一 : cut down,cut up,cut off的区别
cut down,cut up,cut off的区别
cut down
1.削减
He tried to cut down on smoking but failed.
他试图少抽烟,但没成功.
2.缩短
Cut down the article so as to make it fit the space available on the paper.
把文章缩短一些,这样就能排进报纸有限的版面中.
3.砍倒
cut up
1.切开;切碎
Cut up the carrots before you put them into the pot.
把胡萝卜切碎后再放进锅内.
2.抨击
The article was severely cut up by some critics.
那篇文章曾受到一些评论家严厉的抨击.
cut off
1.切除
He had a finger cut off by a machine while working.
他在工作时被机器切掉了一个手指.
2.切断;中断
I was cut off on my line to London.
我打伦敦长途时,电话线被切断了.
3.使死亡
He was cut off in his prime.
他在壮年时过世.
二 : The water supply has been cut off temporarily beca
The water supply has been cut off temporarily because the workers ____ one of the main pipes.
|
| D |
| 句意：“临时切断了水的供给，因为工人正在修理一个主管道。”A表示“过去的过去”，一定要有一个过去的时间做对比，不能单独使用，故不能选A；B现在完成时，表示“过去一个动作对现在的影响或过去一个动作一直持续到现在还有可能再继续下去”，一定和现在有关；在这里，没有表示repair的影响，也没有表示时间的蕴含；C过去时态，单纯的过去，和现在没有关系，不会题意；D现在进行时，说话时正在进行的动作。 【考点定位】考查时态的用法。时态要注意关注语境和时态的基本用法。尤其要注意现在完成时和过去完成时，前者一定和现在有关系，后者是过去的过去，一定要有一个过去的时间作对比。 |
考点：
考点名称：一般现在时一般现在时的概念：
表示通常性、规律性、习惯性的状态或者动作（有时间规律发生的事件）的一种时间状态。
一般现在时的用法：
1）经常性或习惯性的动作，常与表示频度的时间状语连用。常用的时间状语有every...，sometimes，at...，on Sunday等。
例如：I leave home for school at 7 every morning. 每天早上我七点离开家。
2）客观真理，客观存在，科学事实。
例如：The earth moves around the sun. 地球绕太阳转动。
Shang hai lies in the east of China. 上海位于中国东部。
3）表示格言或警句。
例如：Pride goes before a fall. 骄者必败。
注意：此用法如果出现在宾语从句中，即使主句是过去时，从句谓语也要用一般现在时。
例如：Columbus proved that the earth is round. 哥伦布证实了地球是圆的。
4）现在时刻的状态、能力、性格、个性。
例如：I don't want so much. 我不要那么多。
Ann writes good English but does not speak well. 安英语写得不错，讲的可不行。
比较：Now I put the sugar in the cup. 把糖放入杯子。
I am doing my homework now. 我正在做功课。
第一句用一般现在时，用于操作演示或指导说明的示范性动作，表示言行的瞬间动作。
第二句中的now是进行时的标志，表示正在进行的动作的客观状况，所以后句用一般现在时。
一般现在时知识体系：
一般现在时用法拓展：
1、一般现在时表将来：
1）下列动词come, go, arrive, leave, start, begin, return的一般现在时可以表示将来，主要用来表示在时间上已确定或安排好的事情。
例如：The train leaves at six tomorrow morning. 火车明天上午六点开。
—When does the bus star? 汽车什么时候开
—It stars in ten minutes. ？十分钟后。
2）以here, there 等开始的倒装句，表示动作正在进行。
例如：Here comes the bus.=The bus is coming. 车来了。
There goes the bell.=The bell is ringing. 铃响了。
3）在时间或条件句中。
例如：When Bill comes（不是will come）, ask him to wait for me. 比尔来后，让他等我。
I'll write to you as soon as I arrive there. 我到了那里，就写信给你。
4）在动词hope, take care that, make sure that 等的宾语从句中。
例如：I hope they have a nice time next week. 我希望他们下星期玩得开心。
Make sure that the windows are closed before you leave the room. 离开房间前，务必把窗户关了。
2、一般现在时代替一般将来时：
