi-periodic solutions:Newton methods—proposed by Craig and Wayne in[3]and extended by Bourgain(see[1,2])to PDEs in n>1 dimensions—and KAM theory,first developed for one-dimensional Hamiltonian PDEs with Dirichlet boundary conditions(see[10,13]).For PDEs in a dimension higher than1,most of the results are restricted to simplified non-resonant models,obtained by adding a Fourier multiplier to the Laplacian.For example in equation(1)v tt+( +Mσ)2v=f(x,v),(3) where Mσis a linear operator,depending on afinite number—say m—of free parametersσ, which commutes with the Laplacian.A KAM algorithm to prove the existence and stability of quasi-periodic solutions for the non-resonant beam(3)and non-local(s>0)NLS on a torus was proposed in[5].The authors use conservation of momentum and the fact that the nonlinearity is regularizing,to simplify the second-order Melnikov non-resonance condition.The extension to the much more complex local NLS(s=0)is due to Eliasson and Kuksin in[4].To extend the results of[5]to the completely resonant cases(1)and(2),object of this paper,we perform one step of Birkhof normal form to introduce parameters.Consider a HamiltonianH=H(2)(p,q)+H(4)(p,q),H(2)(p,q)=k λ2kp2k+q2k,where H(4)is a polynomial of degree4and theλk are all rational numbers.A step of‘Birkhof normal form’is a symplectic change of variables which reduces the Hamiltonian H toH N=H(2)(p,q)+H(4)res(p,q)+H(6),where H(6)is an analytic function of degree at least6,H(4)res is of degree4and Poisson commutes with H(2).Then one wishes to treat H(2)(p,q)+H(4)res(p,q)as the new unperturbed Hamiltonian and H(6)as a small perturbation.This can work provided that H(2)(p,q)+H(4)res(p,q)is simple enough(possibly completely integrable)and has quasi-periodic motions for large classes of initial data(which now play the role of the parameters σ).An ideal situation is whenλk are non-resonant up to order4so that the normal form is integrable.In the case of the beam equation with a positive‘mass term’(Mσin(3)is substituted by a constantμ>0),Geng and You in[6]proved the existence of quasi-periodic solutions. 2The main point in their paper is to show that the mass term simplifies the normal form so that they can apply the results in [5].In the case of dimension n =2Geng,You and Xu proved the existence of quasi-periodic solutions for equation (2)with s =0by a combination of non-integrable normal form,momentum conservation (in the spirit of [5])and the ideas of Eliasson and Kuksin;indeed they show that in the special dimension 2and in the context of(2)(no explicit spacial dependence)many of the difficulties of [4]simplify.Their approach is based on a non-integrable normal form and hence they do not obtain the linear stability.A related result appears in [8]and [9]where similar techniques allow us to construct wave packets of periodic solutions for the beam and NLS equations in any dimension.The method of these papers was a combination of Lyapunov–Schmidt decomposition and Lindstedt series.In those papers we were able to deal also with Dirichlet boundary conditions for which the normal form is much more complicated.1.2.ResultsDivide the oscillators into two suitable subsets,the tangential and the normal sites,with the strategy of analyzing the equations near the solutions in which the normal sites are at rest and the tangential sites move quasi-periodically on an m -torus.This is done by doubling the action variables in the tangential sites,so each action variable is written as ξi +y i ,where ξi are the free parameters and y i are the symplectic dynamical variables coupled with the angles x i .In theorem 1we prove that for infinitely many choices of the tangential sites,the Hamiltonians of (1)and (2),after a step of normal form,areH =(ω(ξ),y)+Q M (ξ,x,{z k ,¯zk })+P ,where Q is a block-diagonal quadratic form in the ‘normal variables’{z k ,¯zk }with coefficients trigonometric polynomials in x .A main point is that the dimensions of the blocks in Q are uniformly bounded.Finally P is a small perturbation.This result is rather surprising for n >2especially in the case of Dirichlet boundary conditions.Indeed the ‘normal form Hamiltonian’is very complicated,see section 3,and the block-diagonal structure is not easily recognized.The main idea is to transform theorem 1into an abstract combinatorial setting,see lemmas 3.10and 3.11.In the case of periodic boundary conditions one can obtain much finer ly one can prove that there exists a symplectic change of variables which reduces the quadratic normal form to an integrable one (no dependence of the angle variables x );this is discussed in[12].We then specialize to the case n =2and periodic boundary conditions.With this restriction in theorem 2we prove the existence and linear stability of quasi-periodic solutions of (1)and (2).More precisely in proposition 4.1we apply a KAM algorithm,following [7];this implies the existence of quasi-periodic solutions.To prove the stability we reduce the normal form to constant coefficients and show that it is elliptic for appropriate choices of ξ.2.Hamiltonian formalismWe sketch the Hamiltonian structure of the equations,see for instance [5]for a full presentation.We restrict our attention to the invariant subspace U :=⎧⎨⎩u ∈L 2(R ×T n ;C )|u(t,x)= k ∈Z n oddu k (t)e i k ·x ⎫⎬⎭3Z n odd :={k =(k 1,...,k n )|k 1=2h 1,...,k n −1=2h n −1,k n =h n ,h i odd }.It is easily seen that on this subspace u 0=0so − is positive definite and (− )−12= 1.We write the beam equation in complex coordinates by setting √2u(x,t)=(− )12v(x,t)+i (− )−12v t (x,t).We obtaini u t = u +12 1[( 1(u)+ 1(¯u))3].(4)We work on the scale of complex Hilbert spaces ¯ (a,p):= u ={u k }k ∈Z n odd k ∈Z n |u k |2e 2a |k ||k |2p := u 2a,p ∞ ,(5)where a >0,p >n/2.This space is symplectic,with the conjugate variables {i u k ,¯u k },with respect to the complex symplectic form i k d u k ∧d¯u k .While studying the case of Dirichlet boundary conditions,we expand the solutions insin Fourier series;this is equivalent to expanding in the exponentials and restricting to the subspace D := {u k }k ∈Z n odd|u k =−u σi (k) σi (k)=(k 1,...,−k i ,...,k n ),which is invariant for the dynamics.Hence in this case the dynamical variables are {i u k ,¯u k }k ∈Z n odd (+),where Z n odd (+)≡Z n odd ∩Z n +.Let G be the group with 2n elements generated by the reflections σi ;by our assumption on the set Z n odd each orbit of this group has exactly 2n points.We say that two elements h and k are equivalent h ≡k if they are in the same G orbit.Z n odd (+)can be identified with Z n odd with the equivalence relation ≡.It is convenient to denote indifferently by k ∈Z n odd (+)either the point in Z n odd ∩Z n (+)or the orbit k (1),...,k (2n ) ,k (i)∈Z n odd ,k (i)≡k (j),where k (1)∈Z n (+).2.1.The Hamiltonians:periodic boundary conditionsIn (2)we can rescale u to avoid the constant κand pass to the Fourier coefficients and obtainan equation for {u k }:˙u k =i ∂¯u k H ,where H := k ∈Z n odd |k |2u k ¯u k + k i ∈Z n odd k 1+k 3−k 2−k 4=0u k 1¯u k 2u k 3¯u k 4|k 1|s |k 2|s |k 3|s |k 4|s ,(NLS )H :=k ∈Z n odd |k |2u k ¯u k + s i =±1,k i ∈Z n odd s 1k 1+s 2k 2+s 3k 3+s 4k 4=0u s 1k 1u s 2k 2u s 3k 3u s 4k 4|k 1|s |k 2|s |k 3|s |k 4|s ,(Be )where u +k =u k and u −k =¯u k .Since both (2)and(1)do not depend explicitly on the spatial variable,the total momentum M = k ∈Z n oddk |u k |2is a conserved quantity.2.2.The Hamiltonians:Dirichlet boundary conditionsWe restrict to the subspace D and use {i u k ,¯u k }as symplectic variables with k ∈Z n odd (+);weobtain (as usual after rescaling)H := k ∈Z n odd (+)|k |2u k ¯u k +2−n k i ∈Z n odd (+),a i k (a 1)1−k (a 2)2+k (a 3)3−k (a 4)4=0u k (a 1)1¯u k (a 2)2u k (a 3)3¯u k (a 4)4|k 1|s |k 2|s |k 3|s |k 4|s ,(NLS )4H :=k ∈Z n odd (+)|k |2u k ¯u k +2−n k i ∈Z n odd ,s i =±1,a i 4i =1s i k (a i )i =0u s 1k (a 1)1u s 2k (a 2)2u s 3k (a 3)3u s 4k (a 4)4|k 1 k 2 k 3 k 4|.(Be )Recall that k (a)with a =1,...,2n are the elements of the orbit of k and u k (a)=(−1)α(a)u k .The factor 2−n reminds us that if a quadruple appears in the quartic term of H ,then all the 2n elements of the orbit of the quadruple also appear and give the same contribution.We could remove the factor by restricting to the quadruples where say a 4=1(or we could rescale the factor away!).2.3.Analytic HamiltoniansWe consider real Hamiltonians on the space ¯(a,p)which can be formally expanded in the Taylor seriesF =i F (i)(u,¯u)= i (α,β)∈N ∞|α+β|1=i F (α,β) k ∈Z nu αk k ¯u βk k ,where F (i)is a multilinear form of i variables in ¯(a,p).We require that the series is totally convergent in some ball of positive radius,so that by definition the function is analytic.Clearly our Hamiltonians belong to this class and are convergent on any ball B r .For compactness we will write them asH :=k ∈A |k |2u k ¯u k +2−δn ◦ k i ∈A,s i =±1,a i ∈O u s 1k (a 1)1u s 2k (a 2)2u s 3k (a 3)3u s 4k (a 4)4|k 1|s |k 2|s |k 3|s |k 4|s ,(6)where,respectively,for periodic and Dirichlet boundary conditions A is either Z n odd or Z n odd (+),δis either 0or 1and O is either {1}or {1,...,2n }.The expression ◦means that we restrict to 4i =1s i k (a i )i =0and in the case of the NLS also to i s i =0.We will also be interested in the Hamiltonian vector fields X F ={∂¯u k F }k ∈Z n ,where F is ananalytic Hamiltonian in the ball B r .With this notation the Hamilton