Lie–Butcher series, Geometry, Algebra and Computation HansZ.Munthe-KaasandKristofferK.Føllesdal 7 1 0 2 n a J 3 1 Abstract Lie–Butcher (LB) series are formal power series expressed in terms of trees and forests. On the geometric side LB-series generalizes classical B-series ] fromEuclideanspacestoLiegroupsandhomogeneousmanifolds.Onthealgebraic A side,B-seriesarebasedonpre-LiealgebrasandtheButcher-Connes-KreimerHopf N algebra.TheLB-seriesareinsteadbasedonpost-Liealgebrasandtheirenveloping h. algebras. Over the last decade the algebraic theory of LB-series has matured. The t purposeofthispaperistwofold.First,weaimatpresentingthealgebraicstructures a underlyingLBseriesinaconciseandselfcontainedmanner.Secondly,wereviewa m numberofalgebraicoperationsonLB-seriesfoundintheliterature,andreformulate [ theseasrecursiveformulae.Thisispartofanongoingefforttocreateanextensive 1 software library for computations in LB-series and B-series in the programming v languageHaskell. 4 5 6 3 1 Introduction 0 . 1 ClassicalB-seriesareformalpowerseriesexpressedintermsofrootedtrees(con- 0 nectedgraphswithoutanycycleandadesignatednodecalledtheroot).Thetheory 7 1 hasitsoriginsbacktotheworkofArthurCayley[5]inthe1850s,whereherealized : that trees could be used to encode information about differential operators. Being v forgottenforacentury,thetheorywasrevivedthroughtheeffortsofunderstanding i X numericalintegrationalgorithmsbyJohnButcherinthe1960sand’70s[2,3].Ernst r HairerandGerhardWanner[15]coinedthetermB-seriesforaninfiniteformalse- a riesoftheform H.Z.Munthe-Kaas,K.Føllesdal Department of Mathematics, University of Bergen, P.O. Box 7803, N-5020 Bergen, e-mail: [email protected],e-mail:[email protected] 1 2 HansZ.Munthe-KaasandKristofferK.Føllesdal tτ B (α,y,t):=y+ ∑ | | a,τ F (τ)(y), f f σ(τ)h i τ T ∈ wherey Rnisa’base’point, f: Rn Rnisagivenvectorfield,T= , , , ,... ∈ → { } isthesetofrootedtrees, τ isthenumberofnodesinthetree,α: T Raretheco- | | → efficientsofagivenseriesand α,τ Rdenotesevaluationofα atτ.Thebracket h i∈ hintsthatwelaterwanttoconsider α, asalinearfunctionalonthevectorspace h ·i spanned by T. The animal Ff(τ): Rn Rn denotes special vector fields, called → elementarydifferentials,whichcanbeexpressedintermsofpartialderivativesof f. Thecoefficientσ(τ) Niscountingthenumberofsymmetriesinagiventree.This ∈ symmetryfactorcouldhavebeensubsumedintoα,butisexplicitlytakenintothe seriesduetotheunderlyingalgebraicstructures,wherethisfactorcomesnaturally. TheB-seriest B (α,y,t)canbeinterpretedasacurvestartinginy.Bychoosing f 7→ differentfunctionsα,onemayencodeboththeanalyticalsolutionofadifferential equationy(t)= f(y(t))andalsovariousnumericalapproximationsofthesolution. 0 Duringthe1980sand1990sB-seriesevolvedintoanindispensabletoolinanal- ysisofnumericalintegrationfordifferentialequationsevolvingonRn.Inthemid- 1990sinterestroseintheconstructionofnumericalintegrationonLiegroupsand manifolds[18,16],andfromthisaneedtointerpretB-seriestypeexpansionsina differentialgeometriccontext,givingbirthtoLie–Butcherseries(LB-series),which combinesB-serieswithLieseriesonmanifolds.Itisnecessarytomakesomemodi- ficationstothedefinitionoftheseriestobeinterpretedgeometricallyonmanifolds: Wecannotaddapointandatangentvectorasiny+F (τ).Furthermore,itturns f • outtobeveryusefultoregardtheseriesasaTaylor-typeseriesforthemapping f B , rather than a series development of