When, while, before, fter, till, once, as soon as, so long as, by the time, if, in case（that）, unless, even if, whether, the moment, the minute, the day, the year, immediately等引导的时间状语从句，条件句中，用一般现在时代替将来时。
例如：He is going to visit her aunt the day he arrives in Beijing. 他一到北京，就去看他姨妈。
3、一般现在时代替一般过去时：
1）"书上说"，"报纸上说"等。
例如：The news paper says that it's going to be cold tomorrow. 报纸上说明天会很冷的。
2）叙述往事，使其生动。
例如：Napoleon's army now advances and the great battle begins. 拿破仑的军队正在向前挺进，大战开始了
4、一般现在时代替现在完成时：
1）有些动词用一般现在时代替完成时，如hear, tell, learn, write, understand, forget, know, find, say, remember等。
例如：I hear（=haveheard）he will go to London. 我听说了他将去伦敦。
I forget（=have forgotten）how old he is. 我忘了他多大了。
2）用句型"It is…since…"代替"It has been…since…"。
例如：It is（=has been）five years since we last met. 从我们上次见面以来，五年过去了。
5、一般现在时代替现在进行时：
在Here comes…/There goes…等句型里，用一般现在时代替现在进行时。
例如：There goes the bell.铃响了。
时态一致：
1、如果从句所叙述的为真理或相对不变的事实，则用现在时。
例如：At that time, people did not know that the earth moves. 那时，人们不知道地球是动的。
He told me last week that he is eighteen.上星期他告诉我他十八岁了。
2、宾语从句中的，助动词ought, need, must, dare的时态是不变的。
例如：He thought that I need not tell you the truth. 他认为我不必告诉你真相。
将来进行时的概念：
表示将来某时进行的状态或动作，或按预测将来会发生的事情。
例如：She'll becoming soon. 她会很快来的。
I'l lbe meeting him sometime in the future. 将来我一定去见他。
将来进行时的基本用法：
1、将来进行时表示将来某一时间正在进行的动作：
如：Don't phone me between 5 and 6. We'll be having dinner then. 五点至六点之间不要给我打电话，那时我们在吃饭。
When I get home, my wife will probably be watching television. 当我到家时，我太太可能正在看电视。
2、表示按计划或安排要发生的动作：
如：I will be seeing you next week. 我下个星期来看你。
I'll be taking my holidays soon. 不久我将度假了。
We shall be going to London next week.下周我们要去伦敦。
3、将来进行时表示委婉语气：
如：Will you be having some tea? 喝点茶吧。
Will you be needing anything else? 你还需要什么吗？
将来进行时与一般将来时的区别：
（1）两者基本用法不一样：
将来进行时表示将来某时正在进行的动作，一般将来时表示将来某时将要发生的动作：
如：What will you be doing this time tomorrow? 明天这个时候你会在做什么呢？
What will you do tomorrow? 你明天干什么？
（2）两者均可表示将来，但用将来进行时语气更委婉：
如：When will you finish these letters? 你什么什候处理完这些信件？(直接询问，如上司对下属)
When will you be seeing Mr White? 你什么时候见怀特先生？(委婉地询问，如下属对上司)
When will you pay back the money?你什么时候还钱？(似乎在直接讨债)
When will you be paying back the money? 这钱你什么时候还呢？(委婉地商量)
（3）有时一般将来中的will含有“愿意”的意思，而用将来进行时则只是单纯地谈未来情况：
如：Mary won't pay this bill. 玛丽不肯付这笔钱。(表意愿)
Mary won't be paying this bill. 不会由玛丽来付钱。(单纯谈未来情况)
过去完成进行时：
过去完成进行时，就是相对过去的某个时刻来说已经对现在有直接影响并且还在进行的动作。放在间接引语或虚拟语气中时它的时态不能再向前推，向后推是现在完成进行时。过去完成进行时是由"had been + 现在分词"构成。
例如：She had been suffering from a bad cold when she took the exam.
