Pad'e expansion and the renormalization of nucleon-nucleon scattering

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Non-adiabatic Kohn-anomaly in a doped graphene monolayer

a r X i v :c o n d -m a t /0611708v 1 [c o n d -m a t .m e s -h a l l ] 28 N o v 2006Non-adiabatic Kohn-anomaly in a doped graphene monolayerMichele Lazzeri and Francesco MauriIMPMC,Universit´e s Paris 6et 7,CNRS,IPGP,140rue de Lourmel,75015Paris,France(Dated:February 6,2008)We compute,from ﬁrst-principles,the frequency of the E 2g ,Γphonon (Raman G band)of graphene,as a function of the charge doping.Calculations are done using i)the adiabatic Born-Oppenheimer approximation and ii)time-dependent perturbation theory to explore dynamic eﬀects beyond this approximation.The two approaches provide very diﬀerent results.While,the adiabatic phonon frequency weakly depends on the doping,the dynamic one rapidly varies because of a Kohn anomaly.The adiabatic approximation is considered valid in most materials.Here,we show that doped graphene is a spectacular example where this approximation miserably fails.PACS numbers:71.15.Mb,63.20.Kr,78.30.Na,81.05.UwGraphene is a 2-dimensional plane of carbon atoms arranged in a honeycomb lattice.The recent demonstra-tion of a ﬁeld-eﬀect transistor (FET)based on a few-layers graphene sheet has boosted the interest in this system [1,2,3].In particular,by tuning the FET gate-voltage V g it is possible to dope graphene by adding an excess surface electron charge.The actual possibil-ity of building a FET with just one graphene monolayer maximizes the excess charge corresponding to a single atom in the sheet.In a FET-based experiment,graphene can be doped up to 31013cm −2electron concentra-tion [1,2],corresponding,in a monolayer,to a 0.2%valence charge variation.The resulting chemical-bond modiﬁcation could induce a variation of bond-lengths and phonon-frequencies of the same order,which would be measurable.This would realize the dream of tuning the chemistry,within an electronic device,by varying V g .The presence of Kohn anomalies (KAs)[4,5]in graphene could act as a magnifying glass,leading to a variation of the optical phonon-frequencies much larger than the 0.2%expected in conventional systems.On the other hand,the phonon-frequency change induced by FET-doping could provide a much more precise de-termination of the KA,with respect to other experimen-tal settings.KAs manifest as a sudden change in the phonon dispersion for a wavevector q ∼2k F ,where k F is a Fermi-surface wavevector [4].The KA can be deter-mined by studying the phonon frequency as a function of q by,e.g.,inelastic x-ray,or neutron scattering.These techniques have a ﬁnite resolution,in q and energy,which limits the precision on the measured KA dispersion.In graphene,2k F is proportional to V g .This suggests an alternative way to study the KA,that is to measure the phonon frequency at a ﬁxed q and to vary 2k F by chang-ing V g .Within this approach,one could use Raman scat-tering,which has a much better energy and momentum resolution than x-ray and neutron scattering.This ap-proach is feasible for graphene,which has a KA for the Raman-active E 2g Γ-phonon [5](Raman G -band).In this paper,we compute the variation of phonon frequency of the Raman G -band (E 2g mode at Γ)in a graphene monolayer,as a function of the Fermi level.First,the calculations are done using a fully ab-initio approach within the customary adiabatic Born-Oppenheimer approximation.Then,time-dependent perturbation theory (TDPT)is used to go beyond.Ab-initio calculations are done within density func-tional theory (DFT),using the functional of Ref.[6],plane waves (30Ry cutoﬀ)and pseudopotentials [7].The Brillouin zone (BZ)integration is done on a uni-form 64×64×1grid.An electronic smearing of 0.01Ry with the Fermi-Dirac distribution is used [8].The two-dimensional graphene crystal is simulated using a super-cell geometry with an interlayer spacing of 7.5˚A (if not otherwise stated).Phonon frequencies are calcu-lated within the approach of Ref.[9],using the PWSCF code [10].The Fermi-energy shift is simulated by consid-ering an excess electronic charge which is compensated by a uniformly charged back-ground.The dependence of the Fermi energy ǫF on the surface electron-concentration σis determined by DFT (Fig 1).In graphene,the gap is zero only for the two equivalent K and K’BZ-points and the electron energy ǫcan be approximated as ǫπ∗/π(K +k )=±βk for the π∗and πbands,where k is a small vector.Within this approxi-mation,at T =0K temperature σ=sign(ǫF )ǫ2FeV210.361013cm −2(1)where β=5.52eV ·˚A from DFT,sign(x )is the sign ofx and ǫF =0at the πbands crossing.We remark that,from Fig 1,the typical electron-concentration obtained in experiments [1,2]corresponds to an important Fermi-level shift (∼0.5eV).For such shift,the linearized bands are still a good approximation (Fig.1).The dependence of the graphene lattice-spacing a on σ,a (σ),is obtained by minimizing F (σ,A )=[E (σ,A )−E (0,A 0)]/A with respect to A ,where E (σ,A )is the en-ergy of the graphene unit-cell,A is unit-cell area and A 0=5.29˚A 2is the equilibrium A [11]at zero σ.E (σ,A )is computed by DFT letting the inter-layer spacing,L ,tend to inﬁnity in order to eliminate the spurious interac-2Electron concentration (1013 cm -2)[a (σ)-a (0)]/a (0) (10-3)Electrons in the surface-unit-cellElectron concentration (1013 cm -2)F e r m i e n e r g y (e V )FIG.1:Graphene monolayer.Upper panel:ǫF as a function of the surface electron-concentration σfrom DFT calculations and from linearized bands (at T =300K).Lower panel:in-plane lattice spacing a as a function of σ.The ﬁtting function is Eq.2and the dashed line is from Ref.[13].tion between the background and the charged sheet [12].∆a (σ)=[a (σ)−a (0)]/a (0)was determined in Ref.[13]for intecalated graphite on the basis of a semi empiri-cal ing the same functional dependence as in Ref.[13],our DFT calculations are ﬁtted by∆a (σ)=6.748·10−6|σ|3/2+1.64·10−4σ,(2)where σis in units of 1013cm −2.With σ=31013cm −2,the lattice spacing variation is ∼0.05%,which is,as ex-pected,of the same order of the valence-charge variation.Electron concentration (1013 cm -2)-40-2002040F r e q u e n c y s h i f t (c m -1)FIG.2:Frequency of the E 2g Γphonon (Raman G band)as a function of σ:shift with respect to the zero-doping fre-quency.Calculations are done using standard DFT (adia-batic)or TDPT (dynamic),keeping the lattice-spacing con-stant (constant lattice)or varying it according to Eq.2(ex-panded lattice).Points are DFT calculations.Dashed line is from Eq.3.Experiments should be compared with the continuous line.The frequency of the E 2g Γphonon is computed by static perturbation theory of the DFT energy [9],i.e.from the linearized forces acting on the atoms due to the static displacement of the other atoms from their equilib-rium positions.This approach is based on the adiabatic Born-Oppenheimer approximation,which is the standard textbook approach for phonon calculations and is always used,to our knowledge,in the ab-initio frequency calcu-lations.The computed zero-doping phonon frequency is ω0a /(2πc )=1554cm −1,where c is the speed of light.The frequency variation ∆ωwith σis reported in Fig.2.Cal-culations are done keeping the lattice-spacing constant at a (0),or varying it according to Eq.2.In this latter case,∆ωis ﬁtted by ∆ωN kk nm|D (k +q )m,k n |2[˜f (k +q )m −˜f k n ]3 Imposingω=0andδ=0in Eq.4,one obtains thestandard adiabatic approximation[9]and the phonon fre-quency isωǫF a= DǫF q(ω)/M.However,consid-ering dynamic and doping eﬀects as perturbations,atthe lowest order one can insert the adiabatic zero-dopingphonon frequencyω0a in Eq4and obtain the real part ofthe dynamic frequency fromωǫFd=ℜeN k k,n=m|D k m,k n|2[˜f k m−˜f k n]N k k,n|D k n,k n|2δT(ǫk n−ǫF),(6)whereδT(x)=−d f T(x)/(dx).In the dynamic caseFǫF0(ω0a)=2ǫk m−ǫk n+ ω0a+iδ.(7)In Eq.6(adiabatic case),there are two contributions, theﬁrst from inter-band and the second from intra-band transitions(depending onδT and proportional to the den-sity of states atǫF).On the contrary,in Eq.7(dynamic case)only inter-band transitions contribute.The variation ofωǫF withǫF is∆ω=ωǫF−ω0≃DǫF−D02Mω0a(9)∆ωd=ℜe ˜FǫF(ω0a)−˜F0(ω0a)βk−δT(βk−ǫF)−δT(−βk−ǫF)},(11) whereα=2A0 D2Γ /π.Substituting Eq.11into Eq.9 one obtains∆ωa.At any T,∆ωa=0.This re-sult is not trivial and comes from the exact cancella-tion of the inter-band(πtoπ∗,ﬁrst line of Eq.11) and intra-band(πtoπandπ∗toπ∗,second line of Eq.11).For example,at T=0,both contributions to ∆ωa are large and equal toα′|ǫF|and−α′|ǫF|,re-spectively,whereα′= α/(2Mω0aβ2)=4.4310−3and α′/(2π c)=35.8cm−1/(eV).Concluding,an adiabatic calculation ofωǫF does not show any singular behavior inǫF related to the Kohn anomaly,in agreement with the state-of-the-art adiabatic DFT calculations of Fig.2. In the dynamic case˜FǫF(ω0a)=α ¯k−¯k f T(βk−ǫF)−f T(−βk−ǫF)4ln |ǫF|− ω0a|ǫF|+ ω0a4Finally,the Raman G -band has a ﬁnite homogeneous linewidth due to the decay of the phonon into electron-hole pairs.Such EPC broadening can be obtained either from the imaginary part of the TDPT dynamical matrix (Eq.12)or,equivalently,from the Fermi golden rule [16]:γ=π2πc α′ f T − ω0a 2−ǫF (14)where γis the full-width half-maximum (FWHM)in cm −1.At T =0and ǫF =0,one recovers the result of Ref.[16],γ=11.0cm −1.The phonon-phonon scattering contribution to the FWHM is smaller (∼1cm −1[17])and independent of ǫF .The total homogeneous FWHM is re-ported in Fig.3.The FWHM displays a strong doping dependence;it suddenly drops for |σ|∼0.11013cm −2(|ǫF |∼0.1eV).Indeed,because of the energy and mo-mentum conservation,a Γphonon decays into one elec-tron (hole)with energy ω0a /2above (below)the level crossing.At T =0K such process is compatible with thePauli exclusion-principle only if |ǫF |< ω0a /2.-0.6-0.4-0.200.20.40.6Electron concentration (1013 cm -2)-8-4048F r e q u e n c y s h i f t (c m -1)T=300 K T=70 K T=4 Kdynamic+expanded lattice04812P h o n o n l i n e w i d t hF W H M (c m -1)FIG.3:(Color online)Linewidth and dynamic frequency of the E 2g Γmode (Raman G band).See the caption of Fig.1.Concluding,a Kohn anomaly dictates the dependence of the highest optical-phonon on the wavevector q ,in undoped graphene [5].Here,we studied the impact of such anomaly on the q =0phonon,as a function of the charge-doping σ.We computed,from ﬁrst-principles,the phonon frequency and linewidth of the E 2g ,Γphonon (Raman G band)in the σ-range reached by recent FET experiments.Calculations are done using i)the custom-ary adiabatic Born-Oppenheimer approximation and ii)time-dependent perturbation theory to explore dynamic eﬀects beyond this approximation.The two approaches provide very diﬀerent results.The adiabatic phonon fre-quency displays a smooth dependence on σand it is not aﬀected by the Kohn anomaly.On the contrary,when dy-namic eﬀects are included,the phonon frequency and life-time display a strong dependence on σ,due to the Kohn anomaly.The variation of the Raman G -band with the doping in a graphene-FET has been recently measured by two groups [18,19].Both experiments are well de-scribed by our dynamic calculation but not by the more approximate adiabatic one.We remark that the adiabatic Born-Oppenheimer approximation is considered valid in most materials and is commonly used for phonon calcu-lations.Here,we have shown that doped graphene is a spectacular example where this approximation miserably fails.We aknowledge useful discussions with A.M.Saitta,A.C.Ferrari and S.Piscanec.Calculation were done at IDRIS (Orsay,France),project n o 061202.[1]K.S.Novoselov et al.Science 306,666(2004);K.S.Novoselov et al.Nature 438,197(2005).[2]Y.Zhang,Y.W.Tan,H.L.Stormer,and P.Kim,Nature438,201(2005).[3]A.C.Ferrari et al.,Phys.Rev.Lett.97,187401(2006).[4]W.Kohn Phys.Rev.Lett.2,393(1959).[5]S.Piscanec,zzeri,F.Mauri,A.C.Ferrari,and J.Robertson,Phys.Rev.Lett.93,185503(2004).[6]J.P.Perdew,K.Burke,and M.Ernzerhof Phys.Rev.Lett.77,3865(1996).[7]D.Vanderbilt,Phys.Rev.B 41,7892(1990).[8]M.Methfessel and A.T.Paxton Phys.Rev.B 40,3616(1989).[9]S.Baroni,S.de Gironcoli,A.Dal Corso,and P.Gian-nozzi,Rev.Mod.Phys.73,515(2001).[10]S.Baroni et al..[11]Consider a graphene sample of N unit-cells,which is incontact with an electrode of area A e .If N >A e /A ,A is obtained by minimizing the total energy of the sample E (σ,A )A e /A +(N −A e /A )E (0,A 0).This is equivalent to minimizing F (σ,A ).[12]By a simple electrostatic model,the dependence ofthe cell-energy E on L is E (L )/A =π/6σ2L +α0+α1/L +α2/L 2+O (1/L 3).E is computed for L =7.5,15,22.5,30,37.5˚A and the parameters αi ,are ﬁtted to obtain the limit L →∞.[13]L.Pietronero and S.Str¨a ssler Phys.Rev.Lett.47,593(1981).[14]Eqs.4.17a and 4.23of P.B.Allen,in Dynamical Proper-ties of Solids ,Ed.by G.K.Horton and A.A.Maradudin,(North-Holland,Amsterdam,1980),v.3,p.95-196.[15]The dependence of ∆n on ωq and ǫF is neglected,becausethe functional of Eq.4is stationary with respect to ∆n .Notice that there are other,equivalent expressions for the dynamical matrix which are not stationary in ωq ,for which this approximation is not justiﬁed.[16]zzeri,S.Piscanec,F.Mauri,A.C.Ferrari,and J.Robertson,Phys.Rev.B 73,155426(2006).[17]N.Bonini,zzeri,F.Mauri,N.Marzari unpublished .[18]J.Yan,Y.Zhang,P.Kim,and A.Pinczuk (2006),unpub-lished.Results shown the 28Sept.2006at the Graphene Conference,Max Planck Institut,Dresden.[19]S.Pisana,zzeri,C.Casiraghi,K.S.Novoselov,A.K.5 Geim,A.C.Ferrari,F.Mauri,cond-mat/0611714(2006).。