a curvet B (a,y,t). The target f f 7→ 7→ spaceof f B isdifferentialoperators,andwecanremoveexplicitreference f 7→ tothebasepointyfromtheseries. Themapping f B inputsavectorfieldandoutputsaserieswhichmayrep- f • 7→ resent either a vector field or a solution operator (flow map). Flow maps are expressedasaseriesinhigherorderdifferentialoperators.Wewillseethattrees encode first order differential operators. Higher order differential operators are encodedbyproductsoftrees,calledforests.Wewanttoalsoconsiderseriesin linearcombinationsofforests. We will in the sequel see that the elementary differential map τ F (τ) is a f • 7→ universalarrowinaparticulartypeofalgebras.Theexistenceofsuchauniquely definedmapexpressesthefactthatthevectorspacespannedbytrees(withcer- tainalgebraicoperations)isauniversalobjectinthiscategoryofalgebras.Thus thetreesencodefaithfullythegivenalgebraicstructure.Wewillseethatthealge- bracomesnaturallyfromthegeometricpropertiesagivenconnection(covariant derivation) on the manifold. For Lie groups the algebra of the natural connec- tion is encoded by ordered rooted trees, where the ordering of the branches is important.Theorderingisrelatedtoanon-vanishingtorsionoftheconnection. Lie–Butcherseries 3 The symmetry factor σ(τ) in the classical B-series is related to the fact that • severaldifferentorderedtreescorrespondtothesameunorderedtree.Thisfactor isabsentintheLie–Butcherseries. The time parameter t is not essential for the algebraic properties of the series. • SinceFtf(τ)=t|τ|Ff(τ),wecanrecoverthetimefactorthroughthesubstitution f tf. 7→ WearriveatthedefinitionofanabstractLie–Butcherseriessimplyas ∑ α,ω ω, (1) h i ω OF ∈ where OF= I, , , , , , , ,..., , ,... { } denotes the set of all ordered forests of ordered trees, I is the empty forest, and α: OF R are the coefficients of the series. This abstract series can be mapped → downtoaconcretealgebra(e.g.analgebraofdifferentialoperatorsonamanifold) byauniversalmappingω F (ω). f 7→ Wecanidentifythefunctionα: OF Rwithitsseries(1)andsaythataLie– → Butcher series α is an element of the graded dual vector space of the free vector space spanned by the forests of ordered rooted trees. However, to make sense of this statement, we have to attach algebraic and geometric meaning to the vector space of ordered forests. This is precisely explained in the sequel, where we see that the fundamental algebraic structures of this space arise because it is the uni- versalenvelopingalgebraofafreepost-Liealgebra.Hencewearriveattheprecise definition: AnabstractLie–Butcherseriesisanelementofthedualoftheenvelopingalge- braofthefreepost-Liealgebra. Wewillinthispaperpresentthebasicgeometricandalgebraicstructuresbehind LB-seriesinaselfcontainedmanner.Furthermore,animportantgoalforthiswork istoprepareasoftwarepackageforcomputationsonthesestructures.Forthispur- posewehavechosentopresentallthealgebraicoperationsbyrecursiveformulae, ideallysuitedforimplementationinafunctionalprogramminglanguage.Wearein the process of implementing this package in the Haskell programming language. The implementation is still at a quite early stage, so a detailed presentation of the implementationwillbereportedlater. 