她在考试之前一直患重感冒。
过去完成进行时构成：
过去完成进行时是由"hadbeen+现在分词"构成。
如：She had been suffering from a bad cold when she took the exam. 她在考试之前一直患重感。
Had they been expecting the news for some time? 他们期待这个消息有一段时间了吧？
过去完成进行时用法：
1、表示过去某一时间之前一直进行的动作。
过去完成进行时表示动作在过去某一时间之前开始，一直延续到这一过去时间。和过去完成时一样，过去完成进行时也必须以一过去时间为前提。过去完成进行时也是一个相对的时态，上下文中须有明示或暗示的作为参照的过去的时间。
如：I had been looking for it for days before I found it. 这东西我找了好多天才找着。
如：They had only been waiting for the bus a few minutes when it came. 他们只等了几分钟车就来了。
2、表示反复的动作。
如：He had been mentioning your name to me. 他过去多次向我提到过你的名字。
3、过去完成进行时还常用于间接引语中。
如：The doctor asked what he had been eating. 医生问他吃了什么。
I asked where they had been staying all those days. 我问他们那些天呆在哪儿。
4、过去完成进行时之后也可接具有"突然"之意的when分句。
如：I had only been reading a few minutes when he came in. 我刚看了几分钟他就进来了。
She'd only been reviewing her lessons for a short while when her little sister in terrupted her. 她温习功课才一会儿，她妹妹就打断她了。
过去完成进行时和过去完成时的比较：
如：She had cleaned the office, so it was very tidy. 她已经打扫过办公室了，所以很整洁。(强调结果)
She had been cleaning the office, so we had to wait outside. 她一直在打扫办公室，所以我们不得不在外面等着。(强调动作一直在进行)
过去将来时的被动语态的概念：
过去将来一般时的被动语态的主语是第一人称时用should be加及物动词的过去分词构成；主语是第二、三人称时用would be加及物动词的过去分词构成。
例如：He said that something would be needed to finish the task. 他说要完成这项任务，需要某种条件。
过去将来时的被动语态的用法：
过去将来时的被动语态常出现在英语间接引语中，其形式为：
(1)would/should+be+动词的过去分词
(2)was/were+going to be+动词的过去分词.
如：He said the project would be finished by the time we reached there.
She said that some new factories would be built soon in our city.
He thought that your watch was going to be mended after an hour.
He said that something would be needed to finish the task. 他说要完成这项任务，需要某种条件。
It was said that they would be selected by lottery. 据说他们将抽签选出。
三 : CUT-OFF FOR LARGE SUMS OF GRAPHS
Ann.Inst.Fourier,Grenoble
Workingversion–February8,2007
CUT-OFFFORLARGESUMSOFGRAPHS
byBernardYCART(*)
Abstract.IfListhecombinatorialLaplacianofagraph,exp(?Lt)convergestoamatrixwith
identicalcoe?cients.Thespeedofconvergenceismeasuredbythemaximalentropydistance.When
thegraphisthesumofalargenumberofcomponents,acut-o?phenomenonmayoccur:beforesome
instantthedistancetoequilibriumtendstoin?nity;afterthatinstantittendsto0.Asu?cient
conditionforcut-o?isgiven,andthecut-o?instantisexpressedasafunctionofthegapand
eigenvectorsofcomponents.Examplesincludesumsofcliques,starsandlines.
Convergenceabruptepourdegrandessommesdegraphes
R′esum′e.SiLestlelaplaciencombinatoired’ungraphe,exp(?Lt)convergeversunematricedont
touslescoe?cientssont′egaux.Lavitessedeconvergenceestmesur′eeparladistanced’entropie
maximale.Quandlegrapheestlasommed’ungrandnombredecomposantes,unph′enom`enede
convergenceabruptepeutsurvenir:avantuncertaininstantladistancea`l’′equilibretendvers
l’in?ni;apr`escetinstantelletendvers0.Uneconditionsu?santedeconvergenceabrupteest
donn′ee,etl’instantdeconvergenceestexprim′eenfonctiondutrouspectraletdesvecteurspropres
descomposantes.Lessommesdecliques,d’′etoilesetdelignessonttrait′eesenexemple.
1.Introduction
ManybinaryoperationsongraphsresultinagraphwhosesetofverticesistheCartesianproductofthesetsofverticesofthegraphstowhichtheoperationisapplied(seesection2.5p.65ofCvetkovi′cetal.[11]).Oneofthesimplestisthesum:ifG1andG2aretwographs,thentwocouples(x1,y1),(x2,y2)areadjacentinG1+G2ifandonlyifeitherx1=x2andy1,y2areadjacentinG2ory1=y2andx1,x2areadjacentinG1.Let(Gn)n??1beasequenceofgraphs.Allare?nite,undirected,withnoloopormultipleedge,andconnected.Forn??1,let
Sn=G1+···+Gn.
