A note on negative isotropic curvature

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Role of the unstable directions in the equilibrium and aging dynamics of supercooled liquid

a r X i v :c o n d -m a t /0007207v 1 [c o n d -m a t .s o f t ] 12 J u l 2000Role of the unstable directions in the equilibrium and aging dynamics of supercooledliquids.Claudio Donati,Francesco Sciortino and Piero TartagliaDipartimento di Fisica,Universita’di Roma ”La Sapienza”and Istituto Nazionale per la Fisica della Materia,Piazzale AldoMoro 2,I-00185,Roma,Italy(Revised version LM7663:February 1,2008)The connectivity of the potential energy landscape in su-percooled atomic liquids is investigated through the calcu-lation of the instantaneous normal modes spectrum and a detailed analysis of the unstable directions in conﬁguration space.We conﬁrm the hypothesis that the mode-coupling critical temperature is the T at which the dynamics crosses over from free to activated exploration of conﬁguration space.We also report the observed changes in the local connectiv-ity of conﬁguration space sampled during aging,following a temperature jump from a liquid to a glassy state.PACS numbers:64.70.Pf,61.20.Ja,61.20.LcUnderstanding the microscopic mechanism for the in-credible slowing down of the dynamics in supercooled glass forming liquids is one of the hot topics in con-densed matter physics.In recent years,the combined eﬀort of high level experimental techniques [1],compu-tational analysis [2,3]and sophisticated theoretical ap-proaches [4–6]has provided an enormous amount of novel information.In particular,it appears more and more clearly that in addition to the melting temperature and the calorimetric glass transition temperature,T g ,another temperature plays a relevant role.This temperature,lo-cated between 1.2and 2T g depending on the fragility of the liquid,signals a deﬁnitive change in the microscopic processes leading to structural relaxation.Mode Cou-pling Theory (MCT)[4]was ﬁrst in identifying the role of this cross-over temperature,T c .According to MCT,for T ≥T c the molecular dynamics is controlled by the statistics of the orbits in phase space [7],while below T c ,the dynamics becomes controlled by phonon assisted pro-cesses [4].Studies based on disordered mean ﬁeld p -spin models have also stressed the role of such a cross-over temperature [8].Computer simulation studies of realistic models of liq-uids have addressed the issue of the structure of con-ﬁguration space in supercooled states [3].Two diﬀer-ent techniques have provided relevant information on phase space structure:the instantaneous normal mode approach (INM)[9],which focuses on the properties of the ﬁnite temperature Hessian,allowing the calculation of the curvature of the potential energy surface (PES)along 3N independent directions and (ii)the inherent structure (IS)approach [10],which focuses on the lo-cal minima of the potential energy.Diﬀerent from mean ﬁeld models,computer simulation analysis provides a de-scription based on the system’s potential energy,the free energy entering only via the equilibrium set of analyzed conﬁgurations.In principle,the INM approach should be well suited for detecting a change in the structure of the PES vis-ited above and below T c .A plot of the T -dependence of the fraction of directions in conﬁguration space with negative (unstable)curvature,f u ,should reveal the pres-ence of T c .Unfortunately,as it was soon found out [11],anharmonic eﬀects play a non negligible role and several of the negative curvature directions are observed even in crystalline states,where diﬀusivity is negligible.A sim-ilar situation is seen in p-spin models,where the num-ber of negative eigenvalues of the Hessian is not zero at T c [12].Bembenek and Laird [13]suggested inspecting the energy proﬁle along the unstable directions to par-tition the negative eigenmodes in shoulder modes (sh )(i.e.anharmonic eﬀects)and double-well (dw )modes (i.e.directions connecting diﬀerent basins).While for a particular molecular system the fraction of double well modes f dw has been shown to go to zero close to T c [14],for atomic system,f dw is signiﬁcantly diﬀerent from zero even in crystalline and glassy states [13].The existence of dw directions in thermodynamic conditions where the diﬀusivity is zero,i.e.in situations where the system is constrained in a well deﬁned basin,strongly suggests that the two minima joined by the unstable double-well directions may lead to the same inherent structure IS.Such possibility has been demonstrated very clearly in Ref.[15].The aim of this Letter is to present a detailed eval-uation of:(i)the number of escape directions,N escape ,leading to a basin diﬀerent from the starting one;(ii)the number of distinct basins,N distinct ,which are connected on average to each conﬁguration via a dw direction.This analysis,based on a computationally demanding proce-dure,shows that indeed the PES regions visited in ther-mal equilibrium above and below T c are clearly diﬀerent.We also study the evolution of the sampled PES as a function of time following a quench from above to below T c .We show that,in analogy with the equilibrium case,two qualitatively diﬀerent dynamical regimes exist dur-ing aging,related to diﬀerent properties of the sampled PES.The system we study is composed of a binary (80:20)mixture of N=1000Lennard Jones atoms[16].Thedynamics of this system is well described by MCT,with a critical temperature T c equal to0.435[17].We begin by considering the equilibrium case.At each T,for each of the50analyzed conﬁgurations,we calculate the INM spectrum and—by rebuilding the potential en-ergy proﬁle along straight paths following directions with negative curvature—we classify the unstable modes into dw and sh.It is important to notice that the classiﬁ-cation is done by studying the shape of the PES along one eigenvector,i.e.beyond the point in conﬁguration space where the eigenvector was calculated.In princi-ple,to identify a dw or a sh mode,one should follow a curvilinear path.Indeed the use of straight paths guar-antees only the identiﬁcation of the dw modes whose one-dimensional saddle energy is close to the potential energy of the system.As discussed in more length in[18]those dw modes are the ones relevant for describing motion in conﬁguration space.Figure1shows the T dependence of the unstable and dw modes for the studied system.In agreement with the analogous calculation of Ref.