Asymptotic behaviour of some infinite products involving prime

n≥0 αn z n,
then the asymptotic behaviour of αn
as n → ∞ is closely related to that of F (z ) as z → ∞ by the formula αn = 1 2iπ z −n−1 F (z ) dz,
|z |= r
,
(1)
,
(2)
1
uniformly for 1 ≤ m ≤ M log log x, for any ﬁxed M > 0, where r = (m − 1)/(log log x), G and F are entire functions deﬁned by G(z ) = F (z ) = 1 z 1+ Γ(z + 1) p prime p−1 1 z 1+ Γ(z + 1) p prime p 1− 1− 1 p
p prime
log 1 +
z p
+ z log 1 −
1 p
,
(z = −p, p prime).
(5)
Let H0 denote a certain truncated Hankel contour around the origin (counter-clockwise) in the s-plane (see the next section for precise deﬁnition). Throughout this paper, the symbol ε always denotes a small positive quantity whose value may vary from one occurrence to another. Our main result is the following. Theorem 1 The function f satisﬁes − 1 f (z ) = z+1 2iπ πz s log(1/s) ds + R(z ), (1 + s) sin πs (6)

Global Study of Nuclear Structure Functions

Global Study of Nuclear Structure Functions
S. A. Kulagin∗ Institute for Nuclear Research, 117312 Moscow, Russia R. Petti† CERN, CH-1211 Gen´ eve 23, Switzerland
arXiv:hep-ph/0412425v3 22 Oct 2005
Abstract
We present the results of a phenomenological study of unpolarized nuclear structure functions for a wide kinematical region of x and Q2 . As a basis of our phenomenology we develop a model which takes into account a number of diﬀerent nuclear eﬀects including nuclear shadowing, Fermi motion and binding, nuclear pion excess and oﬀ-shell correction to bound nucleon structure functions. Within this approach we perform a statistical analysis of available data on the ratio of the nuclear structure functions F2 for diﬀerent nuclei in the range from the deuteron to the lead. We express the oﬀ-shell eﬀect and the eﬀective scattering amplitude describing nuclear shadowing in terms of few parameters which are common to all nuclei and have a clear physical interpretation. The parameters are then extracted from statistical analysis of data. As a result, we obtain an excellent overall agreement between our calculations and data in the entire kinematical region of x and Q2 . We discuss a number of applications of our model which include the calculation of the deuteron structure functions, nuclear valence and sea quark distributions and nuclear structure functions for neutrino charged-current scattering.