2 GeometryofLie–Butcherseries B-seriesandLB-seriescanbothbeviewedasseriesexpansionsinaconnectionon afibrebundle,whereB-seriesarederivedfromthecanonical(flatandtorsionfree) connection on Rn and LB-series from a flat connection with constant torsion on a 4 HansZ.Munthe-KaasandKristofferK.Føllesdal fibre bundle. Rather than pursuing this idea in an abstract general form, we will provideinsightthroughthediscussionofconcreteandimportantexamples. 2.1 Paralleltransport LetMbeamanifold,F(M)thesetofsmoothR-valuedscalarfunctionsandX(M) thesetofrealvectorfieldsonM.Fort Rand f X(M)letΨt,f: M Mdenote ∈ ∈ → thesolutionoperatorsuchthatthedifferentialequationγ (t)= f(γ(t)),γ(0)=p 0 ∈ M has solution γ(t)=Ψt,f(p). For φ F(M) we define pullback along the flow ∈ Ψt∗,f: F(M)→F(M)as Ψt∗,fφ =φ◦Ψt,f. Thedirectionalderivative f(φ) F(M)isdefinedas ∈ d f(φ)= dt Ψt∗,fφ. (cid:12)t=0 (cid:12) (cid:12) Through this, we identify X(M) with the(cid:12)first order derivations of F(M), and we obtainhigherorderderivationsbyiterating,i.e. f f isthesecondorderderivation ∗ f f(φ):= f(f(φ)). With Iφ =φ being the 0-order identity operator, the set of ∗ all higher order differential operators on F(M) is called the universal enveloping algebraU(X(M)). This is an algebra with an associative product . The pullback ∗ satisfies ∂ ∂tΨt∗,fφ =Ψt∗,ff(φ). By iteration we find that dn Ψ φ = f(f( f(φ)))= f n(φ) and hence the dtn t=0 t∗,f ··· ∗ Taylorexpansionofthepullba(cid:12)ckis (cid:12) (cid:12) t2 Ψt∗,fφ =φ+tf(φ)+2!f∗f(φ)+···=exp∗(tf)(φ), (2) wherewedefinetheexponentialas ∞ tj exp∗(tf):= ∑ f∗j. j! j=0 ThisexponentialisanelementofU(X(M)),ormorecorrectly,sinceitisaninfinite series,inthecompletionofthisalgebra.WecanrecovertheflowΨt,f fromexp∗(tf) by letting φ be the coordinate maps. However, some caution must be exercised, since pullbacks compose contravariantly Ψt,f◦Ψs,g ∗ =Ψs∗,g◦Ψt∗,f, we have that exp∗(sg) exp∗(tf)correspondstothediffeomorphismΨt,f Ψs,g. ∗ (cid:0) (cid:1) ◦ Numericalintegratorsareconstructedbysamplingavectorfieldinpointsneara basepoint.Tounderstandthisprocess,weneedtotransportvectorfields.Pullback ofvectorfieldsis,however,lesscanonicalthanofscalarfunctions.Thedifferential Lie–Butcherseries 5 geometricconceptofparalleltransportofvectorsisdefinedintermsofaconnection. AnaffineconnectionisaZ-bilinearmappingB: X(M) X(M) X(M)suchthat × → (φf)Bg=φ(fBg) fB(φg)= f(φ)g+φfBg forall f,g X(M)andφ F(M).Notethatthestandardnotationforaconnection ∈ ∈ indifferentialgeometryisis∇fg fBg.Ournotationischosentoemphasisethe ≡ operationasabinaryproductonthesetofvectorfields.Thetrianglenotationlooks nicer when we iterate, such as in (3) below. Furthermore, the triangle notation is alsostandardinmuchofthealgebraicliteratureonpre-Liealgebras,aswellasin severalrecentworksonpost-Liealgebras. Thereisanintimaterelationshipbetweenconnectionsandtheconceptofparal- leltransport.Foracurveγ(t) M,letΓ(γ)t denoteparalleltransportalongγ(t), ∈ s meaningthat Γ(γ)t: TM TM isalinearisomorphismofthetangentspaces. • s γ(s)→ γ(t) Γ(γ)s=Id,theidentitymap. • s Γ(γ)u Γ(γ)t =Γ(γ)u. • t ◦ s s Γ dependssmoothlyons,t andγ. • FromΓ,letusconsidertheactionofparalleltransportpullbackofvectorfields,for t∈Rand f ∈X(M)wedenoteΨt∗,f: X(M)→X(M)theoperation Ψt∗,fg(p):=Γ(γ)t0g(γ(t)), forthecurveγ(t)=Ψt,f(p). Anyconnectioncanbeobtainedfromaparalleltransportastherateofchangeofthe paralleltransportpullback.ForagivenΓ wecandefineacorrespondingconnection as d fBg:= dt Ψt∗,fg. (cid:12)t=0 (cid:12) Conversely, we can recoverΓ from B by(cid:12)solving a differential equation. We seek (cid:12) a power series expansion of