Comparesion of Functionalization of Multi-Walled Carbon Nano

a rX iv:mat h /52229v1[mat h.CV]11Fe b25APPROXIMATION DES FONCTIONS LISSES SUR CERTAINES LAMINATIONS ROMAIN DUJARDIN R ´e sum ´e .Nous montrons que sur une lamination par surfaces de Riemann localement plong´e e dans C 2,les fonctions C 1sont localement limites uniformes de fonctions C 1de l’espace ambiant,avec contrˆo le L p des d´e riv´e es le long des feuilles.On en d´e duit qu’un courant domin´e par un courant uniform´e ment laminaire dans C 2est uniform´e ment laminaire.Introduction Les laminations par surfaces de Riemann sont des objets devenus usuels en dynamique holomorphe uni-et multi-dimensionnelle.En dimension complexe 2,toute r´e union de graphes disjoints dans un bidisque forme une telle lamination,souvent appel´e e mouvement holomor-phe.Il est bien connu que l’holonomie d’un mouvement holomorphe n’est en g´e n´e ral pas lisse,ce qui est source de difficult´e s.Le probl`e me se pose en particulier pour la d´e termination de la structure locale des “courants structuraux”(Sullivan [Su],voir en particulier la note p.236)ou “courants dirig´e s”(Berndtsson-Sibony [BS]et Fornæss-Sibony [FS])par une telle lamination.Nous montrons ici le th´e or`e me de r´e gularit´e suivant pour une lamination par surfaces de Riemann plong´e e dans un ouvert Ω⊂C 2(Th´e or`e me 2.3).On consultera le paragraphe 2.1pour la notion de fonction C 1sur une lamination.Th´e or`e me 1.Toute fonction de classe C 1sur une telle lamination est limite uniforme de restrictions de fonctions C 1de l’espace ambiant,avec controle L p des d´e riv´e es le long des feuilles.Il est `a noter que ce r´e sultat ne r´e sulte pas directement des th´e or`e mes usuels de r´e gularit´e des hom´e omorphismes quasiconformes,qui ne traitent que des d´e riv´e es transverses aux feuilles (en particulier nous ne savons pas ´e tendre notre th´e or`e me au cas des laminations par hyper-surfaces dans C n ).Le th´e or`e me a une cons´e quence sur la structure locale des courants laminaires (Th´e or`e me1.1)qui ´e tait `a l’origine de ce travail.Nous obtenons en particulier le r´e sultat suivant (Corol-laire 1.2):Th´e or`e me 2.Si T est un courant uniform´e ment laminaire dans Ω⊂C 2et S est un courant positif ferm´e tel que S ≤T ,alors S est uniform´e ment laminaire.Une motivation pour l’´e tude de la r´e gularit´e des laminations par surfaces de Riemann en dimension 2,et des courants qu’elles supportent,est la question suivante,qui demeure toujours ouverte :2ROMAIN DUJARDINExiste t’il une lamination par surfaces de Riemann dans P2munie d’une mesure transverse invariante?La r´e ponse(n´e gative)`a cette question a´e t´e donn´e e dans des cas particuliers:-par Camacho-Lins Neto-Sad[CLS](voir´e galement Deroin[De])dans le cas o`u la lam-ination est subordonn´e e`a un feuilletage holomorphe singulier.Ces auteurs utilisent la construction de Bott d’une connexion plate dans lefibr´e normal`a la lamination,qui doit pour ceciˆe tre de classe C1.-Fornæss-Sibony[FS]d´e montrent un r´e sultat de non existence de telles laminations en supposant soit une hypoth`e se de r´e gularit´e transverse sur la lamination(voir´e galement [HM]dans ce cas),soit une hypoth`e se de r´e gularit´e sur la mesure transverse.Ceci leur permet essentiellement de consid´e rer le produit T∧T,o`u T est le cycle feuillet´e associ´e, pour parvenir`a une contradiction.Dans les deux cas,c’est un probl`e me de r´e gularit´e de l’holonomie qui empˆe che l’extensionau cas g´e n´e ral.Les difficult´e s soulev´e es sont du mˆe me ordre que celles que nous rencontrerons dans cet article.1.Mouvements holomorphes et courants1.1.Mouvements holomorphes.Dans tout l’article,le cadre d’´e tude sera le suivant:onse donne une famille L de graphes Lα={w=ϕα(z)}au dessus du disque unit´e D dans C2, index´e e par un param`e tre transverseα∈τ,o`uτ⊂{0}×D.On supposera de plus que pour toutα∈τet tout z∈D,|ϕα(z)|≤1−ε<1.En passant`a la clˆo ture de la famille de graphes,on pourra supposer queτest ferm´e e.Dans un article classique,Ma˜n´e,Sad et Sullivan[MSS]ont montr´e que l’application d’holonomieh z,z′:({z}×D)∩L→( z′ ×D)∩Lest continue et quasiconforme.Une telle application est en g´e n´e ral appel´e e mouvement famille de graphes L admet ainsi une structure de lamination par surfaces de Riemann.Concluant une longue s´e rie de travaux,S l odkowski[S l]a d´e montr´e qu’un mouve-ment holomorphe d´efini surτadmet toujours une extension`a un mouvement holomorphe deC tout preuve`a notre avis la plus accessible de ce th´e or`e me est celle de Chirka[C].Apr`e s extension,l’holonomie est d´efinie dans C et quasiconforme,elle v´e rifie donc une´e quation de Beltrami∂h0,z∂w,|µz(w)|<1.Il est connu que z→µz(w)est holomorphe,et donc d’apr`e s le lemme de Schwarz,|µz|≤|z|. En particulier le coefficient de Beltramiµz est arbitrairement petit si le temps de vie du mouvement holomorphe est suffisamment court.Une cons´e quence est que l’application d’holonomie est H¨o ld´e rienne.Il est possible de donner une preuve directe du caract`e re H¨o lder de l’holonomie de la fa¸c on suivante:siα,β∈τsont distincts,la fonctionk:s→−log11+|z|≤k(z)1−|z|,APPROXIMATION SUR LES LAMINATIONS3 on d´e duit que2−2|z|1−|z|≤|ϕα(z)−ϕβ(z)|≤22|z|1+|z|.Un corollaire est que la constante de H¨o lder peutˆe tre choisie arbitrairement proche de1, quitte`a r´e duire le disque de base.1.2.Courants laminaires et dirig´e s.Siµest une mesure positivefinie surτ,la formuleT= τ[Lα]dµ(α)d´efinit un courant positif ferm´e de masse localementfinie dans D2qui est dit uniform´e ment laminaire et subordonn´e`a la lamination L.On peut d´efinir une autre classe de courants associ´e s`a L.Sullivan[Su]d´efinit les courants structuraux–que suivant Sibony,nous appellerons dirig´e s–comme le cˆo ne convexe ferm´e engendr´e par les courants de la forme iδp(dℓ(p)∧de.On en d´e duit ais´e ment que S= t(p),· fd T (p)=fT. Remarque1.3.Le r´e sultat´e tant de nature locale,le th´e or`e me vaut´e galement pour un courant subordonn´e`a une lamination plong´e e dans une surface complexe.La preuve du th´e or`e me1.1que nous pr´e sentons maintenant est en quelque sorte une version rigoureuse du raisonnement heuristique suivantd(fT)=0=⇒f constante le long des feuilles.Elle repose de mani`e re cruciale sur le th´e or`e me d’approximation2.3.4ROMAIN DUJARDINPreuve du th´e or`e me.On pose S=fT.Soitχune fonction continue sur la transversaleτ.On peut prolongerχen une fonction,toujours not´e eχ,constante le long des feuilles.Lemme1.4.Le courantχS est ferm´e.D´e monstration.Soitφune1-forme test.On veut montrer que χS,dφ =0.Supposons pour un instant que l’holonomie soit lisse de classe C1.Dans ce cas on peut´e crireχS,dφ =− S,dχ∧φ = τ Lαfdχ∧φ dµ(α)=0carχest C1et constante le long des feuilles.Dans le cas g´e n´e ral la fonctionχn’est plus C1,et les calculs pr´e c´e dents n’ont plus de sens. D’apr`e s le th´e or`e me2.3,pour tout1≤p<∞χest limite uniforme d’une suite de fonctions lissesχn de sorte que les d´e riv´e es deχn le long de Lαtendent vers0dans L p,uniform´e ment enα.On choisit p>1+εε. Le th´e or`e me est alors une cons´e quence directe de la caract´e risation suivante des courants uniform´e ment laminaires.Lemme 1.5.Soit S un courant positif ferm´e port´e par L,tel que pour toute fonctionχcontinue et constante le long des feuilles,χS soit ferm´e.Alors S est uniform´e ment laminaire. D´e monstration.On veut montrer que le courant S est dans le cˆo ne convexe ferm´e engendr´e par les[Lα].Supposons le contraire.D’apr`e s le th´e or`e me de Hahn-Banach,il existe une forme testφtelle que [Lα],φ <0pour toutαet S,φ >0.On consid`e re alors un recouvrement fini deτpar des ouverts U i de diam`e tre inf´e rieur`a1100.En appliquantit´e rativement le proc´e d´e,on obtient une suite de courants S k positifs ferm´e s de masse1, dont les supports convergent vers une certaine feuille L.Ainsi S k→c[L],avec c>0.Mais S k,φ >0alors que [L],φ <0,ce qui est contradictoire. Remarque1.6.Le lemme1.5a pour cons´e quence directe un r´e sultat dˆu`a N.Sibony[Si]: si L est une lamination transversalement totalement discontinue,et T est un courant positif ferm´e port´e par L,alors T est uniform´e ment laminaire.2.Le th´e or`e me d’approximation2.1.Structure C1.Rappelons d’abord ce qu’on entend par fonction de classe C1sur une lamination.Une fonction f sur L sera dite de classe C1(L)si elle est C1le long des feuilles et que les d´e riv´e es de f le long des feuilles sont transversalement continues.On norme l’espace C1(L)de mani`e re naturelle.APPROXIMATION SUR LES LAMINATIONS5 Il convient de remarquer que mˆe me si L est un feuilletage de classe C1,une fonction C1(L) n’est pas n´e cessairement la restriction d’une fonction C1de l’espace ambiant.On a n´e anmoins dans ce cas un r´e sultat d’approximation[De,Lemme3.23],dont nous incluons une preuve succinte pour la commodit´e du lecteur.Lemme2.1.Si L est de classe C1,toute fonction f∈C1(L)est limite dans C1(L)de restrictions de fonctions C1de l’espace ambiant.D´e monstration.Apr`e s redressement par un diff´e omorphisme de classe C1,on peut supposer que L est la lamination naturelle de D×τ.Si g est une fonction continue surτ,on dispose d’un prolongement continu E(g)de g`a C,o`u E est l’op´e rateur de prolongement de Whitney C(τ)→C(C)(voir[St]).En particulier la fonction(z,α)→E(f(z,·))(α)est d´efinie dans