The Crossover beteween Aslamazov-Larkin and Short Wavelength Fluctuations Regimes in HTS Co

a rXiv:c ond-ma t/97390v11Mar1997The crossover between Aslamazov-Larkin and short wavelength ﬂuctuation regimes in HTS conductivity experiments M.R.Cimberle,C.Ferdeghini,E.Giannini,D.Marr´e ,M.Putti,A.Siri INFM /CNR,Dipartimento di Fisica,Universit`a di Genova,via Dodecaneso 33,Genova 16146,Italy F.Federici,A.Varlamov Laboratorio “Forum”dell’INFM,Dipartimento di Fisica Universit`a di Firenze,Largo E.Fermi 2,50125Firenze,Italy (February 1,2008)We present paraconductivity (AL)measurements in three diﬀerent high temperature supercon-ductors:a melt textured Y Ba 2Cu 3O 7sample,a Bi 2Sr 2CaCu 2O 8epitaxial thin ﬁlm and a highly textured Bi 2Sr 2Ca 2Cu 3O 10tape.The crossovers between diﬀerent temperature regimes in excess conductivity have been analysed.The Lawrence-Doniach (LD)crossover,which separates the 2D and 3D regimes,shifts from lower to higher temperatures as the compound anisotropy decreases.Once the LD crossover is overcome,the ﬂuctuation conductivity of the three compounds shows the same universal behaviour:for ǫ=ln T /T c >0.23all the curves bend down according to the 1/ǫ3law.This asymptotic behaviour was theoretically predicted previously for the high temperature region where the short wavelength ﬂuctuations (SWF)become important.PACS:74.25.-q;74.25.Fy;74.40.+k It is well known that,owing to strong anisotropy,high critical temperature and low charge carrier concentration,thermodynamic ﬂuctuations play an important role in the explanation of the normal state properties of high tempera-ture superconductors (HTS).Just after the realization of high quality epitaxial single crystal samples,the in-plane ﬂuc-tuation conductivity was investigated in detail and the Lawrence-Doniach (LD)crossover between three-dimensional (3D)and two-dimensional (2D)regimes (or at least a tendency to it)was observed in the vicinity of T c in the majority of HTS compounds.Analogous phenomena were observed in magnetic susceptibility,thermoconductivity 1and other properties of HTS.Let us recall that LD crossover takes place in the temperature dependence of in-plane conductivity and it is related to the fact that ﬂuctuative Cooper pairs motions change from 2D to 3D rotations.It takes place at the temperature T LD which is deﬁned by the condition ξc (T LD )≈s ,where ξc is the coherence length and s is the interlayer distance.Nevertheless,the LD crossover in the temperature dependencies of diﬀerent characteristics does not exhaust all possibilities:additional crossovers can be observed in HTS compounds.For instance,another kind of crossover (0D →3D)can take place in c -axis paraconductivity temperature dependence,at the same temperature T LD .It is due to the fact that the pair propagation along c-axis has a zero-dimensional character relatively far from T c and it changes into a three-dimensional rotation in the immediate vicinity of the transition.This eﬀect was predicted 2and observed 3in fully oxygenated YBCO samples while in BSCCO samples it is masked by the increase of resistivity due to ﬂuctution density of states renormalization.Below we will remind the reader of the possible kinds of crossover phenomena taking place in layered superconductors and ﬁnally we will present the experimental evidences of the crossover related to the breakdown of Ginzburg Landau (GL)approximation,due to the importance of short wavelength ﬂuctuations.How the LD crossover appears in the framework of the GL theory can be shown explicitly considering the model of an open electron Fermi surface which,for instance,can be chosen in the form of a “corrugated cylinder”4.In this case the energy spectrum has the formξ(p )=ǫ0(p )+J cos(p ⊥s )−E F ,(1)where ǫ0(p )=p 2/(2m ),p ≡(p ,p ⊥),p ≡(p x ,p y )is a two-dimensional,intralayer wavevector,and J is an eﬀective hopping energy.The Fermi surface is deﬁned by the condition ξ(p F )=0and E F is the Fermi energy.This spectrum is the most appropriate for strongly anisotropic layered materials where J/E F≪1.In the framework of the Ginzburg-Landau theory for an isotropic spectrum,the ﬂuctuation contribution to the free energy of a superconductor above the critical temperature can be presented as the sum over long wavelength ﬂuctuations 5:F =−T qln πTsome dimensional coeﬃcient)and in the case of anisotropic spectrum(1)must be substituted by the more soﬁsticated expression including the additional dependence on q⊥:(v·q)2 = [ξ(p)−ξ(q−p)]2 =1f(ǫ).(4)16¯h sIn the GL region of temperature,whereǫ≪1,f(ǫ)=1/ǫand the result coincides with the well known AL one.In the opposite caseǫ≫1,for clean2D superconductors,f(ǫ)∼1/ǫ3=1/ln3(T/T c)was carried out.In the theoretical consideration it was natural to assume formally the very rigid restrictionǫ≫1for the validity of the latter asymptotic behaviour.Nevertheless,as it will be seen below,in experiments the crossover to this asymptotic behaviour takes place universally for all the samples investigated atǫ∼0.23and this can be attributed to some particularly fast convergence of the integrals in the expression of f(ǫ).The long tails in the in-planeﬂuctuation conductivity of HTS materials have been observed frequently.One of the eﬀorts toﬁt the high temperature paraconductivity with the extended AL theory results was undertaken in13where the deviation of the excess conductivity from AL behaviour was analysed for three Bi2Sr2CaCu2O8epitaxialﬁlms. Very goodﬁt with the formula(4)was found in the region of temperatures0.02<∼ǫ<∼0.14.We show here that the careful analysis of the higher temperature region(just above the edge of the region investigated in13)allows to observe the surprisingly early approaching to the SWF asymptotic regime(at the reduced teperatureǫ∗∼ln(T∗/T c)∼0.23). We have performed resistivity measurement of three diﬀerent HTS compounds:a melt textured YBa2Cu3O7sample (Y123),a Bi2Sr2CaCu2O8(Bi2212)thickﬁlm and a highly textured Bi2Sr2Ca2Cu3O10(Bi2223)tape.The Y123was obtained by melting14;the sample was cut in a nearly regular parallelepipedal shape with a cross section of about 4mm2and a length of7mm.The resistivity measurements were performed from85to330K.The critical temperature,deﬁned as the point where the temperature derivative is maximal,is92K;ρN(100K)=120µΩcm, whereρN is the resistivity in the normal state extrapolated from the high temperature region whereρis linear.The Bi2212ﬁlm was prepared by a liquid phase epitaxy technique15.Theﬁlm has a thickness of about1µm.The resistivity measurements were performed from80to170K.The critical temperature was estimated to be84.2K andρN(100K)=150µΩcm.The Bi2223tape was obtained by means of the power in tube procedure,as described elsewhere16.The thickness of the oxideﬁlament inside the tape was about30µm;theﬁlament turned out to bestrongly textured(rocking angle≈8◦)with the c-axis oriented perpendicular to the tape plane.The resistivity measurements were performed in the range from100to250K,after removing the silver sheathing chemically.The critical temperature was estimated to be108K andρN(100K)=300µΩcm.We ascribe this high value ofρN to diﬀerent causes:ﬁrst,the grain boundaries may determine a resistance in series with the grain resistance;second,the chemical treatment may have damaged the surface of the sample and the eﬀective cross section of the superconductor can be decreased.The excess conductivity was estimated by subtracting the background of the normal state conductivityσN= 1/ρN.The evaluation ofρN was made with particular accuracy;in fact,starting the interpolation at a certain temperature corresponds to forcingσfl to vanish artiﬁcially at such temperature.Therefore,we need to estimateρN at a temperature as large as possible and to verify thatρN does not depend on the temperature range where the interpolation is performed.In the case of Y123sample the resistivity shows a linear behavior from160to330K.In this range we have veriﬁed thatρN does not change by shifting the interpolation temperature region. Therefore,for the Y123sample,the upper limit ofǫat which the excess conductivity may be analysed isǫup≈ln(160/92)=0.55.In an analogous way we obtainǫup≈0.46and0.51for Bi2212and Bi2223,respectively.In Fig.1,in a log-log scale,we plotσﬂ 16hsǫ.The interlayer distance s is considered as a free parameterand it has been adjusted so that the experimental data can follow the1/ǫbehaviour in theǫregion where the AL behavior is expected.We can see that all the curves exhibit the same general behaviour.The region where the2D1/ǫbehaviour is followed,has diﬀerent extension for each compound,depending on its anisotropy,and atǫ≈0.23all the curves bend downward and follow the same asymptotic1/ǫ3behaviour.We discuss now some features in detail:1)The interlayer distance values weﬁnd are the following:for Y123we obtain s≈13˚A which must be comparedwith the YBCO interlayer distance that is about12˚A;for Bi2212we obtain s≈11˚A to be compared with 15˚A,and for Bi2223we obtain s≈25˚A to be compared with18˚A.The diﬀerences in the interlayer distance evaluation are all compatible with the uncertainty on the geometrical factors.We point out that the smallest error is for Y123(about10%)that is a bulk sample with a well deﬁned rger errors are found for the Bi2212thickﬁlm(about30%),for which the evaluation of the thickness is rough,and for the Bi2223tape (about40%)for which an overestimation of the cross section of the tape is possible,as we mentioned above.We conclude that the AL behaviour is well followed.2)On the lowǫvalue side(ǫ<0.2)the three compounds show diﬀerent behaviours due to the diﬀerent extensionof the AL region.The least anisotropic compound,Y123,forǫ<0.1bends going asymptotically to the3D behaviour(1/ǫ0.5)showing the LD crossover atǫ≈0.09;the Bi2223sample starts to bend forǫ<0.03while the most anisotropic Bi2212in the overallǫrange considered shows the2D behaviour.3)On the highǫvalue side,starting from the AL behaviour,the curves show a crossover at aboutǫ=0.23and thenbend downward following the asymptotic1/ǫ3behaviour.At the valueǫ≈0.45all the curves drop indicating the end of the observableﬂuctuation regime.This value is lower than the above reportedǫup values,at which theﬂuctuation conductivity comes out to be zero.To conclude:we have observed in three diﬀerent HTS compounds the universal high temperature behaviour of the in-plane conductivity that manifests itself in the2D regime,once the LD crossover is passed.Beyond the AL regime all the curves reach soon the SWF1/ǫ3regime.For all the compounds the crossover occurs at the same point ǫ≈0.23,which corresponds to T≈1.3T c and,therefore,is experimentally well observable.The universality of the paraconductivity behaviour is much more surprising if we consider that it has been observed in three compounds with diﬀerent crystallograﬁc structure and anisotropy,and moreover prepared by means of very diﬀerent techniques.We gratefully acknowledge the fruitful discussions with Giuseppe Balestrino.4In principle it is possible to start from ellipsoidal Fermi surface,but the crossover in this case can be observed when the increase of anisotropy results in the intersection of the Fermi surface with the Brillouin zone boundary only.This fact implies actually the return to the open Fermi surface.5A.A.Abrikosov,The Fundamentals of the Theory of Metals,North-Holland,Amsterdam,(1988).6A.Varlamov,L.Reggiani,Phys.Rev.B45,1060(1992).7K.Maki,“Gapless Superconductivity”in Superconductivity vol.2edited by R.Parks,N.Y.(1969).8H.Schmidt,Ann.Phys.Lpz216,336(1968).9L.Aslamazov,A.Varlamov,Jour.of Low Temp.Phys.38,223(1980).10F.Federici,A.Varlamov,JETP Letters,64,497-501(1996).11A.Varlamov and L.Yu,Phys.Rev.B44,7078(1991).12L.Reggiani,R.Vaglio,A.Varlamov,Phys.Rev.B44,9541(1991).13G.Balestrino,M.Marineli,ani,L.Reggiani,R.Vaglio,A.Varlamov,Phys.Rev.B46,14919(1992).14M.Marella,G.Dinelli,B.Burtet Fabris,B.Molinas J.Alloy and Compounds189,L23(1992).15G.Balestrino,M.Marinelli,ani,A.Paoletti and P.Paroli J.Appl.Phys.68,361(1990).16B.Hensen,G.Grasso and R.Fl¨u kiger,Phys.Rev.B51,15456(1995).FIG.1.σﬂ 16hsǫ。

