Modulo 2 periodicity of complex Clifford algebras and electromagnetic field

Factors Contributing to Fossilization IntroductionIn the past decades, many researchers and scholars in linguistic circle and the interrelated fields have done a lot of studies of fossilization from different perspectives to discover the causes of fossilization, and a number of different theories have been proposed, among which Selinker’s five psycholinguistic processes, and three models (the biological, interactional and acculturation model), one principle (the multiple effects principle) and Krashen’s in put hypothesis are worth mentioning. To be specific, these theories include:1. Selinker’s five central processesSelinker’s(1972)early explanation of the causes of fossilization consists of five central processes:Language transfer: Learners’IL systems are greatly influenced by their first language, and they cannot produce correct L2 output. Selinker regarded language transfer as the most decisive factor in leading to fossilization.Transfer of training: L2 learners may have done excessive training on certain IL structures that they cannot successfully continue to develop new structures. For instance, if a learner has too much training on the structure containing the verb “be”, he may form the habit of using “be”when it is not necessary.Inappropriate learning strategies: Learners may use inappropriate strategies in their learning progress and thus cause the fossilization in IL, such as translating L1 sentences into L2 sentences directly, etc. Inappropriate communication strategies: When learners are communicating in L2, they may apply some inappropriate strategies so as not to influence the fluency or effect of communication, such as avoidance, simplification, reduction of lexicon.Overgeneralization: This type of fossilization consists mainly of the overgeneralization of some target language rules, like “goed”“teached”.2. The biological causesOne of the most remarkable representatives is Lenneberg. Lenneberg advanced Critical Period Hypothesis in his monumental, The Function of Language, in 1967, believing that there was a neurologically based critical period, ending around the onset of puberty, beyond which complex mastery of a language, first or second, was not possible. Besides Lenneberg, many scholars, including Scovel(1988), Long(1990), Patkowski(1994)are supportive of the biological theory.Lamendella used “sensitive period”to explain the acquisition of second language. Lamendella(1977)also proposed another concept of infrasystem. He holds that while L1 acquisition calls for an infrasystem, L2 acquisition also requires its corresponding infrasystem. If a learner has not developed the infrasystem for acquiring a second language or if thisinfrasystem is underdeveloped, then he or she has to turn to the already-developed infrasystem for mother tongue to acquire the second language. However, the infrasystem for mother tongue is not appropriate for acquiring the second language, after the close of the critical period for primary language acquisition, the L2 learner stands a greater chance of fossilizing far from target-language norms.The Critical Period Hypothesis mostly explains the fossilization of L2 pronunciation, as the available evidence suggests that children do better than adult L2 learners in pronunciation and speaking tests, while adolescent and adult L2 learners are similar to or better than children in the acquisition of grammar and morphemes.3. Social and cultural causesL2 learner’s lack of desire to acculturate is also the reason for fossilization. Schumman(1981)proposed the Acculturation Hypothesis to interpret fossilization from a social-psychological perspective.According to Schumman, acculturation means the social and psychological integration of the learner with the target language group. In Schumman’s Acculturation Hypothesis,acculturation is seen as the determining variable in the sense that it controls the level of linguistic success achieved by second language learners. Stauble(1980)also affirmed the essential roles of social and psychological distance in second language acquisition.4. Vigil&Oller’s interactional modelVigil&Oller presented an early model of fossilization which focused on the role of extrinsic feedback. They expounded their opinions in the following:(1)When the language learners communicate with their teachers and classmates, some incorrect language output sometimes plays the role of input which leads to the learners’ language fossilization.(2)The information transmitted in interpersonal communication includes two kinds of information: one is cognitive information and another is affective information. The former contains facts, assumptions, and beliefs which are expressed in language. The latter is expressed in the form of facial expressions, intonation and gestures etc.Vigil&Oller argued that the interactive feedback received by a learner has a controlling influence on fossilization. Certain types of feedback were said to prompt learners to modify their knowledge of the L2, while other types encouraged learners to stand pat. They suggested that there were cognitive and affective dimensions to feedback. In this scheme, a combination of positive cognitive feedback and negative affective feedback was most likely to promote fossilization, while negative cognitive and positive affective feedback combined to cause learners to modify their linguistic knowledge.According to Han, one problem found in the interactional models isthat there is no way to determine what percentage of cognitive feedback needs to be positive in order to trigger fossilization. Another problematic aspect is the question of whether negative cognitive feedback destabilizes all the rules used to assemble the utterance.5. Krashen’s input hypothesisKrashen believes that most adult second-language learners “fossilize”. He concluded 5 possible causes of fossilization:(1)Insufficient quantity of inputKrashen claims that insufficient input is the most obvious cause of fossilization. Some second-language performers may cease progress simply because they have stopped getting comprehensible input.(2)Inappropriate quality of inputInappropriate quality of input, which means input of the wrong sort, or input filled with routines and patterns, a limited range of vocabulary, and little new syntax, is more subtle than insufficient quantity of input.(3)The affective filterComprehensible input is not sufficient for full language acquisition. To acquire the entire language, including late-acquired elements that do not contribute much to communication, a low affective filter may be necessary. The affective filter is a block that prevents input from reaching the Language Acquisition Device (LAD),and affects acquisition, preventing full acquisition from taking place.(4)The output filterThe output filter is a device that sometimes restrains second-language users from performing their competence (Krashen, 1985).(5)The acquisition of deviant formsThis may occur in two different kinds of situation, both of which are characterized by beginners being exposed nearly exclusively to imperfect versions of the second language. The first situation can be called the “extreme foreign-language” situations. The second situation is that of the performer in the informal environment, where he has communication demands that exceed his second-language competence, and is faced witha great deal of incomprehensible input.6. Multiple effects principleIn a later study, Selinker and Lakshmanan (1992 emphasize the importance of the role of language transfer in fossilization. They raise the question of why “certain linguistic structures become fossilized while others do not” They suggest that the multiple effects principle (MEP) may help explain this. The MEP states that two or more SLA factors, working in tandem, tend to promote stabilization of interlanguage forms leading to possible fossilization. Among various possible SLA factors that have fossiling effects language transfer has been singled out as the principal one.In their paper,Han and Selinker(1997)described a longitudinal case study they made to prove the MEP prediction. We may take what they said in the conclusion par as a summary of the main points of the MEP: What is showed in the case study “brings direct corronoration to the MEP in that language transfer functions as a co-factor in setting multiple effects, and that when it conspires with other SLA processes, there is a greater chance of stabilization of the interlanguage structure”. ConclusionIn summary, factors contributing to language fossilization have been illustrated, whether in terms of empirical studies of not, by different researchers from amount of perspectives This paper has listed a number of reasons from the following six views: Selinker’s five central processes (1972), biologica causes, social and cultural causes, Vigil&Oller’s interactiona model, Krashen’s inpu t hypothesis and Multiple Effects principle.There is no doubt that causes of “cessation” of learners’ might owe to other elements, however, knowing the above six ones, at their least value, inspires some solutions in overcoming the phenomenon of fossilization.References:[1]Selinker.L. Interlanguage[J].International Review of Applied Linguistics,1972.[2]Selinker. L.Fossilization: What we think we know [J].Internet, 1996[3]Lemendella,J.T.General principles of neurofunctional organization and their manifestations in primary and non-primary language acquisition[J]. Language Learning, 1977, (27), 155-196.[4]Vigil,N.&Oller,J. Rule fossilization: A tentative model[J].Language Learning,1976.[5]Ellis，R. The Study of Second Language Acquisition[M]. Oxford: Oxford University Press, 1994.[6]Rod Ellis.Underastanding Second Language Acquisition[M].Shanghai Foreign Language Education Press,1999.[7]Krashen,S. The Input Hypothesis: Issues and Implication[M].London:Longman,1985.[8]李炯英：中介语石化现象研究30年综观[J]，Foreign Language Teaching Abroad, 2003[9]陈慧媛：关于语言僵化现象起因的理论探讨[J]，外语教育与研究,1999.（3）：21-24[10]牛强：过渡语的石化现象及其教学启示[J]，外语与外语教学,2000（4）：28-31。