D×me E est lin´e aire et continu,le prolongement est C1par rapport `a la variable z.Il ne reste qu’`a convoler par un noyau r´e gularisant de variableαpour obtenir l’approximation voulue. Remarque2.2.Par un argument analogue`a la preuve du Th´e or`e me1.1on montre que si L est une lamination par surfaces de Riemann plong´e e satisfaisant la conclusion du lemme,tout courant positif ferm´e dirig´e par L est uniform´e ment laminaire(voir´e galement[De,Proposition 3.22]). notation L d´e signe toujours une lamination par graphes au dessus du disque unit´e.On´e crira L∈L pour dire que L est une feuille(plaque)de L,et∇L d´e signe le gradient le long de L.Pour p>1on d´e signe par W1,p(L)l’espace de“Sobolev”des fonctions continues sur L dont les d´e riv´e es dans les feuilles sont dans L p,et varient continˆu ment au sens o`u l’application L→∇L f est continue.On le munit de la norme∇L f L p,f W1,p= f L∞+supL∈Lqui en fait un espace de Banach.Le th´e or`e me d’approximation est le suivant.Th´e or`e me2.3.Soit1<p<∞et f∈C1(L).Il existe une suite(f n)de fonctions lisses dans D×C telles quef−f n W1,p−→n→∞sur tout compact.La preuve sera d´e coup´e e en´e tapes et occupera le reste de la section.Localisation.Remarquons d’abord qu’en recollant des approximations locales avec une par-tition de l’unit´e,on peut se contenter de r´e soudre le probl`e me dans un voisinage arbitraire {0}×D(0,r)de{0}×D.En particulier le coefficient de dilatation de l’´e quation de Beltrami (1)ci-dessous pourraˆe tre choisi arbitrairement petit.On renormalise ensuite pour consid´e rer `a nouveau une lamination dans D×C.Extension.Grˆa ce au th´e or`e me de S l odkowski,on´e tend la lamination,toujours not´e e L, `a D×C.On peut supposer que les feuilles sont des droites horizontales hors du bidisque6ROMAIN DUJARDINunit´me au§1.1,l’holonomie entre droites verticales v´e rifie maintenant une´e quation de Beltrami(1)∂h z,z′∂w, µz,z′(w) ≤κ=2r1Dans le cas,suffisant pour le th´e or`e me1.1,o`u f est constante le long des feuilles,l’utilisation du Lemme 2.1est superflue.Il suffit d’approcher uniform´e ment f par des fonctions lisses dans la transversale{0}×C et d’´e tendre les fonctions lisses en constantes le long des feuilles.APPROXIMATION SUR LES LAMINATIONS7 est de classe C1par rapport`a la variable(z,α).Puisque h0,zε(α)=ϕεα(z)est holomorphe en z,il suffit de v´e rifier que les d´e riv´e es dans la direction verticale de h0,zε(α)sont continues par rapport`a(z,α).L’argument est le suivant(voir[CG,pp.22-24]ou[A]):soitµzε(α)le coefficient de Bel-trami associ´e`a h0,zε(α);µzε(α)est C1enαet holomorphe en z.En particulier∂h0,zε(α)=eϕ,o`uϕv´e rifie une´e quation de type Beltrami∂αde variable z,dont le second membre est∂enα.Parpailleurs∂h0,zεen la variable(z,α).∂αConvergence en norme W1,p.Onfixe maintenant une feuille L de L,et on va montrer que ∇L fε−∇L f W1,p→0.Le lecteur v´e rifiera ais´e ment que l’uniformit´e en L est cons´e quence de la preuve.Quitte`a redresser par une application holomorphe,on peut toujours supposer que L= L0=(w=0).On pose fε=fε◦Φ−1ε(et de mˆe me pour f)de sorte que fε(z,α)=fε(z,w). Par d´efinition de fε,on afε(z,0)= fε(z,h z,0ε(z,0))= f(z,h z,0ε(z,0)),car h z,0ε(z,0)est le projet´e du point(z,0)sur{0}×C le long de Lε.Cette projection n’est certainement pas injective,puisque Lεest essentiellement parall`e le`a L.Deux cas se pr´e sentent:soit L0est´e galement une feuille de Lε,auquel cas fε(z,0)=f(z,0) et il n’y a rien`a d´e montrer;dans le cas contraire,les intersections de L et Lεsont transverses sauf en au plus un nombre localementfini de points[BLS,Lemme6.4],donc en un point g´e n´e rique de L,L est une transversale holomorphe locale`a Lε.Le point crucial de la preuve est le lemme suivant.Il est peutˆe tre d´e j`a connu des sp´e cialistes, mais nous ne l’avons pas trouv´e dans la litt´e rature.Rappelons que pour toute paire de droitesverticales{z}×C et{z′}×C,l’holonomie h z,z′εest quasiconforme et de dilatation major´e e parκ.Lemme2.4.Soient D1et D2deux transversales holomorphes locales`a une feuille L0εde Lε. Alors l’holonomieh1,2:D1⊂U1−→U2⊂D2est quasiconforme de dilatation major´e e parκ.En d’autres termes,la dilatation ne d´e pend pas des transversales,pourvu qu’elles soient holomorphes.Preuve du lamination Lεest en fait un feuilletage de classe C1,donc h1,2est diff´e rentiable.Pour simplifier les notations,dans la suite de la preuve nous omettrons les indicesε.Si L est une feuille proche de L0on pose pour i=1,2,p i=(z i,w i)=D i∩L,de sorte que h1,2(p1)=p2.Les droites{z1}×C et{z2}×C sont des transversales respectives en p1et p2,donc au voisinage de p1,on peut d´e composer h1,2en h z2,2◦h z1,z2◦h1,z1o`u h1,z1 (resp.h z2,2)envoie localement D1sur{z1}×C(resp.{z2}×C sur D2).8ROMAIN DUJARDINL’observation2est que h1,z1est diff´e rentiable au point p1,et sa diff´e rentielle est la projectionde T p1D1sur{z1}×C parall`e lement`a T p1L.C’est une application C-lin´e aire,et donc soncoefficient de Beltrami au point p1est nul.Il en va de mˆe me pour h z2,2en p2.Ainsi le coefficient de dilatation de h1,2en p1est major´e parκ. Nous pouvons maintenant terminer la preuve du th´e or`e me.On veut majorer f(z,0)− f(z,h z,0ε(z,0))en norme W1,p.D’apr`e s la premi`e re approximation, f est C1,donc il suffit de montrer que les d´e riv´e es deπε:z→h z,0ε(z,0)sont petites en norme L p.D’apr`e s le lemme,au voisinage de tout point de transversalit´e de L={w=0}et Lε,la projectionπεest un hom´e omorphisme quasiconforme de coefficient de dilatation≤κ.Donc hors d’un ensemble discret de points de L,πεv´e rifie une´e quation de Beltrami(2)∂πε∂z,avec|ν|≤κ.On identifie L et D et on´e tendν`a C par z´e ro.Toutes les solutions de cette´e quation sur D sont de la formeπε=uε◦kεo`u kεest l’unique hom´e omorphisme tangent`a l’infini solution de(2)et uεest holomorphe sur kε(D)[DB].En particulier ∇kε L p≤C o`u C ne d´e pend que deκ,et les modules de continuit´e(H¨o lder)kεet k−1εne d´e pendent´e galement que deκ.Par ailleurs,Lεest proche de L,donc les feuilles de Lεissues de L coupent{0}×C pr`e s de0.En particulier l’image deπε,c’est`a dire celle de uεest de petit diam`e tre.Onfixe un disque l´e g`e rement plus petit D(0,1−δ)⊂⊂D.On a alors l’estimationdist(∂(kε(D(0,1−δ)),∂(kε(D)))≥c(κ)δθ(κ),et donc d’apr`e s la formule de Cauchy,les d´e riv´e es de uεsont uniform´e ment petites sur kε(D(0,1−δ)).On en conclut que les d´e riv´e es de kε=uε◦kεsont petites en norme L p sur tout compact.R´e f´e rences[A]Ahlfors,Lars V.Lectures on quasiconformal mappings.Van Nostrand Co.,Inc.,Toronto,Ont.-NewYork-London1966.[BLS]Bedford,Eric;Lyubich,Mikhail;Smillie,John.Polynomial diffeomorphisms of C2.IV:The measure of maximal entropy and laminar currents.Invent.Math.112(1993),77-125.[BS]Berndtsson,Bo;Sibony,Nessim.The2Ceci reste vrai mˆe me si L n’est pas lisse.APPROXIMATION SUR LES LAMINATIONS9 [MSS]Ma˜n´e,Ricardo;Sad,Paulo;Sullivan,Dennis On the dynamics of rational maps.Ann.Sci.´e coleNorm.Sup.16(1983),193-217.[PSW]Pugh,Charles;Shub,Michael;Wilkinson,Amie.H¨o lder foliations.Duke Math.J.86(1997),517-546. [Si]Sibony,munication personnelle.[S l]S l odkowski,Zbigniew.Holomorphic motions and polynomial hulls.Proc.Amer.Math.Soc.111(1991), 347-355.[St]Stein,Elias M.Singular integrals and differentiability properties of functions.Princeton University Press,Princeton,N.J.1970.[Su]Sullivan,Dennis.Cycles for the dynamical study of foliated manifolds and complex manifolds.Invent.Math.36(1976),225-255.UFR de math´e matiques,Universit´e Paris7,Case7012,2place Jussieu,75251Paris cedex05,France.dujardin@math.jussieu.fr。

FROM USAGE TO GRAMMAR:THE MIND’S RESPONSE TO REPETITIONJ OAN B YBEEUniversity of New MexicoA usage-based view takes grammar to be the cognitive organization of one’s experience withlanguage.Aspects of that experience,for instance,the frequency of use of certain constructionsor particular instances of constructions,have an impact on representation that is evidenced inspeaker knowledge of conventionalized phrases and in language variation and change.It is shownthat particular instances of constructions can acquire their own pragmatic,semantic,and phonolog-ical characteristics.In addition,it is argued that high-frequency instances of constructions undergogrammaticization processes(which produce further change),function as the central members ofcategories formed by constructions,and retain their old forms longer than lower-frequency in-stances under the pressure of newer formations.An exemplar model that accommodates bothphonological and semantic representation is elaborated to describe the data considered.