新核心综合学术英语教程第二册_Unit_2

Claudius Ptolemy
Claudius Ptolemy was a Greco-Egyptian writer of Alexandria, known as a mathematician, astronomer, geographer, astrologer, and poet of a single epigram in the Greek Anthology. Ptolemy was the author of several scientific treatises, three of which were of continuing importance to later Islamic and European science. The first is the astronomical treatise now known as the Almagest. The second is the Geography, which is a thorough discussion of the geographic knowledge of the Greco-Roman world. The third is the astrological treatise known in Greek as the Tetrabiblos (―Four books‖), in which he attempted to adapt horoscopic astrology to the Aristotelian natural philosophy of his day.
In this unit, you will
• learn about the concept ―scientific method‖ and its application in science; • research ―verification of theories‖ and falsification of theories‖ on the Internet and find reliable information; • learn words, expressions, and sentence patterns related to the theme and use them in writing and speech; • learn strategies such as listening for introduction (listening), skimming (reading), agreeing and disagreeing (discussing), writing thesis statements (writing), etc; • learn the deductive and inductive method of reasoning; • give an oral presentation on an assigned topic to the class.

Forward stagewise regression and the monotone lasso

1 THE LASSO AND RELATED METHODS
4
Least-angle regression uses a similar strategy, but only enters “as much” of a predictor as it deserves: the coeﬃcient of the predictor is increased only up to the point where some other predictor has as much correlation with the current residual. Then this new predictor is entered, and the process is continued. Algorithm 1 gives more details. Algorithm 1 Least Angle Regression 1. Standardize the predictors to have mean zero and variance 1. Start ¯ , β1 , β2 , . . . βp = 0. with the residual r = y − y 2. Find the predictor xj most correlated with r. 3. Move βj from 0 towards its least-squares coeﬃcient xj , r , until some other competitor xk has as much correlation with the current residual as does xj . 4. Move (βj , βk ) in the direction deﬁned by their joint least squares coefﬁcient of the current residual on (xj , xk ), until some other competitor xl has as much correlation with the current residual. 5. Continue in this way until all p predictors have been entered. After p steps, we arrive at the full least-squares solutions.

Approximation in C N Norm Levenberg

Surveys in Approximation Theory Volume 2, 2006. pp. 92–140. Copyright c 2006 Surveys in Approximation Theory. ISSN 1555-578X All rights of reproduction in any form reserved.
arXiv:math/0611249v1 [math.CA] 8 Nov 2006
Approximation in CN
Norm Levenberg
3 November 2006
Abstract. This is a survey article on selected topics in approximation theory. The topics either use techniques from the theory of several complex variables or arise in the study of the subject. The survey is aimed at readers having an acquaintance with standard results in classical approximation theory and complex analysis but no apriori knowledge of several complex variables is assumed. MSC: 32-02, 41-02 1 Introduction and motivation . . . . . . . . . 2 Polynomial hulls and polynomial convexity . . . . 3 Plurisubharmonic functions and the Oka-Weil theorem . . 4 Quantitative approximation theorems in C . . . . 5 The Bernstein-Walsh theorem in CN , N > 1 . . . . 6 Quantitative Runge-type results in multivariate approximation ¯ 7 Mergelyan property and solving ∂ . . . . . . . 8 Approximation on totally real sets . . . . . . 9 Lagrange interpolation and orthogonal polynomials . . . 10 Kergin interpolation . . . . . . . . . . 11 Rational approximation in CN . . . . . . . . 12 Markov inequalities . . . . . . . . . . 13 Appendix on pluripolar sets and extremal psh functions . 14 Appendix on complex Monge-Amp` ere operator . . . 15 A few open problems . . . . . . . . . . References . . . . . . . . . . . . . 92 96 97 103 105 109 111 115 118 121 125 128 130 134 135 136

TNPM全员规范化生产维修知识讲座

TNPM全员规范化生产维修知识讲座
？ What is TPM ?
全效率 O E E
全系统 Whole System
全员 All Members
Participating
TPM大行动 TPM ACTION 空间、时间全系统 SPACE，TIME，THE WHOLE SYSTEM 设备管理靠全员 ALL MEMBERS PARTICIPATING 提高效率才成功 SUCCESS RELIES ON OEE INCLRESING
为什么在中国企业安装TPM会更困难？
WHY IS IT MORE DIFFICULT TO INSTALL TPM IN CHINESE COMPANIES？
历史文化不同；
DIFFERENT HISTORY AND CULTURE；
企业员工平均素质不同；
DIFFERENT AVERAGE PERSONNEL
company or trans-trade, standard is true to every company. 2 Normalization is dynamic and changeable, the normalization can be alternated along with the progress of personnel and equipment. However, the standard is stable in a longer period.
2 Willingness: company culture, education and promoting, but sometime, distant water cannot quench present thirst.

发动机悬置软垫刚度对汽车舒适性的影响分析

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根据上式计算得表２结果。
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A cosmological constant from degenerate vacua

[6]. Coleman further incorporated ﬂuctuations of the spacetime topology in terms of the “wormhole” conﬁgurations and found a double exponential dependence [7]. One should note, however, that the expectation values obtained in these approaches should be regarded as giving an average over the time in the history of the universe [1]. So they may not be necessarily related with the values we observe today. We may rather interpret it as predicting a vanishing cosmological constant in some ground state where the universe spends most of the time in history. 1 More recently, a number of higher dimensional models have been proposed in which the maximally symmetric solution of the three-brane must have a vanishing four-dimensional cosmological constant [8]. Since our Universe has not settled in a maximally symmetric state, it is diﬃcult to understand implications of these results on current values of cosmological parameters in our Universe, but we are tempted to interpret them as predicting a vanishing cosmological constant again in some ultimate ground state. In the present Letter we argue the possible origin of the small but ﬁnite cosmological constant without introducing any small numbers under the hypothesis that the cosmological constant vanishes in the true ground state with lowest possible energy density. In other words, we attempt to solve the second problem under the assumption that the ﬁrst one is solved in the true ground state by arguing that we have not fallen into that state. Other proposed solutions to the second problem, such as quintessence [13] (see also [14] for earlier work) or meV-scale false vacuum energy [15], are also based on such a hypothesis. Our starting point is that the energy eigenvalue of the true ground state of a theory with two or more degenerate perturbative vacua, which cannot be transformed from one another without costing energy, is smaller than that of a quasi-ground state localized around one of these states in ﬁeld space by an exponentially small amount. Here, by perturbative vacua we mean a state with the lowest energy density without taking possible tunneling eﬀect to another perturbative vacuum into account. Our hypothesis is that the cosmological constant vanishes not in these degenerate perturbative vacua but in the absolute ground state with quantum tunneling eﬀects taken into account, because there are both classical and quantum contributions to the cosmological constant and what we observe is their sum. For illustration let us ﬁrst consider an abstract ﬁeld theory model whose perturbative vacuum states are classiﬁed into two distinct categories labeled by |+ and |− with +|− = 0 at the lowest order. We also assume that, although the transition from |+ to |− is classically forbidden, there is an instanton solution which describes quantum tunneling from |+ to |− and vice versa. By nature the instanton is localized in space and (Euclidean) time with a ﬁnite Euclidean action S0 . Then the true ground state, |S , with this tunneling eﬀect taken into account, is given by the symmetric superposition of |+ and |− , namely, |S = |+ + |− √ , 2 (1)