a r X i v :h e p -t h /0102190v 1 27 F eb 2001Generalized WDVV equations for B r and C r pure N=2Super-Yang-Mills theoryL.K.Hoevenaars,R.MartiniAbstractA proof that the prepotential for pure N=2Super-Yang-Mills theory associated with Lie algebrasB r andC r satisfies the generalized WDVV (Witten-Dijkgraaf-Verlinde-Verlinde)system was given by Marshakov,Mironov and Morozov.Among other things,they use an associative algebra of holomorphic diffter Ito and Yang used a different approach to try to accomplish the same result,but they encountered objects of which it is unclear whether they form structure constants of an associative algebra.We show by explicit calculation that these objects are none other than the structure constants of the algebra of holomorphic differentials.1IntroductionIn 1994,Seiberg and Witten [1]solved the low energy behaviour of pure N=2Super-Yang-Mills theory by giving the solution of the prepotential F .The essential ingredients in their construction are a family of Riemann surfaces Σ,a meromorphic differential λSW on it and the definition of the prepotential in terms of period integrals of λSWa i =A iλSW ∂F∂a i ∂a j ∂a k .Moreover,it was shown that the full prepotential for simple Lie algebras of type A,B,C,D [8]andtype E [9]and F [10]satisfies this generalized WDVV system 1.The approach used by Ito and Yang in [9]differs from the other two,due to the type of associative algebra that is being used:they use the Landau-Ginzburg chiral ring while the others use an algebra of holomorphic differentials.For the A,D,E cases this difference in approach is negligible since the two different types of algebras are isomorphic.For the Lie algebras of B,C type this is not the case and this leads to some problems.The present article deals with these problems and shows that the proper algebra to use is the onesuggested in[8].A survey of these matters,as well as the results of the present paper can be found in the internal publication[11].This paper is outlined as follows:in thefirst section we will review Ito and Yang’s method for the A,D,E Lie algebras.In the second section their approach to B,C Lie algebras is discussed. Finally in section three we show that Ito and Yang’s construction naturally leads to the algebra of holomorphic differentials used in[8].2A review of the simply laced caseIn this section,we will describe the proof in[9]that the prepotential of4-dimensional pure N=2 SYM theory with Lie algebra of simply laced(ADE)type satisfies the generalized WDVV system. The Seiberg-Witten data[1],[12],[13]consists of:•a family of Riemann surfacesΣof genus g given byz+µz(2.2)and has the property that∂λSW∂a i is symmetric.This implies that F j can be thought of as agradient,which leads to the followingDefinition1The prepotential is a function F(a1,...,a r)such thatF j=∂FDefinition2Let f:C r→C,then the generalized WDVV system[4],[5]for f isf i K−1f j=f j K−1f i∀i,j∈{1,...,r}(2.5) where the f i are matrices with entries∂3f(a1,...,a r)(f i)jk=The rest of the proof deals with a discussion of the conditions1-3.It is well-known[14]that the right hand side of(2.1)equals the Landau-Ginzburg superpotential associated with the cor-∂W responding Lie ing this connection,we can define the primaryfieldsφi(u):=−∂x (2.10)Instead of using the u i as coordinates on the part of the moduli space we’re interested in,we want to use the a i .For the chiral ring this implies that in the new coordinates(−∂W∂a j)=∂u x∂a jC z xy (u )∂a k∂a k )mod(∂W∂x)(2.11)which again is an associative algebra,but with different structure constants C k ij (a )=C k ij(u ).This is the algebra we will use in the rest of the proof.For the relation(2.7)weturn to another aspect of Landau-Ginzburg theory:the Picard-Fuchs equations (see e.g [15]and references therein).These form a coupled set of first order partial differential equations which express how the integrals of holomorphic differentials over homology cycles of a Riemann surface in a family depend on the moduli.Definition 6Flat coordinates of the Landau-Ginzburg theory are a set of coordinates {t i }on mod-uli space such that∂2W∂x(2.12)where Q ij is given byφi (t )φj (t )=C kij (t )φk (t )+Q ij∂W∂t iΓ∂λsw∂t kΓ∂λsw∂a iΓ∂λsw∂a lΓ∂λsw∂t r(2.15)Taking Γ=B k we getF ijk =C lij (a )K kl(2.16)which is the intended relation (2.7).The only thing that is left to do,is to prove that K kl =∂a mIn conclusion,the most important ingredients in the proof are the chiral ring and the Picard-Fuchs equations.In the following sections we will show that in the case of B r ,C r Lie algebras,the Picard-Fuchs equations can still play an important role,but the chiral ring should be replaced by the algebra of holomorphic differentials considered by the authors of [8].These algebras are isomorphic to the chiral rings in the ADE cases,but not for Lie algebras B r ,C r .3Ito&Yang’s approach to B r and C rIn this section,we discuss the attempt made in[9]to generalizethe contentsof the previoussection to the Lie algebras B r,C r.We will discuss only B r since the situation for C r is completely analogous.The Riemann surfaces are given byz+µx(3.1)where W BC is the Landau-Ginzburg superpotential associated with the theory of type BC.From the superpotential we again construct the chiral ring inflat coordinates whereφi(t):=−∂W BC∂x (3.2)However,the fact that the right-hand side of(3.1)does not equal the superpotential is reflected by the Picard-Fuchs equations,which no longer relate the third order derivatives of F with the structure constants C k ij(a).Instead,they readF ijk=˜C l ij(a)K kl(3.3) where K kl=∂a m2r−1˜C knl(t).(3.4)The D l ij are defined byQ ij=xD l ijφl(3.5)and we switched from˜C k ij(a)to˜C k ij(t)in order to compare these with the structure constants C k ij(t). At this point,it is unknown2whether the˜C k ij(t)(and therefore the˜C k ij(a))are structure constants of an associative algebra.This issue will be resolved in the next section.4The identification of the structure constantsThe method of proof that is being used in[8]for the B r,C r case also involves an associative algebra. However,theirs is an algebra of holomorphic differentials which is isomorphic toφi(t)φj(t)=γk ij(t)φk(t)mod(x∂W BC2Except for rank3and4,for which explicit calculations of˜C kij(t)were made in[9]we will rewrite it in such a way that it becomes of the formφi(t)φj(t)=rk=1 C k ij(t)φk(t)+P ij[x∂x W BC−W BC](4.3)As afirst step,we use(3.4):φiφj= Ci·−→φ+D i·−→φx∂x W BC j= C i−D i·r n=12nt n2r−1 C n·−→φ+D i·−→φx∂x W BCj(4.4)The notation −→φstands for the vector with componentsφk and we used a matrix notation for thestructure constants.The proof becomes somewhat technical,so let usfirst give a general outline of it.The strategy will be to get rid of the second term of(4.4)by cancelling it with part of the third term,since we want an algebra in which thefirst term gives the structure constants.For this cancelling we’ll use equation(3.4)in combination with the following relation which expresses the fact that W BC is a graded functionx ∂W BC∂t n=2rW BC(4.5)Cancelling is possible at the expense of introducing yet another term which then has to be canceled etcetera.This recursive process does come to an end however,and by performing it we automatically calculate modulo x∂x W BC−W BC instead of x∂x W BC.We rewrite(4.4)by splitting up the third term and rewriting one part of it using(4.5):D i·−→φx∂x W BC j= −12r−1 D i·−→φx∂x W BC j= −D i2r−1·−→φx∂x W BC j(4.6) Now we use(4.2)to work out the productφkφn and the result is:φiφj= C i·−→φ−D i2r−1·r n=12nt n D n·−→φx∂x W BC j +2rD i2r−1·rn=12nt n −D n·r m=12mt m2r−1[x∂x W BC−W BC]j(4.8)Note that by cancelling the one term,we automatically calculate modulo x∂x W BC −W BC .The expression between brackets in the first line seems to spoil our achievement but it doesn’t:until now we rewrote−D i ·r n =12nt n 2r −1C m ·−→φ+D n ·−→φx∂x W BCj(4.10)This is a recursive process.If it stops at some point,then we get a multiplication structureφi φj =r k =1C k ij φk +P ij (x∂x W BC −W BC )(4.11)for some polynomial P ij and the theorem is proven.To see that the process indeed stops,we referto the lemma below.xby φk ,we have shown that D i is nilpotent sinceit is strictly upper triangular.Sincedeg (φk )=2r −2k(4.13)we find that indeed for j ≥k the degree of φk is bigger than the degree ofQ ij5Conclusions and outlookIn this letter we have shown that the unknown quantities ˜C k ijof[9]are none other than the structure constants of the algebra of holomorphic differentials introduced in [8].Therefore this is the algebra that should be used,and not the Landau-Ginzburg chiral ring.However,the connection with Landau-Ginzburg can still be very useful since the Picard-Fuchs equations may serve as an alternative to the residue formulas considered in [8].References[1]N.Seiberg and E.Witten,Nucl.Phys.B426,19(1994),hep-th/9407087.[2]E.Witten,Two-dimensional gravity and intersection theory on moduli space,in Surveysin differential geometry(Cambridge,MA,1990),pp.243–310,Lehigh Univ.,Bethlehem,PA, 1991.[3]R.Dijkgraaf,H.Verlinde,and E.Verlinde,Nucl.Phys.B352,59(1991).[4]G.Bonelli and M.Matone,Phys.Rev.Lett.77,4712(1996),hep-th/9605090.[5]A.Marshakov,A.Mironov,and A.Morozov,Phys.Lett.B389,43(1996),hep-th/9607109.[6]R.Martini and P.K.H.Gragert,J.Nonlinear Math.Phys.6,1(1999).[7]A.P.Veselov,Phys.Lett.A261,297(1999),hep-th/9902142.[8]A.Marshakov,A.Mironov,and A.Morozov,Int.J.Mod.Phys.A15,1157(2000),hep-th/9701123.[9]K.Ito and S.-K.Yang,Phys.Lett.B433,56(1998),hep-th/9803126.[10]L.K.Hoevenaars,P.H.M.Kersten,and R.Martini,(2000),hep-th/0012133.[11]L.K.Hoevenaars and R.Martini,(2000),int.publ.1529,www.math.utwente.nl/publications.[12]A.Gorsky,I.Krichever,A.Marshakov,A.Mironov,and A.Morozov,Phys.Lett.B355,466(1995),hep-th/9505035.[13]E.Martinec and N.Warner,Nucl.Phys.B459,97(1996),hep-th/9509161.[14]A.Klemm,W.Lerche,S.Yankielowicz,and S.Theisen,Phys.Lett.B344,169(1995),hep-th/9411048.[15]W.Lerche,D.J.Smit,and N.P.Warner,Nucl.Phys.B372,87(1992),hep-th/9108013.[16]K.Ito and S.-K.Yang,Phys.Lett.B415,45(1997),hep-th/9708017.。