[13],even at the lowest tem-perature where equilibration is feasible within our com-puter facilities,f dw is signiﬁcantly diﬀerent from zero. To estimate roughly the number of double wells which do not contribute to diﬀusion,we have calculated the IS associated with the T=0.446equilibrium conﬁgurations, and we have heated them back to various temperatures below T c.The corresponding f u and f dw are also shown in Fig.1.Both these quantities depend linearly on the temperature.If we extrapolate f dw for the equilibrium and for these non-diﬀusive cases,weﬁnd that the two curves cross close to the MCT critical temperature T c. Thus T c seems to be the temperature at which the num-ber of directions leading to diﬀerent basins goes to zero, leaving only local activated processes as residual channels for structural relaxation[19].To estimate in a less ambiguous way the number of diﬀerent basins which can be accessed from each con-ﬁguration we apply the following procedure:(i)calcu-late the dw directions via INM calculation;(ii)follow each straight dw direction climbing over the potential energy barrier and down on the other side until a new minimum is found;(iii)perform a steepest descent path starting from the new minimum found along the dw di-rection,asﬁrst suggested by Gezelter et al.[15];(iv)save the resulting IS conﬁguration;(v)repeat(ii-iv)for each dw mode.This procedure produces a list of IS s which can be reached from the initial conﬁguration crossing a one-dimensional dw.We note that our procedure only guarantees that the two starting conﬁgurations for the quenches are on diﬀerent sides of the double well,since —as discussed in[18]—there is some arbitrariness in the location of the two minima.We calculate the relative distance d ijd ij=N N l=1(x i l−x j l)2+(y i l−y j l)2+(z i l−z j l)2(1)between all IS pairs in the list to determine the num-ber of distinct basins,N distinct,connected to the starting conﬁguration.Here x k m,y k m,and z k m are the coordinates of the m-th particle in the k-th IS of the list(IS k).We also calculate the distance d0i—where0indicates the IS associated to the starting conﬁguration—to enumer-ate the number of escape directions,N escape,leading to a basin diﬀerent from the original one(IS0).In Fig.2(upper panel)we show a plot of the distribu-tions of the distances d0i between the starting IS and the minima identiﬁed with this procedure for diﬀerent tem-peratures.At high temperatures,the distribution has a single peak centered approximately at d=0.3.For lower temperatures,the peak moves to the left and decreases in height.At the same time,a second but distinct peak, centered around d≈10−4,appears.Since our sample is composed of1000particles,an average distance of the order of10−4means that,also in the case that only one particle has a diﬀerent position between the two diﬀerent conﬁgurations,this particle has moved less that0.003in-terparticle distance.Thus,we consider the IS i and j as coincident if d ij<10−2.5.The upper panel of Fig.3reports the T-dependence of the number of dw directions N dw,the number of escape directions and the number of distinct basins which can be reached by following a dw direction.Weﬁnd that at high temperature nearly all dw directions leads to a dif-ferent IS,thus fully contributing to the diﬀusion process. On lowering the temperature,a large fraction of the di-rections leads to the same minimum,and,close to T c, almost all dw directions lead to intra-basin motion.A basin change becomes a rare event.Data in Fig.3sup-port the view that T c is associated to a change in the PES sampled by the system,and,consequently,in the type of dynamics that the system experiences.Above T c the sys-tem can change basins in conﬁguration space by moving freely along an accessible dw direction,while below T c the thermally driven exploration of conﬁguration space favors the exploration of the interior of the basins.Dif-fusion in conﬁguration space,i.e.basin changes,requires an local activated process.We next turn to the study of out-of-equilibrium dy-namics.Conﬁgurations are equilibrated at high tem-perature(T=5.0)and at time t w=0are brought to low temperature T f=0.2by instantaneously changing the control temperature of the thermostat.Within100 molecular dynamics steps,the kinetic energy of the sys-tem reaches the value corresponding to T f.Conﬁgura-tions are recorded for diﬀerent waiting times t w after the quench.We then repeat the same analysis done for the equilibrium conﬁgurations,as described before.Figure2(lower panel)shows the distribution of the dis-tances d0i between the minima IS0and IS i.For short t w the distribution has a single peak centered approximately at d0i=0.3,as found in the equilibrium conﬁgurations at high temperature.As the system ages,the peak shifts to smaller distances and a second distinct broad peak centered at approximately d0i=5×10−4appears.The