the parallel transport pullback. Just like the case of scalars,itholdsalsoforpullbackofvectorfieldsthat ∂ ∂tΨt∗,fg=Ψt∗,ffBg, henceweobtainthefollowingTaylorseriesofthepullback t2 t3 Ψt∗,fg=g+tfBg+ 2 fB(fBg))+3!fB(fB(fBg)))+···. (3) Recallthatinthecaseofpullbackofascalarfunction,weused f(g(φ))=(f g)(φ) ∗ toexpressthepull-backintermsofexp (tf).Whetherornotwecandosimilarlyfor ∗ vectorfieldsdependsongeometricpropertiesoftheconnection.Wewouldliketo extendBfromX(M)toU(X(M))suchthat fB(gBh)=(f g)Bhandhence(3) ∗ 6 HansZ.Munthe-KaasandKristofferK.Føllesdal becomesΨt∗,fg=exp∗(tf)Bg. However, this requires that f B(gBh)−gB(f B h)= f,g Bh,where f,g := f g g f istheJacobibracketofvectorfields. ∗ − ∗ ThecurvaturetensoroftheconnectionR: X(M) X(M) End(X(M))isdefined ∧ → as J K J K R(f,g)h:= fB(gBh) gB(fBh) f,g Bh. − − Thus,weonlyexpecttofindasuitableextensionofBtoU(X(M))ifBisflat,i.e. whenR=0. J K In addition to the curvature, the other important tensor related to a connection isthetorsion.GivenB,wedefineanF(M)-bilinearmapping : X(M) X(M) · × → U(X(M))as f g:= f g fBg. (4) · ∗ − Theskew-symmetrisationofthisproductcalledthetorsion T(f,g):=g f f g X(M), · − · ∈ andif f g=g f wesaythatBistorsionfree. · · The standard connection on Rn is flat and torsion free. In this case the algebra X(M),B forms a pre-Lie algebra (defined below). This gives rise to classical { } B-series. More generally, transport by left or right multiplication on a Lie group yieldsaflatconnectionwheretheproduct isassociative,butnotcommutative.The · resulting algebra is called post-Lie and the series are called Lie–Butcher series. A thirdimportantexampleistheLevi–Civitaconnectiononasymmetricspace,where isaJordanproduct,T =0andRisconstant,non-zero.Thisthirdcaseisthesubject · offorthcomingpapers,butwillnotbediscussedhere. 2.2 TheflatCartanconnectiononaLiegroup LetGbeaLiegroupwithLiealgebrag.ForV gandp GweletVg:=TR V p ∈ ∈ ∈ T G. There is a 1–1 correspondence between functions f C∞(G,g) and vector p ∈ fieldsξ X(G)givenasξ (p)= f(p)p.Leftmultiplicationwithq Ggivesrise f f ∈ ∈ toaparalleltransport Γ : T G T G:Vp Vqp. q p qp → 7→ Thistransportisindependentofthepathbetween pandqpandhencegivesriseto a flat connection. We express the corresponding parallel transport pullback on the spaceC∞(G,g)as (Γq∗f)(p)= f(qp) whichyieldstheflatconnection d (fBg)(q)= g(exp(tf(q))q). dt (cid:12)t=0 (cid:12) (cid:12) Thetorsionisgivenas[22] (cid:12) Lie–Butcherseries 7 T(f,g)(p)= [f(p),g(p)] . g − Thetwooperations fBgand[f,g]:= T(f,g)turnC∞(G,g)intoapost-Liealge- − bra,seeDefinition3below.ThisisthefoundationofLie–Butcherseries. WecanalternativelyexpresstheconnectionandtorsiononX(G)viaabasis E j { } forg.Let∂ X(G)betherightinvariantvectorfield∂ (p)=E p.ForF,G X(G), j j j ∈ ∈ whereF= fi∂,G=gj∂ and fi,gj F(G),wehave i j ∈ FBG= fi∂i(gj)∂j F G= figj∂∂ i j · T(F,G)= figj(∂∂ ∂ ∂). i j j i − WereturntoBdefinedonC∞(G,g).LetU(g)bethespanofthebasis{Ej1Ej2···Ejk}, whereE E E U(g)correspondstotherightinvariantk-thorderdifferential j1 j2··· jk ∈ operator∂ ∂ ∂ U(X(G)).OnU(g)wehavetwodifferentassociativeprod- jk··· j2 j1 ∈ ucts, the composition of differential operators f g and the ’concatenation prod- ∗ uct’ f g = f g f Bg which is computed as the concatenation of the basis, · ∗ − fiE gjE = figjEE . The general relationship between these two products and i j i j · BextendedtoU(g)isgivenin(28)–(31)below.Inparticularwehave fB(gBh)=(f g)Bh, ∗ whichyieldstheexponentialformoftheparalleltransport Ψt∗,fg=exp∗(tf)Bg, whereexp (tf)isgivingustheexactflowof f. ∗ Wecanalsoformtheexponentialwithrespecttotheotherproduct, t2 t3 exp·(tf)=I+tf+ f f+ f f f+ . 