a r X i v :c o n d -m a t /0211142v 1 [c o n d -m a t .s u p r -c o n ] 7 N o v 2002Europhysics Letters PREPRINT µ+SR study of carbon-doped MgB 2superconductors K.Papagelis 1,J.Arvanitidis 1,K.Prassides 1,A.Schenck 2,T.Takenobu 3and Y.Iwasa 31School of Chemistry,Physics and Environmental Science,University of Sussex,Brighton BN19QJ,UK 2Institute for Particle Physics,ETH Zurich,CH-5232Villigen PSI,Switzerland 3Institute for Materials Research,Tohoku University,Sendai 980-8577and CREST,Japan Science and Technology Corporation,Kawaguchi 332-0012,Japan PACS.76.75.+i –Muon spin rotation and relaxation.PACS.74.70.Ad –Metals,alloys and binary compounds (including A15,Laves phases,etc.).PACS.74.20.-z –Theories and models of superconducting state.Abstract.–The evolution of the superconducting properties of the carbon-doped MgB 2superconductors,MgB 2−x C x (x =0.02,0.04,0.06)have been investigated by the transverse-field muon spin rotation (TF-µ+SR)technique.The low-temperature depolarisation rate,σ(0)at 0.6T which is proportional to the second moment of the field distribution of the vortex lattice decreases monotonically with increasing electron doping and decreasing T c .In addition,the temperature dependence of σ(T )has been analysed in terms of a two-gap model.The size of the two superconducting gaps decreases linearly as the carbon content increases.Introduction.–The recent discovery of superconductivity at ∼40K in MgB 2[1]has opened new prospects for the understanding of the microscopic origin of high-T c supercon-ductivity.In contrast to the complex crystal structures,multicomponent nature and compli-cations due to magnetism and strong electron-electron correlations of the high-T c cuprates,MgB 2has a simple hexagonal crystal structure (AlB 2-type)comprising close-packed Mg 2+layers alternating with graphite-like boron layers and sp electrons involved in the supercon-ducting process.Several attempts [2]have been made to introduce dopants in the Mg layers in order to explore the relationship between T c ,crystal structure and doping level.The reported experiments invariably lead to a decrease in T c .Successful substitution at the B sites can be also achieved by carbon doping [3],which results in a significant contraction of the a lattice parameter,but affects little the interlayer separation.T c also decreases with increasing doping.Experimental studies of the superconducting properties of MgB 2show deviations from thosecalculated with the standard BCS theory [4,5].Various experiments,including scanning tun-neling microscopy (STM)[6],point-contact spectroscopy [7],specific heat measurements [8],optical [9]and Raman spectroscopy [10]suggested the existence of a secondary superconduct-ing gap,implying that the simple one-band approach for MgB 2superconductivity must be extended.On the basis of the electronic structure,the existence of multiple gaps has been cEDP Sciences2EUROPHYSICS LETTERS invoked in order to explain the magnitude of T c in MgB2[11].In addition,solution of the full Eliashberg equations at low temperature yields different gap values for the different parts of the Fermi surface(∼1.8meV for3D sheets and∼6.8meV for2D sheets)[12].Although there is much support for the applicability of the multiband description to MgB2,there is still some debate in the literature,in particular since some tunneling and NMR measurements show only a single gap[13].µ+SR measurements have also made significant contributions to the understanding of the superconducting properties of MgB2,mainly through the precise determination of the pene-tration depth,λfrom the zero-temperature extrapolated value of theµ+spin depolarisation rate,σ(0).Early experiments[14]proposed that the low temperature magnetic penetration depth of MgB2shows a quadratic temperature dependence and were interpreted in terms of unconventional superconductivity with an energy gap that has nodes at certain points in the k space.However,a systematic study by Niedermayer et al.[15]reported that the temperature evolution of the depolarisation rate,σin polycrystalline MgB2could be well interpreted in terms of a two-gap model.In the present paper we employ the transverse-field(TF)variant of theµ+SR technique to characterise the superconducting properties of carbon-doped MgB2systems(MgB2−x C x, x=0.02,0.04,0.06).Analysis of the experimental data in terms of a two-gap model reveals the effect of electron doping on the superconducting gaps and the anisotropic properties of the compounds.Experimental.–Polycrystalline MgB2−x C x(x=0.02,0.04,0.06)samples were prepared by reaction of Mg,amorphous B and carbon black at900◦C for2h,as described elsewhere[3]. Theµ+SR measurements were performed with the GPS spectrometer on theπM3muon beamline at the Paul Scherrer Institute(PSI),Switzerland.Pressed sample pellets were attached with low-temperature varnish on a Ag sample-holder placed on the stick of a He continuousflow cryostat operating down to1.8K.After cooling the sample in an external field,H ext to temperatures below T c in order to induce a homogenousflux line lattice,positive muons(100%spin-polarised)with their initial spin polarisation transverse to the externalfield were implanted in the solid sample.The implanted muons come to rest at an interstitial site and act as highly sensitive local magnetic probes.In the presence of localfields,B loc,theµ+ spin undergoes Larmor precession with frequency,ωµ=γµB loc,whereγµ/2π=13.553kHz/G is the muon gyromagnetic ratio.The time evolution of theµ+spin polarisation,Pµ(t)is measured by monitoring the positrons preferentially emitted along theµ+spin direction at the instant of muon decay.For type II superconductors,Pµ(t)is an oscillatory function with decreasing amplitude and the damping of theµ+precession signal provides a measure of the inhomogeneity of the magneticfield,∆B in the vortex state and hence of the magnetic penetration depth,λ.Results and Discussion.–For polycrystalline samples in the vortex state,the TFµ+ spin polarisation function is approximately Gaussian(Pµ(t)∼exp(−12). In the case of anisotropic(λc/λab>>4)type II superconductors,σis related to the in-planemagnetic penetration depth,λab via the relation[16]:σ[µs−1]=7.086×104λ−2ab [nm−2].The anisotropy of the upper criticalfield in single crystalline MgB2was reported as6(at5 K)[17]or4.3[18],while for polycrystalline samples and thinfilms,the reported values span a wide range.The above equation holds in the London limit(κ=(λ/ξ)>>1)and in the absence of pinning-induced distortions of the vortex lattice.The former restriction is valid for MgB2(ξab≈7nm[19],λab≈95nm[15]),while the latter is addressed below.The LondonK.Papagelis et al.:µ+SR study of carbon-doped MgB2superconductors3Fig.1–Temperature dependence of theµ+spin depolarisation rate,σat H ext=0.6T for MgB2−x C x (x=0.02,0.04,0.06).The open circles correspond to MgB1.98C0.02,the triangles to MgB1.96C0.04 and the squares to MgB1.94C0.06.The solid lines arefits to the two-gap model.The inset shows the fit of the experimental data for MgB1.98C0.02,assuming an isotropic single gap model(dashed line) or a T2-law(dotted line).model predicts that the second moment of the magneticfield distribution of a perfect vortex lattice should be independent of H ext forλ>L where L is the distance between vortices.In order to check the influence of pinning on the depolarisation rate,σ,we have measured its field dependence at5K.Our results show thatσ(H)increases almost linearly up to50mT, displays a peak at∼80mT and then decreases rapidly,reaching a plateau at higherfields (∼0.6T)in excellent agreement with the reported data for MgB2[15].We thus performed our measurements at an externalfield of0.6T in order to ensure formation of an ideal vortex lattice and avoid underestimation ofλ(T).Fig.1presents the extracted temperature dependence of the TF-µ+SR depolarisation rate at H ext=0.6T for the MgB2−x C x(x=0.02,0.04,0.06)samples.The values ofσ(T)are derived after subtraction of the normal state temperature-independent depolarisation rate,σback(σ2(T)=σ2meas(T)−σ2back).In the case of MgB1.98C0.02,σ(T)increases monotonically as the temperature decreases below∼35K and reaches a plateau at T≤6K,remaining almost constant at lower temperatures.Analogous behaviour is observed for the other two compositions,except that as the doping level increases,the low-temperature plateau smooths4EUROPHYSICS LETTERS out and the extrapolated value,σ(0)shifts to lower values.The experimentalσ(T)dependence is reproduced well by means of a two-gap model,while attempts tofit the experimental data with a T2-law or an isotropic one-gap model led to unsatisfactory results(inset infig.1).The two-gap model is based on the existence of two discrete superconducting gaps,∆1and∆2,at T=0K,both closing at T c and each associated with a different energy band.By assuming that the coupling between the two bands(i.e.due to impurity or phonon scattering)is sufficiently weak(vide infra),σ(T)can be expressed as[8,5,15]:σ(T)=σ(0)−(γ1/γ)δσ(∆1,T)−(γ2/γ)δσ(∆2,T)(1) where2σ(0)δσ(∆,T)=ε2+∆(T)2/k B T))−1(3)Each band is characterised by a partial Sommerfeld constant,γi(γ1+γ2=γ,whereγis the total Sommerfeld constant).As the Sommerfeld constant is proportional to the density-of-states at the Fermi level,the ratiosγi/γdetermine the partial N i(ǫF)for the two bands.The temperature dependence of the band gaps is taken from BCS theory,i.e.