observed behavior is similar to the equilibrium one,on substituting T with t w.The lower panel of Fig.3shows N dw,N escape and N distinct as functions of t w.At small t w,following a dou-ble well direction always brings to a basin that is distinct from the original one,while at long times all dw direc-tions leads to intra-basin motion.Again,the behavior is very reminiscent of the equilibrium one,on substituting T with t w.From these results,we suggest that two qual-itatively diﬀerent dynamical processes control the aging processes,respectively for short and long t w[20].For short waiting times the system is always close to saddle points that allow it to move from one basin to a distinct one.As the system ages,it is conﬁned to wells that are deeper and deeper,and the number of basins that it can visit by following a dw direction decays to zero.Thus, for long t w the dynamics of aging proceeds through local activated processes,that allow the system to pass over potential energy barriers.As shown in Fig.3,beyond t w=105the number of distinct basins is almost zero, which,in analogy with the equilibrium results,supports the view that a cross-over from MCT-like dynamics to hopping dynamics may take place at a cross-over time during the aging process.In summary,this Letter conﬁrms the hypothesis that T c can be considered as the T at which dynamics crosses over from the free-exploration to the locally activated exploration of conﬁguration space.In this respect,theanalogy between the slowing down of dynamics of liquids and the slowing down of the dynamics in meanﬁeld p-spin model is strengthened.This Letter also shows that changes in dynamical processes associated with changes in the explored local connectivity of conﬁguration space are observed even during the aging process,in the time window accessed by molecular dynamics experiments. MCT-based models of aging[21]could be able to de-scribe the early part of the aging process—i.e.the saddle-dominated dynamics—but may not be adequate for describing the locally activated dynamical region. We acknowledgeﬁnancial support from the INFM PAIS98,PRA99and Iniziativa Calcolo Parallelo and from MURST PRIN98.We thank W.Kob,G.Parisi and T.Keyes for discussions.mσ2AA/48ǫAA.Both atoms have unitary mass m.The simulations have been run for up to5·106MD steps in the NVT ensemble (Nose-Hoover thermostat).One MD step is0.02. [17]W.Kob and H.C.Andersen,Phys.Rev.Lett.73,1376(1994);Phys.Rev.E52,4134(1995).[18]Motion along a straight direction in conﬁguration spacewill always be associated with a fast rise of the energy proﬁle,since every direction will always describe pairs of atoms moving close enough to probe the repulsive part of the pair potentials[11].In systems with steep repulsive potentials,such an unphysical rise of the potential energy proﬁle along the straight eigenmode direction sets in very early.In other words,the eigenvector direction changes very rapidly as the system moves in conﬁguration space and very soon the energy proﬁle diﬀers from the proﬁle evaluated along the straight eigenvector approximation.The major eﬀects of such an artiﬁcial rise in energy are[11](i)the transformation of dw direction in sh directionsand(ii)the arbitrary location of the two minima along the dw direction and the extremely low value of the one-dimensional barriers along the dw.In particular,eﬀect(i)sets in when,in the studied direction,the system islocated far from the saddle and eﬀect(ii)sets in when thesystem is located close to the saddle point[T.Keyes and W.-X.Li,J.Chem.Phys.111,5503(1999)].Notwith-standing these potential pitfalls in the classiﬁcation pro-cedure,the classiﬁcation helps in understanding the dy-namical changes taking place in the liquid as a function of T andρ.Indeed the only saddles which are relevant for the dynamical behavior of the system are the ones which are explored by the system,i.e.the ones located ina potential energy range V±∆V—where V is the aver-age potential energy of the system and∆V is a measure of the potential energyﬂuctuations.The accessed saddles are included in the type(ii)classiﬁcation in the above list.As a result,the classiﬁcation performed along straight di-rections retains its validity,even if no physical meaning can be attributed to the calculated one-dimensional en-ergy proﬁle.Of course,it would be more appropriate to re-calculate the eigenvectors at each step and follow the curved one-dimensional energy proﬁle.Such a procedure could still suﬀer from the frequent mode-crossing events which are known to characterize the liquid dynamics[M.Buchener and T.Dorfm¨u ller,J.Mols.Liquids,65/66, 157(1995).].[19]Such estimate of the free-exploration to activated dy-namics cross-over temperature could be biased by the arbitrary choice of the IS conﬁguration(the T=0.446 conﬁgurations).[20]W.Kob et al.,Europhys.Letts.49,590(2000).[21]tz,J.Phys.:Condens.Matter,in the press. FIG.1.T-dependence of f u(squares)and f dw(circles). Theﬁlled symbols are calculated from equilibrium conﬁgura-tions,the empty symbols from out-of-equilibrium conﬁgura-tions below T c(see text).FIG.2.a)Distribution of the distances d≡d0i between the IS0and the i-th IS from equilibrium conﬁgurations at diﬀerent temperatures;b)same quantity for diﬀerent t w val-ues during the aging process after a temperature jump from T i=5.0to T f=0.2.FIG.3.a)Number of double wells N dw,of escape direc-tions N escape and of distinct basins N distinct for equilibrium conﬁgurations as a function of T;b)Same quantities for dif-ferent t w values after a temperature jump from T i=5.0to T f=0.2.Time is measured in molecular dynamics steps.T0.050.10.15fFIG.1.C.Donati et alP (d )Log 10(d)P (d )FIG.2.C.Donati etalNt wNFIG.3.C.Donati et al。