Factorization theorems, effective field theory, and

CCUTH-96-07 Factorization theorems,eﬀectiveﬁeld theory,and nonleptonic heavy meson decaysTsung-Wen YehDepartment of Physics,National Cheng-Kung University,Tainan,Taiwan,R.O.C.Hsiang-nan LiDepartment of Physics,National Chung-Cheng University,Chia-Yi,Taiwan,R.O.C.January9,1997PACS numbers:13.25.Hw,11.10.Hi,12.38.Bx,13.25.FtAbstractThe nonleptonic heavy meson decays B→D(∗)π(ρ),J/ψK(∗)and D→K(∗)πare studied based on the three-scale perturbative QCD factorization theorem developed recently.In this formalism the Bauer-Stech-Wirbel pa-rameters a1and a2are treated as the Wilson coeﬃcients,whose evolution from the W boson mass down to the characteristic scale of the decay processes is determined by eﬀectiveﬁeld theory.The evolution from the characteristic scale to a lower hadronic scale is formulated by the Sudakov resummation. The scale-setting ambiguity,which exists in the conventional approach to nonleptonic heavy meson decays,is moderated.Nonfactorizable and non-spectator contributions are taken into account as part of the hard decay subamplitudes.Our formalism is applicable to both bottom and charm de-cays,and predictions,including those for the ratios R and R L associated with the B→J/ψK(∗)decays,are consistent with experimental data.1I.INTRODUCTIONThe analysis of exclusive nonleptonic heavy meson decays has been a chal-lenging subject because of the involved complicated QCD dynamics.These decays occur through the HamiltonianH=G F√2V ij V∗kl(¯q l q k)(¯q j q i),(1)with G F the Fermi coupling constant,V’s the Cabibbo-Kabayashi-Maskawa (CKM)matrix elements,q’s the relevant quarks and(¯q q)=¯qγµ(1−γ5)q the V−A current.Hard gluon corrections cause an operator mixing,and their renormalization-group(RG)summation leads to the eﬀective HamiltonianH eﬀ=G F√2V ij V∗kl[c1(µ)O1+c2(µ)O2],(2)where the four-fermion operators O1,2are written as O1=(¯q l q k)(¯q j q i)and O2=(¯q j q k)(¯q l q i).The Wilson coeﬃcients c1,2,organizing the large loga-rithms from the hard gluon corrections to all orders,describe the evolution from the W boson mass M W to a lower scaleµwith the initial conditions c1(M W)=1and c2(M W)=0.The simplest and most widely adopted approach to exclusive nonleptonic heavy meson decays is the Bauer-Stech-Wirbel(BSW)model[1]based on the factorization hypothesis,in which the decay rates are expressed in terms of various hadronic transition form factors.Employing the Fierz transfor-mation,the coeﬃcient of the form factors corresponding to the external W boson emission is a1=c1+c2/N,and that corresponding to the internal W boson emission is a2=c2+c1/N,N being the number of colors.The form factors may be related to each other by heavy quark symmetry,and be mod-elled by diﬀerent ansatz.The nonfactorizable contributions which can not be expressed in terms of hadronic transition form factors,and the nonspectator contributions from the W boson exchange(or annihilation)are neglected.In this way the BSW method avoids the complicated QCD dynamics.Though the BSW model is simple and gives predictions in fair agreement with experimental data,it encounters several diﬃculties.It has been known that the large N limit of a1,2,ie.the choice a1=c1(M c)≈1.26and a2= c2(M c)≈−0.52,with M c the c quark mass,explains the data of charm decays[1].However,the same large N limit of a1=c1(M b)≈1.12and2a2=c2(M b)≈−0.26,M b being the b quark mass,does not apply to the bottom case.Even after including the c1,2/N term so that a1=1.03and a2=0.11,the BSW predictions are still insuﬃcient to match the data.To overcome this diﬃculty,parametersχ,denoting the corrections from the nonfactorizableﬁnal-state interactions,have been introduced[2].They lead to the eﬀective coeﬃcientsa eﬀ1=c1+c21N+χ1,a eﬀ2=c2+c11N+χ2.(3)χshould be negative for charm decays,canceling the color-suppressed term1/N,and be positive for bottom decays in order to enhance the predictions.Unfortunately,the mechanism responsible for this sign change has not beenunderstood completely.Furthermore,in such a framework theoretical predic-tions depend sensitively on the choice of the scaleµfor the Wilson coeﬃcients:Settingµto2M b or M b/2gives rise to a more than20%diﬀerence.Equivalently,one may regard a1,2as free parameters,and determine themby dataﬁtting.However,the behavior of the transition form factors involvedin nonleptonic heavy meson decays requires an ansatz[4]as stated above,such that the extraction of a1,2from experimental data becomes model-dependent.On the other hand,it was found that the ratio a2/a1from anindividualﬁt to the CLEO data of B→D(∗)π(ρ)[3]varies signiﬁcantly[4].It was also shown that an allowed domain(a1,a2)exists for the threeclasses of decays¯B0→D(∗)+,¯B0→D(∗)0and B−→D(∗)0,only when the experimental errors are expanded to a large extent[5].Moreover,it has been very diﬃcult to explain the two ratios associatedwith the B→J/ψK(∗)decays[6],R=B(B→J/ψK∗)B(B→J/ψK),R L=B(B→J/ψK∗L)B(B→J/ψK∗),(4)simultaneously in the BSW framework,where B(B→J/ψK)is the branch-ing ratio of the decay B→J/ψK.It was argued that the inclusion of nonfac-torizable contributions is essential for the resolution of this controversy[7]. Such contributions have been analyzed in[7,8]based on the Brodsky-Lepage approach to exclusive processes[9],in which the full Hamiltonian in Eq.(1) was employed.It was found that the nonfactorizable internal W-emission amplitudes are of the same order as the factorizable ones.However,this3naive perturbative QCD(PQCD)approach can not account for the destruc-tive interference between the external and internal W-emission contributions in charm decays.This is obvious from the fact that the coeﬃcient associated with the internal W emissions is a2=1/N in both bottom and charm decays, and thus does not change sign.Recently,a modiﬁed PQCD formalism has been proposed following the series of works[10,11,12,13,14,15],where the PQCD formalism constructed from the full Hamiltonian H was shown to be applicable to the B→D(∗)de-cays[14]in the fast recoil region ofﬁnal-state hadrons[12].It was recognized that nonleptonic heavy meson decays involve three scales:the W boson mass M W,the typical scale t of the decay processes,and the hadronic scale of or-derΛQCD.Accordingly,the decay rates are factorized into three convolution factors:the“harder”W-emission function,the hard b quark decay subampli-tude,and the nonperturbative meson wave function,which are characterized by M W,t andΛQCD,respectively.Radiative corrections then produce two types of large logarithms ln(M W/t)and ln(t/ΛQCD).In this three-scale fac-torization theorem ln(M W/t)are summed to give the evolution from M W down to t described by the Wilson coeﬃcients a1,2(t),and ln(t/ΛQCD)are summed into a Sudakov factor[16],which describes the evolution from t to the lower hadronic scale.The former has been derived in eﬀectiveﬁeld theory,and the latter has been implemented by the resummation technique [11].This modiﬁed PQCD formalism isµ-independent,ie.RG-invariant,and thus the scale-setting ambiguity existing in conventional eﬀectiveﬁeld the-ory is moderated[15].As the variable t runs to below M b and M c,the con-structive and destructive interferences involved in bottom and charm decays, respectively,appear naturally.Furthermore,not only the factorizable,but the nonfactorizable and nonspectator contributions are taken into account and evaluated in a systematic way.With the inclusion of the nonfactorizable contributions,weﬁnd that a1,2restore their original role of the Wilson coef-ﬁcients,instead of being treated as the BSW free parameters.The branching ratios of various decay modes B→D(∗)π(ρ)and D→K(∗)π,and the ratios R and R L associated with the B→J/ψK(∗)decays can all be well explained by our formalism.In Sec.II we derive the three-scale PQCD factorization theorem,con-centrating on the separation of the contributions characterized by diﬀerent scales.The incorporation of the Sudakov resummation is brieﬂy reviewed.4In Sec.III the decays B→D(∗)π(ρ)are investigated to demonstrate the im-portance of the nonfactorizable contributions.We then apply the formalism to the decays D→K(∗)πin Sec.IV,and show that the internal W-emission amplitude can become suﬃciently negative in charm decays.In Sec.V we compute the decay rates of B→J/ψK(∗)andﬁnd that the predictions for R and R L match the data simultaneously.Section VI is the conclusion, where possible further improvements and applications of our approach are proposed.II.THREE-SCALE F ACTORIZATION THEOREMS In this section we construct the modiﬁed PQCD formalism,that embod-ies both eﬀectiveﬁeld theory and factorization theorems.The motivation comes from the fact that the Wilson coeﬃcients c1,2of the eﬀective Hamilto-nian in Eq.