equations are i˙u k =X F .We will restrict our attention to the vector fields which map B r →¯ a,¯p with ¯p=p +s >p .If|X F |r :=sup u ∈B r X F a,¯p /r <12,(7)the symplectic change of variables generated by F is well defined and analytic say from B r/2→B r .Note that condition (7)is not verified by the quadratic part of our Hamiltonians but only by the higher order terms.3.Normal formFor small u (i.e. u a,p < 1)we perform a standard step of the Birkhof normal form removing all the terms of order 4of H which do not Poisson-commute with the quadratic part H (2):= k |k |2|u k |2.5In fact the monomial u s 1k 1u s 2k 2u s 3k 3u s 4k 4is an eigenvector with respect to {H (2),−}with the eigenvalue i (s 1|k 1|2+s 2|k 2|2+s 3|k 3|2+s 4|k 4|2).Thus,we perform the symplectic change ofvariables H →e {F,−}(H ),generated by the flow ofF :=−i2−δn ◦i s i |k i |2=0u s 1k (a 1)1u s 2k (a 2)2u s 3k (a 3)3u s 4k (a 4)4(|k 1|2−|k 2|2+|k 3|2−|k 4|2)|k 1|s |k 2|s |k 3|s |k 4|s .For sufficiently small,this is a well-known analytic change of variables (cf [1,5])¯(a,p)⊃B →B 2 ⊂¯ (a,p)(where B denotes as usual the open ball of radius )which brings (6)to the formH N :=k ∈A |k |2u k ¯u k +2−δn ◦ RESu s 1k (a 1)1u s 2k (a 2)2u s 3k (a 3)3u s 4k (a 4)41s 2s 3s 4s +P (6)(u),(8)where ◦RES is the sum restricted to s 1|k 1|2+s 2|k 2|2+s 3|k 3|2+s 4|k 4|2=0,4i =1s i k (a i )i =0,(9)with the further restriction i s i =0for the NLS.P (6)(u)is analytic of degree at least 6inu k |k |−s .In a small ball B ,P (6)(u)is perturbative with respect to the terms of degrees 2and 4.Lemma 3.1.In the case of periodic boundary conditions F commutes with M so that H N still satisfies momentum conservation:namely a monomial k u αk k ¯u βk k appears in H N only ifk k(αk −βk )=0.Inthe case of Dirichlet boundary conditions we obtain the much weaker restriction that k k(a k )(αk −βk )=0for some choice of a k =1,...,2n .Lemma 3.2.The resonance condition (9)can be verified for the quadruples k 1,...,k 4only if i s i =0.Proof.We restate (9)without loss of generality ass 1|k 1|2+s 2|k 2|2+s 3|k 3|2+s 4|k 4|2=0,k 1+k 2+k 3+k 4=0,k i ∈Z n odd .If all s i are positive or negative,then we have no solution,in the same way if i s i =±2,we have|k 1|2+|k 2|2+|k 3|2=|k 1+k 2+k 3|2↔ k 1,k 2+k 3 + k 2,k 3 =0which is impossible since the restriction k i ∈Z n odd implies that the left-hand side is an odd integer.The previous lemma shows that the resonance condition is the same for the beam and NLS equation.For (k 1,k 2,k 3,k 4)|k i ∈Z n odd ,denoteP :={(k 1,k 2,k 3,k 4)|k 1+k 3=k 2+k 4,|k 1|2+|k 3|2=|k 2|2+|k 4|2}.Our resonance condition states that (k 1,k 2,k 3,k 4)∈P for periodic boundary conditions and that k (a 1)1,k (a 2)2,k (a 3)3,k (a 4)4 ∈P for Dirichlet boundary conditions.Trivial computations show that (k 1,k 2,k 3,k 4)∈P is equivalent tok 1+k 3=k 2+k 4,(k 1−k 2,k 3−k 2)=0,(10)6so that the integer vectors k 1,k 2,k 3,k 4form the vertices of a rectangle .In P some of the k i may coincide;we wish to evidence this structure.Definition 3.3.P .In the case of periodic boundary conditions we say that a rectangle is degenerate if it reduces to a segment or to a point.We call P the subset of P of non-degenerate rectangles.D.In the case of Dirichlet boundary conditions we say that a rectangle is degenerate if either it degenerates to a segment or to a point or if the vertices of the rectangle are all equivalent.As before we call P the set of non-degenerate rectangles.We can rewrite our Hamiltonians asH N = k ∈A |k |2u k ¯u k +3δn k |u k |4|k |4s +2δn +1 k 1=k 2∈A |u k 1|2|k 1|2s |u k 2|2|k 2|2s+2−δnk i ∈A,a i ∈O (k (a 1)1,k (a 2)2,k (a 3)3,k (a 4)4)∈P u k (a 1)1¯u k (a 2)2u k (a 3)3¯u k (a 4)41s 2s 3s 4s +P (6)(u),(11)where A =Z n odd ,δ=0and O ={1}in the case of periodic boundary,while A =Z n odd (+),δ=1,O ={1,...,2n }in the case of Dirichlet boundary.Note that in the case of Dirichlet boundary the sum k (a 1)1,k (a 2)2,k (a 3)3,k (a 4)4 ∈P contains terms in which the k i are pairwise equivalent.3.4.Elliptic-action angle variablesWe are interested in non-symmetric domains D ⊂B where some of the variables u k (the tangential sites)are bounded away from zero—and hence can be passed in action angle variables—while all the other variables are much smaller.We first partition our index spacesZ n odd =S ∪S c ,Z n odd (+)=S ∪S c ,(12)respectively,for periodic and Dirichlet boundary conditions.The set S :=(v 1,...,v m )is called tangential sites and S c the normal sites .In the case of Dirichlet boundary conditions we will denote by v (a)i 2n a =1the orbit of v i ∈S .We set u k :=z k for k ∈S c ,u v i := ξi +y i e i x i for i =1,...m ;(y,x)×(z,¯z)∈R m ×T m × (a,p)where we denote by (a,p)the subspace of ¯ (a,p)ׯ (a,p)generated by the indices in S c and w =(z,¯z)(considered as the row vectors)are the corresponding coordinates;the symplectic form is d y ∧d x +id z ∧d¯z .We consider the domain A α×D(ρ,r):= ξ:12r α ξi r α × x,y,w :x ∈T m ρ,|y | r 2,|w |a,p r ⊂R m ×T m ρ×C m × (a,p).(13)Here,0<α<12,0<r <1,0<ρ<12are the parameters.T m ρdenotes the open subset of the complex torus,where Re (x)∈T m ,|Im (x)|<ρ.We are interested in the functions F (y,x,w)which are analytic in D(ρ,r).Following[11]we use Lipschitz sup-norm for the dependence on ξ:F λρ,r =sup A α×D(ρ,r)|F |+λsup D(ρ,r)sup ξ,η∈A αξ=η|F (ξ)−F (η)||ξ−η|,7where λ∝r α/max i (|v i |2).For the Hamiltonian vector fieldsX F ={∂y F,−∂x F,J d w F },we use the weighted Lipschitz norm (cf (7))|X F |ρ,r :=sup A α×D(ρ,r)|∂y F |λ+r −2|∂x F |λ+r −1 d w F λa,¯p .3.5.Choice of the tangential sitesWe impose the following constraint on the set S.Constraint 1.Any three elements v i ,v j ,v l ∈S satisfy v (a)i −v (b)j,v l −v (b)j =0,only if they are the vertices of a degenerate rectangle,i.e.in the unavoidable cases v (a)i =v (b)j or v (b)j =v l or v i ≡v j ≡v l .Constraint 2.We choose v i so that |v i |=|v j |for all i =j and i νi v i =0wheni |νi | 8,ν=0.The Hamiltonian (8)can be written asH N =H 0+P (3)(z,y ;ξ,x)+P (6)(z,y ;ξ,x),where (14)(1)the term P (3)collects all the terms of H N −P (6)in which at least three k indices are in S cand it is of degree at least 3(and at most 4)in z,¯z;(2)by constraint 2,P (6)at z =0does not contain any term of degree 9in the elements u v i which is non-constant in x ;(3)by constraint 1,no non-degenerate rectangles P contain more than two elements of S ,hence setting I i =ξi +y i :H 0:=m i =1 |v i |2I i +3δn |v i |4s I 2i +2δn +2 i<j I i I j |v i |2s |v j |2s +2δn +2 i ;k ∈S c I i |z k |2|v i |2s |k |2s + k ∈S c |k |2|z k |2+4∗ i =j ;h,k ∈S c I i I j e i (x i −x j )|v i |s |v j |s |h |s |k |s z h ¯z k +2∗∗ i<j ;h,k ∈S c I i I j e −i (x i +x j )|v i |s |v j |s |h |s |k |s z h z k +2∗∗ i<j ;h,k ∈S c I i I j )e i (x i +x j )|v i |s |v j |s |h |s |k |s ¯z h ¯z k .Here, ∗denotes the constraint h (a 1),k (a 2),v (a 3)i ,v (a 4)j ∈P for some a 1,...,a 4∈O .In the same way∗∗denotes the constraint h (a 1),v (a 2)i ,k (a 3),v (a 4)j ∈P for some a 1,...,a 4∈O .Remark 3.6.Let us give a geometric interpretation to the constraints:P.In periodic boundary ∗means that k belongs to the hyperplane H i,j defined by x −v i ,v i −v j =0and h =k +v j −v i (hence h ∈H j,i ).In the same way ∗∗means that k and h belong to the sphere S i,j of the equation x −v j ,x −v i =0.D.In Dirichlet boundary ∗means that k belongs to the hyperplane H (i,a),(j,b)(with a,b ∈{1,...,2n })defined by x −v (a)i ,v (a)i −v (b)j =0and h (c)=k +v (b)j −v (a)i for some c .In the same way ∗∗means that k and h belong to the sphere S (i,a),(j,b)of the equations x −v (b)j,x −v (a)i =0and h (c)=−k +v (a)i +v (b)j .8Conservation laws.The conservation of M in the new variables implies that the monomials appearing in H in the case of periodic boundary are of the formzα¯zβy c e i(x,ν),i v iνi+k∈S c(αk−βk)k=0,(15)whereν=(ν1,...,νm),νi∈Z andα,βare the multi-indices in N and|α|,|β|are the sum of their coordinates.3.7.Final form of the HamiltonianWe note that our Hamiltonian is analytic in the domain Aα×Dρ,r;we define the degree of a monomial y i z j¯z l by2i+j+l.We drop in formula(14)the constant part(depending only on the parametersξ)and we separate H=N+P,where N is a‘quadratic normal form’namely contains only the terms of H0which are of degree 2:N:=(ω(ξ),y)+kk|z k|2+Q M(w),(16) whereωi(ξ):=|v i|2+2·3δn ξii+2δn+2j=iξjj i,k(ξ)=|k|2+2δn+2jξjj.(17)Finally the quadratic form isQ M(ξ,x;w)=41 i=j mk∈H i,jh=k+v j−v iξiξj e i(x i−x j)|v i|s|v j|s|h|s|k|s z h¯z k+4i<jk∈S i,j,h<k,h=−k+v j+v iξiξj|v i|s|v j|s|h|s|k|se−i(x i+x j)z h z k+e i(x i+x j)¯z h¯z k,(18)wherea couple(h,k)∈S i,j is given;we decide arbitrarily an ordering h<k with the convention that on S j,i the ordering is reversed.For the perturbation,due to constraint2we have the boundsP ρ,r Cr min(5α,1+52α,4),|X P|ρ,s C r min(52α,2).