工程光学英语

Transverse magnification (Lateral magnification) is defined as the ratio of image size, y', to object size, y.
Axial magnification (longitudinal magnification) is defined as the ratio of a short length, or depth, in the image measured along the axis, to the conjugate length in the object. In the air, it is equal to the square of the transverse magnification.
upright or inverted, real or virtual) 8. Concave and convex mirror
7
Some Concept
1. Vertex is the point where the optic intersects the surface.
2. Object distance is the distance from vertex to the object.
2
4. The Law of Refraction
The angle made by the incident ray, that made by the refracted ray, and the surface normal at the point of incidence in a refractive process obey the expression

Open and closed CDM isocurvature models contrasted with the CMB data

a r X i v :a s t r o -p h /0108422v 2 7 D e c 2001HIP-2001-48/TH astro-ph/0108422Phys.Rev.D 65,0230XXOpen and closed CDM isocurvature models contrasted with the CMB dataKari Enqvist ∗Department of Physical Sciences,University of Helsinki,and Helsinki Institute of Physics,P.O.Box 64,FIN-00014University of Helsinki,FinlandHannu Kurki-Suonio †and Jussi V¨a liviita ‡Department of Physical Sciences,University of Helsinki,P.O.Box 64,FIN-00014University of Helsinki,Finland(August 27,2001)We consider pure isocurvature cold dark matter models in the case of open and closed universes.We allow for a large spectral tilt and scan the 6-dimensional parameter space for the best ﬁt to the COBE,Boomerang,and Maxima-1data.Taking into account constraints from large-scale structure and big bang nucleosynthesis,we ﬁnd a best ﬁt with χ2=121,which is to be compared to χ2=44of a ﬂat adiabatic reference model.Hence the current data strongly disfavor pure isocurvature perturbations.I.INTRODUCTIONThe recent measurements of the cosmic microwave background (CMB)temperature ﬂuctuations by the Boomerang [1,2]and Maxima-1[3,4]balloon experiments and the DASI interferometer [5]have widely been re-garded as indicating that we live in a Ω=1universe.This is so because the ﬁrst acoustic peak is found at the multipole ℓ≃200,implying a ﬂat universe.The ﬁrmness of such a conclusion is,however,based on certain tacit as-sumptions.In particular,when ﬁtting the acoustic peak positions,one often assumes that the primordial pertur-bations are adiabatic and that the spectrum is nearly scale invariant.If perturbations are adiabatic,the relative abundances of particle species are equal to their thermal equilibrium values.This is the case in the simplest,one-ﬁeld inﬂa-tion models but it is not a generic feature of inﬂation.More generally,perturbations can be either adiabatic or nonadiabatic;the latter would be perturbations in the particle number densities,or entropy perturbations,and are called isocurvature perturbations.Because no generally accepted theory of inﬂation ex-ists,it is natural to consider both adiabatic and isocur-vature perturbations as being equally probable.This is the generic situation when more than one ﬁeld is excited during inﬂation,such as is the case in double inﬂation [6]or in the minimally supersymmetric standard model with ﬂat directions [7].One should also note that in the pre-big-bang scenario,which has been proposed as an alternative to the inﬂationary universe,pre-big-bang axion ﬁeld ﬂuctuations give rise to an isocurvature per-turbation spectrum [8].Purely isocurvature Ω=1per-turbations are,however,not consistent [9–11]with theobservational data,but an admixture of (uncorrelated or correlated)adiabatic and isocurvature perturbations cannot be ruled out [11–14].However,if we do not insist on a ﬂat universe,the situation could be diﬀerent.Recently,it was pointed out [15]that in the general (Gaussian)case the scalar power spectrum is a 5×5matrix P ij (k )= A i (k )A j (−k ) ,where i,j label one adi-abatic and four isocurvature modes [cold dark matter (CDM),baryon,neutrino density,and neutrino velocity]and their correlations.Here we shall focus on a purely isocurvature primordial perturbation in the CDM which has the power spectrumP S (k )=B[16,17].Thus we stress that we are using a phenomeno-logical power-law spectrum,which does not necessarily follow from any particular inﬂation model.We shall re-turn to this point later in this paper.After the clear detection of the acoustic peak around ℓ≃200it became evident that the adiabatic modelsﬁt well to the data[1,2,4,5,18,19].However,this should not be taken as a proof that all pure isocurvature models are ruled out.Some unconventional combination of cosmo-logical parameters,e.g.,Ω=1and a spectrum with a large tilt,could at least in principle give an equally good ﬁt as do the adiabatic models.Pure isocurvature models have two well-recognized problems:excess power at low multipoles and a peak structure that is roughly speaking out of phase byπ/2 when compared to the adiabatic one[20].Since the angu-lar power in the low multipole region was measured quite ﬁrmly by the Cosmic Background Explorer(COBE),χ2ﬁtting forces the overall normalization constant in pure isocurvature models to be smaller than in the adiabatic case,which leads to too little power at higher multipoles. The easiest and perhaps the only way to compensate for this is to introduce a large spectral tilt.Moreover,since ﬂat adiabatic modelsﬁt the observed peak atℓ≃200 well,it is obvious that theℓ≃200peak falls between the ﬁrst and second peaks of anyﬂat isocurvature model. Accordingly,in our earlier study[11],the best-ﬁtﬂat isocurvature model was found to have a largeχ2=116 for30data points and6parameters whereas the best adiabatic