WeareinterestedhereinasymptoticpropertiesofSn.Sumsofidenticalcopiesofagivengraphsometimesappearintheliteratureas“Cartesianpowers”([2,5]).Thedenomination“sum”thatweusefollowing[10,11],iscoherentwithspectralproperties.
WeshallonlydealherewiththecombinatorialLaplacian(seeColindeVerdi`ere[8]):ifG=(V,E)isagraph,itisde?nedasL=D?A,whereDisthediagonalmatrixofdegreesandAtheadjacencymatrix.Thematrix?Listhein?nitesimalgeneratorofthecontinuoustimerandomwalkonthegraph(seeforinstanceC?inlar[7]).Ifx∈Visavertex,thedistributionattimetoftherandomwalk,startingatxattime0isthex-throwofthematrixexp(?Lt).Asttendstoin?nity,itconvergestotheuniformdistributiononV.Themaximalentropydistancebetweenrowsofexp(?Lt)andtheuniformdistribution(De?nition2.1),measuresthespeedofconvergence.Manyotherdistancescouldhavebeenchosen,leadingtosimilarresults(see[3]).Asttendstoin?nity,thedistancetoequilibriumdecays
2BERNARDYCART
asae?2ρt,whereρisthegap(smallestpositiveeigenvalueofL),andadependsontheeigenvectorsassociatedtothegap-eigenvaluesofL(Lemma2.2).
Itisawell-knownfactthattheLaplacianofG1+G2istheKroneckersumoftheLaplaciansofG1andG2(chap.12ofBellman[4]);itsspectrumcontainsallpossiblesumsofeigenvaluesofG1andG2.ThecontinoustimerandomwalkonG1+G2canbewrittenasacouplewhosecoordinatesareindependentcopiesoftherandomwalksonG1andG2respectively.ThemaximalentropydistancetoequilibriumforG1+G2isthesumofdistancesforG1andG2(Lemma2.3).Asntendstoin?nity,thedistancetoequilibriumforSn=G1+···+Gnmaydecayquitesteeply,exhibitingasocalledcut-o?phenomenon(De?nition2.4):beforesomeinstanttn,thedistanceisveryhigh,anditabruptlydropsdownaftertn.Intherandomwalkinterpretation,tnis“the”instantatwhichthewalkreachesitsequilibrium.Thecut-o?phenomenonofsteepconvergence,?rstidenti?edbyAldousandDiaconis[1],hasbeenobservedonmanystochasticprocesses(see[3,14]andreferencestherein).Ourmainresult,Theorem3.1,givesanexplicitexpressionfortn,andsu?cientconditionsforcut-o?attimetnforSn.ThisresultisrelatedtoTheorem3of[3],obtainedinasomewhatdi?erentsetting.Itcouldbeextendedtootherkindsofsymmetricoperators,suchasweightedLaplacians[6,9,12].
TheprobabilitytransitionmatrixofthediscretetimerandomwalkonagraphGisP=I?1
ky∈V??log(kpx,y(t)).
Themaximalentropydistancedecaysase?2ρt,whereρisthegap(smallestpositiveeigenvalue).Theequivalentcanbeexpressedintermsoftheeigenvectorsassociatedtoeigenvaluesequaltothegap.Letλ1=0<λ2??···??λkbetheeigenvaluesofL.Denotebyv1,...,vkeigenvectorssuchthatLvi=λiviand(v1,...,vk)isanorthonormalbasisfortheEuclideannorm.Asusual,v1istheconstantvectorwhosecoordinatesareallequaltok?1/2.
Lemma2.2.—
t→+∞limd(t)
CUT-OFFFORSUMSOFGRAPHS3
with
a=k
k
Thus
log(kpx,y(t))=
=??+k??i=2vi(x)vi(y)e?λit.??log1+kk??k??i=2k??i=2vi(x)vi(y)e?λit??,vi(x)vi(y)e?λit
k2
?
ky∈V??log(kpx,y(t))=?y∈Vi=2k????vi(x)vi(y)e?λit
k+
ky∈V??log(kpx,y(t))=ke?2ρt
4BERNARDYCART
Sincetheeigenvectorviisnon-null,vi(x)cannotvanishforeveryx∈V.(Foragivenx∈V,itmayhoweverhappenthatvi(x)=0forallisuchthatλi=ρ.)Thisimpliesthattheorderofd(t)isexactlye?2ρtandnotsmaller.??Wewillconsiderthreeexamplesofgraphswithkvertices:theclique,thestarandtheline,respec-tivelydenotedbyKk,TkandLk.ThespectraldecompositionoftheirLaplacianiseasytocompute.ColindeVerdi`ere[8]mentionsthatthespectrumofthelinegraphisthe?rsteverpublished,byLagrangein1867.WesummarizebelowthevaluesofρandaforKk,TkandLk.