(19) Theorem1.For all choices of tangential sites S satisfying constraints1and2,the quadratic form Q M(w)is block diagonal with blocks of uniformly bounded dimension.Remark3.8.In the case of n=2,the block structure of Q M is obvious since the lines H(i,a),(j,b)may intersect only at afinite number of points.In dimension n>2the intersection is an affine subspace so this simple proof fails and it is indeed very hard(especially in the case of Dirichlet boundary conditions)to have a geometric picture of which the couples(k,h)contribute non-trivially to Q M.We therefore proceed in a more combinatorial way.Proof.To understand the block structure of(18)it is useful to relate the quadratic form to a graph.9Definition 3.9.Graph representation.Given h,k in Z n odd (resp.in Z n odd (+)for Dirichlet boundary conditions)if there exist v i ,v j ∈S resp .v (a)i ,v (b)j ∈S such that ∗holds,we connect the points with a black edge labeled as (v i ,v j ) resp . v (a)i ,v (b)j .Given h,k in Z n odd (resp in Z n odd (+))if h,k satisfy ∗∗for some v i ,v j ∈S resp .v (a)i ,v (b)j ∈S we connect them with a red edge labeled as (v i ,v j ) resp . v (a)i ,v (b)j .This defines a graph S in Z n odd (resp in Z n odd (+)).We define a sequence of points k 1,...,k N to be a path if all k 1,k i +1are the edges of the graph.It is obvious that the red edges are finite in number;indeed only the integer points on a finite number of spheres may be connected by a red edge.The following is a standard lemma in graph theory.Lemma 3.10.The block-diagonal blocks of the quadratic form Q M (w)are given by the connected components of the above-described graph.Hence the quadratic form Q M (w)is block diagonal with blocks of uniformly bounded dimension if and only if the length of paths of black edges with no loops is uniformly bounded.To identify a path in S with no red edges,one may give a starting point k ∈Z n odd and a list of the connected edges v (a 1)i 1,v (b 1)j 1 ,..., v (a N )i N ,v (b N )j N .We consider the set of distinct couples v (a)i ,v (b)j as an alphabet and we call a list of connected black edges in S a word.Finally we say that a word has no loops if the corresponding path in S has no loops.We are now in the notations of appendix A4of [8],we apply lemmas A4.1–A4.3and we obtain the following.Lemma 3.11.There exists K such that if a word has a length k K,then the word contains a loop.The value of K depends only on the number of letters of the alphabet.This completes the proof.4.n =2with periodic conditionsAs we have stated in remark 3.8that the case n =2is particularly simple.In the case of periodic boundary conditions Geng,You and Xu have shown in [7]that one may impose (rather complicated)arithmetic conditions on the set S so thatConstraint 3.All the intersection points x /∈S such thatx ∈∪i,j,l,m H i,j ∩H l,mor x ∈∪i,j,l,m S i,j ∩H l,m or x ∈∪i,j,l,m S i,j ∩S l,m are not integer vectors.Given any m ∈Z ,there exist S satisfying constraints 1–3,with |S |=m ,such that in the Hamiltonian (16),each k ∈S c may belong to at most one element of the list {H i,j ,S i,j }.We have obtained a normal form Hamiltonian which is essentially identical to that of [7].A key point in [7]is the study of the block-diagonal homological equation associated with the normal form N .Let us define some spaces (by convention we denote z k =z +k ,¯z k =z −k )F 0,1:=Span (e i ν·x z k ,e −i ν·x ¯z k ),where i νi v i +k =0;F 0,2:=Span e i ν·x z σk z τh ,where i νi v i +σk −τh =0.10We study the operator ad(N):={N,·}on the spaces F 0,1and F 0,2.A direct computation shows that i ad(N)on F 0,1is block diagonal with blocks of dimension at most 2×2.If k /∈∪i,j (S i,j ∪H i,j ),we have the usual diagonal term (ω,ν)+ k for e i ν·x z k such that l νl v l =−k .Let ¯ωi =ωi −|v i |2and ¯ k = k −|k |2.We have 2×2blocks ((ω,ν)+|k |2−|v i |2)I +⎛⎝−¯ωi +¯ k 4σ√ξi ξj |v i |s |v j |s |h |s |k |s 4√i j |v i ||v j ||h k |σ(−¯ωj +¯ h )⎞⎠(20)connecting e i (ν−e i )·x z k and e i (ν−σe j )·x z σh ,where i <j ,h =σ(k −v i )+v j and if σ=+,k ∈H ij ,and if σ=−,k ∈S ij h <k .Clearly there are only a finite number of non-symmetric matrices (since there are a finite number of h,k ∈S i,j )and if |k |is large enough,the off-diagonal terms are irrelevant.By definition |k |2−|v i |2=σ(|h |2−|v j |2)and by momentum conservation lv l νl =−(k −v i )=−σ(h −v j ).(21)Note that,given ν,constraints 1–3imply that there is at most one element in ∪i,j H i,j ∪S i,j and one couple (h,k)with k ∈H i,j (resp.S i,j )i <j ,satisfying (21).Let us denote by λν,k ,λν,h the two eigenvalues of (20);since ad(N)is symplectic,the eigenvalues −λν,k ,−λν,h appear in the block e −i (ν−e i )·x z −k ,e −i (ν−σe j )·x z −σh ).To study ad(N)on F 0,2,we note that F 0,1⊗F 0,1→F 0,2surjectively via the map e i ν·x z σk z τh =e i σν1z σk ·e i τν2z τh for all ν1,ν2such that ν1+k =ν2+h =0and σν1+τν2=ν.Then—by the Leibnitz rule—ad(N)|F 0,2is induced by ad(N)|F 0,1⊗I +I ⊗ad(N)|F 0,1,henceblock diagonal with blocks of dimension at most 4×4.Finally,the eigenvalues of ad(N)|F 0,2are the sum of two eigenvalues of ad(N)|F 0,1and the eigenvectors are the product of two eigenvectors of ad(N)|F 0,1.Proposition 4.1.For all γsmall enough and τlarge enough,there exists a positive measure Cantor set A γ,τ⊂A α(with |A α\A γ,τ|=O(γr α))such that,for all ξ∈A γ,τwe have the following.(i)The operator ad(N)|F 0,1is regular semi-simple,with the eigenvalues satisfying|λν,k |>γr α|ν|τ,|λν1,k 1±λν2,k 2|>γr α1+|ν1±ν2|τ,for all (ν,k)=(ν ,k ),satisfying the usual momentum conservation if k or k /∈∪i,j (S i,j ∪H i,j )and (21)otherwise.(ii)The kernel of ad(N)on F 0,1×F 0,2is the set of functions of the form Q(ξ,x ;w)=k O k |z k |2+ 1 i =j mk ∈H i,j h =k +v j −v i e i (x i −x j )a ijkh z h ¯z k +i<j k ∈S i,j ,h =−k +v j +v i b ijhk (e −i (x i +x j )z h z k +e i (x i +x j )¯z h ¯z k ),(22)such that ad(Q)is simultaneously diagonalizable with ad(N)on F 0,1.Sketch of the Proof.(i)A direct computation shows that λν,k are all different functions of ξ;note that this holdstrue also for the two eigenvalues of a single block.The measure estimates on the Cantor set follow (recall that for |k |large the off-diagonal terms in (20)are irrelevant).(ii)Follows by standard linear algebra,see [12]for details.11。
Hierarchical cluster analysis
Chapter 7Hierarchical cluster analysisIn Part 2 (Chapters 4 to 6) we defined several different ways of measuring distance (or dissimilarity as the case may be) between the rows or between the columns of the data matrix, depending on the measurement scale of the observations. As we remarked before, this process often generates tables of distances with even more numbers than the original data, but we will show now how this in fact simplifies our understanding of the data. Distances between objects can be visualized in many simple and evocative ways. In this chapter we shall consider a graphical representation of a matrix of distances which is perhaps the easiest to understand – a dendrogram, or tree – where the objects are joined together in a hierarchical fashion from the closest, that is most similar, to the furthest apart, that is the most different. The method of hierarchical cluster analysis is best explained by describing the algorithm, or set of instructions, which creates the dendrogram results. In this chapter we demonstrate hierarchical clustering on a small example and then list the different variants of the method that are possible.ContentsThe algorithm for hierarchical clusteringCutting the treeMaximum, minimum and average clusteringValidity of the clustersClustering correlationsClustering a larger data setThe algorithm for hierarchical clusteringAs an example we shall consider again the small data set in Exhibit 5.6: seven samples on which 10 species are indicated as being present or absent. In Chapter 5 we discussed two of the many dissimilarity coefficients that are possible to define between the samples: the first based on the matching coefficient and the second based on the Jaccard index. The latter index counts the number of ‘mismatches’ between two samples after eliminating the species that do not occur in either of the pair. Exhibit 7.1 shows the complete table of inter-sample dissimilarities based on the Jaccard index.Exhibit 7.1 Dissimilarities, based on the Jaccard index, between all pairs ofseven samples in Exhibit 5.6. For example, between the first two samples, A andB, there are 8 species that occur in on or the other, of which 4 are matched and 4are mismatched – the proportion of mismatches is 4/8 = 0.5. Both the lower andupper triangles of this symmetric dissimilarity matrix are shown here (the lowertriangle is outlined as in previous tables of this type.samples A B C D E F GA00.50000.4286 1.00000.25000.62500.3750B0.500000.71430.83330.66670.20000.7778C0.42860.71430 1.00000.42860.66670.3333D 1.00000.8333 1.00000 1.00000.80000.8571E0.25000.66670.4286 1.000000.77780.3750F0.62500.20000.66670.80000.777800.7500G0.37500.77780.33330.85710.37500.75000The first step in the hierarchical clustering process is to look for the pair of samples that are the most similar, that is are the closest in the sense of having the lowest dissimilarity – this is the pair B and F, with dissimilarity equal to 0.2000. These two samples are then joined at a level of 0.2000 in the first step of the dendrogram, or clustering tree (see the first diagram of Exhibit 7.3, and the vertical scale of 0 to 1 which calibrates the level of clustering). The point at which they are joined is called a node.We are basically going to keep repeating this step, but the only problem is how to calculated the dissimilarity between the merged pair (B,F) and the other samples. This decision determines what type of hierarchical clustering we intend to perform, and there are several choices. For the moment, we choose one of the most popular ones, called the maximum, or complete linkage, method: the dissimilarity