model hadχ2=22.Thus we have two possibilities for a better isocurva-ture model.Theﬁrst is to lower the total energy density parameter so much that the position of theﬁrst isocur-vature peakﬁts to the observed peak atℓ≃200,which means that we have to allow for an open universe(Ω<1). The other possibility is to increase the total energy den-sity parameter so much that the position of the second isocurvature peakﬁts theℓ≃200peak[21],implying a closed universe(Ω>1).In this case theﬁrst isocurva-ture peak atℓ≃60...100should eﬀectively disappear. In fact,a large spectral tilt would have precisely this ef-fect since it would decrease the relative power at lowℓ. The purpose of the present paper is to study these pos-sibilities systematically toﬁnd out if CDM isocurvature models are indeed completely ruled out by the presently available CMB data.II.METHODS AND RESULTSIn order to compare the isocurvature models with adiabatic ones we choose one representative well-ﬁtted adiabatic model(n adi,Ωm,ΩΛ,ωb,ωc,τ)= (0.98,0.38,0.62,0.021,0.13,0);cf.[1].Using the same data sets and algorithm as for isocurvature models, we getχ2=44for this adiabatic“reference”model. Fig.3(b)conﬁrms that this modelﬁts well both the low ℓpart of the angular power spectrum and the acoustic peaks.Our starting point for analyzing isocurvature models is a large grid with the following free parameters:•n iso=1.00...7.00(60values)•Ωm=0.06...2.31(16values)•ΩΛ=−1.00...1.10(14values)•ωb=0.001...0.100(10values)•ωc=0.01...1.60(15values),whereΩm is the total matter density,ΩΛis the vacuum energy density,ωb=h2Ωb is the baryon density,andωc= h2Ωc is the CDM density.The sixth free parameter is the overall normalization factor B of Eq.(1).The Hubble constant h is not a free parameter,since h2Ωm=ωm=ωb+ωc.We use a top-hat prior h=0.45...0.90and assumeτ=0for the optical depth due to reionization. The angular power spectrum of all the models in the grid was calculated by CAMB[22]assuming isocurvature CDM initial conditions.We use theχ2method to compare models and data, because it allows us to quickly search a large parameter space.This method is approximate[17]and we do not attempt precise estimates for cosmological parameters or conﬁdence levels.As will be seen,the conclusion is clear enough in ruling out the isocurvature models so that it is not necessary to go to a full maximum likelihood analysis [23].Using the latest Boomerang data[1],together with Maxima-1[3]and COBE data[24]we calculateχ2for each model.The resulting best-χ2contours in the (Ωm,ΩΛ)plane are presented in Fig.1by gray levels. The best-ﬁt model turns out to haveχ2=80with (n iso,Ωm,ΩΛ,ωb,ωc)=(2.00,2.11,−1.00,0.020,1.40). From Fig.1(a)we see that the best-ﬁt isocurvature mod-els lie along two bands in the(Ωm,ΩΛ)plane,the left band corresponding to open universes,and the right cor-responding to closed universes.In the best-ﬁt models the spectral index falls in the range n iso=2...3.A detailed examination of the various pure isocurva-ture models allows us to conclude that the main prob-lems are the spacings of the higher acoustic peaks and the slope in the(lowℓ)Sachs–Wolfe region.COBE mea-sured a close-to-ﬂat Cℓspectrum,but the isocurvature models have a signiﬁcant positive slope arising from the large primordial blue spectral tilt needed to get enough power at higher multipoles.In the best-ﬁt open models the prominent peak in the CMB data isﬁtted by theﬁrst acoustic peak of the isocur-vature model.Fig.1(a)shows that in the best-ﬁt open region theﬁrst peak lies in the range150<∼ℓ<∼230. Since the data do not show a high second peak,these models need a small baryon densityωb to boost up the ﬁrst peak and suppress the second peak.(In the adia-batic case,adding more baryons enhances odd acousticnear to the lower right corner.The contours for deviation from the bestﬁt are as follows:white∆χ2<10;light gray 10<∆χ2<40;medium gray40<∆χ2<100;and dark gray∆χ2>100.(a)Dashed lines show the position(ℓ)of theﬁrst acoustic peak and solid lines the second peak.(b) Solid lines give the values ofσ8Ω0.56m,and the dotted area is that allowed by the LSS constraint0.43<σ8Ω0.56m<0.70.peaks over even[20],but in the isocurvature case increas-ingωb boosts even peaks.)Actually,all the best-ﬁt open models have a baryon density ofωb=0.001,which is the smallest value in the grid.However,even assuming such an unphysically low baryon density as0.0005only gives about half of the power needed toﬁt theﬁrst peak,so not scanning belowωb<0.001seems justiﬁed.In the best-ﬁt closed models theℓ≃200peak in the CMB data isﬁtted by the second isocurvature peak,which lies,according to Fig.1(a),in the range8mχ2=103.The contours for deviation from the bestﬁt are as follows:white∆χ2<35;light gray35<∆χ2<140;medium gray140<∆χ2<350;and dark gray∆χ2>350.The upper left corner corresponds to the closed models where the second acoustic peakﬁts the prominent peak in the Cℓdata.(b)The best-ﬁt physical region using theﬁne grid.The solid contours show the baryon densityωb.The best-ﬁt model hasχ2=121 and the gray levels are as follows:white∆χ2<6;light gray 6<∆χ2<30,medium gray30<∆χ2<60,and dark gray ∆χ2>60.225<∼ℓ<∼265.As one might expect(see,e.g.,[25]for an adiabatic analogy),now the ratio of theℓ≃200peak to the higher multipole Cℓ’s in the dataﬁxesωb near the value0.02in the whole best-ﬁt band.In contrast one ob-tains almost no restriction forωc.This is consistent with Fig.1,whereΩm can be seen to be able to take almost any value,which is then compensated byΩΛto produce the correct peak position.According to Fig.3(a)the best isocurvature model (χ2=80)does badly with the COBE region as well aswith COBE(⋄),Boomerang(•),and Maxima-1(◦)data.