Graph
k
StarTk
2(1?cos(π/k))cos2(π/(2k))(k?1)/2
Theentropydistanceisparticularlywelladaptedtosumsofgraphs.
Lemma2.3.—LetG1,G2betwographs.Letd1andd2bethemaximalentropydistancesofG1andG2respectively.ThemaximalentropydistanceofG1+G2isd1+d2.
Proof.—ThattheKullbackdistancebetweentensorproductsisthesumofKullbackdistancesbetweencomponentsisawell-knownfact(seeforinstanceLemma3.3.10p.100in[13]).ItcanbeeasilycheckedusingDe?nition2.1.Letx=(x1,x2)andy=(y1,y2)betwoverticesofG1+G2.Onehas
px,y(t)=px1,y1(t)px2,y2(t),
Thus,
?1
ki??log(kipxi,yi(t)).
??yi∈Vi
Hencetheresult,bytakingthemaximumoverallvertices(x1,x2).
Hereisthede?nitionofcut-o?forasequenceofgraphs.
Definition2.4.—Forn=1,2,...,letHnbeagraphanddnbethemaximalentropydistanceofHn.Let(tn)beasequenceofpositivereals,tendingto+∞.Thesequenceofgraphs(Hn)hasacut-o?at(tn)ifforc>0:
c<1=?
c>1=?n→∞
n→∞limdn(ctn)=+∞limdn(ctn)=0.
Thisde?nitionmatchestheusualde?nitionforcut-o?ofstochasticprocesses(see[3]andreferencestherein).
Our?rstexampleisthesumofcopiesofagivengraph.LetGbeagraph,anddbeitsmaximalentropydistance.Forn=1,2,...,letGnbeisomorphictoG.ThemaximalentropydistanceofthesumSn=G1+···+Gnisnd.ItfollowsfromLemma2.2thatthesequence(Sn)hasacut-o?atlog(n)/(2ρ),whereρisthegapofG.Actuallyinthisexample,theconvergencetakesplaceinawindowoftimeoforder1,sinceforallu∈R:??log(n)limndn→∞
CUT-OFFFORSUMSOFGRAPHS5
3.Cut-o?forasumofgraphs
Let(Gn)n??1beasequenceofgraphs.Forn??1,letknbethenumberofverticesofGn,letdnbethemaximalentropydistanceofGn,andletρnandanbe,asinLemma2.2,suchthat
dn(t)(3.1)limt→∞
2ρ(i,n)
Theorem3.1.—Assumethat:
(1)theconvergencein(3.1)isuniforminn,
(2)
(3.3)n→∞;i=1,...,nOurmainresultgivesconditionsunderwhichacut-o?occursat(tn).??.limρ(1,n)tn=+∞,
(3)thereexistsaconstantαsuchthat0<α<1andfornlargeenough,
(3.4)?i=2,...,n,a(i,n)??αA(i?1,n).
Then(Sn)hasacut-o?at(tn),de?nedby(3.2).
Proof.—Bythe?rsthypothesis,thereexistst0suchthatfort??t0andforalln,
1
2ρ(i?n,n)
Onehas:
σn??in??i=1?.a(i,n)e?2ρ(i,n)ctn
?2ρ(i?,n)ctnn??A(i?e,n)n
tnn,n)=e2(1?c)ρ(i?
??e2(1?c)ρ(1,n)tn
SUBMITTEDARTICLE:CUTOFFSUMS.TEX
6
For0<c<1,theresultfollowsby(3.3).BERNARDYCART
Letnowcbelargerthan1.Foralll=1,...,n?1,onehas:
σn??A(l,n)e?2ρ(1,n)ctn+
i=l+1
n??n??a(i,n)e?2ρ(i,n)ctn
??A(l,n)e?2ρ(1,n)ctn+ca(i,n)A?