between the merged pair and the others will be the maximum of the pair of dissimilarities in each case. For example, the dissimilarity between B and A is 0.5000, while the dissimilarity between F and A is 0.6250. hence we choose the maximum of the two, 0.6250, to quantify the dissimilarity between (B,F) and A. Continuing in this way we obtain a new dissimilarity matrix Exhibit 7.2.Exhibit 7.2 Dissimilarities calculated after B and F are merged, using the‘maximum’ method to recomputed the values in the row and column labelled(B,F).samples A(B,F)C D E GA00.62500.4286 1.00000.25000.3750(B,F)0.625000.71430.83330.77780.7778C0.42860.71430 1.00000.42860.3333D 1.00000.8333 1.00000 1.00000.8571E0.25000.77780.4286 1.000000.3750G0.37500.77780.33330.85710.37500Exhibit 7.3 First two steps of hierarchical clustering of Exhibit 7.1, using the ‘maximum’ (or ‘complete linkage’) method.The process is now repeated: find the smallest dissimilarity in Exhibit 7.2, which is 0.2500 for samplesA and E , and then cluster these at a level of 0.25, as shown in the second figure of Exhibit 7.3. Then recomputed the dissimilarities between the merged pair (A ,E ) and the rest to obtain Exhibit 7.4. For example, the dissimilarity between (A ,E ) and (B ,F ) is the maximum of 0.6250 (A to (B ,F )) and 0.7778 (E to (B ,F )).Exhibit 7.4 Dissimilarities calculated after A and E are merged, using the ‘maximum’ method to recomputed the values in the row and column labelled (A ,E ).samples(A,E)(B,F)CDG(A,E)00.77780.4286 1.00000.3750(B,F)0.777800.71430.83330.7778C 0.42860.71430 1.00000.3333D1.00000.8333 1.000000.8571G 0.37500.77780.33330.85710In the next step the lowest dissimilarity in Exhibit 7.4 is 0.3333, for C and G – these are merged, as shown in the first diagram of Exhibit 7.6, to obtain Exhibit 7.5. Now the smallest dissimilarity is 0.4286, between the pair (A ,E ) and (B ,G ), and they are shown merged in the second diagram of Exhibit 7.6. Exhibit 7.7 shows the last two dissimilarity matrices in this process, and Exhibit 7.8 the final two steps of the construction of the dendrogram, also called a binary tree because at each step two objects (or clusters of objects) are merged. Because there are 7 objects to be clustered, there are 6 steps in the sequential process (i.e., one less) to arrive at the final tree where all objects are in a single cluster. For botanists that may be reading this: this is an upside-down tree, of course!1.00.50.0B F B F A E1.00.50.0Exhibit 7.5 Dissimilarities calculated after C and G are merged, using the‘maximum’ method to recomputed the values in the row and column labelled (C,G).samples(A,E)(B,F)(C,G)D(A,E)00.77780.4286 1.0000(B,F)0.777800.77780.8333(C,G)0.42860.77780 1.0000D 1.00000.8333 1.00000Exhibit 7.6 The third and fourth steps of hierarchical clustering of Exhibit 7.1, using the ‘maximum’ (or ‘complete linkage’) method. The point at which objects (or clusters of objects) are joined is called a node.Exhibit 7.7 Dissimilarities calculated after C and G are merged, using the‘maximum’ method to recomputed the values in the row and column labelled (C,G).samples(A,E,C,G)(B,F)D samples(A,E,C,G,B,F)D (A,E,C,G)00.7778 1.0000(A,E,C,G,B,F)0 1.0000 (B,F)0.777800.8333D 1.00000D 1.00000.83330B F A EC G1.00.50.0B F A EC G1.00.50.0Exhibit 7.8 The fifth and sixth steps of hierarchical clustering of Exhibit 7.1, using the ‘maximum’ (or ‘complete linkage’) method. The dendrogram on the right is the final result of the cluster analysis. In the clustering of n objects, there are n–1 nodes (i.e. 6 nodes in this case).Cutting the treeThe final dendrogram on the right of Exhibit 7.8 is a compact visualization of the dissimilarity matrix in Exhibit 7.1, based on the presence-absence data of Exhibit 5.6. Interpretation of the structure of data is made much easier now – we can see that there are three pairs of samples that are fairly close, two of these pairs ((A,E) and (C,G)) are in turn close to each other, while the single sample D separates itself entirely from all the others. Because we used the ‘maximum’ method, all samples clustered below a particular level of dissimilarity will have inter-sample dissimilarities less than that level. For example, 0.5 is the point at which samples are exactly as similar to one another as they are dissimilar, so if we look at the clusters of samples below 0.5 – i.e., (B,F), (A,E,C,G) and (D) – then within each cluster the samples have more than 50% similarity, in other words more than 50% co-presences of species. The level of 0.5 also happens to coincide in the final dendrogram with a large jump in the clustering levels: the node where (A,E) and (C,G) are clustered is at level of 0.4286, while the next node where (B,F) is merged is at a level of 0.7778. This is thus a very convenient level to cut the tree. If the branches are cut at 0.5, we are left with the three clusters of samples (B,F), (A,E,C,G) and (D), which can be labelled types 1, 2 and 3 respectively. In other words, we have created a categorical variable, with three categories, and the samples are categorized as follows:A B C D E F G2 1 23 2 1 2Checking back to Chapter 2, this is exactly the objective which we described in the lower right hand corner of the multivariate analysis scheme (Exhibit 2.2) – to reveal a categorical variable which underlies the structure of a data set.B F A EC G1.00.50.0B F A EC G D1.00.50.0Maximum, minimum and average clusteringThe crucial choice when deciding on a cluster analysis algorithm is to decide how to quantify dissimilarities between two clusters. The algorithm described above was characterized by the fact that at each step, when updating the matrix of dissimilarities, the maximum of the between-cluster dissimilarities was chosen. This is also known as complete linkage cluster analysis, because a cluster is formed when all the dissimilarities (‘links’) between pairs of objects in the cluster are less then a particular level. There are several alternatives to complete linkage as a clustering criterion, and we only discuss two of these: minimum and average clustering.The ‘minimum’ method goes to the other extreme and forms a cluster when only one pair of dissimilarities (not all) is less than a particular level – this is known as single linkage cluster analysis. So at every updating step we choose the minimum of the two distances and two clusters of objects can be merged when there is a single close link between them, irrespective of the other inter-object distances. In general, this is not a suitable choice for most applications, because it can lead to clusters that are quite heterogeneous internally, and the usual object of clustering is to obtain homogeneous clusters.The ‘average’ method is an attractive compromise where dissimilarities are averaged at each step, hence the name average linkage cluster analysis. For example, in Exhibit 7.1 the first step of all types of cluster analysis would merge B and F. But then calculating the dissimilarity between A, for example, and (B,F) is where the methods distinguish themselves. The dissimilarity between A and B is 0.5000, and between A and F it is 0.6250. Complete linkage chooses the maximum: 0.6250; single linkage chooses the minimum: 0.5000; while average linkage chooses the average: (0.5000+0.6250)/2 = 0.5625.Validity of the clustersIf a cluster analysis is performed on a data matrix, a set of clusters can always be obtained, even if there is no actual grouping of the objects, in this case the samples. So how can we evaluate whether the three clusters in this example are not just any old three groups which we would have obtained on random data with no structure? There is a vast literature on validity of clusters (we give some references in the Bibliography, Appendix E) and here we shall explain one approach based on permutation testing. In our example, the three clusters were formed so that internally in each cluster formed by more than one sample the between-sample dissimilarities were all less than 0.5000. In fact, if we look.at the result in the right hand picture of Exhibit 7.8, the cutpoint for three clusters can be brought down to the level of 0.4286, where (A,E) and (C,G) joined together. As in all statistical considerations of significance, we ask whether this is an unusual result or whether it could have arisen merely by chance. To answer this question we need an idea of what might have happened in chance results, so that we can judge our actual finding. This so-called “null distribution” can be generated through permuting the data in some reasonable way, evaluating the statistic of interest, and doing this many times (or for all permutations if this is feasible computationally) to obtain a distribution of the statistic. The statistic of interest could be that value at which the three clusters are formed, but we need to choose carefullyhow we perform the permutations, and this depends on how the data were collected. We consider two possible assumptions, and show how different the results can be.The first assumption is that the column totals of Table Exhibit 5.6 are fixed; that is, that the 10 species are present, respectively, 3 times in the 7 samples, 6 times, 4 times, 3 times and so on. Then the permutation involved would be to simply randomly shuffle the zeros and ones in each column to obtain a new