∆(t)=∆(0)δ(t) whereδ(t)is the normalised BCS gap at the reduced temperature,t=T/T c[20].Thefitted parameters are summarised in Table I.For comparison the corresponding values for MgB2[15]are also included.The evolution of the depolarisation rateσ(0)with T c and of the superconducting gap sizes,∆1(0)and∆2(0)with carbon content,x are shown in Fig.2.σ(0)decreases monotonically with increasing carbon content and decreasing T c(dσ(0)/d T c=−0.25µs−1K−1).The∼11%decrease inσ(0)at x=0.06reflects an increase of∼6%in the in-plane magnetic penetration depth,λab(0).In addition,as the carbon content increases,the two gap sizes at0K shift to smaller values at nearly the same rate(d∆i/d x=−20.4meV,i= 1,2).γ1/γwhich is a measure of the partial N(ǫF)for the band with the larger gap also shows a tendency to increase with increasing electron doping.The superfluid plasma frequency in the ab plane,ωsf p(=c/λab(0))which is related to the charge density of the superfluid condensate at0K is also included in Table I.Table I–Extracted parameters for MgB2−x C x(x=0.02,0.04,0.06).σ(0)is the low-temperature µ+spin depolarisation rate,λab(0)is the in-plane magnetic penetration depth,∆1and∆2are the superconducting gaps at0K,γ(γ1)is the(partial)Sommerfeld constant,ωsf p is the superfluid plasma frequency,n s is the superconducting carrier density and m∗ab/m e is the effective mass enhancement of the B layers.The data for MgB2are from Ref.[15]and the T c values from Ref.[3].xσ(0)∆1γ1/γn s(m∗ab/m e)−1(µs−1)(meV)(×1021cm−3)07.9 6.0(0)0.6(2) 3.150.027.5(1) 5.8(4)0.7(2) 2.99(4)0.047.3(1) 5.2(2)0.8(1) 2.91(4)0.067.0(1) 4.8(3)0.8(2) 2.79(3)K.Papagelis et al.:µ+SR study of carbon-doped MgB2superconductors5Fig.2–Relationship between T c and low-temperatureµ+spin depolarisation rate,σ(0)(upper panel). Evolution of the superconducting gap sizes,∆1and∆2at T=0K with carbon content,x(lower panel).The closed symbols correspond to the larger gap,∆1associated with the2Dσsheets and the open ones to the smaller gap,∆2associated with the3Dπsheets.In the London model at the clean limit,the magnetic penetration depth,λis related to the superconducting carrier density,n s and the effective mass,m∗byσ∝1m∗c2(16EUROPHYSICS LETTERS The electronic band structure of MgB2has been extensively investigated[22,23,24].The valence band of MgB2is made up predominantly of B2p states,which form two distinct sets of bands ofσ(p x,y)andπ(p z)type whose k dependence differs considerably.The most pronounced dispersion for the B p x,y states is along theΓ-K direction of the Brillouin zone (BZ),while aflat zone is formed in the k z direction(Γ-A),reflecting the2D character of the boron lattice.Theseσbands are partially unoccupied creating a hole-type conduction band which gives rise to two2D light-hole and heavy-hole sheets forming coaxial cylinders along theΓ-A BZ direction.The strong coupling of these holes to the optical bond stretching modes drives the superconductivity in MgB2[25].In addition,Bπstates dominate at the bottom of the conduction band,while the Fermi surface associated with these bands consists of a3D tubular network.These bands exhibit maximum dispersion along theΓ-A direction of the BZ.Theoretical calculations estimate that44%of N(ǫF)comes from the2Dσcylindrical sheets and the rest from the3Dπsheets[12].For the superconducting state of MgB2,the gap is nonzero everywhere on the Fermi surface and the gap values are grouped in two distinct regions[12].The larger superconducting gap,∆1on the2Dσcylindrical sheets has an average value of∼6.8meV,while the smaller one,∆2associated with the3Dπ-sheets an average value of∼1.8meV.Although the electronic structure and the superconducting behaviour of MgB2has been studied theoretically in detail,there is no systematic investigation concerning the effect of doping.In the framework of the rigid band model,it is expected[24]that doping of the B sublattice with C should lead to a shift of the Fermi level to higher energies in the region of the DOS minimum(pseudogap).A rigid band model estimate gives a value of∼0.16electrons/cell in order tofill theσbands[23].Electrons are also added to theπbands that lie in the same energy range,leading to a total doping level∼0.25electrons/cell.Thus it is expected that electron doping should lead to both superconducting gaps decreasing gradually.However, the differences in the slope−dln T c/d V between high-pressure experiments(∼0.28˚A−3)on MgB2[26]and chemical substitution[3](∼0.37˚A−3),imply that the effect of carbon doping is more complicated due to the B layer contraction and theoretical studies beyond the rigid band approximation are necessary.Carbon substitution should also lead to an increase in the interband impurity scattering, which should result in the size of theσ-andπ-gaps converging to the same value.However, theoretical calculations[27]have shown that the particular electronic structure of MgB2results in extremely weakσπimpurity scattering,even for low quality samples and in the presence of Mg vacancies,Mg-substitutional impurities and B-site substitutions by N or C.The dominant mechanism for impurity scattering is due to intraband scattering of theσandπbands with the scattering rate inside theπbands greater than that of theσbands.Intraband scattering does not change T c and the gap values but influences the penetration depth(Anderson’s theorem)[28].Our experimental data can be described well with a two-gap model in which impurity scattering is essentially ignored.This indicates that interband impurity scattering is relatively weak and both superconducting gaps are preserved,at least up to x=0.06.Carbon doping also reduces both superconducting gaps with this effect relatively more pronounced for the smaller gap,∆2associated with the3D sheets.The error inγ1/γ2(∼N1(ǫF)/N2(ǫF))is quite large and hence it is difficult to extract reliably its dependence on doping.Nonetheless, there is a tendency forγ1/γ2to increase with increasing doping level.In any case,systematic theoretical studies are necessary to shed more light on this issue.Conclusions.–In conclusion,analysis of the temperature dependence of the TF-µ+SR depolarisation rate for MgB2−x C x(x=0.02,0.04,0.06)shows that increasing electron doping is accompanied by a decrease of the low-temperatureσ(0)and the spectral weight,n s/m∗,K.Papagelis et al.:µ+SR study of carbon-doped MgB2superconductors7 consistent with the decreasing density-of-states at the Fermi level.Within a two-gap model, both superconducting gap sizes decrease almost linearly with the smaller gap affected more on doping,while the interband scattering remains relatively weak up to at least x=0.06.∗∗∗We thank C.Niedermayer and I.I.Mazin for helpful discussions,PSI for provision of beamtime and A.Amato,K.Brigatti,ppas and I.Margiolaki for help with the exper-iments.We acknowledge support from the Royal Society and the Marie Curie Fellowship programme of the EU”Improving the Human Research Potential”under contract numbers HPMF-CT-2001-01435(K.Papagelis)and-01436(J.Arvanitidis).REFERENCES[1]Nagamatsu J.et al.,Nature,410(2001)63.[2]Slusky J.S.et al.,Nature,410(2001)343;Tampieri A.et al.,Solid State Commun.,121(2002)497;Zhao Y.G.et al.,Phyica C,361(2001)91;Cimberele M.R.et al.,Supercond.Sci.Technol.,15(2001)43.[3]Takenobu T.et al.,Phys.Rev.B,64(2001)134513.[4]Bud’ko S.L.et al.,Phys.Rev.Lett.,86(2001)1877;Hinks D.G.et al.,Nature,411(2001)457.[5]Bouquet F.et al.,Phys.Rev.Lett.,87(2001)047001.[6]Giubileo F.et al.,Phys.Rev.Lett.,87(2001)177008;Iavarone M.et al.,Phys.Rev.Lett.,89(2002)187002.[7]Szabo P.et al.,Phys.Rev.Lett.,87(2001)137005;Schmidt H.et al.,Phys.Rev.Lett.,88(2002)127002.[8]Wang Y.et al.,Phyica C,355(2001)179;Bouquet F.et al.,Europhys.Lett.,56(2001)856[9]Kuz’menko A.B.et al.,Solid State Commun.,121(2002)479.[10]Chen X.K.et al.,Phys.Rev.Lett.,87(2001)157002.[11]Liu A.Y.et al.,Phys.Rev.Lett.,87(2001)087005.[12]Choi H.J.et al.,Phys.Rev.B,66(2002)020513.[13]Gonnelli R.S.et al.,Phys.Rev.Lett.,87(2001)097001;Rubio-Bollinger G.et al.,Phys.Rev.Lett.,86(2001)5582;Kotegawa H.et al.,Physica C,378-381(2002)25.[14]Panagopoulos C.et al.,Phys.Rev.B,64(2001)094514.[15]Niedermayer C.et al.,Phys.Rev.B,65(2002)094512.[16]Barford W.and Gunn J.M.F.,Phyica C,156(1988)515.[17]Angst M.et al.,Phys.Rev.Lett.,88(2002)167004.[18]Takahashi K.et al.,Phys.Rev.B,66(2002)012501.[19]De Lima O.F.et al.,Phys.Rev.Lett.,86(2001)5974.[20]Muhlschlegel B.et al.,Z.Phys.,155(1959)313.[21]Uemura Y.J.et al.,Phys.Rev.Lett.,66(1991)2665.[22]Kortus J.et al.,Phys.Rev.Lett.,86(2001)4656;Ravindran P.et al.,Phys.Rev.B,64(2001)224509;Belashchenko K.D.et al.,Phys.Rev.B,64(2001)092503.[23]An J.M.et al.,Phys.Rev.Lett.,86(2001)4366.[24]Medvedeva N.I.et al.,Phys.Rev.B,64(2001)020502.[25]Kong Y.et al.,Phys.Rev.B,64(2001)020501.[26]Prassides K.et al.,Phys.Rev.B,64(2001)012509;Vogt T.et al.,Phys.Rev.B,63(2001)220505.[27]Mazin I.I.et al.,Phys.Rev.Lett.,89(2002)107002.[28]Golubov A.A.et al.,Phys.Rev.B,66(2002)054524.。