*1.U SAGE-BASED GRAMMAR.The observance of a separation between the use of lan-guage and its internalized structure can be traced back to de Saussure’s well-known distinction between LANGUE and PAROLE(1915[1966]:6–17),which was adhered to by American structuralists and which made its way into generative grammar via Chomsky’s distinction between competence and performance(Chomsky1965).In American struc-turalism and in generative grammar,the goal of studying langue/competence was given highest priority and the study of language use in context has been considered to be less relevant to the understanding of grammar.Other goals for linguistic research which do not isolate the study of language structure from language use,however,have been pursued through the last few decades by a number of functionalist researchers(for instance,Greenberg1966,Givo´n1979,Hopper&Thompson1980,Bybee1985)and more recently by cognitive linguists as well,all working to create a broad research paradigm under the heading of USAGE-BASED THEORY(Barlow&Kemmer2000,Lan-gacker2000,Bybee2001).While all linguists are likely to agree that grammar is the cognitive organization of language,a usage-based theorist would make the more specific proposal that grammar is the cognitive organization of one’s experience with language.As is shown here, certain facets of linguistic experience,such as the frequency of use of particular in-stances of constructions,have an impact on representation that we can see evidenced in various ways,for example,in speakers’recognition of what is conventionalized and what is not,and even more strikingly in the nature of language change.The proposal presented here is that the general cognitive capabilities of the human brain,which allow it to categorize and sort for identity,similarity,and difference,go to work on the language events a person encounters,categorizing and entering in memory these experi-ences.The result is a cognitive representation that can be called a grammar.This grammar,while it may be abstract,since all cognitive categories are,is strongly tied to the experience that a speaker has had with language.In addition to presenting evidence that specific usage events affect representation,I also address the issue of the type of cognitive representation that is necessary to accom-*This article is an expanded version of the Presidential Address of January8,2005,presented at the annual meeting of the LSA in Oakland,California.I am grateful to Sandra Thompson and Rena Torres-Cacoullos for many discussions of the phenomena treated here.In addition,the questions and comments after the presentation in Oakland in2005from Ray Jackendoff,Mark Baker,Larry Horn,and Janet Pierrehumbert stimulated improvements in the article.711712LANGUAGE,VOLUME82,NUMBER4(2006)modate the facts that are brought to light in this usage-based perspective.I argue for morphosyntax,as I have for phonology,that one needs an exemplar representation for language experience,and that constructions provide an appropriate vehicle for this type of representation.2.C ONVERGING TRENDS IN LINGUISTIC THEORY.In recentyears many researchers have moved toward a consideration of the effect that usage might have on representation. One practice that unites many of these researchers is a methodological one:it is common now to address theoretical issues through the examination of bodies of naturally occur-ring language use.This practice has been in place for decades in the work of those who examine the use of grammar in discourse with an eye toward determining how discourse use shapes grammar,notably Givo´n,Thompson,Hopper,and DuBois(e.g. DuBois1985,Givo´n1979,H opper&Thompson1980,Ono etal.2000,Thompson& Hopper2001).In addition,researchers in sociolinguistic variation,such as Labov, Sankoff,and Poplack(bov1972,Poplack2001,Poplack&Tagliamonte1999, 2001,Sankoff&Brown1976),have always relied on natural discourse to study the inherentvariat ion in language use.The importance of usage-and text-based research,always important to traditional historical linguistics,is especially emphasized in functionalist work on grammaticiza-tion,for example,Bybee2003a,b,Hopper&Traugott1993,and Poplack&Tagliamonte 1999.In fact,the study of grammaticization has played a central role in emphasizing the point that both grammatical meaning and grammatical form come into being through repeated instances of language use.This line of research along with the discourse research mentioned above indeed seeks to explain the nature of grammar through an examination of how grammar is created over time,thus setting a higher goal for linguis-tic explanation than that held in more synchronically oriented theory,which requires only that an explanatory theory provide the means for adequate synchronic description (Chomsky1957).Of course,one major impetus for the shift to analysis of natural language use is the recent availability of large electronic corpora and means of accessing particular items and patterns in such corpora.Through the work of corpus linguists,such as John Sinclair (1991),computational linguists,such as Dan Jurafsky and colleagues(e.g.Jurafsky et al.2001,Gregory et al.1999),and those who are proposing probabilistic or stochastic grammar,such as Janet Pierrehumbert(e.g.2001)and Rens Bod(1998),access to the nature and range of experience an average speaker has with language is now within our grasp.Studies of words,phrases,and constructions in such large corpora present a varying topography of distribution and frequency that can be quite different from what our intuitions have suggested.In addition,the use of large corpora for phonetic analysis provides a better understanding of the role of token frequency and specific words and collocations in phonetic variation.At the same time a compatible view of language acquisition has been developing. The uneven distribution of words and constructions in speech to children is mirrored somewhat in the course of acquisition:children often produce their first instances of grammatical constructions only in the context of specific lexical items and later general-ize them to other lexical items,leading eventually to productive use by the child;see work by Tomasello,Lieven,and their colleagues(e.g.Lieven et al.2003,Savage et al.2003,Tomasello2003).3.F INDINGS.As linguists turn their attention to natural language use,they find a fascinating new source of insights about language.One finding that seems to holdFROM USAGE TO GRAMMAR:THE MIND’S RESPONSE TO REPETITION713 across many studies and has captured the interest of researchers is that both written and spoken discourse are characterized by the high use of conventionalized word sequences, which include sequences t hatwe mightcall formulaic language and idioms,butalso conventionalized collocations(sometimes called‘prefabs’;Erman&Warren2000). Idioms are conventionalized word sequences that usually contain ordinary words and predictable morphosyntax,but have extended meaning(usually of a metaphorical na-ture),as in these examples:pull strings,level playing field,too many irons in the fire. Idioms are acknowledged to need lexical representation because of the unpredictable aspects of their meaning,but as Nunberg and colleagues(1994)point out,they are not completely isolated from related words and constructions since many aspects of their meaning and form derive from more general constructions and the meaning of the component words in other contexts.Idioms provide evidence for organized storage in which sequences of words can have lexical representation while still being associated with other occurrences of the same words,as schematized in this diagram from Bybee 11998.F IGURE1.The relation of an idiom to its lexical components.Idioms have a venerable history in linguistic study,but prefabs or collocations have attracted less attention through recent decades(but see Bolinger1961,Pawley&Syder 1983,Sinclair1991,Biber etal.1999,Erman&Warren2000,and Wray2002).Prefabs are word sequences that are conventionalized,but predictable in other ways,for exam-ple,word sequences like prominent role,mixed message,beyond repair,and to need help.In addition,phrasal verbs(finish up,burn down)and verb-preposition pairings (interested in,think of,think about),which are pervasive in English as well as other languages,can be considered prefabs,though in some cases their semantic predictability could be called into question.These conventional collocations occur repeatedly in dis-course and are known to represent the conventional way of expressing certain notions (Erman&Warren2000,Sinclair1991,Wray2002).Erman and Warren(2000)found that what they call prefabricated word combinations constitute about55%of both spoken and written discourse.Speakers recognize prefabs as familiar,which indicates that these sequences of words are stored in memory despite being largely predictable in form and meaning.The line between idiom and prefab is not always clear since many prefabs require a metaphorical stretch for their interpretation.The following may be intermediate exam-ples,where at least one of the words requires a more abstract interpretation:break a habit,change hands,take charge of,give(someone/something)plenty of time,drive 1See Barlow2000for an interesting discussion of the way a conventionalized expression can undergo permutations that demonstrate that its compositionality is also maintained.714LANGUAGE,VOLUME82,NUMBER4(2006)(someone)crazy.I bring up these intermediate cases to demonstrate the gradient nature of these phenomena;the lack of a clear boundary between idioms and prefabs would also suggest that both types of expression are stored in memory.What we see instantiated in language use is not so much abstract structures as specific instances of such structure that are used and reused to create novel utterances.This point has led Hopper(1987)to propose grammar as emergent from experience,mutable, and ever coming into being rather than static,categorical,and fixed.Viewed in this way,language is a complex dynamic system similar to complex systems that have been identified,for instance,in