(a) Best-ﬁt isocurvature model of Fig.1(solid line)and best-ﬁt open model with LSS constraint(dashed line).(b)Best phys-ical isocurvatureﬁt from theﬁne grid(solid line)and the adi-abatic reference model(dashed line).Note that up toℓ=25 theℓaxis is logarithmic.after the prominent peak.This peak isﬁtted quite well by the second acoustic peak while theﬁrst acoustic peak appears as a small shoulder aroundℓ≃80.The considerations so far rely on the CMB data only. However,as is well known,when discussing isocurva-ture models it is essential to include also the large-scale structure(LSS)data.As we will see,rough mea-sures are already very eﬀective in constraining the mod-els.Therefore we make use of the the amplitude of the rms massﬂuctuations in an8h−1Mpc sphere only,de-noted asσ8,which the LSS data restricts to the range0.43<σ8Ω0.56m <0.70[26].The contours ofσ8Ω0.56mareshown in Fig.1(b).Apart from the upper left corner of the(Ωm,ΩΛ)plane,the best-ﬁt closed models appeargive a far too largeσ8Ω0.56m>∼1.5.This is natural, we need a large n iso to do away with theﬁrst peakisocurvature shoulder”)atℓ≃60...100and to getpower at higher multipoles.A large n iso evidentlyto a largeσ8.To compensate for this,one woulda smallΩm.We have checked that the smallermwe have,the larger n iso is allowed for by the LSS con-In particular,the upper left corner closed models Fig.1b obey the LSS constraint,although they have rather large spectral index n iso≃3.1.On the other hand,the best-ﬁt open models tend toa slightly too smallσ8Ω0.56m.These models have a small n iso 2.1,for the following reasons.(1) these modelsﬁt theﬁrst isocurvature peak to the ≃200peak in the data,they do not need a large n iso eliminate thisﬁrst peak.(2)The smaller scales do not as large a boost from n iso,since power is provided the second peak where the data requires it.Because this smaller n iso these modelsﬁt the COBE region We have repeated the analysis of minimizingχ2but with the LSS constraint.As one might expect,this most of the best-ﬁt closed models,leaving only with a smallΩm and a largeΩΛ;see the upper leftof Fig.2(a).The reason for this shifting of theclosed-model region to the opposite corner in the m,ΩΛ)plane is easy to rge n iso leadsa largeσ8,and hence the prior0.43<σ8Ω0.56m<0.70Ωm to be small,which in turn implies a largeΛin order to adjust the peak position.After imposing the LSS constraint,the best-ﬁt model no longer a closed one but an open model at the cor-of the parameter space withωb=0.001andΩΛ= 1.00.Thisﬁt hasχ2=103and(n iso,Ωm,ΩΛ,ωb,ωc)= .05,0.71,−1.00,0.001,0.16).Fig.3(a)shows that the acoustic peak atℓ≃170is too low toﬁt the data.It is clear that theﬁt would further improve if one allowed for even smallerωb andΩΛ.However,such a smallωb is in clear conﬂict with big bang nucleosynthesis(BBN). There is some debate in the BBN community[27]on how small anωb could be acceptable.After imposing a very conservative lower limit,ωb≥0.003,our best-ﬁt open model is already signiﬁcantly worse than the best-ﬁt closed models.Moreover,the best-ﬁt open models have a very small,even a negative,ΩΛ.This region of the (Ωm,ΩΛ)plane is disfavored by the observed supernova redshift-distance relationship[28].Thus we conclude that the best candidates for pure isocurvature models are the remaining best-ﬁt closed models.These models satisfy the LSS constraint and have an acceptableωb.They lie in the region of smallΩm and largeΩΛ.We scanned this region with aﬁner grid.The resulting best-χ2contours in the (Ωm,ΩΛ)plane are shown in Fig.2(b)along with the baryon density of these models.The best“physically acceptable”isocurvatureﬁt has(n iso,Ωm,ΩΛ,ωb,ωc)= (2.80,0.12,0.97,0.015,0.074).Theﬁt remains very bad, however,withχ2=121for40data points and6pa-rameters,to be compared toχ2=44of theﬂat adia-batic reference model.Because of the highχ2of the best ﬁt,it is unnecessary to consider the LSS spectrum in a more detailed way.The badness of theﬁt is mainly due to the COBE and Boomerang data;see Fig.3(b).The COBE contribution toχ2is2.4per COBE data point, the Boomerang contribution is4.2per data point,while the Maxima contribution remains at1.7.The slope of the best-ﬁt model is the reason for the poorﬁt to COBE,and although the prominent peak in the data isﬁtted quite well,the“ﬂat adiabatic”peak structure of the second and third peaks in the Boomerang data leads to a con-ﬂict with the isocurvature peak structure.As mentioned earlier,the power-law form for the power spectrum is not necessarily the most natural one in open and closed models due to the eﬀect of spatial curvature. The curvature scale in the models studied is compara-ble to the Hubble scale,or larger.Thus its eﬀect is ex-pected to be reﬂected in the COBE region of the power spectrum,but not in the Boomerang/Maxima region. To assess the signiﬁcance of this problem,we repeated our analysis without the COBE data points.The re-sults remained essentially unchanged.Without the8 COBE points we gotχ2=70for the best-ﬁt model,χ2=91for the best-ﬁt with LSS constraint,χ2=89for the best physically acceptableﬁt from the reﬁned grid, andχ2=40for the adiabatic reference model.Hence the Boomerang data alone are suﬃcient to rule out pure isocurvature models and our conclusions do not depend on the question of the eﬀect of spatial curvature on the power spectrum.Actually,since the main discriminant is the relative po-sitions of the three peaks in the Boomerang data,which show an“adiabatic”instead of an“isocurvature”pat-tern,our conclusion should be independent of the shape of the primordial power spectrum as long as the observed peaks are indeed due to acoustic oscillations and do not represent features of the primordial