华南师范大学硕士学位论文答辩合格证明学位申请人童叠豳向本学位论文答辩委员会提交题为童兰垦熊．塑堑丞鱼塞猃型窒的硕士论文，经答辩委员会审议，本论文答辩合格，特此证明。

学位论文答辩委员会委员(签名)主席：兰P垒伊尹论文指导老师(签名)：w叮年6月扩日中文摘要类比推理的研究是心理学领域一个非常重要的内容，其研究成果不仅能够提示人类认知加工过程，而且对其他领域如教育领域的作用也越来越明显。

前人的研究从不同角度、采用不同的研究方法对影响类比推理的因素进行了系统地考察，这些研究结果具有一定的价值和意义。

然而类比推理研究中一个十分重要的领域是对于类比推理结果的研究，目前对这一领域的研究相对比较薄弱。

许多研究都强调了类比映射对于文本表征变化的重要作用，但是很少有研究精确地指出映射所产生的结果。

本研究将通过3个实验对类比推理的一种可能的结果进行探讨，即探讨类比推理的结果如何整合进目标信息的表征。

本研究所采用的技术模型如下：被试首先阅读目标信息，再阅读一个潜在的类比源(类比组)或是填充材料(无类比组)，然后进行再认测试，测试中一些句子是曾在目标信息文本中出现过的义本项目，一些足未在目标信息的文体中出现过并且跟文本内容无关的无关项哥，一些是末在目标信息的文本中出现过但可以通过类比推理得出来的类比推理项目。

如果类比推理的结果能够整合进目标信息的表征，则阅读了源信息的被试错误地认为类比推理项目曾出现在目标信息中。

实验1通过对1sabelle Blanchette和Keyin Dunbar(2002)的研究的进一步分析，对他们的实验材料进行了修改，排除了记忆混淆现象对先前实验结果的可能解释，确证了类比组被试与无类比组被试成绩的差异是由于类比推理的结果整合进目标信息的表征而导致的。

实验2探讨在映射阶段类比推理整合进目标信息的表征的具体过程。

实验2包括两个小实验，实验2a探讨类比推理整合进目标信息的表征时问题的结构特征和表面特征哪一个更重要。

辽宁省大连市滨城高中联盟2024-2025学年高三上学期10月月考英语试卷一、听力选择题1．Where are the speakers most probably?A．On the train.B．In a restaurant.C．In a bookstore.2．What are the speakers probably going to do?A．Buy a coat.B．Take a trip.C．Attend a party.3．How much should the man pay?A．$1.B．$3.C．$4.4．What does the man think of the movie?A．Disappointing.B．Exciting.C．Interesting.5．What are the speakers talking about?A．Advantages of online books.B．Their favorite books.C．The future of books.听下面一段较长对话，回答以下小题。

6．What is the man most probably?A．A host.B．A teacher.C．An exchange student. 7．What once bothered the man?A．The way people say goodbye.B．The way people send invitations.C．The way people start a conservation.听下面一段较长对话，回答以下小题。

8．What happened to the woman most probably?A．She got hurt while skiing.B．She got hurt by lifting heavy things.C．She was hit while taking some tests.9．Where does the conversation probably take place?A．In the doctor’s office.B．In the ski field.C．In the drugstore.听下面一段较长对话，回答以下小题。

a r X i v :a s t r o -p h /0310026v 1 1 O c t 2003SF2A 2003bes,D.Barret and T.Contini (eds)MODULATION OF THE X-RAY FLUXES BY THE ACCRETION-EJECTION INSTABILITY Varni`e re,P.1,Muno,M.2,Tagger,M.3and Frank,A 4Abstract.The Accretion-Ejection Instability (AEI)has been proposed to explain the low frequency Quasi-Periodic Oscillation (QPO)observed in low-mass X-Ray Binaries.Its frequency,typically a fraction of the Keple-rian frequency at the disk inner radius,is in the right range indicated by observations.With numerical simulation we will show how this instability is able to produce a modulation of the X-ray flux and what are the character-istic of this modulation.More simulations are required,especially 3D MHD simulations.We will briefly present a new code in development:AstroBear which will allow us to create synthetic spectra.1Introduction:a brief presentation of the AEI and relation to observation The Accretion-Ejection Instability [5]is a spiral instability similar to the galactic spiral but driven by magnetic stress instead of self-gravity.This instability affects the inner region of the disk when the plasma β=8πp/B 2is of the order of one,i.e.there is equipartition between the gas and magnetic pressure.It forms a quasi-steady spiral pattern rotating in the disk at a frequency of the order of a few times the orbital frequency at the inner edge of the disk.This spiral density wave is coupled with a Rossby vortex it creates at the corotation radii (corotation between the spiral wave and the gas in the diks).This coupling permits the energyand angular momentum to be stored at the corotation radius allowing accretion to proceed in the inner part of the disk.Contrary to MRI based accretion models the AEI does not heat-up the disk as the energy is transported by waves and not deposited locally.The Rossby vortex twists the foot points of the field liness.In the presence ofa low density corona this torsion will propagate as an Alfven wave transporting energy and angular momentum store in the vortex.This will put energy into the238SF2A2003corona where it might power a wind or a jet[7].In order to apply the AEI to the phenomena occuring in microquasars we have two observables that we used:the presence of jet;thecharacteristic of the Quasi-Periodic Oscillation(QPO).We already have compared some of the properties of the AEI with observations,mainly the relation between the inner radius of the disk from spectralfit and the QPO frequency[3],[6].This leads us to try to compare more QPO characteristics with the observations.2AEI and QPOIn order to compare the AEI to observations we aim to produce synthetic spectra from numerical simulation.We used the code presented in[2]and add an energy equation to compute the heating of the disk at spiral shocks.The idea was to use the spiral shock to heat the disk and create a thicknenning along the ing an hydrostatic approxmation to compute the local thickness of the disk we get the inner region of the accretion disk appears as infigure1.We see that along the spiral arm the disk gets thicker(the z coordinates)and hotter(the color scale) than other locations in the disk.Fig.1.Snapshot of the inner disk,the heigh represent the local thickness of the disk and the color represent the temperature(lighter color meaning higher temperature) In collaboration with M.Muno we have computed the X-rayflux coming from such an accretion disk.We obtain a modulation of the observedflux.This mod-ulation is coming from geometrical effect related to the inclination angle of the source such as seen on the left graph offigure2.We see that the rms amplitude(∼5%)is too small to explain the observations (as much as20%sometimes).Several phenomena could increase the observed rms amplitude of the modulation.Indeed,the simulation did not take into account relativistic effects.Another effect is the fact that the disk thickness is computed as posteriori and not evolve in the simulation.The hydrostatic equilibrium is a good“first approximation”and allows us to prove that the AEI is able to modulate the X-rayfluxes but in order to compare with observation we need simulations of the disk taking into account the three dimensions.Modulation of the X-ray Fluxes by the AEI239Fig.2.left:maximun rms amplitude obtain for different inclination of the system.right: evolution of the rms amplitude of the modulation as function of time.3A New3D MHD AMR code:AstroBearNumerical simulation is a tool which allows us to study phenomena and also to compare theory with observation by the mean of synthetic observation/spectra in their non-linear behaviour.If we want to study in more detail the accretion and ejection we need to have a3D MHD code.Including Adaptativ Mesh Refinement (AMR)in the code will allow us to have a good resolution with less numerical cost. Using AMR is not always useful depending on the phenomena study,i.e.AMR is not interesting for turbulence but it will be really useful to study jet’s knots or spiral waves.We are working on such a3D AMR MHD code using a Godunov-type methods for the base scheme.This method projects the solution of the eigenfunction of the Riemann problem associated.Astro-Bear use a package called BEARCLAW.BEARCLAW is a general pur-pose software package for solving time dependent partial differential equations (automatic adaptive mesh refinement,parallel execution).AstroBear is a module meant for astrophysical purpose,especially the accretion-ejection phenomena.At the moment the hydrodynamics module fully operationnal in3D and we are test-ing the MHD.We have done the standard1D MHD test to check the ability of the code to resolve each waves.When going to2D one needs to take care of the divergence of the magneticfield.Several method exist currently in AstroBear we utilize three of them,the8th-waves,the GLM and the projection method.Our interest is to find the one that is the most adapted depending on the problem we want to solve. We will also add the choice between several Riemann solver,each of them having different weakness and strength.Infigure3show two standard2D tests.On the left is a cloud-shock interaction where a shock wave propagates through a media having a high density cloud in it. To do this test we used the8th-waves method for the divergence constraint.This method does not“clean”the divergence mkdir at each step but advect it with the240SF2A2003flow.It does not required a new step but add the non-vanishing divergence term in the source step.It is the fastest method and well adapted.Thefigure on the Fig.3.2D test of the MHD solver with two differents method for the divergence.left:a cloud-shock interaction using the8-waves method.right:the Orzag&Tang vortex using the projection method.right is the Orzag&Tang vortex.This time we have used the projection method in order to“clean”the divergence at each time step.This method requires us to solve a Poisson equation and cost a lot in term of efficiency.4conclusionUsing numerical simulation we gave the proof of principle that the AEI is able to modulate the X-rayflux and we aim to continue this study by creating synthetic spectra which can be directly compare with observation.In order to do this we need a code able to fully simulate an accretion disk. We have done thefirst step toward this goal by developing and testing the MHD module of AstroBear.ReferencesCaunt,S.&Tagger,M.,2001,A&A,367,1095A.Dedner,F.Kemm,D.Kroner,C.D.Munz,T.Schnitzer,M.Wesenberg,2002,JComp-Phys,175,645-673.Rodriguez,J.,Varni`e re,P.,Tagger.M.,and Durouchoux,P.,2002,A&A,387,487-496. K.G.Powell,P.L.Roe,T.J.Linde,T.I.Gombosi,D.L de Zeeuw,1999,JCompPhys,154, 284-309.Tagger,M.,and Pellat,R.,1999,A&A,349,1003Varni`e re,P.,Rodriguez,J.and Tagger.M.,2002,A&A,387,497-506.Varni`e re,P.and Tagger.M.,2002,A&A,394,329-338.This figure "varniere_fig4.jpeg" is available in "jpeg" format from: /ps/astro-ph/0310026v1This figure "varniere_fig5.jpeg" is available in "jpeg" format from: /ps/astro-ph/0310026v1。