2 · 3! · · ··· Whatisthegeometricmeaningofthis?Wesaythatavectorfieldgisparallelalong f iftheparalleltransportpullbackofgalongtheflowof f isconstant,andwesay that g is absolutely parallel if it is constant under any parallel transport. Infinites- imally we have that g is parallel along f if f Bg=0 and g is absolutely parallel if f Bg=0 for all f. InC∞(G,g) the absolutely parallel functions are constants g(p)=V, which correspond to right invariant vector fields ξ X(G) given as g ∈ ξ (p)=Vp.Theflowofparallelvectorfieldsarethegeodesicsoftheconnection. g Ifgisabsolutelyparallel,wehaveg g=g g+gBg=g g,andmoregenerally ∗ · · gn =gn,henceexp (g)=exp(g).If f(p)=g(p)atapoint p G,thentheyde- ∗ · ∗ · ∈ finethesametangentatthepoint.Hence fn(p)=gn(p)foralln,andweconclude · · that exp(f)(p)=exp(g)(p)=exp (g)(p). Thus, the concatenation exponential · · ∗ exp(f)ofageneralvectorfield f producestheflowwhichinagivenpointfollows · thegeodesictangentto f atthegivenpoint. 8 HansZ.Munthe-KaasandKristofferK.Føllesdal On a Lie group, we have for two arbitrary vector fields represented by general functions f,g C∞(G,g)that ∈ (exp·(tf)Bg)(p)=g(exp(tf(p))p). (5) 2.3 Numericalintegration Lie–Butcherseriesanditscousinsaregeneralmathematicaltoolswithapplications innumerics,stochasticsandrenormalisation.Theproblemofnumericalintegration onmanifoldsisaparticularapplicationwhichhasbeenanimportantsourceofin- spiration.Wediscussasimpleillustrativeexample. Example1(Lie–trapezoidal method). Consider the classical trapezoidal method. For a differential equation y0(t)= f(y(t)), y(0)=y0 on Rn a step from t =0 to t=hisgivenas h K= (f(y )+f(y )) 0 1 2 y =y +K. 1 0 Consider a curve y(t) G evolving on a Lie group such that y(t)= f(y(t))y(t), 0 ∈ where f C∞(G g)andy(0)=y .IntheLie-trapezoidalintegratorastepfrom 0 ∈ → y toy y(h)isgivenas 0 1 ≈ h K= (f(y )+f(y )) 0 1 2 y =exp (K)y , 1 g 0 whereexp : g GistheclassicalLiegroupexponential.Wecanwritethemethod g → asamappingΦtrap: X(M) Diff(G)fromvectorfieldstodiffeomorphismsonG, → givenintermsofparalleltransportonX(M)as 1 K= (f+exp·(K)Bf) (6) 2 Φtrap(f):=exp·(K). (7) Tosimplify,wehaveremovedthetimesteph,butthiscanberecoveredbythesub- stitution f hf.NotethatwepresentthisasaprocessinU(X(M)),withoutaref- 7→ erencetoagivenbasepointy0.ThemethodcomputesadiffeomorphismΦtrap(f), which can be evaluated on a given base point y . This absence of an explicit base 0 pointfacilitatesaninterpretationofthemethodasaprocessintheenvelopingalge- braofafreepost-Liealgebra,anabstractmodelofU(X(M))tobediscussedinthe sequel. Lie–Butcherseries 9 Abasicproblemofnumericalintegrationistounderstandinwhatsenseanumer- icalmethodΦ(tf)approximatestheexactflowexp (tf).Theorderoftheapproxi- ∗ mationiscomputedbycomparingtheLB-seriesexpansionofΦ(tf)andexp (tf), ∗ andcomparingtowhichorderint thetwoseriesagree. Thebackwarderror ofthemethodisdefinedasamodifiedvectorfield f such h that the exact flowof f interpolates the numericalsolution at integer times1. The h combinatorialdefinitionofthebackwarderroris e e exp∗(fh)=Φ(hf). The backward error is an important teool which yields important structural infor- mation of the numerical