(2)are explicitlyµ-dependent.Since physical quantities such as the decay rates,which are expressed as the products of c1,2with the matrix elements of the four-fermion operators O1,2,do not depend onµ,the latter should contain aµdependence to cancel that of the former.However,such a cancellation has never been implemented in any previous analysis of nonlep-tonic heavy meson decays.As stated in the Introduction,the BSW method employs the factorization hypothesis[1],under which the matrix elements of O1,2are factorized into two hadronic matrix elements of the(axial)vector currents(¯q q).Since the current is conserved,the hadronic matrix elements have no anomalous scale dependence,and thus theµdependence of the Wil-son coeﬃcients remains.To remedy this problem,µshould be chosen in such a way that the factorization hypothesis gives dominant contributions. However,the hadronic matrix elements involve both a short-distance scale associated with the heavy quark and a long-distance scale with the mesons. Naively settingµto the heavy quark mass will lose large logarithms contain-ing the small scale.It is then quite natural that theoretical predictions are sensitive to the value ofµ[17,18].We shall show that the cancellation of theµdependence is explicit in our formalism.We begin with the idea of the conventional PQCD factorization theorem for the B→D transition form factors,which describe the amplitude5of a b quark decay into a c quark through the current operator(¯c Lγµb L).Ra-diative corrections to these form factors are ultraviolet(UV)ﬁnite,because the current is not renormalized.However,the corrections give rise to infrared (IR)divergences at the same time,when the loop gluons are soft or collinear to the light partons in the mesons.These IR divergences should be sepa-rated from the full radiative corrections and grouped into nonperturbative soft functions.The separation of IR divergences in one of the higher-order diagrams is demonstrated by Fig.1(a),where the bubble represents the lowest-order de-cay subamplitude of the B meson.This diagram is reexpressed into two terms:Theﬁrst term,with proper eikonal approximation for quark prop-agators,picks up the IR structure of the full diagram.The second term, containing an IR subtraction,isﬁnite.Theﬁrst term,being IR sensitive, is absorbed into a meson wave functionφ(b,µ),if the radiative correction is two-particle reducible,or into a soft function U(b,µ),if the radiative correc-tion is two-particle irreducible as shown in Fig.1(a).Here b is the conjugate variable of the transverse momentum k T carried by a valence quark of the meson,and thus can be regarded as the transverse extent of the meson.It will become clear later that the scale1/b serves as an IR cutoﬀof the as-sociated loop integral.The function U corresponds,in some sense,to the nonfactorizableﬁnal-state interactions in the literature of nonleptonic heavy meson decays[2].The second term,being IRﬁnite,is absorbed into the hard decay subamplitude H(t,µ)as a higher-order correction,where t is its typical scale.The above factorization procedure is graphically described by Fig.1(b), where the diagrams in theﬁrst parentheses contribute to H,and that in the second parentheses toφor U.Below we shall neglect U,because of the pair cancellation between the diagram in Fig.1(a)and the diagram with the right end of the gluon attaching the lower quark line.The combination of these two diagrams leads to an integrand proportional to a factor1−e i l T·b,l T being the transverse loop momentum.It is then apparent that U is unimportant, if the main contributions to the form factors came from the small b region.It will be shown that the Sudakov factor mentioned in the Introduction exhibits a strong suppression at large b,and thus justiﬁes the neglect of U.On the other hand,U involves complicated colorﬂows.Hence,we leave its discussion to a separate work[19].Though the full diagrams are UVﬁnite,the IR factorization introduces6UV divergences intoφand H,which have opposite signs.This observation hints a RG treatment of the factorization formula derived above.Letγφbe the anomalous dimension ofφ.Then the anomalous dimension of H must be−γφ.We have the RG equationsµddµφ=−γφ=−µddµH,(5)whose solutions are given byφ(b,µ)=φ(b,1/b)exp−µ1/bd¯µ¯µγφ(αs(¯µ)),(6)H(t,µ)=H(t,t)exp−tµd¯µ¯µγφ(αs(¯µ)).(7)Equation(6)describes the evolution ofφfrom the IR cutoﬀ1/b to an arbi-trary scaleµ,and Eq.(7)describes the evolution of H fromµto the typical scale t.The physics characterized by momenta smaller than1/b is absorbed into the initial conditionφ(b,1/b),which is of nonperturbative origin.After the RG treatment,the large logarithms ln(t/µ)in H are grouped into the exponent,and thus the initial condition H(t,t)can be computed by perturba-tion bining Eqs.(6)and(7),the factorization formula becomes free of theµdependence as indicated byH(t,µ)φ(b,µ)=H(t,t)φ(b,1/b)exp−t1/bd¯µ¯µγφ(αs(¯µ)).(8)The eﬀective Hamiltonian H eﬀin Eq.(2)can be constructed in a similar way.Consider the nonleptonic b quark decays through a W boson emission up to O(αs).We reexpress the full diagram,which does not possess UV divergences because of the current conservation and the presence of the W boson propagator,into two terms as shown in Fig.2(a).Theﬁrst term, obtained by shrinking the W boson line into a point,corresponds to the local four-fermion operators O1,2appearing in H eﬀ,and is absorbed into the hard decay subamplitude H(t,µ).This subamplitude is characterized by momenta smaller than the W boson mass M W,that is,by the typical scale t of the heavy meson decays,since gluons in H do not”see”the W boson.The second term,characterized by momenta of order M W due to the subtraction7term,is absorbed into a”harder”function H r(M W,µ)(not a amplitude).Note that the factorization in H is not complete yet,because it still containsIR divergences,ie.the contributions characterized by the hadronic scale.We then obtain the O(αs)factorization formula shown in Fig.2(b),wherethe diagrams in theﬁrst parentheses contribute to H r,and those in the secondparentheses to H.This formula should be interpreted as a matrix relationbecause of the mixing between the operators O1and O2,or equivalently,the four-fermion vertex should be regarded as the linearly combined opera-tors O1±O2,which evolve independently.The four-fermion vertex in the denominator means that H r does not carry Dirac and color matrix struc-tures.Similarly,UV divergences are introduced in the above factorizationprocedure,when the W boson line is shrunk,and thus both H and H r needrenormalization.The RG improved factorizaton formula is written asH r(M W,µ)H(t,µ)=H r(M W,M W)H(t,t)expM Wtd¯µ¯µγHr(αs(¯µ)),(9)withγHrthe anomalous dimension of H r.It is easy to identify the exponential in Eq.(9)as the Wilson coeﬃcient,implying that the scaleµin the Wil-son coeﬃcient should be set to the hard scale t.The function H r(M W,M W)can now be safely taken as its lowest-order expression H(0)r =1,since thelarge logarithms ln(M W/µ)have been organized into the exponent.Note that the appropriate activeﬂavor number should be substituted intoαs(¯µ), when¯µevolves from M W down to t.The continuity conditions for the tran-sition ofαs(¯µ)between regions with diﬀerent activeﬂavor numbers[20]are understood.We are now ready to construct the three-scale factorization theorem by combining Eqs.(8)and(9).Start with the nonleptonic heavy meson decay amplitude up to O(αs)without integrating out the W boson.The IR sensitive functions areﬁrst factorized according to Fig.3(a),such that the diagrams in theﬁrst parentheses are characterized by momenta larger than the IR cutoﬀ. Employing Fig.2(b)to separate H r,we arrive at the factorization formula described by Fig.3(b).The diagrams in the last parentheses are identiﬁed as the hard decay subamplitude H.It is obvious that its anomalous dimension is given byγH=−γφ−γH r.Applying the RG analysis to each convolution8factor,we deriveH r(M W,µ)H(t,µ)φ(b,µ)=c(t)H(t,t)φ(b,1/b)exp−t1/bd¯µ¯µγφ(αs(¯µ)),(10)where the Wilson coeﬃcient c(t)represents the exponential in Eq.(9).The cancellation of theµdependences among the three convolution factors is ex-plicit.The two-stage evolutions from1/b to t and from t to M W have been established.We emphasize that the Wilson coeﬃcient appears as a con-volution factor of the three-scale factorization formula,which is,however,a constant coeﬃcient(once its argumentµis set to a value)in the conventional approach of eﬀectiveﬁeld theory.In the leading logarithmic approximation c1,2are given,in terms of the combination c±(µ)=c1(µ)±c2(µ),byc±(µ)=αs(M W)αs(µ)−6γ±33−2n f,(11)with the constants2γ+=−γ−=−2,and n f the number of active quark ﬂavors.Below we shall employ the more complicated two-loop expressions of c1,2presented in the Appendix A,that include next-to-leading logarithms [20].At last,we explain how to incorporate the Sudakov factor into the above factorization formula.The RG solution in Eq,(6)sums only the single log-arithms contained in the meson wave functionφ.In fact,there exist also double logarithms coming from the overlap of collinear and soft divergences. Hence,an extra large scale P,the meson momentum,should be added into φas an argument.The scale P is associated with the collinear divergences, while the small scale1/b is associated with the soft divergences as stated at the beginning of this section.Before reaching Eq.