presence-absence matrix with exactly the same column totals as before. Performing the compete linkage hierarchical clustering on this matrix leads to that value where the three cluster solution is achieved, and becomes one observation of the null permutation distribution. We did this 9999 times, and along with our actual observed value of 0.4286, the 10000 values are graphed in Exhibit 7.9 (we show it as a horizontal bar chart because there are only 15 different values observed of this value, shown here with their frequencies). The value we actually observed is one of the smallest – the number of permuted matrices that generates this value or a lower value is 26 out of 10000, so that in this sense our data are very unusual and the ‘significance’ of the three-cluster solution can be quantified with a p -value of 0.0026. The other 9974 randompermutations all lead to generally higher inter-sample dissimilarities such that the level at which three-cluster solutions are obtained is 0.4444 or higher (0.4444 corresponds to 4 mistmatches out of 9.Exhibit 7.9 Bar chart of the 10000 values of the three-cluster solutions obtained by permuting the columns of the presence-absence data, including the value we observed in the original unpermuted data matrix.The second and alternative possible assumption for the computation of the null distribution could be that the column margins are not fixed, but random; in other words, we relax the fact that there were exactly 3 samples that had species sp1, for example, and assume a binomial distribution for each column, using the observed proportion (3 out of 7 forspecies sp1) and the number of samples (7) as the binomial parameters. Thus there can be 0 up to 7 presences in each column, according to the binomial probabilities for eachspecies. This gives a much wider range of possibilities for the null distribution, and leads us to a different conclusion about our three observed clusters. The permutation distributionlevel frequency0.800020.7778350.75003630.714313600.70001890.666729670.625021990.60008220.571413810.55552070.50004410.444480.4286230.400020.37501is now shown in Exhibit 7.10, and now our observed value of 0.4286 does not look sounusual, since 917 out of 10000 values in the distribution are less than or equal to it, giving an estimated P -value of 0.0917.Exhibit 7.10 Bar chart of the 10000 values of the three-cluster solutionsobtained by generating binomial data in each column of the presence-absence matrix, according to the probability of presence of each species.So, as in many situations in statistics, the result and decision depends on the initialassumptions. Could we have observed the presence of species s1 less or more than 3 times in the 7 samples (and so on for the other species)? In other words, according to thebinomial distribution with n = 7, and p = 3/7, the probabilities of observing k presences of species sp1 (k = 0, 1, …, 7) are:0 1 2 3 4 5 6 7 0.020 0.104 0.235 0.294 0.220 0.099 0.025 0.003If this assumption (and similar ones for the other nine species) is realistic, then the cluster significance is 0.0917. However, if the first assumption is adopted (that is, the probability of observing 3 presences for species s1 is 1 and 0 for other possibilities), then the significance is 0.0028. Our feeling is that perhaps the binomial assumption is more realistic, in which case our cluster solution could be observed in just over 9% of random cases – this gives us an idea of the validity of our results and whether we are dealing with real clusters or not. The value of 9% is a measure of ‘clusteredness’ of our samples in terms of the Jaccard index: the lower this measure, the more they are clustered, and the hoihger the measure, the more the samples lie in a continuum. Lack of evidence oflevel frequency0.875020.857150.8333230.8000500.7778280.75002010.71434850.7000210.666712980.625011710.60008950.571419600.55554680.500022990.44441770.42865670.40001620.37501070.3333640.300010.2857120.250030.20001‘clusteredness’ does not mean that the clustering is not useful: we might want to divide up the space of the data into separate regions, even though the borderlines between them are ‘fuzzy’. And speaking of ‘fuzzy’, there is an alternative form of cluster analysis (fuzzy cluster analysis, not treated specifically in this book) where samples are classified fuzzily into clusters, rather than strictly into one group or another – this idea is similar to the fuzzy coding we described in Chapter 3.Clustering correlations on variablesJust like we clustered samples, so we can cluster variables in terms of their correlations, or distances based on their correlations as described in Chapter 6. The dissimilarity based on the Jaccard index can also be used to measure similarity between species – the index counts the number of samples that have both species of the pair, relative to the number of samples that have at least one of the pair, and the dissimilarity is 1 minus this index.Exhibit 7.11 shows the cluster analyses based on these two alternatives, for the columns of Exhibit 5.6, using the graphical output this time of the R function hclust for hierarchical clustering. The fact that these two trees are so different is no surprise: the first one is based on the correlation coefficient takes into account the co-absences, which strengthens the correlation, while the second does not. Both have the pairs (sp2,sp5) and (sp3,sp8) at zero dissimilarity because these are identically present and absent across the samples. Species sp1 and sp7 are close in terms of correlation, due to co-absences – sp7 only occurs in one sample, sample E , which also has sp1, a species which is absent in four other samples. Notice in Exhibit 7.11(b) how species sp10 and sp1 both join the cluster (sp2,sp5) at the same level (0.5).Exhibit 7.11 Complete linkage cluster analyses of (a) 1–r (1 minus the correlation coefficient between species); (b) Jaccard dissimilarity between species (1 minus the Jaccard similarity index). The R function hclust which calculates the dendrograms places the object (species) labels at a constant distance below its clustering level.(a) (b)s p 1s p 7s p 2s p 5s p 9s p 3s p 8s p 6s p 4s p 100.00.51.01.52.0H e i g h ts p 4s p 6s p 10s p 1s p 2s p 5s p 7s p 9s p 3s p 80.00.20.40.60.81.0H e i g h tClustering a larger data setThe more objects there are to cluster, the more complex becomes the result. In Exhibit 4.5 we showed part of the matrix of standardized Euclidean distances between the 30 sites of Exhibit 1.1, and Exhibit 7.12 shows the hierarchical clustering of this distance matrix, using compete linkage. There are two obvious places where we can cut the tree, at about level 3.4, which gives four clusters, or about 2.7, which gives six clusters. To get an ideaExhibit 7.12 Complete linkage cluster analyses of the standardized Euclidean distances of Exhibit 4.5.of the ‘clusteredness’ of these data, we performed a permutation test similar to the one described above, where the data are randomly permuted within their columns and the cluster analysis repeated each time to obtain 6 clusters. The permutation distribution of levels at which 6 clusters are formed is shown in Exhibit 7.13 – the observed value in Exhibit 7.12 (i.e., where (s2,s14) joins (s25,s23,s30,s12,s16,s27)) is 2.357, which is clearly not an unusual value. The estimated p -value according to the proportion of the distribution to the left of 2.357 in Exhibit 7.13 is p = 0.3388, so we conclude that these samples do not have a non-random cluster structure – they form more of a continuum, which will be the subject of Chapter 9.s s 2s 23s 3016s 27s s s 2 4 classes6 classes7-11Exhibit 7.13 Estimated permutation distribution for the level at which 6clusters are formed in the cluster analysis of Exhibit 7.12, showing the valueactually observed. Of the 10000 permutations, including the observed value,3388 are less than or equal to the observed value, giving an estimated p -valuefor clusteredness of 0.3388.SUMMARY: Hierarchical cluster analysis1. Hierarchical cluster analysis of n objects is defined by a stepwise algorithm whichmerges two objects at each step, the two which have the least dissimilarity.2. Dissimilarities between clusters of objects can be defined in several ways; forexample, the maximum dissimilarity (complete linkage), minimum dissimilarity (single linkage) or average dissimilarity (average linkage).3. Either rows or columns of a matrix can be clustered – in each case we choose theappropriate dissimilarity measure that we prefer.4. The results of a cluster analysis is a binary tree, or dendrogram, with n – 1 nodes. Thebranches of this tree are cut at a level where there is a lot of ‘space’ to cut them, that is where the jump in levels of two consecutive nodes is large.5. A permutation test is possible to validate the chosen number of clusters, that is to seeif there really is a non-random tendency for the objects to group together. 6-clu ster level f r e q u e n c y 2.0 2.5 3.002004006008001000(observed value)。