flow operator f Φ(hf). The backward error analysis 7→ is fundamental in the study of geometric properties of numerical integration algo- rithms[9,14]. Yetanotherproblemisthenumericaltechniqueofprocessingavectorfield,i.e. we seek a modified vector field f such that Φ(f )=exp (f). An important tool h h ∗ intheanalysisofthistechniqueisthecharacterizationofasubstitutionlaw.What happenstotheseriesexpansionofeΦ(hf)if f isreeplacedbyamodifiedvectorfield f expressedintermsofaseriesexpansioninvolving f? h The purpose of this essay is not to pursue a detailed discussion of numerical aenalysis of integration schemes. Instead we want to introduce the algebraic struc- turesneededtoformalizethestructureoftheseriesexpansions.Inparticularwewill presentrecursiveformulasforthebasicalgebraicoperationssuitableforcomputer implementations. We finally remark that numerical integrators are typically defined as families of mappings, given in terms of unspecified coefficients. For example the Runge– Kutta family of integrators can be defined in terms of real coefficients a s { i,j}i,j=1 and b s as { j}j=1 s Ki=exp·(∑ai,jKj)Bf, fori=1, ,s ··· j=1 s ΦRK(f)=exp·(∑bjKj). j=1 In a computer package for computing with LB-series we want the possibility of computingseriesexpansionsofsuchparametrizedfamilieswithoutspecifyingthe coefficients.Thisisaccomplishedbydefiningthealgebraicstructuresnotoverthe concrete field of real numbers R, but instead allowing this to be replaced by an abstractcommutativringwithunit,suchase.g.theringofallrealpolynomialsin theindeterminates a s and b s . { i,j}i,j=1 { j}j=1 1Technicalissuesaboutdivergenceofthebackwarderrorvectorfieldisdiscussedin[1]. 10 HansZ.Munthe-KaasandKristofferK.Føllesdal 3 AlgebraicstructuresofLie–Butchertheory We give a concise summary of the basic algebraic structures behind Lie–Butcher series. 3.1 Algebras Allvectorspacesweconsiderareoverafield2kofcharacteristic0,e.g.k R,C . ∈{ } Definition1(Algebra). An algebra A, is a vector space A with a k-bilinear { ∗} operation : A A A.A iscalledunitalifithasaunitIsuchthatx I=I x ∗ × → ∗ ∗ forallx A.The(minus-)associatoroftheproductisdefinedas ∈ a (x,y,z):=x (y z) (x y) z. ∗ ∗ ∗ − ∗ ∗ Iftheassociatoris0,thealgebraiscalledassociative. Definition2(Liealgebra).ALie-algebraisanalgebra g,[, ] suchthat { · ·} [x,y]= [y,x] − [[x,y],z]+[[y,z],x]+[[z,x],y]=0. Thebracket[, ]iscalledthecommutatororLiebracket.Anassociativealgebra · · A, giverisetoaLiealgebraLie(A),where[x,y]=x y y x. { ∗} ∗ − ∗ A connection on a fibre bundle which is flat and with constant torsion satisfies the algebraic conditions of a post-Lie algebra [22]. This algebraic structure first appearedinapurelyoperadicsettingin[27]. Definition3(Post-Liealgebra).Apost-Liealgebra P,[, ],B isaLiealgebra { · · } P,[, ] togetherwithabilinearoperationB: P P P suchthat { · ·} × → xB[y,z]=[xBy,z]+[x,yBz] (8) [x,y]Bz=aB(x,y,z)−aB(y,x,z). (9) Apost-LiealgebradefinesarelationshipbetweentwoLiealgebras[22]. Lemma1.Forapost-LiealgebraP thebi-linearoperation x,y =xBy yBx+[x,y] (10) − definesanotherLiebracket. J K Thus,wehavetwoLiealgebrasg= P,[, ] andg= P, , relatedbyB. { · ·} { · · } 2Inthecomputerimplementationswearerelaxingthistoallowkmoregenerallytobeacommu- J K tativering,suchase.g.polynomialsinasetofindeterminates.Inthislattercasethek-vectorspace shouldinsteadbecalledafreek-module.Wewillnotpursuethisdetailinthisexposition.