(5),one performs the resummation for these double logarithms,and obtainφ(P,b,µ)=φ(b,µ)exp[−s(P,b)].(12) e−s is the Sudakov factor,which exhibits a strong suppression in the large b re-gion.The single-scale wave functionφ(b,µ)discussed above is then identiﬁed as the initial condition of the resummation for the two-scale wave function φ(P,b,µ).For the detailed derivation of Eq.(12),refer to[11,12].9In summary,the large logarithms ln(M W /t )are grouped into the Wil-son coeﬃcients c 1,2,and ln(tb )are organized by the resummation technique and by the RG bining Eqs.(10)and (12),we derive the ﬁnal expression of the three-scale factorization formulaH r (M W ,µ)H (t,µ)φ(x,P,b,µ)=c (t )H (t,t )φ(x,b,1/b )×exp −s (P,b )− t1/b d ¯µ¯µγφ(αs (¯µ)),(13)where the momentum fraction x associated with a valence quark of the meson has been inserted.III.The B →D (∗)π(ρ)DecaysIn the conventional BSW approach the branching ratios of the exclusive nonleptonic heavy meson decays are parametrized only by the factorizable contributions from the external and internal W emissions as stated in the Introduction.The associated nonfactorizable contributions,which can not be expressed in terms of hadronic form factors,are ignored.The nonspectator contributions from the W -exchange (or annihilation)diagrams,which may be factorizable or nonfactorizable,are not included either.However,the naive PQCD analysis based on the full Hamiltonian in Eq.(1)has shown that the nonfactorizable contributions are comparable to the factorizable ones for the internal W emissions and for the W exchanges [8].In this section we shall investigate the importance of the nonfactorizable and nonspectator contributions to exclusive nonleptonic heavy meson decays employing the more sophiscated three-scale PQCD factorization theorem de-veloped in the previous section [15].We evaluate the branching ratios of the B →D (∗)π(ρ)decays,taking into account the factorizable,nonfactoriz-able and nonspectator contributions,and letting the Wilson coeﬃcients c 1,2evolve according to eﬀective ﬁeld theory.In this framework the external W emissions also give nonfactorizable contributions.The relevant eﬀective10Hamiltonian is given byH eﬀ=G F √2V cb V ∗ud [c 1(µ)O 1+c 2(µ)O 2],(14)with the four-fermion operators O 1=(¯du )(¯c b )and O 2=(¯c u )(¯db).The full Hamiltonian H is then a special case with the choice c 1=1and c 2=0.We ﬁrst study the B →D (∗)πdecays.The analysis of the B →D (∗)ρdecays is similar.The factorizable external W -emission amplitudes deﬁne the B →D (∗)transition form factors ξthrough the hadronic matrix elements,D (P 2)|V µ|B (P 1) =M B M D [ξ+(η)(v 1+v 2)µ+ξ−(η)(v 1−v 2)µ], D ∗(P 2)|V µ|B (P 1) =i M B M D ∗ξV (η) µναβ ∗νv 2αv 1β, D ∗(P 2)|A µ|B (P 1) = M B M D ∗[ξA 1(η)(η+1) ∗µ−ξA 2(η) ∗·v 1v µ1−ξA 3(η) ∗·v 1v µ2].(15)P 1(P 2),M B (M D (∗))and v 1(v 2)are the momentum,the mass,and the veloc-ity of the B (D (∗))meson,satisfying the relation P 1=M B v 1(P 2=M D (∗)v 2). ∗is the polarization vector of the D ∗meson.The velocity transfer v 1·v 2in two-body nonleptonic decays takes the maximal value η=(1+r 2)/(2r )with r =M D (∗)/M B .In the rest frame of the B mesonP 1and P 2are expressed asP 1=(M B /√2)(1,1,0T )and P 2=(M B /√2)(1,r 2,0T )[12].For the analysis below,we deﬁne k 1(k 2)the momentum of the light valence quark in the B (D (∗))meson.k 1may have a minus component k −1,giving the momentum fraction x 1=k −1/P −1,and small transverse components k 1T .k 2may have alarge plus component k +2,giving x 2=k +2/P +2,and small k 2T .The pion thencarries the momentum P 3=P 1−P 2,whose nonvanishing component is only P −3.One of its valence quark carries the fractional momentum x 3P 3,and small transverse momenta k 3T .In the inﬁnite mass limit of M B and M D (∗)the form factors ξi with i =+,−,V ,A 1,A 2,and A 3obey the relationsξ+=ξV =ξA 1=ξA 3=ξ,ξ−=ξA 2=0.(16)ξis the so-called Isgur-Wise (IW)function [21],which is normalized to unity at zero recoil η→1by heavy quark symmetry.ξi include the contributions from the hadronic matrix element of O 1shown in Fig.4(a)and from the color-suppressed matrix element of O 2in11Fig.4(b).Therefore,their factorization formulas involve the Wilson coef-ﬁcient a1=c1+c2/N.We deﬁne the form factorsξ(∗)int for the internal W-emission diagrams,which include the factorizable contributions from the matrix elements of O2in Fig.4(c),and from the color-suppressed matrix element of O1in Fig.4(d).These form factors then contain the Wilson co-eﬃcient a2=c2+c1/N.Similarly,we deﬁne the form factorsξ(∗)exc for theW-exchange diagrams,which include the factorizable contributions from the matrix elements of O2in Fig.4(e),and from the color-suppressed matrix element of O1in Fig.4(f).Hence,these form factors also contain the Wilson coeﬃcient a2.For the nonfactorizable contributions to the B→D(∗)πdecays,the pos-sible diagrams are exhibited in Fig.5,which correspond to those in Fig.4. Figs.5(a),5(c),and5(e)do not contribute at O(αs)simply because a trace of odd number of color matrices vanishes.Hence,all the nonfactorizable contributions come from Figs.5(b),5(d),and5(f),denoted by the ampli-tudes M(∗)b ,M(∗)d,and M(∗)f,respectively,and are thus color-suppressed.Their expressions are more complicated,and can not be written in terms ofhadronic form factors.The amplitudes M(∗)b for the nonfactorizable externalW emissions depend on the Wilson coeﬃcient c2/N.They have the same expressions for the charged and neutral B meson decays,because replacing the spectator¯u quark in the B−meson by the¯d quark does not change the Feynman rules.The amplitudes M(∗)dfor the nonfactorizable internal Wemissions contain the Wilson coeﬃcient c1/N.M(∗)f for the nonfactorizableW exchanges involve the Wilson coeﬃcient c1/N.The decay rates of B→D(∗)πhave the expressionΓi=1128πG2F|V cb|2|V ud|2M3B(1−r2)3r|M i|2,(17)where i=1,2,3and4denote the modes B−→D0π−,¯B0→D+π−, B−→D∗0π−and¯B0→D∗+π−,respectively.With the above form factors and the nonfactorizable amplitudes,the decay amplitudes M i are written as M1=fπ[(1+r)ξ+−(1−r)ξ−]+f Dξint+M b+M d,(18)M2=fπ[(1+r)ξ+−(1−r)ξ−]+f Bξexc+M b+M f,(19)M3=1+r2r fπ[(1+r)ξA1−(1−r)(rξA2+ξA3)]12+f D∗ξ∗int+M∗b+M∗d,(20)M4=1+r2r fπ[(1+r)ξA1−(1−r)(rξA2+ξA3)]+f Bξ∗exc +M∗b+M∗f,(21)where f B,f D(∗),and fπare the B meson,D(∗)meson,and pion decay con-stants.We have made explicit that the charged B meson decays B−→D(∗)0π−contain the external and internal W-emission contributions,and the neutral B meson decays¯B0→D(∗)+π−contain the external W-emission andW-exchange contributions.In the considered maximal recoil region with P+2 M D(∗)/√2 P−2,weregard the D(∗)meson as a light meson for simplicity[12].Double logarithms contained in the B meson,D(∗)meson and pion wave functionsφB,φD(∗), andφπ,respectively,are organized into the corresponding Sudakov factors using the resummation technique[12,16,22]:φB(x1,P1,b1,µ)=φB(x1,b1,1/b1)exp[−S B(µ)],φD(∗)(x2,P2,b2,µ)=φD(∗)(x2,b2,1/b2)exp[−S D(∗)(µ)],φπ(x3,P3,b3,µ)=φπ(x3,b3,1/b3)exp[−Sπ(µ)],(22) with the exponentsS B(µ)=s(x1P−1,b1)+2 µ1/b1d¯µ¯µγ(αs(¯µ)),S D(∗)(µ)=s(x2P+2,b2)+s((1−x2)P+2,b2)+2µ1/b2d¯µ¯µγ(αs(¯µ)),Sπ(µ)=s(x3P−3,b3)+s((1−x3)P−3,b3)+2µ1/b3d¯µ¯µγ(αs(¯µ)).(23)The quark anomalous dimensionγ=−αs/π,is related toγφ=2γintroduced in Sec.II.The spatial extents b i of the mesons are the Fourier conjugate vari-ables of k iT.We shall neglect the intrinsic b dependence of the wave functions, denoted by the argument b,and the O(αs(1/b))corrections,denoted by the argument1/b.That is,we assumeφ(x,b,1/b)=φ(x).The wave functions φi(x),i=B,D,D∗,andπ,satisfy the normalization10φi(x)dx=f i2√.(24) 13The exponent s is written as [23]s (Q,b )=Q1/b dµµ ln Q µ A (αs (µ))+B (αs (µ)) ,(25)where the anomalous dimensions A to two loops and B to one loop are given byA =C F αs π+ 679−π23−1027n f +83β1ln e γE 2 αs π 2,B =23αs πln e 2γE −12 ,(26)with C F =4/3the color factor and γE the Euler constant.The two-loop expression of the running coupling constant,αs (µ)π=4β0ln(µ2/Λ2)−16β1β30ln ln(µ2/Λ2)ln 2(µ2/Λ2),(27)will be substituted into Eq.(25),with the coeﬃcientsβ0=33−2n f 3,β1=153−19n f 6,(28)and the QCD scale Λ≡ΛQCD .Combined with the evolution of the hard decay subamplitudes H ,the variables µin Eq.(23)are replaced by the hard scales t as shown in Eq.(13),leading to the RG invariant Sudakov exponents S B (t ),S D (∗)(t )and S π(t ).Since large logarithms have been organized,we compute H to lowest order by sandwiching Figs.4and 5with the matrix structures (P 1+M B )γ5/√2N from the initial B meson,with γ5(P 2+M D)/√, (P 2+M D ∗)/√,and γ5P 3/√2N from the ﬁnal D meson,D ∗meson,and pion,respectively.The expressions of all the form factors and nonfactorizable amplitudes for the B →D (∗)πdecays are listed below.The form factors ξi ,i =+,A 1and A 3,and ξj ,j =−and A 2,derived from Figs.4(a)and (b),are given by ξi =16πC F √rM 2B 10dx 1dx 2 ∞0b 1db 1b 2db 2φB (x 1)φD (∗)(x 2)a 1(t )×αs (t )[(1+ζi x 2r )h (x 1,x 2,b 1,b 2,m )+(r +ζ i x 1)h (x 2,x 1,b 2,b 1)]14。