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a r X i v :0706.2227v 1 [a s t r o -p h ] 15 J u n 2007Mon.Not.R.Astron.Soc.000,000–000(0000)Printed 1February 2008(MN L A T E X style file v1.4)The Norma Cluster (ACO 3627):I.A Dynamical Analysisof the Most Massive Cluster in the Great AttractorP.A.Woudt 1⋆,R.C.Kraan-Korteweg 1,J.Lucey 2,A.P.Fairall 1,S.A.W.Moore 21Department of Astronomy,University of Cape Town,Private Bag X3,Rondebosch 7701,South Africa 2Departmentof Physics,University of Durham,Durham DH13LE,United Kingdom2007May 22ABSTRACTA detailed dynamical analysis of the nearby rich Norma cluster (ACO 3627)is pre-sented.From radial velocities of 296cluster members,we find a mean velocity of 4871±54km s −1and a velocity dispersion of 925km s −1.The mean velocity of the E/S0population (4979±85km s −1)is offset with respect to that of the S/Irr population (4812±70km s −1)by ∆v =164km s −1in the cluster rest frame.This offset increases towards the core of the cluster.The E/S0population is free of any detectable substructure and appears relaxed.Its shape is clearly elongated with a po-sition angle that is aligned along the dominant large-scale structures in this region,the so-called Norma wall.The central cD galaxy has a very large peculiar velocity of 561km s −1which is most probably related to an ongoing merger at the core of the cluster.The spiral/irregular galaxies reveal a large amount of substructure;two dynamically distinct subgroups within the overall spiral-population have been identified,located along the Norma wall elongation.The dynamical mass of the Norma cluster within itsAbell radius is 1−1.1×1015h −173M ⊙.One of the cluster members,the spiral galaxy WKK 6176which recently was observed to have a 70kpc X-ray tail,reveals numer-ous striking low-brightness filaments pointing away from the cluster centre suggesting strong interaction with the intracluster medium.Key words:galaxies:clusters:individual:Norma cluster (ACO 3627)–galaxies:elliptical and lenticular,cD –galaxies:individual:WKK 6176–galaxies:kinematics and dynamics1INTRODUCTIONThe observed velocity flow field of galaxies in the nearby Universe is largely dominated by the Great Attractor (Dressler et al.1987;Lynden-Bell et al.1988;Tonry et al.2000)and the ∼3times more distant Shapley superclus-ter (Hudson et al.2004).Both are extended overdensities in the large-scale mass distribution of the local Universe and both are thought to contribute significantly to the peculiar motion of the Local Group (LG)(Lucey,Radburn-Smith &Hudson 2005;Kocevski &Ebeling 2006).The relative contri-bution of the Great Attractor and the Shapley supercluster to the motion of the LG,however,remains poorly deter-mined (cf.Erdo˘g du et al.2006;Kocevski &Ebeling 2006)and is still a matter of debate.The Shapley supercluster (SCL)is clearly visible as an overdensity in the distribution of Abell clusters (Scaramella et al.1989;Einasto et al.1997;Proust et al.2006),whereas⋆E-mail:Patrick.Woudt@uct.ac.za the Great Attractor (GA)and its location was identifiedfirst and foremost from the systematic peculiar velocities of galaxies streaming towards this apex (e.g.Lynden-Bell et al.1988),and confirmed later from reconstructed mass-density fields of the local Universe (Dekel 1994;Kolatt,Dekel &Lahav 1995;Erdo˘g du et al.2006).However,no sig-nificant overdensity was obvious in the distribution of galax-ies or Abell clusters at the position of the GA (Lynden-Bell &Lahav 1988).This is not surprising given the location of the GA with respect to the Zone of Avoidance (ZOA).Kolatt et al.(1995)locate the approximate centre of the ex-tended GA overdensity at (ℓ,b,v )∼(320◦,0◦,4000km s −1)based on their smoothed reconstructed mass-density field.The uncertainty in this position is ∼17◦as a result of the applied smoothing of 1200km s −1;this smoothing is neces-sary given the sparsely sampled data (Kolatt et al.1995).Towards such low Galactic latitudes,the view of the extra-galactic light distribution is increasingly reduced by the dust and stars in the Milky Way.As a result,a large part of the GA overdensity is hidden from view by the Milky Way and early attempts to quantify the nature and extent of the GAc0000RAS2Woudt etal.Figure 1.An overview of the large-scale structures in the Great Attractor region between 3000<v hel <7000km s −1.The Norma cluster is located at (ℓ,b,v )=(325.3◦,−7.2◦,4871km s −1).Other major clusters in this vista are the Pavo II cluster at (ℓ,b,v )=(332.3◦,−23.6◦,4167km s −1),the Centaurus cluster at (ℓ,b,v )=(302.4◦,+21.6◦,3418km s −1),the Hydra cluster at (ℓ,b,v )=(269.6◦,+26.5◦,3777km s −1),and the low-latitude CIZA J1324.7–5736and Cen-Crux clusters,at (ℓ,b,v )=(307.4◦,+5.0◦,5700km s −1)and (305◦,+5◦,6214km s −1),respectively.The solid contour marks a line of equal Galactic foreground extinction (A B =3.m 0,from Schlegel,Finkbeiner &Davis 1998).(e.g.Dressler 1988;Hudson 1993a,1993b;Rowan-Robinson et al.1990;Jahoda &Mushotsky 1989)were unsatisfactory:the clear (and significant)mismatch between the inferred mass of the GA and the visible galaxy distribution could not be understood.A deep optical galaxy search at low Galactic latitudes in the GA region (Woudt &Kraan-Korteweg 2001)has lifted part of the veil of the Milky Way.Close to the pre-dicted centre of the GA,the Norma cluster (ACO 3627:Abell,Corwin &Olowin 1989)has been identified as the most massive cluster in the GA region (Kraan-Korteweg et al.1996;Woudt 1998).Abell et al.(1989)classified this cluster as an irregular (I)cluster with Bautz-Morgan type I (Bautz &Morgan 1970).They furthermore classify it as a richness-class 1cluster with ‘59?’galaxies in the magni-tude interval m 3and m 3+2,where m 3corresponds to the magnitude of the third brightest galaxy in the clus-ter.Independently,X-ray observations of the Norma cluster from ROSAT (B¨o hringer et al.1996)and ASCA (Tamura et al.1998)confirm the massive nature of this cluster.From our deep optical galaxy survey in the ZOA in the general GA region,and our follow-up redshift survey (Fairall,Woudt &Kraan-Korteweg 1998;Woudt,Kraan-Korteweg &Fairall 1999;Woudt et al.2004),a clearer view of the obscuredGA overdensity has emerged.The Norma cluster is the cen-tral cluster in a web of connected filaments and wall-like structures (Woudt 1998;Kraan-Korteweg &Lahav 2000;Radburn-Smith et al.2006),analogous to the structures ob-served in and around major mass concentrations in the Λ-CDM (cold dark matter)Millenium simulation (Springel et al.2005).One of the most prominent newly identified structures is a great-wall-like structure with the Norma cluster at its cen-tre which we dubbed the Norma supercluster (Woudt 1998;Fairall et al.1998),a wall of galaxies which runs nearly par-allel to the Galactic Plane (Kraan-Korteweg &Lahav 2000;Radburn-Smith et al.2006)connecting the Pavo II cluster with the Norma cluster and continuing across (and nearly parallel to)the Galactic Plane to the more distant Vela over-density (Kraan-Korteweg,Fairall &Balkowski 1995)via the Cen-Crux cluster (Woudt 1998).Fig.1gives a clear overview of the dominant large-scale structures in the Great Attractor region.Support for the prominence of the Norma SCL has come from various complementary multiwavelength stud-ies at lower Galactic latitudes such as the detection of sev-eral further clusters embedded in the Norma SCL.An X-ray search for highly obscured clusters in the ZOA (Ebel-c0000RAS,MNRAS 000,000–000The Norma Galaxy Cluster (ACO 3627):I.3Figure 2.The distribution in Galactic coordinates of optically-detected galaxies (Woudt &Kraan-Korteweg 2001)around the Norma cluster.The contours indicate lines of equal Galactic reddening (from the DIRBE/IRAS reddening maps,Schlegel et al.1998)of E (B −V )=0.242,0.363and 0.726mag,respec-tively.Assuming a standard Galactic reddening law (Cardelli et al.1989),these values correspond to A B =1.0,1.5and 3.0mag,respectively.The dashed circle marks the Abell radius of the Norma cluster.ing,Mullis &Tully 2002)revealed the second most mas-sive cluster in the Norma SCL,namely CIZA J1324.7–5736.This cluster is ∼50%–70%less massive than the Norma cluster (Radburn-Smith et al.2006)and is located at (ℓ,b,v )∼(307.4◦,+5.0◦,5700km s −1).Deep near-infrared observations (Nagayama et al.2004)furthermore uncov-ered a low-mass cluster around PKS 1343-601at (ℓ,b,v )∼(309.7◦,+1.7◦,3900km s −1),also within the Norma SCL.Apart from this significant collection of clusters,a general overdensity along the Norma SCL is also clearly present in the Parkes deep H I multibeam ZOA survey (Kraan-Korteweg et al.2005).In the first of a series of papers investigating the Norma cluster,we present a detailed dynamical analysis of this clus-ter,the most massive cluster in the Great Attractor over-density,centrally located in a cosmic web of filaments and wall-like structures.Figure 2shows the distribution of the optically-detected galaxies (Woudt &Kraan-Korteweg 2001)in the general di-rection of the Norma cluster,where the Abell radius (R A ≡1.