biology(Lindblom et al.1984,Larsen-Freeman1997).It does not have structure a priori,but rather the apparent structure emerges from the repetition of many local events(in this case speech events).I describe here some data that help us understand what some of the properties of an emergent,usage-based gram-mar mightbe.4.G OALS OF THE ARTICLE.There are a number of important consequences of the fact that speakers are familiar with certain multiword units.For the present article I focus on the implications of the fact that the use of language is lexically particular; certain words tend to be used in certain collocations or constructions.My goal is to explore the implications of this fact for cognitive representation.I discuss a series of cases in which there is evidence that lexically particular instances of constructions or word sequences are stored in memory and accessed as a unit.I further discuss facts that show that the frequency of use of such lexically particular collocations must also be a part of the cognitive representation because frequency is a factor in certain types of change.I argue that in order to represent the facts of usage,as well as the facts of change that eventually emerge from this usage,we need to conceive of grammar as based on constructions and as having an exemplar representation in which specific instances of use affect representation.The model to be proposed,then,uses a type of exemplar representation with constructions as the basic unit of morphosyntax(see §§6–9and13).After discussing further aspects of the approach taken,five types of evidence are discussed.First,evidence for the importance of frequency in the developing autonomy of new constructions in grammaticization is presented.Second,I discuss the effects of context and frequency of use on the development of conventionalized collocations and grammatical constructions.Third,I briefly treat phonological reduction in high-frequency phrases.Fourth,I turn to the organization of categories within constructions where it is seen that in some cases high-frequency exemplars serve as the central members of categories.Finally,the fact that high-frequency exemplars of constructions can resist change is taken as evidence that such exemplars have cognitive representation.5.F REQUENCY EFFECTS ON PROCESSING AND STORAGE.Before turning to the evidence,I briefly review three effects of token frequency that have been established in recent literature.First,high-frequency words and phrases undergo phonetic reduction at a faster rate than low-and mid-frequency sequences(Schuchardt1885,Fidelholtz1975,Hooper 1976,Bybee&Scheibman1999,Bybee2000b,2001).This REDUCING EFFECT applies to phrases of extreme high frequency like I don’t know,which shows the highest rate of don’t reduction(Bybee&Scheibman1999),and also to words of all frequency levels undergoing gradual sound change,such as English final t/d deletion or Spanish [L]deletion,both of which affect high-frequency words earlier than low-frequency words(Bybee2001,2002,Gregory et al.1999).The explanation for this effect is that theFROM USAGE TO GRAMMAR:THE MIND’S RESPONSE TO REPETITION715 articulatory representation of words and sequences of words is made up of neuromotor routines.When sequences of neuromotor routines are repeated,their execution becomes more fluent.This increased fluency is the result of the establishment of a new routine, as when a group of words comes to be processed as a single unit(Anderson1993, Boyland1996).In the new routine articulatory gestures reduce and overlap as the routine is repeated.A second effect of token frequency(the CONSERVING EFFECT)relates to the morpho-syntactic structure of a string.High-frequency sequences become more entrenched in their morphosyntactic structure and resist restructuring on the basis of productive pat-terns that might otherwise occur.Thus among English irregular verbs the low-frequency verbs are more likely to regularize(weep,weeped)while the high-frequency verbs maintain their irregularity(keep,kept).My proposal to explain this tendency(Hooper 1976,Bybee1985)is that frequency strengthens the memory representations of words or phrases,making them easier to access whole and thus less likely to be subject to analogical reformation.This effect applies to syntactic sequences as well,allowing higher-frequency exemplars to maintain a more conservative structure(Bybee& Thompson1997).In§15the example of the maintenance of the older type of negation in English(no-negation)with high-frequency constructions is discussed.The third effect(AUTONOMY)is related to the second one.Autonomy refers to the factt hatmorphologically complex forms(or st rings of words)of high frequency can lose their internal structure as they become autonomous from etymologically related forms(Bybee1985).This can be seen for example in the way that words with deriva-tional affixes become less transparently related to their base forms as they become more frequent(Bybee1985,Hay2001).Hay(2001)argues that the semantic opacity of words like dislocate is due to the fact that their complex forms are more frequent than the bases from which they were originally derived.The effect applies to inflection only in cases of extreme high frequency,where it leads to suppletion.Thus went was formerly the past tense of wend but(for unknown reasons)itincreased in frequency and moved away from wend,joining go to become the past tense of that verb.This effect also applies in grammaticization when sequences that are originally complex (such as be going to)lose their semantic and syntactic transparency and move away from other instances of the words be,go,and to.In discussing these effects here and elsewhere,I refer to high and low frequency and to extreme high frequency without specifying exactly what these values mean in numerical terms.Thus,the conserving effect applies to high-frequency items but auton-omy appears to affect only strings of extreme high frequency.The reducing effect appears to be graded in that the higher the frequency of the string,the greater its reduction.The phenomenon discussed in§14in which higher-frequency items form the centers of categories requires that the item not be so high in frequency as to be autonomous.The impossibility at the moment of specifying ranges for extreme high, medium,and low is only a function of the state of our knowledge.As more empirical studies appear,absolute frequency ranges for each phenomena will eventually be speci-fiable.6.C ONSTRUCTION-BASED REPRESENTATIONS.For the phenomena to be examined here, cognitive representations based on constructions turn out to be highly effective.Several versions of grammar in terms of constructions have been discussed in the literature. Proposals have been made by Fillmore and Kay(e.g.Fillmore etal.1988,Kay& Fillmore1999),Goldberg(1995,2003),Lakoff(1987),Langacker(1987),and Croft716LANGUAGE,VOLUME82,NUMBER4(2006)(2001).All of these proposals agree on a basic point:Cognitive representations of grammar are organized into constructions which are partially schematic,conventional-ized sequences of morphemes with a direct semantic representation.According to Goldberg,all of the following constitute constructions:(i)idioms with fixed lexical content:go great guns;(ii)idioms that are partially filled:jogϽsomeone’sϾmemory; (iii)constructions with some fixed material:he made his way through the crowd;and (iv)fully abstract constructions:they gave him an award.It is interesting to note that almost all constructions contain some explicit morphologi-cal material,tying them fairly concretely to specific words or morphemes(e.g.way and the possessive pronoun in(iii)).The ditransitive construction in(iv)contains no specific phonological material that identifies it as the ditransitive.Only the word order signals this.However,it should be noted that only a small class of verbs can occur in this construction so that it also has a grounding in lexical items.In fact,the continuum in(i)through(iv)shows examples from the most lexically explicit to the most schematic.Prefabs can also be considered to be instances of con-structions that are lexically filled.Given the high use of prefabs and idioms in natural speech,it appears that a good deal of production(and perception)refers to sequences of prespecified lexical choices rather than to abstract grammar.For this reason,a model that builds a grammar from specific instances of language use,such as an exemplar model or a connectionist model,seems appropriate.For present purposes,I focus on representation in an exemplar model.7.E XEMPLAR REPRESENTATION.Several versions of exemplar theory have been pro-posed in the psychology literature on categorization(Nosofsky1988,Goldinger1996). The version of exemplar theory adopted here has found its way into linguistics as a means of representing phonetic variation(K.Johnson1997,Pierrehumbert2001,2002). In this model,every token of experience is classified and placed in a vast organizational network as a part of the decoding process.The major idea behind exemplar theory is that the matching process has an effect on the representations themselves;new tokens of experience are not decoded and then discarded,but rather they impact memory representations.In particular,a token of linguistic experience that is identical to an existing