(i,n).
i=l+1
Inthelastinequality,thesumisaRiemannsumforthe(decreasing)functionx→x?c.Hence:
??A(n,n)?2ρ(1,n)ctnx?cdxσn??A(l,n)e+
A(l,n)
??A(l,n)e?2ρ(1,n)ctn+1
c?1
Otherwise,??.
tnn,n)A(1,n)<e2ρ(1,n)tn??e2ρ(i?=A(i?.n,n)
Letln>1besuchthat
A(ln?1,n)<e2ρ(1,n)tn??A(ln,n)=A(ln?1,n)+a(ln,n).
Applying(3.6)tol=ln?1yields
σn??e2ρ(1,n)(1?c)tn+1
????tn1?c??
??2(1?c)ρ(1,n)tn??e1+c?111?a(ln,n)e?2ρ(1,n)
CUT-OFFFORSUMSOFGRAPHS7
Sumsofcliques
ThemaximalentropydistancedofthecliqueKkis:
(3.9)d(t)=?k?1
k??k?1log1+
((k?1)/2)e?kt
tendsto1asttendstoin?nity,uniformlyink.Sothe?rsthypothesisofTheorem3.1issatis?ed.Observethatforallk,
log(Nk(n)(k?1)/2)tn??
=+∞,k
whichimposesthatbkshouldgrowfasterthanexponentiallyink.
Wewillcheckthatifbkgrowsatmostexponentiallyink,thenthereisnocut-o?.Theradiusof??convergenceoftheseriesbkzkisnolargerthan1.Supposeitisequaltoe?2t0forsomepositivet0.Considertheseries∞??k?1bk
k=2
2e?2kt
tendsto+∞fort<t0,toa?nitevaluefort>t0.SodoesDn(t),andthereisnocut-o?.Iftheradiusofconvergenceis1,thenDn(t)remainsboundedforallpositivet.
Sumsofstars
ThemaximalentropydistanceofthestarTkis:
(3.10)d(t)=?k?2
k?1e?t+1
k
Thistimetheconvergenceisnotuniforminkandthe?rsthypothesisofTheorem3.1holdsonlyif(kn)isbounded.Ifitisunbounded,theremaystillbeacut-o?for(Sn).Itmaybeattheinstanttnde?nedby(3.2),orelsewhere.Forinstance,ifkn=n+1,thereisacuto?at(logn),asexpected.Butifkn=2n,thereisacuto?atn/(log2),andnotn/(2log2)as(3.2)wouldleadtothink.
Sumsoflines
LetGn=Lkn.Recallthatρn=2(1?cos(π/kn))andan=cos2(π/(2kn)).If(kn)isunbounded,thenthegapρ(1,n)tendstozeroandtheconvergencein(3.1)isnotuniform.However,inthisparticularcase,thereexistsapositiveconstantbsuchthatforalln,
ane?2ρnt(1?be?ρnt)??dn(t)??ane?2ρnt(1+be?ρnt).??k(k?2)log1+k?1e?kt??.
SUBMITTEDARTICLE:CUTOFFSUMS.TEX
8
Usingthede?nition(3.5)ofσn,onehas:BERNARDYCART
(1?be?ρ(1,n)ctn)σn??Dn(ctn)??(1+be?ρ(1,n)ctn)σn.
SoDn(ctn)canbereplacedbyσnprovidedρ(1,n)tntendstoin?nity.Since1/2<an<1,thethirdhypothesisofTheorem3.1issatis?ed.Therefore(3.3)aloneisasu?cientconditionofcut-o?forasumoflines.If(kn)growsatmostpolynomiallyinn,then(3.3)holds.Butif(kn)growsexponentiallyinn,thenρ(1,n)tnremainsbounded.
References
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[4]R.Bellman,Introductiontomatrixanalysis,McGraw-Hill,London,1960.¨sser,“Edge-isoperimetricproblemsforCartesianpowersofregulargraphs”,Theor.Com-[5]S.Bezrukov&R.Elsa
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′,M.Doob,I.Gutman&A.Torgaˇ[10]D.Cvetkovicsev,Recentresultsinthetheoryofgraphspectra,Ann.Discrete
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[15]B.Ycart,“Cuto?forsamplesofMarkovchains”,ESAIMProbab.Stat.3(1999),p.89-107.
BernardYCART
LJK,CNRSUMR5224,Universit′eJosephFourier,Grenoble,France
Bernard.Ycart@ujf-grenoble.fr
ANNALESDEL’INSTITUTFOURIER
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