Reduction Modulo2and3of Euclidean Lattices,IIJacques MartinetLaboratoire A2X,UMR5465associ´e e au C.N.R.S.&Universit´e Bordeaux1Abstract.This paper is a continuation of Reduction Modulo2and3ofEuclidean Lattices([7],Journal of Algebra,2002).R´e sum´e.Reduction modulo2et3des r´e seaux euclidiens(II).Cet article faitsuite`a l’article[7]Reduction Modulo2and3of Euclidean Lattices,paru en2002au Journal of Algebra.1.Introduction.Let E be an n-dimensional Euclidean space with scalar product x·y,and let L be the set of lattices(discrete subgroups of rank n)in E.For a latticeΛ∈L, we denote by minΛits minimal norm:minΛ=min x∈Λ {0}x·x,and by detΛthe determinant of the Gram matrix(e i·e j)of any Z-basis(e1,e2,...,e n)ofΛ.In this paper,we still study short representatives for classes modulo2and3of a given lattice,indeed only modulo2from Section3onwards.Section2is devoted to root lattices modulo2and3and Section3to laminated lattices modulo2.In Section4,we give a few complements which may apply to odd lattices.In Section5, we show how to attach to classes modulo2containing not too large vectors a lattice of codimension1having a relatively large minimum.Notably,the Leech lattice produces the23-dimensional“equiangular”integral lattice of minimum5described in[9]whose set of minimal vectors constitutes a spherical(tight)5-design;see also[1].We now recall some results which are proved in[7].We denote byΛa lattice, and we set n=dimΛ,m=minΛ,and for t>0,we denote by S t the set of norm tvectors inΛand we set s t=12|S t|.We also denote by m the norm of the secondlayer ofΛ:m =min N(x)>m N(x).1991Mathematics Subject Classification.11H55,11H56.Key words and ttices,short vectors,reduction.I thank the authors of the PARI system(H.Cohen et al.),and especially Christian Batut for his specific lattice programs.12J.MARTINETFor lattices modulo 2,the basic identity,involving non-zero vectors x and y =x +2z ≡x mod 2Λ,is N (y )+N (x )=2 N (z )+N (x +z ) .(1)Provided that y =±x ,this implies N (y )+N (x )≥4m ,with equality if and only if z and x +z =y −z are minimal.Proposition 1.1.If x =0and y =±x ,we have N (y )+N (x )≥4m ,and equality holds if and only if z and x +z =y −z are minimal.We then havex ·z =−N(x )2,and x and y are orthogonal.Proof.The first part is clear.If N (z )=N (x +z )=m ,then2x ·z +N (x )=0and y ·x =N (x )+2x ·z =0.A complete set T of shortest representatives for non-zero classes modulo 2yields a weighted formula of the kindx ∈T 1w (x )=2n −1(2)where w (x )=|{y ∈Λ|N (y )=N (x )and y ≡x mod 2Λ}|.In [7],we essentially considered vectors of norm N ≤2m .The weight w (x )is equal to 1if 0<N (x )<2m and belongs to the interval [1,n ]if N (x )=2m ;some estimations for w beyond norm 2m will be proved in the next sections.This implies the inequality 0<t<2ms t +s 2m n ≤2n −1and various other inequalities of the same kind related to better bounds for w under various hypotheses.For lattices modulo 3,the basic identity,involving non-zero vectors x and y =x +3z ≡x mod 3Λ,is N (y )+2N (x )=3 2N (z )+N (x +z ) ,(3a )together with its companion identity obtained by exchanging x and y :N (x )+2N (y )=3 2N (z )+N (y −z ) .(3b )We shall this time enumerate the classes modulo 3up to sign (i.e.,we now consider the set T of pairs ±C of classes modulo 3).The weighted formula now takes the form x ∈T 1w (x )=3n −12.(4)In [7],Theorem 3.13,we proved for the weight the following results:•w (x )=1if 0<N (x )<m +m or m +m <N (x )<2m +m ;•w (x )=1or 3if N (x )=m +m ;•1≤w (x )≤n +1if N (x )=2m +m .[w =3(resp.w =n +1)corresponds to a configuration A 2(resp.A ∗n +1.)]In the sequel,weighted formulae will be displayed in the following form:for norms N where the weight may take values larger than 1,an expression such as (a 1+a 2+···)w +b 1+b 2+···w+···REDUCTION MODULO 2AND 3OF EUCLIDEAN LATTICES,II 3means that a 1,a 2,...are the number of pairs of vectors in the various norm N orbits with weight w ,etc.As noticed in [7](Proposition 2.9and Table I),lattices having mod 2repre-sentatives of norm N <2m constitute an open set in L .The corresponding mod 3result applies to lattices having representatives of norm N <2m +m ;this results from [7],Propositions 3.7and 3.8;it notably applies to A n (n ≤3)and D 4;see next section.Finally,we shall essentially consider only irreducible lattices;indeed,the bound 2m for classes modulo 2does not hold for reducible lattices to within the three exceptions A 1⊥A 1,A 2⊥A 1and A 2⊥A 2(up to scale).It is easy to verify that the similar list for mod 3lattices reduces to A 1⊥A 1and A 2⊥A 1.2.Root Lattices Modulo 2and3.In this section,we consider irreducible root lattices,indeed lattices isometric to A n (n ≥1),D n (n ≥4)or E n (n =6,7,8).We have m =2,and disregarding the trivial cases of A 1and A 2,m =4,hence 2m +m =4m =8.We denote by (ε0,ε1,...,εn )(resp.(ε1,...,εn ))the canonical basis for Z n +1(resp.Z n )and set A n ={x ∈Z n +1| x i =0}and D n ={x ∈Z n | x i ≡0mod 2}.Proposition 2.1.Up to signs,shortest representatives for classes modulo 2or 3of A n or D n are the vectors which are of one of the following forms:(1)εi 1±εi 2±···±εi k ,of norm 2k ,for A n and D n ,modulo 2and 3.(2)2εi ,of norm 4,for D n modulo 2.(3)2εi 1+···+2εi +εi +1+···+εi +k −εi +k +1−···−εi 3 +2k ( >0),of norm 6 +2k ,for A n modulo 3.(4)2εi 1±εi 2±···±εi k (k >0),of norm k +4,for D n modulo 3.Moreover,the weights of the vectors above are 1in case (1),n in case (2), 3+k in case (3),and k in case (4).Proof.(Sketch.)If x = j a j εj ∈Λ(Λ=A n or D n )has some largecomponent a i ,consider a transformation of the form x →x ±2(εi −εj )or x →x ±3(εi −εj )if Λ=A n ,x →x ±2(εi ±εj )or x →x ±3(εi ±εj )if Λ=D n (and then also x →x ±4εi or x →x ±6εi if x has a single component).That the vectors listed above are among the shortest representatives in their class is easy to verify;we leave to the reader the calculation of the weights.For exceptional lattices,we have:Proposition 2.2.For E 6mod 2,E 7mod 3and E 8mod 2and 3,all classes possess representatives of norm ≤2m =4or 2m +m =8.For E 7mod 2(resp.E 6mod 3),there is one missing class,whose smallest representatives have norm6(resp.12);this is the set of minimal vectors in 2E ∗7(resp.3E ∗6),of weight 28(resp.27).Proof.Modulo 2,we use the general bound w ≤n and its refinement w ≤n −1(proved in [7]before Theorem 2.4)which applies to lattices such that (2m )n det(Λ)is not a square,together with the fact that the sum must not exceed 2n −1.This immediately gives us the following three formulae for vectors of norm 2and 4:E 6mod 2:36+1355=63=26−1;4J.MARTINETE7mod2:63+3786=126=(27−1)-1;E8mod2:120+10808=255=28−1.The same device applies for classes modulo3.For integral lattices,the bound w≤n+1can be refined to w≤n whenever the scaled copy of A∗m to norm 2m+m (here,8)is not integral;this applies to E6and E7.We also need the fact that norm6vectors in E7share out among two orbits(in all other cases,primitive vectors of norm N≤8constitute a single orbit).These two orbits have s 6=28 and s 6=1008pairs of vectors;thefirst one is2S(E∗7).These remarks show that the weighted formulae for vectors of norm N≤8are:E6mod3:36+135+3603+4326=363=36−12-1;E7mod3:63+378+28+10083+20167=1093=37−12;E8mod3:120+1080+33603+86409=3280=38−12.There remains to characterize the two missing classes.For E7mod2,considerx∈2S(E∗7).We have N(x)=22·32=6.Let y≡x mod2in E7.We haveN(y)≡N(x)≡2mod4.Hence if y were shorter than x,it would have norm2. But pairs of norm2vectors in E7constitute an orbit of63>28different classes mod2,a contradiction.Similarly,the27pairs of vectors in3S(E∗6),of norm3243=12,cannot becongruent mod3to a shorter vector,for such a vector would have norm6(because y≡x mod3E6⇒N(y)≡N(x)mod3),and norm6vectors constitute an orbitof3602=120>27vectors.