exemplar is mapped onto that exemplar,strengthening it.Tokens that are similar but not identical(differing in slight ways in meaning,phonetic shape,pragmat-ics)to existing exemplars are represented as exemplars themselves and are stored near similar exemplars to constitute clusters or categories.Thus the phonetic shape of a word mightconsistof a setof phonet ic exemplars t hatare very similar t o one anot her.2 Exemplar clusters can also be arranged hierarchically.A set of exemplars that are judged to be similar phonetically and represent the same meaning are clustered together and are represented as a word or phrase.Constructions emerge when phrases that bear some formal similarity as well as some semantic coherence are stored close to one another.According to K.Johnson(1997),phonetic exemplars are tagged for an array of information about their occurrence:phonetic context,semantic and pragmatic informa-tion,other linguistic context,and social context.Thus an exemplar is an extremely2In some versions of exemplar representation,exemplars are scattered randomly through space.Only when categorization of a new exemplar is necessary are they organized by similarity(Chandler2002,Skousen 1989).Because linguistic categorization takes place so often I propose that linguistic categories(in the form of phonetic,morphosyntactic,semantic/pragmatic characteristics)are more entrenched in the sense that frequently used categorizations have an impact on neurological organization.FROM USAGE TO GRAMMAR:THE MIND’S RESPONSE TO REPETITION717 complex item.Not only does the phonetic side of a word or phrase take different forms and thus require multiple exemplars,but the semantic and contextual sides also encompass ranges of variation.It is necessary,then,to expand the theory to encompass exemplars that differ by any of the facets of a linguistic sign.That is,the particular semantic interpretations associated with tokens of use also contain details and nuances that lend themselves to exemplar representation.When meanings are the same,they are mapped onto the same exemplar,but when there are differences,separate exemplars are created.Similarly,the memory for contexts of use for words and constructions can be organized in exemplar clusters.These points are further illustrated in§13.In applying this kind of model to linguistic data,it is important to bear in mind just which aspects of the model are helpful to our understanding of language.Thus I focus on the following factors because they help us understand how constructions come into being and change over time.a.Exemplar representations allow specific information about instances of useto be retained in representation.b.Exemplar representations provide a natural way to allow frequency of useto determine the strength of exemplars.c.Exemplar clusters are categories that exhibit prototype effects.They are or-ganized in terms of members that are more or less central to the category,rather than in terms of categorical features.At first it might seem rather implausible to suppose that every token of language use encountered by a speaker/hearer has an effect on cognitive representation.Therefore it is important to consider how these notions can be applied to language.First,we have to move beyond the goal of structural and generative linguistics to try to establish which features or forms are stored in the lexicon and which are not(cf. Langacker’s rule/list fallacy(1987))and take a more probabilistic view of representa-tion.A phrase that is experienced only once by an adult is likely to have only a minute impact on representation compared to all of the accumulated exemplars already existing. Compare this to a young child whose experience is much more limited:each new token of experience has a greater impact on his/her representations.In addition,given a highly organized network of morphemes,words,phrases,and constructions,it will be difficult to distinguish between specific storage(storing the relatively low-frequency phrase such as beige curtains as a unit)and distributed storage(mapping the two words onto existing exemplars of these words)because both types of processing are occurring at the same time.Only when a sequence is repeated will access to it as a unit rather than by its parts become more efficient(Boyland1996,Haiman1994,Hay2001).Thus the question of storage or nonstorage will always be a probabilistic one,based on the experience of the language user.Second,human memory capacity is quite large.Nonlinguistic memories are detailed and extensive,suggesting a strong memory capacity for experienced phenomena(Gol-dinger1996,K.Johnson1997).In particular it is interesting to consider how memory for repeated experiences is represented,because an important feature of linguistic expe-rience is the regular repetition of phonological strings,words,and constructions.Con-sider a repeated experience,such as walking from your office on campus to your classroom.You probably take much the same route each time,and although many of the perceptual details are not important,you register them anyway.Was there a pigeon on the path?Did you see your colleague coming from the other direction?These details are registered in memory so that if they are repeated,it is noted and eventually you718LANGUAGE,VOLUME82,NUMBER4(2006)may come to expect to see a pigeon on the path or your colleague returning from his class.At the same time,memories decay over time.If in fact you never see your colleague again on that path,you may forget the one time you saw him/her there.Both of these properties of memory—the build up of strength in repeated memories and the loss of nonrepeated memories—are important for explaining linguistic phenomena. Third,it is clear that linguistic memories represented as exemplars can undergo considerable reorganization,particularly when change is ongoing in a language.Exam-ples of reorganization of constructions are presented in§§10–12,but for now I mention a phonetic example.Consider the case of a sound change occurring at a word boundary which sets up variation among the forms of a word,such as in the final syllable of a Spanish word ending in[s]in dialects where it is reduced to[h]before a consonant but maintained before a vowel.In a case like this,the phonetically conditioned variation is not maintained.Rather,the more frequent variant,the one used before a consonant, is extended to use before a vowel.This gives evidence for considerable reorganization within the exemplar cluster resulting in a smaller range of variation(Bybee2000a, 2001).(For experiments that model category formation from exemplars,see Pierrehumbert2001,2002,Wedel2004.)The cases I discuss here provide evidence that linguistics needs a model that allows particular instances of use to affect representation.It is important to note that there are several models that have been computationally tested that have this property;for in-stance,connectionist models work on the principle that detailed information is the basis of more general patterns.Such models have been tested on a variety of cognitive phenomena,but it may be that language will present both the toughest testing ground for such theories and the best source of data for our understanding of categorization and memory,especially memory for repeated events.8.E XEMPLAR REPRESENTATION OF CONSTRUCTIONS.As mentioned above,exemplars of words or phrases that are similar on different dimensions are grouped together in cognitive representation.From such a grouping a construction can emerge.For example, an exemplar representation of a partially filled construction would have experienced tokens mapping onto the constant parts of the construction exactly,strengthening these parts,while the open slots would not match exactly.If there are similarities(in particu-lar,semantic similarities)among the items occurring in the open slot,a category for these items would begin to develop.Thus in a Spanish example that I discuss further in§14,adjectives following the verb quedarse‘to become’,such as quieto,tranquilo, or inmo´vil,would be categorized together due to their semantic coherence,and this category would then predict novel uses of quedarseםadjective(Bybee&Eddington 2006).(1)Example:Spanish quedarseםADJECTIVE‘become ADJ’quedarse͕tranquilo‘tranquil’quieto‘quiet,still’inmo´vil‘still,immobile’͖Further details about the mapping of form and meaning in constructions is presented in§13.9.E FFECTS OF REPETITION ON PARTICULAR INSTANCES OF CONSTRUCTIONS.The next three sections of this article contain a discussion of the effects of repetition and context of use on particular exemplars of constructions.The examples provided are intended as empirical evidence that specific instances(exemplars)of constructions are part of the cognitive representation of language and that frequency of use has an impact on。