′7/z )of the Norma cluster is indicated by the dashed circle.At the redshift of the Norma cluster (see Sect.3),the Abell radius corresponds to an angular radius of 1.75◦.Assum-ing a Hubble constant of H 0=73km s −1Mpc −1and the cosmological concordance model (assumed throughout this paper),the Abell radius corresponds to a physical size of 2.0Mpc (the cosmology-corrected angular scale at this dis-tance is 1.16Mpc per degree).Contours of equal Galactic foreground extinction,taken from the DIRBE/IRAS Galac-tic reddening map (Schlegel,Finkbeiner &Davis 1998),areoverlayed on the galaxy distribution in Fig.2(A B =1.0,1.5and 3.0mag,respectively;Cardelli,Clayton &Mathis 1989)and show that the Galactic foreground extinction within the Abell radius of the Norma cluster is moderate,A B ≤1.5mag.Within the Abell radius,there are 603optically-detected galaxies with observed diameters in excess of 12′′(Woudt &Kraan-Korteweg 2001)and 219(near-infrared-detected)galaxies in the extended source catalogue (XSC)of the 2Micron All-Sky Survey (2MASS,Skrutskie et al.2006).The 2MASS galaxies in the Norma cluster represent a subset of the 603optically-detected galaxies,although not all the 2MASS galaxies have an optical counterpart;165of the 2192MASS galaxies (75%)were also found by Woudt &Kraan-Korteweg (2001).For the brighter 2MASS galaxies (10′′-aperture K s -band <12.5mag),the overlap between 2MASS and the optical survey is excellent:97%of the 2MASS galax-ies have an optical counterpart.It should be noted that at the position of the Norma cluster (ℓ,b ∼325◦,−7◦)star-crowding is the primary limiting factor,not the Galactic foreground extinction.The star-crowding leaves a Zone of Avoidance imprint on the 2MASS XSC catalogue near the Galactic Bulge (Kraan-Korteweg &Jarrett 2005)and the Norma cluster is located on the edge of this Zone of Avoid-ance.At moderate extinction (A B ≤3mag),but in the pres-ence of severe star-crowding,optical surveys still retrieve the most complete galaxy distribution in the Zone of Avoidance (Kraan-Korteweg &Jarrett 2005).We have obtainted 129new redshifts of galaxies within the Abell radius of the Norma cluster using the 2dF spec-trograph at the Anglo-Australian Observatory.These new observations are presented in Section 2.In Section 3all the redshifts obtained to date are combined and a detailed dy-namical analysis of the cluster based on 296cluster members is presented.In Section 4,we discuss a few individual galax-ies in the Norma cluster of dynamical interest.22DF SPECTROSCOPYSpectra were obtained with the 2dF facility (Lewis et al.2002)on the 3.9m Anglo-Australian Telescope.Full de-tails of the observing 2dF setup used for observations are given in Table 1.As the main objective was to measure the velocity disperisons of the cluster’s early-type galaxies the 1200V gratings were used in each of the 2dF spectro-graphs.These gave a FWHM resolution of ∼125km s −1at Mg b which is sufficient to determine velocity dispersions down to ∼60km s −1.In all,three fibre configurations were observed.Spectra were extracted from the raw data frames,wavelength calibrated and sky-subtracted using the AAO 2dfdr software package †.Redshifts were determined via cross-correlation for the absorption line spectra and/or the direct measurement of emission lines.The 2dF spectroscopic observations focussed on the de-termination of accurate velocity dispersions of early-type galaxies in the Norma cluster for a Fundamental Plane anal-ysis of the cluster.The primary target list therefore con-†.au/2df/software.html#2dfdrc0000RAS,MNRAS 000,000–0004Woudt et al.Table 1.2dF setupused.Figure 3.A comparison of the 2dF velocities with previously determined velocities.sisted of known bright ellipticals in the cluster (Woudt &Kraan-Korteweg 2001).However,we used the spare fibres of the 2dF spectrograph to extend the redshift coverage of the Norma cluster.Galaxies were primarily selected from the optical catalogue of Woudt &Kraan-Korteweg (2001)and the 2MASS XSC,indicated by ‘WKK’and ‘2MASX J’,respectively in Table 2.Additional galaxies were identified on deep R C images taken with the ESO/MPG 2.2-m tele-scope and the Wide Field Imager (see Sect.4).These are identified as ‘ZOA J’in Table 2.Redshifts were obtained for 182galaxies,53of which had a previous measurement.For 76galaxies,multiple mea-surements were obtained to gauge the internal accuracy of the 2dF spectrograph.Table 2shows a representative sam-ple of the results obtained from the 2dF spectroscopy.The full table is available online.Figure 3shows a comparison of the measured 2dF he-liocentric velocities with measurements from the literature.The vast majority of these previous measurements were ob-tained in the course of our ZOA redshift survey (SAAO:Woudt et al.1999;MEFOS:Woudt et al.2004).The overall agreement is very good:v 2dF −v lit =−6±17km s −1with a dispersion of σext ,all =124km s −1(based on 51galaxies).Only one galaxy revealed a discrepent heliocen-tric velocity;for WKK 6329,the 2dF spectroscopy resulted in v =4749±35km s −1as compared to the previously low signal-to-noise value for this galaxy of 2477±250km s −1(Woudt et al.1999).We then compared the 2dF results with a subset of the literature sample,namely those for which redshifts were ob-tained with the MEFOS multi-fibre spectrograph (Woudt et al.2004).This subset has the most accurate redshifts avail-able for the Norma cluster.There are 16galaxies incommonFigure 4.Galaxies with known redshifts as a function of distance to the central cD galaxies (WKK 6269)in the Norma cluster.The Abell radius (R A )is indicated by the vertical dotted line,were the velocity centroid and the upper and lower 3σlimits are indicated by the horizontal dotted line.The E/S0galaxies in the Norma cluster are plotted as filled circles and the S/Irr galaxies in the Norma cluster are shown as crosses.Galaxies deemed non-members are indicated by the open circles.between 2dF and MEFOS (the filled circles in Fig.3).The agreement is again excellent,with a lower rms (σext ,MEFOS =31km s −1)than the previous comparison (which included the SAAO measurements),v 2dF −v MEFOS =+12±8km s −1.Given the primary goal of obtaining accurate velocity dispersions from the 2dF spectroscopy,we have observed a large number of galaxies repeatedly to gauge the internal uncertainty:69galaxies were observed twice and 7galaxies had three independent velocity measurements.For these re-peated observations we find σint =33km s −1over the entire range of observed velocities.This is comparable to the ex-ternal comparison with the MEFOS spectroscopy.Based on these independent evaluations,we have assigned a standard error of 35km s −1to each of the 2dF velocities.3DYNAMICAL ANALYSIS 3.1Cluster membershipWith the new 2dF observations described above,radial ve-locities are now available for 305galaxies within the Abell radius of the Norma cluster for the velocity range 0–9500km s −1.The velocity distribution of these galaxies as a function of distance to the Norma cluster centre is shown in Fig.4.The centre of the cluster was taken as the cD galaxy WKK 6269,a strong wide-angle-tail radio-continuum source (Jones &McAdam 1992,1996)located at the peak in the 0.7–10keV ASCA map of the Norma cluster (fig-ure 1of Tamura et al.1998).The velocity centroid (see Sect.3.2.3)of the Norma cluster is 4871km s −1and is marked by the central horizontal dashed line.The veloc-ity limits for cluster membership are taken as ±3times the velocity scale/dispersion (925km s −1)around the velocity centroid;these limits are shown as the upper and lower hor-izontal dashed line in Fig.4.Nine galaxies are distinct outliers (open circles in Fig.4)and have been rejected from our subsequent analysis.This leaves 296likely cluster members,of which 107have been classified as elliptical or lenticular (E/S0:filled circles inc0000RAS,MNRAS 000,000–000The Norma Galaxy Cluster(ACO3627):I.5 Table2.A representative sample of the results of the2dF spectroscopy.ZOA J16070347-6113587160703.465–611358.7415868WKK5916160750.369–611006.843053WKK5920160752.618–603112.954762WKK5926160808.744–611244.3715856ZOA J16081355-6109377160813.548–610937.6536402MASX J16082135-6044498160821.312–604450.2029813ZOA J16083012-6039511160830.118–603951.086040WKK5958160901.326–605203.7015708WKK5964160906.402–605907.704711ZOA J16091138-6108285160911.377–610828.45156743R A,2N)nearestneighbours,where N represents the total number of galaxiesin the sample;often only the10nearest galaxies are usedin this analysis.These local parameters are then comparedwith the global mean( v )and standard deviation(σ)of allthe galaxies in the sample.For each galaxy,δi is calculatedwhereδi is given byδ2i= N nn+13Abell radius(=0.67Mpc)Figure6shows the optical image of the central34×34arcminof the Norma cluster,obtained with the2.2-m MPG/ESOtelescope at la Silla and the Wide Field Imager during threenights in1999May(see also Section4).The area displayedc 0000RAS,MNRAS000,000–0006Woudt et al.Figure6.An optical colour image of the central∼0.66Mpc×0.66Mpc of the Norma cluster.The white contours show the X-ray subgroup identified from ROSAT observations(reproduced from B¨o hringer et al.1996)overlayed on the optical galaxy distribution.The central cD galaxy(WKK6269)coincides with the peak in the X-ray emission.The inset shows the distribution of the identified galaxies in thisfield of view in equatorial coordinates(E/S0cluster members:red dots,S/Irr cluster members:blue dots,WKK galaxies with no redshift information:large black dots,small galaxies identified from the Wide Field Image data:small black dots).For reference when comparing thisfigure with the distribution in Galactic coordinates,a line of equal Galactic latitude(b=−7.5◦)is drawn as a diagonal dashed line in the inset.in Fig.6corresponds to∼1.6times the core radius(R c;King1966)of the Norma cluster,where R c,opt=10.′4±1.′1(optical:Kraan-Korteweg et al.1996)and R c,X=9.′95±1.