power spectrum it-self.III.SUMMARYWe have surveyed a large space of parameters for pure isocurvature models,and allowed for both open and closed universes,toﬁnd out whether there are any pure isocurvature models thatﬁt the current CMB data better than or at least equally as well as theﬂat adiabatic model. There are none.We conclude that,even if one ignores the high-z supernova data,pure isocurvature CDM mod-els,including the ones with a heavily tilted spectrum,are completely ruled out by the present CMB and LSS data. Incidentally,the isocurvature models do not do too badly with the Maxima-1data.The main CMB problems are with the COBE and the Boomerang data.To have suﬃ-cient smaller-scale power,and to suppress theﬁrst peak and boost the second peak in the closed models,a large blue tilt is needed.This leads to a slope in the Sachs–Wolfe region and reduces the largest-scale power below the level observed by COBE.The most signiﬁcant prob-lem,however,is with the Boomerang data.Boomerang shows a second and a third peak with a spacing that cor-responds to aﬂat universe,whereas the position of the ﬁrst peak in the data cannot beﬁtted byﬂat isocurvature models.ACKNOWLEDGMENTSThis work was supported by the Academy of Finland under the contracts101-35224and47213.We thank Alessandro Melchiorri for a useful communication,Elina Sihvola for technical help,and the Center for Scientiﬁc Computing(Finland)for computational resources.We acknowledge the use of the Code for Anisotropies in the Microwave Background(CAMB)by Antony Lewis and Anthony Challinor.[15]M.Bucher,K.Moodley,and N.Turok,Phys.Rev.D62,083508(2000);astro-ph/0007360.[16]K.M.G´o rski,B.Ratra,R.Stompor,N.Sugiyama,andA.J.Banday,Astrophys.J.Suppl.Ser.114,1,(1998);B.Ratra and P.J.E.Peebles,Phys.Rev.D52,1837(1995); B.Ratra and P.J.E.Peebles,Astrophys.J.Lett.432,L5(1994); D.H.Lyth and A.Woszczyna, Phys.Rev.D52,3338(1995);M.Zaldarriaga,U.Sel-jak,and E.Bertschinger,Astrophys.J.494,491(1998);A. 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掌握观察的技巧英文作文

掌握观察的技巧英文作文英文：Observation is a crucial skill in our daily lives. It helps us to understand and interpret the world around us. To master observation, one needs to pay attention todetails and use all their senses. Here are some tips that can help:1. Be present in the moment: Focus on what is happening around you and try to be fully present in the moment. This means being aware of your surroundings and paying attention to what is happening.2. Use your senses: Use all your senses to observe the world around you. This means not just relying on your eyes but also on your ears, nose, and touch.3. Look for patterns: Look for patterns in the things you observe. This means looking for similarities anddifferences in the things you see and trying to make connections between them.4. Take notes: Take notes of the things you observe. This means writing down your observations and thoughts so that you can refer to them later.5. Practice: Practice observing the world around you. This means making a conscious effort to observe things and to be aware of your surroundings.中文：观察是我们日常生活中至关重要的技能。

最全的ansys英语单词表1∕3

Aabbreviation 节略/缩写abnomal不正常的abort异常中断abridged删节的ABS (absolute value) 绝对值absorb吸引acceptable可忍受的accidentally偶然地/意外地accretion 增长activation energy 活化能activate 激活active当前的/活跃的Activation overpotential volts 电势active center活性中心accurate精确的accuracy精确性acoustic 声actuator传动装置/致动器ADAMS(Automatic Dynamic Analysis of Mechanical Systems)即机械系统动力学自动分析Adapt/Iso-Value adaption 等值自适应adaptive自适应adaptive descent自适应下降adiabatic绝热的Adiabatic temperature of burnt products 燃烧产物的绝热温度adius 心adjacent邻近的admittance 导纳advection热之水平对流aerosol浮质(气体中的悬浮微粒，如烟，雾等)Aggressiveness factor 力口速因子aix-har谐单元algorithm运算法则align排列/使成一行Aligned with surface与已有表面平行Aligned with view plane与视图平面平行Align Vertices 校准顶点Align next user-positioned与下一个用户位置对齐ASM (algebraic stress model)代数应力模型allocation 分酉己ALP热膨胀系数alpha开端/最初Alter更改alternate交替的alternating stress交变应力/反复应力ambient环境的amines 胺amplitude 幅度analogy 模拟analytical solution 解析解Anisotropic各向异性Anisotropic Hardening 各向异性强化angle角度angular有角(度)的animate动画/使有生命的Animation sequences 动画序歹U Animation frames硬拷贝文件annotation注释文字annular环流的annulus圆环/扇形anthracite 无烟煤apparent显然的/外观上的/近似的antisymmetric 反对称的append附加apply应用approximation 近似值arbitrary任意的arc-length 弧长area moment of inertia 面积惯性矩arc弧archive 合并arsenic砷酸盐array排列/矩阵Area-weighted average 面积加权平均arrowhead 箭头Arrowhead spacing 箭头间距Arruda-Boyce本构模型artificial 仿真的ascend上升ASCII (American Standard Code for Information Interchange) 美国信息交换标准码Aspect Ratio 纵横比assembly 部件assign分配/指派associativity 关联性assume彳假设assumption 假设atomization 雾化attched to添加到At specified loc通过特定点（自己指定）定义局部坐标系attribute归属/特性At wp origin通过以前定义的工作平面的原点为中心定义局部坐标系attribute 属性automatic 自动的Auto range自动范围avg（average）平均AX/AY/AZ矢量磁位差约束（磁分析类）axial轴向的/轴的axe削减axis轴/坐标轴axi-symmetric 轴对称的axi-symmetric shell 轴对称壳体axi-symmetric stress condition 轴对称应力状态axi-symmetric circular elastic plates 轴对称弹性圆板Bbanded绑扎banded contour map波段等高线图Bandwidth带宽；频宽；带宽值bar条形Bars柱状图Backflow 回流Backflow total temperature 回流总温Backflow direction specification method 回流方向定义方法batch file批处理文件battlement 城垛式BCs（Boundary Condition symbol）边界条件标志beam 梁behavior 特性bending 弯曲Bilinears双线性binary二进制/二元的biography 经历biot螺栓bisection对分法bituminous coal 烟煤blank空白Blank entire cell when在满足下列条件时即去除网格单元Blank when删除文件Blank from front从前面剖切Blank from back从后面剖切Blend Volumes修整体的边Blatz-Ko Blatz-Ko 模型block长方体blow-off water 排污水blowing devices鼓风(吹风)装置body force体积力body load体载荷boiler plant锅炉装置(车间)booleans布尔运算/合并，交叉或者删除体Both lines and flood 画线并填充Boundary name pattern 边界名称样式Bounded有界平面bounds界限box-behnken响应面法Box color边框的颜色brief简要的/摘要brick砖状物Brownian rotation 布朗转动buffered缓冲/减轻bulk膨胀bulk density堆积密度burner