(Incidentally,this shows that all norm10vectors in E6are congruent mod3to a norm4vector.)minated Lattices Modulo2.These lattices,that we shall consider only in the range1≤n≤24,were defined inductively by Conway and Sloane;see[3],Chapter6for a precise definition.They have minimum4.There is one lattice in each dimension,denoted byΛn,except for n=11,12,13where there are two,three,and three lattices respectively,charac-terized by their kissing number,and denoted by an extra superscript min,mid or max.The aim of this section is to show that the list of laminated lattices for which it was proved in[7],Section2,that all classes modulo2contain representatives of norm N≤8is actually complete up to dimension24.minated lattices of dimension n≤24possessing represen-tatives of norm N≤8for all classes modulo2are those of dimension n≤6, 8≤n≤10and n=24.Before proceeding to the proof,we state and prove a lemma:Lemma3.2.Let L be an integral lattice of minimum3and letΛ=L even be its even part.Assume that there exist in L two non-orthogonal pairs of minimal vectors.ThenΛhas minimum4and contains a class modulo2of minimum12.REDUCTION MODULO2AND3OF EUCLIDEAN LATTICES,II5 proof of3.2.We have min L even≥4,and if x,y are non-orthogonal,non-proportional minimal vectors in L,we have x·y=±1,hence N(x∓y)=4,whence N(L even)=4.Let e∈S(L)and let f=2e.Since[L:Λ]=2,we haveΛ= L,e = L,f2 ,henceL Λ= x2|x≡f mod2Λ.Since min L=min L Λ=3,we have N(x)≥12on the whole class of f modulo2. Since N(f)=12,this completes the proof of the lemma.proof of Theorem3.1.In dimensions n≤8,the laminated lattices are scaled copies of root lattices,namely A n(n=1,2,3),D n(n=4,5)and E n (n=6,7,8),and Theorem3.1follows from the results of Section2.For n=24,Λ24 is the Leech lattice,and the result is a theorem of Conway(see[3],Chapter12). The case of dimensions9and10is dealt with in[7],Section2,Table III.We are thus left with the18laminated lattices of dimension n∈[11,23].We now show how Lemma3.2can be used to deal with16of them.Recall that O23stands for the unimodular23-dimensional lattice of minimum3. The lattice Z⊥O23can be defined as a Kneser–neighbour ofΛ24through a norm4 vector,which shows that(O23)even is isometric toΛ23.Now,it is shown in[2]that the antilamination s of O23(the descending chain of the densest cross–sections)pro-duce a unique lattice(denoted by O n)in dimensions23to14.Since the antilamina-tions ofΛ23also produce theΛn series in these dimensions,we have(O n)even Λn for14≤n≤23.Then wefind two13-dimensional lattices,which allows againto deal withΛmax13andΛmin13(Λmid13is a dead–end for laminated lattices).We caneven consider dimensions12and11.Explicitly,in the notation of[2],we haveΛmax 13 (O13b)even,Λmin13(O13a)even,Λmax12(O12b)even,Λmid12(O12a)even,andΛmax11 (O11)even.As forΛmid13,it is also the even part of an integral norm3lattice,discovered by Plesken and Pohst([8]),indeed the lattice with s=84of their list.Lemma3.2shows that all the sixteen lattices listed above contain a class of minimum12.To complete the proof of Theorem3.1,it suffices to consider the two latticesΛmin 11andΛmin12.We have shown using PARI-GP that vectors of norm N≤8do notrepresent all classes.For the sake of completeness,we display below the weightedformulae for vectors of norm N≤8for the four latticesΛ9,Λ10,Λmin11andΛmin12.The notation is that of section2.Thefirst two numbers are s4and s6;we then give for each weight the numbers of vectors in a given orbit with this weight.(Note that two congruent vectors may belong to different orbits.)Λ9:136+128+1+89+560+5128+4484=511.Λ10:168+384+3+24+768+288+48+192+1152=1023.Λmin 11:216+816+549+10328+9605+19204+3843=1967.Λmid 12:312+1728+12+969+768+192+248+768+30725+ 1536+23044=3903.6J.MARTINETWe observe that there are (211−1)−1967=80missing classes in the case ofΛmin 11and (212−1)−3903=192in the case of Λmin 12.4.Odd Lattices Modulo 2.In this section,we consider as previously a lattice Λof dimension n and mini-mum m .Our aim is to study the contribution of norm 2m +1vectors.Such vectors of course do not exist if Λis even.In the proposition below,the rˆo le of the dual ofan A k lattice resembles the one it plays for norm 2m +m vectors with respect toΛmod 3.Theorem 4.1.Let Λbe integral.(1)Vectors of norm 2m +1(if any)are minimal in their class modulo 2.(2)If min Λis odd,each class contains at most 2m +2pairs of such vectors,and when this bound is attained,their configuration is that of S (A ∗2m +1).Proof.Let x ∈S 2m +1(Λ),and let y ≡x mod 2Λ,say,y =x +2z .By Proposition 1.1,we have N (y )≥2m −1.Since y ≡x mod 2=⇒N (y )≡N (x )mod 4,we must have N (y )≥N (x ),which proves the first part of Theorem 4.1.Suppose now that N (y )=N (x )=2m +1.Writing −y =x −2(x +z ),we see that changing y into −y amounts to exchanging z and −(x +z ).In the sequel,we shall assume that N (z )=m +1and N (x +z )=m .With this choice we have x ·z =−(m +1),hence x ·y =2m +1−2(m +1)=−1.Lemma 4.2.Let ±x 1,...,±x r ,r ≥2be a system of norm 2m +1vectors in Λbelonging to the same class modulo 2Λ.Then for a convenient choice of the x i among x i ,−x i ,the scalar products x i ·x j ,j =i all have the same value,namely +1if m is even,and −1if m is odd.proof of 4.2.For i =2,...,r ,define z i by x i =x 1+2z i .Taking x =x 1and y =x i in the calculation we made in the course of the proof of Theorem 4.1,we see that we may choose the signs of the x i so that x 1·x i =−1for all i ≥2.We then havex i ·x j =(x 1+2z i )·(x 1+2z j )=2m +1−4(m +1)+4z i ·z jhencez i ·z j =2m +3+x i ·x j4for 2≤i <j ≤r .We must have x i ·x j +2m +3≡0mod 4,whence the result for2≤i <j ≤n .Negating x 1if m is even yields the desired result in all cases. End of proof of 4.1.Since m is odd,we may assume by Lemma 4.2that x i ·x j =−1for all pairs (i,j )with j =i .Since N (x 1+···+x r )=r (2m +1)−2 r 2=r (2m +2−r )≥0,we have r ≤2m +2,and x 1,...,x r generate a canonical section of A ∗2m +1scaledto norm 2m +1.Since the vectors x i +x j 2belong to Λ,we are done.Example 4.3.Let Λbe E ∗7scaled to minimum 3.This is an integral lattice,whose norms are the positive integers congruent to 0or −1modulo 4.We have s 3=28,s 4=63and s 7=288,hences 3+s 4+s 78=28+63+36=127=27−1.REDUCTION MODULO2AND3OF EUCLIDEAN LATTICES,II7 Theorem4.1hence shows that the shortest vectors in classes modulo2ofΛare those of norm3,4,and7.ttices of Codimension1.In this section,we assume thatΛis integral.We explain how to construct lattices of dimension n−1from a vector e∈Λof normµin the range m≤µ<2m. (Everything also works ifµ=2m,but the configuration of minimal vectors of the lattices we are going to construct are then uninteresting orthogonal configurations.) We denote by C the class of e modulo2.Lemma5.1.Let e be as above and let x≡e mod2Λ.Then one of the following conditions holds:(1)x is proportional to e.