′0(X-ray;B¨o hringer et al.1996)for the Norma cluster.Interms of the Abell radius,Fig.6displays the inner∼1The Norma Galaxy Cluster(ACO3627):I.7Figure7.The normalised velocity distribution of cluster mem-bers(cross-hatched histogram)within R<0.67Mpc(upperpanel),R<1.35Mpc(middle panel)and R<2.02Mpc(lowerpanel).In each of the panels,the velocity distribution of the E/S0population is shown by the solid histogram.on the deep R C-band images taken with the Wide Field Im-ager.The central cD galaxy(WKK6269,see the discussionin Sect.3.4on the peculiar velocity of this galaxy)is indi-cated by the black-encircled red dot.Within R≤13Abell radius(=1.35Mpc)The inner23R A-sample,but thisis largely due to an increase in the velocity centroid of theS/Irr population(although the velocity scale of the S/Irrgalaxies has not changed).The values for the biweight ve-locity centroids and scales of the various samples are givenin Table3.c 0000RAS,MNRAS000,000–0008Woudt et al.Table3.An investigation into substructuring in the Norma cluster.R A(=0.67Mpc)23All galaxiesC BI(km s−1)4951±1324962±974979±85S BI(km s−1)964901877N5386107∆∗44.8(75%)83.7(52%)107.7(47%)∆ 50049.684.4106.9Spiral and irregular galaxies∗The percentages given after each value of∆reflect the percentage of Monte Carlo simulations which showed a higher amount of subclustering than the actual observed sample.Percentages below10%indicate significant subclustering.Figure8.Results from the Dressler-Shectmanδ-test:The distribution in Galactic coordinates of galaxies and their measuredδi.The left panels show the results for the inner0.67Mpc(upper panel:E/S0galaxies,lower panel:S/Irr galaxies),the middle panels show the results for the inner1.35Mpc(again separated by morphological classification),and the right panels show the results within the entire Abell radius(2.02Mpc).The symbol sizes are proportional to the value of eδi,where large circles indicate significant deviations from the local mean velocity or local mean velocity dispersion.In the left panels,the location of the X-ray subcluster(B¨o hringer et al.1996) is marked by the solid lines.If present,the arrows in the top right-hand corner of the panels indicate the direction of the position angle of the distribution if significant elongation is detected.c 0000RAS,MNRAS000,000–000The Norma Galaxy Cluster(ACO3627):I.9 The offset in the biweight velocity centroid between theE/S0and S/Irr sample remains albeit slightly lower and is222km s−1with a significance of S V=1.8.The velocitycentroid of the E/S0galaxies has not changed by extendingthe sample to a larger radius(see Table3),although itsbiweight velocity scale is somewhat smaller;S BI=901kms−1for R<23R A.In terms of their spatial distribution,the elliptical andspiral populations now both reveal significant elongation and have position angles of102◦and101◦,respectively,in the equatorial coordinate frame.This corresponds to position angles of146◦and145◦in the Galactic coordinate frame. The latter are again indicated by arrows in the top-right of the middle panels in Fig.8.Theδ-test now clearly reveals substructure in the S/Irr sample(only8%of the MC simu-lations show a larger degree of substructure).Interestingly, theδ-test shows that the E/SO population is completely free of any detectable substructure.In Fig.8,the results from theδ-test of the combined sample out to R<23R A region(inner dashed circle).The centre of Norma A is approximately at right ascension and declination16h12m00s,−61◦04′40′′(J2000.0).Based on thesefive galaxies,wefind a mean velocity of4453km s−1 (which is418km s−1less than the mean of the cluster,cor-responding to411km s−1in the cluster rest frame).Norma A has a velocity dispersion of312km s−1,which is much smaller than the velocity scale of the cluster(925km s−1).A second dynamically distinct group of galaxies (‘Norma B’)is found further from the core of the cluster,centred around WKK5751(other galaxies in-clude WKK5718,WKK5779,WKK5783,WKK5796and WKK5813).This group is indicated by the large solid circle in Fig.9in the region210Woudt etal.Figure 9.The distribution in Galactic coordinates of the 296cluster members within the Abell radius.Encircled dots represent spiral galaxies with δi ≥2.25(based on the δ-test of the entire S/Irr galaxy population).Two spiral-rich subgroups have been identified and are marked by the solid circles.surements:v cD =5448±35km s −1,see Table 2).WKK 6269is also known as PKS B1610-608(one of the 20strongest extragalactic radio sources)and is a textbook example of a wide-angle-tail (WAT)radio galaxy (Jones &McAdam 1992,1996).Such WAT morphology either reflects the motion of the cD galaxy through the cluster and its interaction with the intracluster medium via ram pressure (Owen &Rud-nick 1976),or indicates the presence of a cluster-subcluster merger (Burns 1998).Whether WKK 6269is at rest with respect to the po-tential well of the cluster can be assessed from its peculiar velocity,i.e.the difference between the velocity centroid of the cluster (C BI )and the velocity of the individual cD galaxy (v cD ).The peculiar velocity has to be corrected by a factor (1+z )to ensure the velocity difference is in the cluster rest frame and considered independently for the varying values in the three regarded spheres.In Table 3we also list the values of the peculiar veloc-ity of WKK 6269as determined within the various spheres.It ranges from ∼650km s −1within a radius of 0.67Mpc to ∼550km s −1within 2.02Mpc.The statistical signifi-cance of this peculiar velocity ranges from 6.5to 7.5(S V ≡|v cD −C BI |/N gal )and the small error in the measure-ment of the heliocentric velocity of WKK 6269,makes this large peculiar velocity highly significant.It is exception-ally large when compared to other cD galaxies in clusters (Oegerle &Hill 2001),but not without precedent (Pimbblet et al.2006).Note that the other cD galaxy in the Norma cluster (WKK 6312)has an even larger velocity offset.B¨o hringer et al.(1996)identified an X-ray subgroupclose the centre of the Norma cluster (see Fig.6).This sub-group (dubbed ‘Norma minor’)is fairly massive;Tamura et al.(1998)estimate that the mass of this subgroup could add up to ∼50%to the total mass of the cluster.A comparison of the 843MHz radio continuum emission of PKS B1610-608(Jones &McAdam 1992)with the X-ray contours of this central subgroup (shown in Fig.11)shows that the radio lobes of WKK 6269are closely aligned with the X-ray subgroup.The large observed peculiar velocity of the cD galaxy in the Norma cluster is most likely caused by this ongoing merger.Based on the compactness of the X-ray subcluster,B¨o hringer et al.(1996)argued that the merger has not pro-gressed very far yet,and that most of the main component of the cluster is still undisturbed by the collision.This is consistent with simulations of cluster mergers (Pinkney et al.1996),which show that large peculiar velocities can be reproduced in the event of large-scale mergers at the time of core-crossing.If this merger takes place close to the plane of the sky,it would also explain the non-results of the sta-tistical tests.These are least sensitive to mergers occuring perpendicular to the line-of-sight.Therefore,the X-ray mor-phology –in combination with the large peculiar motion of the central cD galaxy –strongly suggests a recent or com-mencing merger at the core of the cluster.3.5Dynamical mass estimateFor the determination of the dynamical mass of the Normacluster,we have used both the virial theorem (M VT )and the projected mass estimator (M PME ),see equations 21and 22of Pinkney et al.(1996).The use of the bi-weight velocity centroid and scale (Beers et al.1990)in the virial theorem (instead of the velocity mean and standard deviation)leads to a more robust mass estimate (M R VT ).The latter is more robust against the effects of contamination by the inclusion of possible non-members in the analysis.The projected mass estimator (Bird 1995),on the other hand,is sensitive to the presence of (spatially-separated)subclusters due to its pro-portionality to the projected distance between galaxy i and the cluster centroid (R ⊥,i )(see equation 22in Pinkney et al.1996).The presence of a spatially-separated subcluster (e.g.in a premerger configuration)would result in a system-atic offset with respect to the cluster centroid;this leads to larger values of R ⊥,i and thus to a significantly larger mass estimate.For a full discussion of the appropriate use of these dynamical mass estimators we refer to Pinkney et al.(1996)and Bird (1995).The three dynamical mass estimates (M VT ,M R VT and M PME )determined within the three radial limits (using the combined samples of N =129,239and 296galaxies,re-spectively)are given in Table 4.On average,M R VT is ∼5%larger than M VT .The projected mass estimate,however,is generally about 50%larger than M VT and indicates the presence of a spatially-distinct subcluster (projected on the plane of the sky)presumably in the early stages of merging (Pinkney et al.1996).This is consistent with our previous in-dications of subclustering,particularly in the form of Norma minor (the X-ray subgroup).B¨o hringer et al.(1996)and Tamura et al.(1998)both give an estimate of the gravitational mass of the Norma cluster based on ROSAT and ASCA X-ray observations,re-c0000RAS,MNRAS 000,000–000。