Altered Resting Brain Function and Structurein Professional Badminton PlayersXin Di,1,2Senhua Zhu,2,3Hua Jin,1Pin Wang,1Zhuoer Ye,1Ke Zhou,4Yan Zhuo,4and Hengyi Rao 1–3AbstractNeuroimaging studies of professional athletic or musical training have demonstrated considerable practice-dependent plasticity in various brain structures,which may reflect distinct training demands.In the present study,structural and functional brain alterations were examined in professional badminton players and com-pared with healthy controls using magnetic resonance imaging (MRI)and resting-state functional MRI.Gray mat-ter concentration (GMC)was assessed using voxel-based morphometry (VBM),and resting-brain functions were measured by amplitude of low-frequency fluctuation (ALFF)and seed-based functional connectivity.Results showed that the athlete group had greater GMC and ALFF in the right and medial cerebellar regions,respectively.The athlete group also demonstrated smaller ALFF in the left superior parietal lobule and altered functional con-nectivity between the left superior parietal and frontal regions.These findings indicate that badminton expertise is associated with not only plastic structural changes in terms of enlarged gray matter density in the cerebellum,but also functional alterations in fronto-parietal connectivity.Such structural and functional alterations may reflect specific experiences of badminton training and practice,including high-capacity visuo-spatial processing and hand-eye coordination in addition to refined motor skills.Key words:amplitude of low frequency fluctuation;badminton athlete;cerebellum,fronto-parietal network;func-tional connectivity;voxel-based morphometryIntroductionThe human brain shows remarkable plasticity in re-sponse to learning and experience,even after a relatively short time of training.For example,after 7days of juggling training,significant changes in gray matter (GM)intensity could be detected in multiple brain regions,including the fronto-parietal attention network,the middle temporal (MT)area,and the hippocampus (Boyke et al.,2008;Dragan-ski et al.,2004;Driemeyer et al.,2008).Skilled professionals with multiple years of training and practice,such as athletes and musicians,have also demonstrated significant changes in multiple brain regions (for reviews,see Nakata et al.,2010;Schlaug,2001;Yarrow et al.,2009).For example,the cerebel-lum has been shown to be larger in musicians (Gaser and Schlaug,2003;Han et al.,2009;Hutchinson et al.,2003)and basketball players (Park et al.,2009)compared with normal controls,which may reflect refined motor skills in these pro-fessionals.Additional changes in multiple cortical areas in-volving motor,auditory,and visual-spatial processes have been observed in keyboard musicians (Gaser and Schlaug,2003;Han et al.,2009),which may be associated with more subtle and complex patterns of finger dexterity in music prac-tice.GM alterations in sensorimotor and premotor regions have also been revealed in professional divers (Wei et al.,2009)and ballet dancers (Ha¨nggi et al.,2010),while addi-tional parietal changes have been found in skilled golfers,which may reflect more attention and motor control support-ing golf practice (Ja¨ncke et al.,2009).These diverse findings from athletes and musicians suggest that brain structure changes observed in professionals may depend on the spe-cific motor and cognitive processes involved in skill training over time.Compared with structural changes in skilled professionals,evidence supporting functional brain alterations in elite ath-letes or musicians is relatively ing functional mag-netic resonance imaging (fMRI),two recent studies examined neural activations during preshot routines of expert golfers1Center for Studies of Psychological Application,South China Normal University,Guangzhou,China.2Department of Psychology,Sun Yat-Sen University,Guangzhou,China.3Department of Neurology,Center for Functional Neuroimaging,University of Pennsylvania,Philadelphia,Pennsylvania.4State Key Laboratory of Brain and Cognitive Science,Institute of biophysics,Chinese Academy of Sciences,Beijing,China.BRAIN CONNECTIVITY Volume 2,Number 4,2012ªMary Ann Liebert,Inc.DOI:10.1089/brain.2011.0050225(Milton et al.,2007)and archers(Kim et al.,2008).The results demonstrated that professionals tended to recruit a more fo-cused and efficient organization of visual and motor net-works directly related to task requirements,whereas novices tended to engage more brain regions related to high-level cognitive control functions.However,findings from these task-related fMRI studies depend heavily on the task demands and specific cognitive or motor paradigms used in the studies,which makes the comparison of results across studies difficult.The use of resting-state functional connectivity fMRI(fc-fMRI)that examines functional brain changes have been emerging since thefirst report of functional connectivity be-tween bilateral motor cortices(Biswal et al.,1995).Resting-state fc-fMRI measures the synchronization of low-frequency blood oxygen level dependent(BOLD)oscillations between spatially distinct brain regions in the absence of tasks.There-fore,it provides a complimentary method to the conventional task-related fMRI for detecting functional brain changes asso-ciated with training and experience.Since resting-state fc-fMRI does not involve any particular motor or cognitive task re-quirements,it may reflect a cumulative effect of experience over time(Lewis et al.,2009).Recent fc-fMRI studies have sug-gested that motor training altered resting-state brain activity (Xiong et al.,2009;Albert et al.,2009).For example,Albert et al.(2009)examined resting-state functional brain changes during motor training using independent component analysis (ICA)and found that motor learning,but not motor perfor-mance,significantly modulated resting-state functional con-nectivity in the fronto-parietal and cerebellar networks. However,to our knowledge,the effects of long-term training on resting-state brain function remain largely unknown.In this study,we used MRI to examine both structural and resting-state functional brain alterations in professional rac-quet players,specifically badminton players.To the best of our knowledge,structural and functional characteristics of the badminton players’brain have not been explored using neuroimaging methods.Similar to other racquet sports,bad-minton requires refined hand-eye coordination and visuo-spatial ability(Ward et al.,2002;Shim et al.,2005),which may result in structural and functional brain changes that are different from other sports that have been studied,for ex-ample,golf,diving,or basketball.We predict that profes-sional badminton players will show significant structural and functional alterations in brain regions related to visuo-spatial processing and hand-eye coordination.A group of professional badminton players and matched nonathlete con-trols were recruited in the present study.Standard voxel-based morphometry(VBM)(Ashburner and Friston,2000) wasfirst conducted to examine the structural differences be-tween the two groups.The amplitudes of low-frequencyfluc-tuations(ALFF;Zang et al.,2007)of resting-state fMRI were then used to measure regional properties of the brain’s intrin-sic neural activity and compared between the two groups. Finally,the seed-based functional connectivity analyses(Fox et al.,2005)of resting-state fMRI were conducted to character-ize functional integration,using regions that have been shown to be different between professional players and controls from the VBM and ALFF analyses.We chose to use the hypothesis-driven,seed-based functional connectivity analysis instead of data-driven methods such as ICA in order to determine the specific brain regions and networks that are related to culti-vated visuo-spatial processing and hand-eye coordination skills after long-term badminton training and practice. MethodsSubjectsTwenty badminton players(10males)were recruited for the study.All subjects were members of professional teams or professional university teams in Beijing and had completed at least3years of badminton training(mean=8.9years,range from3to16years).Subjects had either completed college-level education or were current college students during the study.The control group consisted of18age-and gender-matched college students(9men).No control subjects had formal training or experience in badminton or other racquet sports(Table1).All athletes and controls were right handed. Written informed consent was obtained from all subjects at the beginning of the study.MRI scanAll subjects were scanned using a Siemens3T Trio scanner located in the Beijing Center for Magnetic Resonance Imaging, Beijing,China.Subjectsfirst completed a task-independent, resting-state scan,during which time they were asked to open their eyes to look at a blank screen.No additional task was required.The resting scan lasted6min,with a TR of 2sec.As a result,180images were obtained for each subject. BOLD images were obtained using an echo-planar imaging se-quence with a12-channel coil.Acquisition parameters were as follows:TE=30ms;TR=2000ms;flip angle=90o;32inter-leaved axial slices;slice thickness=4mm,no gap;image ma-trix=64·64;FOV=200·200mm;bandwidth=2232.At the end of the scan,a high spatial resolution,T1-weighted magne-tization prepared rapid gradient-echo(MP-RAGE)structural image was obtained.Scan parameters were as follows:TR= 2530ms;TE=1.64;flip angle=7.0;176slices;slice thick-ness=1mm;image matrix=256·256;FOV=256·256mm. VBM analysisStandard VBM analysis was conducted using SPM8soft-ware(www.fi/spm/,Wellcome Department of Cognitive Neurology,UK,implemented in Matlab 7.7,Math Works,Natick,MA).For each subject,the high-resolution structural image wasfirst segmented into different brain