(2)N(x)>4m−µ.(3)N(x)=4m−µand x is orthogonal to e.Proof.Write x=e+2z.Proposition1.1shows that if x is not proportionalto e,then N(x)≥4m−N(e),and that if equality holds,then e·z=−N(e)2,whichimplies e·x=e·(e+2z)=0.Lemma5.2.L=C∪2Λis a lattice of determinant22n−2det(Λ).Proof.We have C∪2Λ=2Λ∪(e+2Λ).Hence L is a lattice containing2Λto index2,which shows that det(L)=2−2det(2Λ).By Lemma5.1,min L=µand S(L)={±e}.To obtain a lattice with a larger minimum,we consider L e=(R e)⊥∩L.Proposition5.3.L e=(R e)⊥∩L is an(n−1)-dimensional lattice of minimum M≥4m−µand determinant22n−2µdet(Λ)ifµis even,and22nµdet(Λ)ifµis odd.Proof.Only the last assertion needs a proof.Given a primitive vector e ∈L∗, the determinant of L =L∩(R e )⊥is det(L )=det(L)N(e )(see[6],Proposi-tion1.3.4;N(e )is the determinant of the1-dimensional lattice(R e )∩L∗).Here wemust determine a generator e of R e∩L∗.We have L= 2Λ,e and(2Λ)∗=12Λ∗,henceL∗= y2y∈Λ∗,y·e≡0mod2.Ifµis even,e∈L∗;ifµis odd,2e∈L∗.Since e is primitive inΛ(because N(e)<4m),a congruence e·y≡0mod a may not hold onΛ∗for an integer a>1. This shows that ifµis even(resp.odd),e(resp.2e)is primitive in L∗.This completes the proof of the proposition.Proposition5.4.LetΛe=L e ifµ≡1mod2,Λe=1√2L e ifµ≡2mod4,andΛe=12L e ifµ≡0mod4.ThenΛe is an integral lattice,and we havedet(Λe)=22nµdet(Λ)ifµ≡1mod2,det(Λe)=2n−2µdet(Λ)ifµ≡2mod4, and det(Λe)=µ4det(Λ)ifµ≡0mod4.Proof.The assertions concerning the determinant ofΛe follow immediately from Proposition5.3.It thus suffices to prove that x,y∈L e=⇒x·y≡0 mod(4,µ).Write x=2z(resp.x=e+2z)if x∈2Λ(resp.x/∈2Λ),and similarly y=2t or y=e+2t.If both x and y belong to2Λ,then x·y≡0mod4.If,say,8J.MARTINETx∈2Λand y/∈2Λ,we again have x·y=2(z·e)+4z·t=4z·t≡0mod4. Finally,in the remaining case,we havex·y=µ+2e·z+2e·t+4z·t and2(e·z)≡2(e·t)≡−µmod4, hence x·y≡−µmod4.We now give some examples.In all cases we shall consider,the minimum ofΛe is equal to the lower bound given in Proposition5.3.If m=2and ifΛis even,the only possible choice isµ=2.Take forΛan irreducible root lattice of dimension n≥2.Then norm6vectors inΛbelong to one or two classes modulo2,and exactly one such class C contains a norm2 vector e.WhenΛis isometric to A n(n≥2),D n(n≥4),E6,E7,and E8,Λe has minimum3,and s(Λe)is equal to n−1,2(n−2),10,16,and28respectively. The lattice corresponding to E8is a scaled copy of E∗7,and E7and E6yield lattices similar to Coxeter’s D+6and A25.If m=4,we may chooseµ=4orµ=6,obtaining in general a latticeΛe of minimum M e=3ifµ=4and M e=5ifµ=6(and no other value ifΛis even).Theorem5.5.LetΛbe an integral lattice of minimum4.(1)Ifµ=4,then minΛe≥3and det(Λe)=det(Λ).(2)Ifµ=6,Λe is an integral lattice whose norm5vectors have mutualscalar products±1.In particular,directions of norm5vectors constitutean equiangular family of lines.Proof.Ifµ=4,the result is an immediate consequence of Proposition5.4. Let nowµ=6,and let x=e+2z and y=e+2t(y=±x)be two norm10vectors in L e.We have x·y=−6+4z·t(see the proof of Proposition5.4).Since z and t are minimal inΛ,we have z·t∈{4,2,1,0,−1,−2,−4}.Since y=±x,we have|x·y|≤5,which implies z·t=2or1,hence x·y=±2.This proves that non-proportional norm5vectors inΛe have scalar product±1,hence that they generate an equiangular family of lines.We now consider the important special case ofΛ=Λ24(the Leech lattice).Corollary5.6.LetΛbe the Leech latticeΛ24.(1)Ifµ=4,Λe is the unimodular lattice O23(min=3,s=2300).(2)Ifµ=6,Λe is the integral lattice of minimum5with s=276(=23·242)which is dual(up to scale)to the lattice M23[2]of[9],Table19.2.Proof.Ifµ=4,Λe is a unimodular lattice of minimum M≥3,hence iso-metric to O23([3],Table16.7).Ifµ=6,we use the fact thatΛe has minimum5and kissing number |a10||a6|=276,where we denote as in[7]by a6(resp.a10)the unique orbit of vectors of norm6 (resp.10)inΛ24.Theorem9.1of[9](and the results of[5]on equiangular families of lines)now shows thatΛe,as an integral lattice of minimum5with equiangular directions of minimal vectors and maximal possible value of s,is similar to M23[2]∗.[Forµ=4,since |b12||a4|=2300(notation of[7]),we recover the equality s(O23)=2300.]REDUCTION MODULO 2AND 3OF EUCLIDEAN LATTICES,II 9Remark 5.7.The integral scaled copies L of M 23[2]∗(of minimum 5)and L of M ∗23(of minimum 15)which occur in Table 19.2of [9]have the same configurations of minimal vectors.Indeed,L contains to index 2a lattice isometric to √3L .The successive layers of L (resp.L )have norms 5,8,9,12,...(resp.15,20,24,...).This shows that L contains a class modulo 3of minimum 60,which producesvectors of norm 3·609=20in L .Similarly,using the parity class of O 23(of minimum 15;see [4],where Elkies proves a much more general result),we obtain after rescaling an integral lattice of minimum 12.This lattice is indeed proportional to Λ∗23.References[1] E.Bannai,A.Munemasa,B.Venkov,The Nonexistence of Certain Tight SphericalDesigns (with an appendix by Y.-F.S.Peterman),preprint.[2] C.Batut,J.Martinet,A Catalogue of Perfect Lattices ,http://www.math.u-bordeaux.fr/ martinet.[3]J.H.Conway,N.J.A.Sloane,Sphere Packings,Lattices and Groups ,Grundlehren 290,Springer-Verlag,Heidelberg (1988).Third edition:1993.[4]N.Elkies,Lattices and codes with long shadows ,Math.Res.Lett.2(1995),643–651.[5]P.W.H.Lemmens,J.J.Seidel,Equiangular lines ,J.Algebra 24(1973),494–512.[6]J.Martinet,Perfect Lattices in Euclidean Spaces ,Grundlehren 327,Springer-Verlag,Heidelberg (2003).(English corrected and updated edition of a French version,Masson(now Dunod),Paris,1996.)[7]J.Martinet,Reduction Modulo 2and 3of Euclidean Lattices ,J.Algebra 251(2002),864–887.[8]W.Plesken,M.Pohst,Constructing integral lattices with prescribed minimum.I ,Math.Comp.45(1985),209–221and S5–S16.[9] B.Venkov,R´e seaux et “designs”sph´e riques (Notes by J.Martinet),in R´e seaux eucli-diens,designs sph´e riques et groupes,L’Ens.Math.,Monographie 37,J.Martinet,ed.,Gen`e ve (2001),10–86.J.Martinet A2X,Institut de Math ´e matiques Universit ´e Bordeaux 1351,cours de la Lib ´eration F–33405TALENCE cedexE-mail address :martinet@math.u-bordeaux.fr。