学术英语(社科)_Unit 7

→ The most dominant assumption describes an ideal world in which convergence is a good thing and will happen at any rate.
Do the tasks on pages 142-143.
Unit 7
Change in Journalism
Unit Contents
Lead-in Text A Text B Text C Listening Writing
Unit 7
Change in Journalism
• Lead-in
– Lead-in task
Unit 7
Change in Journalism
Unit 7
Change in Journalism
Text A
Critical reading and thinking of Text A
Text Analysis
Supporting details: (Paras. 3-4) 1) The growth of a global economic system has rewritten the financial basis for journalism and the media in general. 2) deregulation 3) The end of Cold War has unleashed a variety of cultural and political forces
Lead-in
Task Watch the video clip and complete the sentences according to what you have heard.

柚皮素标准物质的研制及不确定度评定

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图1柚皮素的分子结构 Fig. 1 Molecelar structure of narinoediv
收稿日期：221 -06 -24；修回日期：201 -03 -1 基金项目：国家重点研发专项(221YFF0201105 -2)；国家现代农业(柑桔)产业技术体系建设专项(CARS-26)；国家农产品质量
安全风险评估重大专项(GJFP201004, GJFP2618618) *通讯作者：苏学素，副教授，研究方向：生物有机药物合成，E - maii: suxuesu@ 163. com

Complex valued Ray-Singer torsion

Complex valued Ray–Singer torsion
3
we apply this result to the Laplacians ∆E,g,b and therewith complete the proof of the anomaly formula. We restrict the presentation to the case of vanishing Euler–Poincar´ e characteristics to avoid geometric regularization, see [11] and [12]. With minor modiﬁcations everything can easily be extended to the general situation. This is sketched in Section 9. The analytic core of the results, Theorem 7.1 and its corollaries Propositions 6.1 and 6.2, are formulated and proved without any restriction on the Euler–Poincar´ e characteristics. Let us also mention the series of recent preprints [3, 4, 5, 6, 7]. In these papers Braverman and Kappeler construct a “reﬁned analytic torsion” based on the odd signature operator on odd dimensional manifolds. Their torsion is closely related to the analytic torsion proposed in this paper. For a comparison result see Theorem 1.4 in [7]. Some of the results below which partially establish Conjecture 5.1, have ﬁrst appeared in [7], and were not contained in the ﬁrst version of this paper. The proofs we will provide have been inspired by [7] but do not rely on the results therein. Recently, in October 2006, two preprints [15] and [33] have been posted on the internet providing the proof of Conjecture 5.1. In [15] Witten–Helﬀer–Sj¨ ostrand theory has been extended to the non-selfadjoint Laplacians discussed here, and used along the lines of [10], to establish Conjecture 5.1 for odd dimensional manifolds, up to sign. Comments were made how to derive the conjecture in full generality on these lines. A few days earlier, by adapting the methods in [2] to the non-selfadjoint situation, Su and Zhang in [33] provided a proof of the conjecture. The deﬁnition of the complex valued analytic torsion was sketched in [14]. We thank the referees for useful remarks and for pointing out several sign mistakes.

Orthogonal Polynomials

Polynomials 7 Some questions leading to classical orthogonal Electrostatics . . . . . . . . . . . . . . . . . . . Polynomial solutions of eigenvalue problems . . Harmonic analysis on spheres and balls . . . . Approximation theory . . . . . . . . . . . . . . 8 Heuristics polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
V. Totik
72
with special functions, combinatorics and algebra, and it is mainly devoted to concrete orthogonal systems or hierarchies of systems such as the Jacobi, Hahn, Askey-Wilson, . . . polynomials. All the discrete polynomials and the q analogues of classical ones belong to this theory. We will not treat this part; the interested reader can consult the three recent excellent monographs [39] by M. E. H. Ismail, [28] by W. Gautschi and [6] by G. E. Andrews, R. Askey and R. Roy. Much of the present state of the theory of orthogonal polynomials of several variables lies also close to this algebraic part of the theory. To discuss them would take us too far from our main direction; rather we refer the reader to the recent book [24] by C. F. Dunkl and Y. Xu. The other part is the analytical aspect of the theory. Its methods are analytical, and it deals with questions that are typical in analysis, or questions that have emerged in and related to other parts of mathematical analysis. General properties ﬁll a smaller part of the analytic theory, and the greater part falls into two main and extremely rich branches: orthogonal polynomials on the real line and on the circle. The richness is due to some special features of the real line and the circle. Classical real orthogonal polynomials, sometimes in other forms like continued fractions, can be traced back to the 18th century, but their rapid development occurred in the 19th and early 20th century. Orthogonal polynomials on the unit circle are much younger, and their existence is largely due to Szeg˝ o and Geronimus in the ﬁrst half of the 20th century. B. Simon’s recent treatise [80, 81] summarizes and greatly extends what has happened since then. The connection of orthogonal polynomials with other branches of mathematics is truly impressive. Without even trying to be complete, we mention continued fractions, operator theory (Jacobi operators), moment problems, analytic functions (Bieberbach’s conjecture), interpolation, Pad´ e approximation, quadrature, approximation theory, numerical analysis, electrostatics, statistical quantum mechanics, special functions, number theory (irrationality and transcendence), graph theory (matching numbers), combinatorics, random matrices, stochastic processes (birth and death processes; prediction theory), data sorting and compression, Radon transform and computer tomography. This work is a survey on orthogonal polynomials that do not lie on the unit circle. Orthogonal polynomials on the unit circle—both the classical theory and recent contributions—will be hopefully dealt with in a companion article. This work is meant for non-experts, and it therefore contains introductory materials. We have tried to list most of the actively researched ﬁelds not directly connected with orthogonal polynomials on the unit circle, but because of space limitation we have only one or two pages on areas where dozens of papers and several books had been published. As a result, our account is necessarily incomplete. Also, the author’s personal taste and interest is reﬂected in the survey, and the omission of a particular direction or a set of results reﬂects in no way on the importance or quality of the omitted works. For further backgound on orthogonal polynomials, the reader can consult

AND M. REDIVO--ZAGLIA y

Zb xiPk (x) d a
Dedicated to Ted Rivlin on the occasion of his 70th birthday
(x) = 0;
for i = 0; : : : ; k ? 1
where is a positive Borel measure on a; b]. Pk is the polynomial of degree k belonging to the family of orthogonal polynomials on a; b] with respect to the measure . It is well{known that the zeros of Pk are all real, distinct and in a; b] and that they interlace with the zeros of Pk+1 . Recently, several generalizations of the notion of orthogonality appeared in the literature. This paper is devoted to the properties of their zeros. The de nitions and the rst results on the number of their real zeros will be given in Section 1. Since these rst results do not give a complete information on the zeros, some additional results will be proved in Section 2. The last section will be devoted to some vector generalizations of Chebyshev polynomials. The results presented in this paper are far to be complete. They are only given here as a starting point for further investigations.