tissue types using a standard template provided in the SPM software.The unified segmentation model implemented in SPM8(Ashburner and Friston,2005)was used to segment the structural image into GM,white matter(WM),and cere-brospinalfluid(CSF)in standard Montreal Neurological Institute(MNI)space.The tissue probability template was the modified version of the International Consortium for Brain Mapping tissue probabilistic atlases,which was rmation for the Two Groups of SubjectsAthletes Controls Statistics No.of subjects(male)20(10)18(9)Year of age(STD)22.5(4.57)20.7(4.25)p=0.234 Years of badmintontraining8.85(3.25)0226DI ET AL.provided by the SPM8new segment routine.We used the normalized GM images that were not modulated and repre-sented gray matter concentration (GMC).For computational simplicity,the image was re-sampled at a resolution of 1.5·1.5·1.5mm 3in the new segmentation step.The result-ing images were visually checked to make sure there were no misclassifications.Finally,the GMC images were smoothed using a Gaussian filter with a 6-mm full width at half maximum (FWHM)kernal.Preprocessed GMC images were then entered into a sec-ond-level,two-sample t-test model to compare group level differences.Total Gray matter volume (GMV),gender,and age were added to the model as covariates.Since the smooth-ness of GMC images is nonstationary (Ashburner and Friston,2000),nonstationary cluster extent correction was adopted to correct multiple comparisons (Worsley et al.,1999).The am-plitude threshold was first set at p <0.001,and then,the clus-ter extent was corrected at p <0.05using the nonstationarity toolbox (Hayasaka et al.,2004).The clusters identified in the group analysis were labeled using the Talairach Daemon (Talairach and Tournoux,1988;Lancaster et al.,2000)after ac-counting for the discrepancy between MNI space and Talair-ach space (Lancaster et al.,2007).This analysis revealed two clusters in the right cerebellum that were defined as the seed regions for further functional connectivity analysis.Resting-state fMRI data preprocessingPreprocessing of resting-state functional images was com-pleted using SPM8.First,the two initial resting-state images were discarded for each subject,leaving a total of 178images per subject.All functional images were motion corrected using the realign function.Functional images from two athletes and two controls were discarded,because at least one of the motion parameter estimates was larger than 3mm or 3de-grees.As a result,images from 18athletes and 16controls were included in the subsequent analysis.For each subject,the functional images were first registered to the subject’s own high-resolution structural image.The structural image was then normalized to the T1template.Next,all functional images were normalized to the standard MNI space using the parameters obtained from normalizing the subject’s struc-tural images.Finally,all normalized functional images were smoothed using a Gaussian filter with an 8mm FWHM kernal.ALFF analysisWe used the ALFF to index regional properties of the brain’s intrinsic neural activity.ALFF analysis was completed using the resting-state fMRI data analysis toolkit,(REST)v1.4(/forum/REST_V1.4,State Key Laboratory of Cognitive Neuroscience and Learning,Beijing,China;Song et al.,2011).For each subject,linear trends were removed from all the functional images.An ideal band-pass filter be-tween 0.01and 0.08Hz was used to filter the detrended func-tional images.An ALFF map was then calculated for each subject.A normalized ALFF map (mALFF)was also obtained by dividing the whole-brain mean from the original ALFF map.Finally,the mALFF maps were entered into a second-level,two-sample t-test model to compare group-level differ-ences using SPM8.Subject gender and age were added into the group model as covariates.An amplitude threshold was first set at p <0.001,and the cluster extent was corrected at the false discovery rate,p <0.05.The clusters identified in the group level analysis were also labeled using the Talairach Daemon (Lancaster et al.,2000;Talairach and Tournoux,1988)after accounting for the discrepancy between MNI space and Talairach space (Lancaster et al.,2007).This analy-sis revealed clusters in the medial cerebellum and left parietal lobe,which were also defined as the seed regions for further functional connectivity analysis.Functional connectivity analysisFunctional connectivity analysis was conducted using the functional connectivity toolbox,v.12.p (Chai et al.,2012;/swg/software.htm).For each subject,four seed regions were defined,including the posterior cerebellum (Fig.3A)and the anterior cerebellum area (Fig.3C)from the VBM analysis,as well as the medial cerebellum (Fig.3E)and the left superior parietal lobule (Fig.3G)from the ALFF anal-ysis.Before the modeling,the fifth-order eigenvector of WM and the first fifth-order eigenvector of CSF were re-moved from the preprocessed resting-state fMRI images to minimize physiological noise that was probably due toFIG. rger gray matter concentration in the athlete group than the control group.Threshold was set as cluster level false discovery rate (FDR)corrected p <0.05.L:left;R:right.Table 2.Clusters Showed Larger Gray Matter Concentration in the Athlete Group than in the Control GroupMNI coordinatesRegions Cluster sizeCorrected p Peak t x y z R.Cerebellum,Posterior Lobe,Cerebellar Tonsil 5970.001 5.39424À67À45R.Cerebellum,Posterior Lobe,Cerebellar Tonsil 4.76417À63À44R.Cerebellum,Anterior Lobe1410.0224.92824À54À30R.Cerebellum,Anterior Lobe,Culmen3.62132À49À26Threshold was p <0.001with a cluster-level nonstationary correction of p <0.05.R,right;L,left;MNI,Montreal Neurological Institute.BADMINTON PLAYER’S BRAIN227respiratory and cardiac factors.Six rigid-body head-motion parameters were also regressed out using linear regression analyses.The imaging data were also temporally filtered using a 0.01–0.08Hz band-pass filter.Voxel-wise correla-tion analysis was then conducted,and a beta contrast map was generated for each seed region and each subject.These individual seed correlation maps were then entered into two types of group-level analyses.The first group-level analysis used one-sample t-tests to determine the spe-cific intrinsic network connected to each of the seed regions.This is an important step,because subdivisions of a brain structure might be connected to different networks (e.g.,in cerebellum,Habas et al.,2009).The second group-level analysis used two-sample t-tests to compare differences be-tween the two groups.Since anti-correlation between the task-positive network and the default mode network re-mains controversial (Fox et al.,2009;Murphy et al.,2009),we restricted our results within the task-positive network by applying a positive correlation mask of the same seed re-gions.The masks were defined by using an ‘‘or’’operation on the two groups’correlation maps using a threshold of p <0.001.Results VBM resultsThe mean total GMV of the athlete group was 699.1mL (SD =60.2),while the mean total GMV of the control group was 717.7mL (SD =59.3).There was no difference in total GMV observed between the two groups (t =À0.96,p =0.35).A voxel-wise comparison revealed greater GMC in two right cerebellum clusters for the athlete group than the con-trol group (Fig.1and Table 2).One cluster was located in the posterior right cerebellar tonsil (lobule 8),and the other region was located in the anterior right cerebellar hemisphere (lobule 6).However,no region showed greater GMC for the control group than the athlete group.The voxel-wise comparison of GMV showed no differences between the two groups using a threshold of p <0.05with nonstationary correction.However,greater GMV in similar cerebellum clusters was observed using a more liberal thresh-old of p <0.001and cluster size >141voxels,(Table 3).ALFF resultsLarger ALFFs were also found in the cerebellum for the athlete group than the control group (Fig.2and Table 4),in-cluding the anterior part of the culmen,and the medial por-tion of the anterior lobe (vermis lobule 6).In contrast,there was a cluster in the left superior parietal lube (BA 7/19)that revealed larger ALFFs in the control group than the ath-lete group.Functional connectivity analysisFunctional connectivity analysis for all subjects revealed that seeding regions belonged to distinct functional networks (Fig.3).The right posterior cerebellum seed showed positive functional connectivity to the left dorsal-lateral prefrontal cor-tex,left parietal cortex,and left temporal lobe (Fig.3B);while the right anterior cerebellum and the medial part of theTable 3.Clusters Showed Larger Gray Matter Volume in the Athlete Groupthan in the Control Group (R:right;L:left)MNI coordinatesArea#voxels corrected p Peak t x y z R.Cerebellum,Posterior Lobe 2740.008 4.7124À64À44R.Cerebellum,Anterior Lobe 1490.031 4.6823À52À32R.Cerebellum,Anterior Lobe 3.9124À42À32L.Cerebellum,Posterior Lobe 2120.016 4.31À27À78À36L.Cerebellum,Posterior Lobe3.74À30À81À27R.Superior Temporal Gyrus,BA 411820.0214.1950À374R.Superior Temporal Gyrus,BA 134.0942À283Threshold was p <0.001with a cluster extent of 141voxels.The clusters in bold corresponded to the clusters reported in Gray matter con-centration analysis (Table 2).FIG.2.Different amplitude of low-frequency fluctuation between the athlete group and the control group.Threshold was set as cluster level FDR corrected p <0.05.Red:athlete >control;Blue:control >athlete.L:left;R:right.228DI ETAL.FIG.3.Positive (red)and negative (blue)correlation map of all the subjects with the four seeding regions.(A,C,E),and G display the four seed regions,and (B,D,F,H)represent corresponding correlation maps.Threshold was set as cluster level FDR corrected p <0.05.L:left;R:right.Table 4.Clusters Revealed Different Amplitude of Low-Frequency Fluctuation Between GroupsMNI coordinatesLabelCluster sizeCluster p Peak t x y z Athlete >control No Gray Matter3050.004 4.77À4À30À30L.Cerebellum,Anterior Lobe,Culmen 4.56À10À36À28No Gray Matter4.51À10À22À30L.Cerebellum,Anterior Lobe 2160.0124.66À6À60À26R.Cerebellum,Anterior Lobe4.2814À52À28R.Cerebellum,Anterior Lobe,Fastigium 4.166À58À24Control >athleteL.Superior Parietal Lobule,BA 73540.0015.59À38À6854L.Precuneus,BA 195.14À30À7448L.Middle Temporal Gyrus,BA 193.69À42À8032Threshold was p <0.001with a cluster-level false discovery rate correction of p <0.05.BA,Brodmann’s area.BADMINTON PLAYER’S BRAIN229cerebellum seeds showed positive connectivity with bilateral sensorimotor areas (Fig.3D,F).The left superior parietal lob-ule seed showed connectivity with typical task-positive net-works,including the bilateral dorsal lateral prefrontal cortex,the bilateral parietal cortex,the bilateral MT lobe,and the bilateral posterior cerebellum (Fig.3H).No differences were found in functional connectivity maps of both groups when the right posterior and the right anterior cerebellum regions were used as seeds,respectively.The me-dial cerebellum seed region revealed less functional connec-tivity with the right anterior cingulate cortex in the athlete group than the control group (Fig.4).However,this cluster was mainly located in the WM;therefore,it was not further discussed.The left parietal seed region showed greater functional con-nectivity with the right anterior cingulate cortex (BA 32/24)and the left middle frontal cortex (BA 6),and weaker func-tional connectivity with the left inferior frontal cortex (BA 9)and the bilateral middle frontal cortex (BA 9)in the athlete group than the control group (Fig.5and Table 5).Using a positive correlation map of all subjects as an inclusive mask,we further confirmed that the two left middle frontalcortex clusters,which showed different functional connectiv-ity with the left superior parietal lobule between the two groups,belonged to the positive correlation network (Fig.6).DiscussionThe two main findings of the present study are as follows:First,badminton athletes demonstrated greater GMC in the right anterior and posterior lobes of the cerebellum and greater ALFFs in the medial portion of the cerebellum.Sec-ond,badminton athletes showed smaller ALFFs in the left su-perior parietal lobule and altered functional connectivity between the left superior parietal lobule and frontal regions.The role of the cerebellumThe greater cerebellum GMC in badminton players is con-sistent with other cross-sectional studies of athletes,such as basketball players (Park et al.,2009)and keyboard musicians (Hutchinson et al.,2003).Structural expansion may result from microstructural changes at the neuronal level (Anderson et al.,1994;Kleim et al.,1997;Kleim et al.,1998).Changes in the cerebellum were also observed in the ALFF analysis,which is consistent with the recent finding that short-term motor training enhances resting-state independent cerebellar component activity (Albert et al.,2009).There is also evidence that task-related neural responses,as measured by fMRI,may be shaped by motor training (Doyon et al.,2002;Jenkins et al.,1994;Jueptner et al.,1997).These findings suggest that the cerebellum plays a key role in long-term professional bad-minton training and practice.However,the current cross-sectional study cannot rule out the alternative explanation that individuals with larger cerebellar density,and altered in-trinsic activity might be more likely to master the fine motor dexterity that is necessary for elite badminton skills.Future studies using a longitudinal design are needed to answer these questions.It is noteworthy that Hutchinson et al.(2003)and Park et al.(2009)only measured the global volume of the cerebellum or cerebellar lobules.The voxel-wise method adopted in the present study enabled the determination of region-specific anatomical differences within the cerebellar ing functional connectivity analysis,we further demonstrated that regions of the cerebellum which showed structural differ-ences with VBM analysis were not restricted solely to theFIG. 4.Cluster showed reduced functional connectivity with the medial cerebellum seed region in the athlete group than in the control group.Threshold was set as cluster level FDR corrected p <0.05.L:left;R:right.FIG.5.Cluster showed different functional connectivity with the left parietal seed region between groups.Threshold was set as cluster level FDR corrected p <0.05.Red:athlete >control;Blue:control >athlete.L:left;R:right.230DI ETAL.motor network.The right posterior cerebellum region is func-tionally connected to the left dorsal-lateral prefrontal cortex,the left parietal cortex,and the left temporal lobe,suggesting that the posterior cerebellum region belongs to the left fronto-parietal executive network (Habas et al.,2009).Therefore,the expansion of the cerebellum resulting from motor training may not only support motor control functions,but also in-volve other nonmotor functions such as visuo-motor coordi-nation and executive control.It is also worthy to point out that the nonmodulated GMC from VBM analysis did not compensate for the effect of spa-tial normalization;therefore,it cannot reflect absolute volumechanges (Mechelli et al.,2005).Instead,the GMC findings might reflect the differences of relative concentration or the density of cerebellum in the two groups.However,the GMV analysis after modulation showed very similar results,supporting the modulation of badminton training on the cere-bellum structure.The role of the fronto-parietal networkThe present study found,for the first time,that the function of the fronto-parietal network is altered in athletes.First,re-gional ALFFs in the left superior parietal lobule were smaller in the athlete group.The superior parietal lobule is thought to act as a relay that passes information from visual-processing areas to motor-processing areas,regions which support visuomotor coordination (Caminiti et al.,1996;Wise et al.,1997).Enhanced neural activity was observed in the superior parietal lobule after short-term motor training (Jenkins et al.,1994;Jueptner et al.,1997;Sakai et al.,1998).However,lower ALFFs appear to be inconsistent with a recent finding that the strength of the fronto-parietal network increased after 11min of visuomotor training (Albert et al.,2009).We argue that this finding may reflect different consequences of training phases between short-term training in Albert et al.(2009)and long-term athletic training in the present study.This notion is consistent with Xiong et al.(2009),who showed that region-specific brain activation increases first,then decreases,during a longer period of motor training.In addition to the amplitude of fluctuation,our results re-veal altered connectivity within the fronto-parietal network.The fronto-parietal network,which is supported by the un-derlying superior longitudinal fasciculus (Makris et al.,2005),plays a key role in hand-eye coordination (Battaglia-Mayer et al.,2001;Marconi et al.,2001),visual-guided reach-ing (Battaglia-Mayer et al.,2003;Burnod et al.,1999),and grasping (Grol et al.,2007).This is consistent with the notion that badminton athletes have higher visuomotor skills than individuals who do not play racquet sports.In addition,this network is reported in key regions to support the antici-pation of badminton serving (Wright et al.,2010;Wright and Jackson 2007).Our results suggest that functional connectivity has been rewired within the fronto-parietal network (Fig.6)inTable 5.Clusters Revealed Different Functional Connectivity with the Parietal Seed Region Between GroupsMNI coordinatesLabelBA Cluster sizeCluster p Peak t x y z Athlete >controlR.Anterior Cingulate 32355<0.001 5.87103814R.Anterior Cingulate 32 4.8114426R.Anterior Cingulate 24 4.64123020R.Medial Frontal Gyrus 102020.004 5.72644À12L.Middle Frontal Gyrus 61140.025 4.47À362244R.Cingulate Gyrus 311200.025 4.28À2636Control >athleteL.Inferior Frontal Gyrus 92960.001 5.25À461030L.Middle Frontal Gyrus 9 4.41À541432L.Middle Frontal Gyrus 9 3.78À462024R.Middle Frontal Gyrus91030.0484.96601434Threshold was p <0.001with a cluster-level false discovery rate correction of p <0.05.FIG.6.Illustration of different functional connectivity be-tween the athlete group and controls within the left fronto-parietal network.Arrow in cyan represents enhanced functional connectivity between the left superior parietal lobule (green)and the left middle frontal gyrus (BA6)(red)in the athlete group than in the control group.Arrow in purple represents lower functional connectivity between the left superior parie-tal lobule (green)and the left middle frontal gyrus (BA9)(blue)in the athlete group than in the control group.BADMINTON PLAYER’S BRAIN231。

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