assembly燃烧器组件Ccalc(calculation)计算capability 性能capacitor/capacitance 电容cap/capping 覆盖captrue俘获/捕捉carbon monoxide COcarbonate碳酸盐Carpet plot 烟丝图carry-over loss 飞灰损失CART(cartesian)笛卡尔坐标case算例casing 箱/壳catalisis 催化catalog 目录Celsius [亦作Centigrade](温度)摄氏度[符号C-]cent &rad通过定义圆心和半径的方法定义圆弧centr &cornr输入矩形的几何中心点坐标以及一个顶点坐标来绘制centroid 形心CFD(computational fluid dynamics) 计算流体动力学chaboche chaboche 模型channeled有沟的/有缝的char焦炭/炭charge电荷/负载charge-voltage 充电电压chart统计图表Cholesky乔里斯基Cholesky decomposition/Cholesky Decorrelation 乔里斯基分解chosen挑选出来的circular循环的circulation circuit 循环回路circumferential velocity 圆周速度circumscr夕卜接圆classification 类另ijclinkering 熔渣Cylinder 圆柱体Soild cylinder ：通过圆柱底面圆心和半径，以及圆柱的长度定义Hollow cylinder （空心圆柱）：通过空心圆柱底面圆心和内外径，以及长度定义Partial cylinder （部分圆柱）：通过空心圆柱底面圆心和内外径，以及圆柱开始和结束角度、长度来定义By end pts & z :通过圆柱底面直径两端的坐标和圆柱长度来定义By dimensions :通过圆柱内外径、圆柱两底面z坐标、起始和结束角度来定义Clip剪切clipped截尾的clipped Gaussian distribution 截尾高斯分布closure （模型的）封闭cloud of particles 颗粒云cluster颗粒团CMS（Content Management System）内容管理系统coal off-gas煤的挥发气体Coal calculator 煤粉计算Coal dry density煤的干燥密度Coal particle煤粉颗粒Coal as-received HCV煤粉收到的高热值coarse粗糙的coarse grid疏网格，粗网格coarsening网格粗化coaxial同轴的coefficient 系数coefficient of restitution 回弹系数/恢复系数coercive高压的Coherence 粘附cohesive内聚力的/凝聚性的coil线圈coincide 一致coincident 一致的coke 碳collapse崩溃/屈服Collapse factor合并因子/皱缩因子column 柱/歹ijcombine 复合Combined Kinematic and Isotropic Hardening 混合型强化combustion 燃烧common log常用对数compatible 兼容的complex复合的component 构件composite 复合Composition PDF transport combustion model 组分概率密度输运燃烧模型Composition PDF transport model概率密度输运燃烧模型compress 压缩Computational fluid dynamics general notation system 计算流体力学通用记号系统CON(continued)连续的concatenate 连接concentrate 集中concentration 浓度concrete 混凝土condensation 冷凝/凝结conduct 导电conduction热传导/电导conductivity 传导性conductivity matrix 传导矩阵conductor 导体confidence 信任configuration 构型/配置confined flames 有界燃烧confirmation 证实/确认concatenate 多联体cone圆锥Connect Edges Disconnect About Real Edge连接实际的和(或者)虚拟的边Connect Faces Disconnect Faces 合并实面和虚面conservation守恒不灭conservation equation 守恒方程conserved scalars 守恒标量connectivity 连通性considerably 相当地consistent 固定Console控制台constant 常数constraint 约束contact elements 接触单元contamination ^污染contingency偶然/可能性/意外事故/可能发生的附带事件contour 轮廓contours等值线convect对流传热contraction收缩因子contribution 贡献convection 对流convergence 收敛convergence indicator 收敛精度Convergence tolerance 收敛公差convergence value 收敛值conversion 转换convolve 卷coordinate 坐标Copper扫面整个体积饿的指定的源面的网络节点类型correl相关的correlation相关系数correspond符合/对应corrosion 腐蚀/锈couple/coupled/coupling 耦合creep蠕变criteria准则/标准critical 临界critically 精密地Critical rate of strain 临界应变速率cross-correlation 互关联cross product 向量积cross-sectional 截面CS坐标系ctr中点cuit剪切/全脱胶丝cum distrib(cumulative distribution) 累积分布cumulative 累力口cumulativeDF(CDF,cumulative distribution function)累积分布函数cupl耦合curr(current)电流/电流约束(磁分析类)current analysis 电流分析curtain wall护墙/幕墙curvature圆弧/曲率curve曲线/曲线图表curve fitting曲线拟合custom定制cyano鼠(基)，深蓝，青色cyclic循环的cylinder 圆柱CYLIN(cylindrical)柱坐标系cylindrical Y Y-柱坐标系cyclone 旋风cyclone separator旋风分离器［除尘器］Ddamp阻尼damper阻尼器damping阻尼系数dashed虚线DB (dependent)相关dead zones 死区debug调试decide解决decimation 抽取decimation factor 取样因子decompose 分解decouple解藕的defy使成为不可能default恢复默认设置/系统默认值degeneracy 简并性degenerate 简并的degress 角度default默认/不履行/缺省deflection 挠度deformed变形的demography 统计density 密度deposition 沉积Depth blanking 深度剖切derivative with respect to 对…的导数derive导出derivative 导数descend 下降deselect取消选定design cycle设计流程designXplorer多目标优化模块Des options分离涡模型选项desposit积灰，结垢deterministic 宿命的deterministic approach 确定轨道模型deviation 偏差devoid 缺乏devolatilization析出挥发分，液化作用determine 限定diagnostic 诊断Diagonal ratio 对角比率differentiate 微分diffusion 扩散Diffusion energy source考虑组分扩散过程引起的焓输运作用diffusivity扩散系数/热量扩散率digenmode固有模式/本征模Digital elevation map数字高度吐格式Digital exchange format 数字交换格式digitize数字化digonal二角（的）/对角的/二维的dilute 稀的dimensionality 维度dimension 维度diminish 减少diode二极管direct numerical simulation 直接数值模拟Direction specification method 方向定义方法directory 目录discharge 释放discrete离散的Discrete phase model 离散相模型discretization ［数］离散化discrete phase分散相，不连续相discipline 练习Disconnect Vertices 分离顶点discontinuity间断（性）/不连续（性）dispersion 弥散displacement 位移/变形display 展示Display mesh after reading 读入后显示网格dissector 扩流锥dissipation 损耗dissociate thermally 热分解dissociation 分裂Distributed memory on a cluster 集群分布式内存distribution of air 布风divide除以division商/除法divisndmx最大位移/最大变形DOF自由度Domain域；区域；范围；领地Domain extents计算域范围dominant显性的/支配的dot line 虚线dot product 点积Double precision 双精度drag coefficient牵引系数，阻力系数drag and drop 拖放drag force 曳力Draw profiles画轮廓线drift velocity 漂移速度driving force驱［传，主］动力droplet 液滴drum锅筒dry-bottom-furnace 固态排渣炉dry-bottom冷灰斗，固态排渣duct 管dump 渣坑dust-air mixture 一次风dynamic动态的Dynamic pressure 动压duct导管/输送管EEBU---Eddy break up漩涡破碎模型eddy涡旋Eddy dissipation model 涡耗散模型Eddy-dissipation concept model 涡耗散概念模型edge边缘Edge ratio边比率eff（effusion）泄流effective有效的effluent废气，流出物eigen特征eigen bulking特征膨胀elapse经过elapsed time运行时间elastic有弹性的elbow弯头electromag 电磁electromechanic电动机械的electromagnet 电磁体electro-static 电磁electro-staic precipitators 静电除尘器elem （element）单元ellipse 椭圆elliptic椭圆形的EMAG 一整套用于静态、交流、瞬态低频电磁场分析的完整工具emanate散发，发出，发源，［罕］发散，放射embrasure 喷口，枪眼Embed graphics windows 嵌入图形窗口EMF电场耦合值约束（电子类）emissivity ［物］发射率empirical 经验的EMT（electrical-metallic tubing）电子金属管enclosure附件/外壳endothermic reaction 吸热反应energies 能enforced 执行enhance 增，涨Enhanced wall treatment增强型壁面处理enlarge 扩大ensemble 组，群，全体enthalpy热含量/焓值entity实体entrain 携带，夹带entrained-bed 携带床EPEQ累加的等效塑性应变EPPL COMP构件塑性应变分量EPTO COMP构件总应变EPTO eqv等效应变EQ/EQN （equation）反应式/等式Equiangle skew通过单元夹角计算的歪斜率Equisize skew通过单元大小计算的歪斜率equilibrium化学平衡equiv.相等的equivalent 当量eqv平均erase擦除ESCIMO——Engulfment （卷吞）Stretching （拉伸）Coherence （粘附）etc等等evaluation 评价，估计，赋值evaporation 蒸发（作用）Eulerian approach 欧拉迫近Eulerian model 欧拉模型Everything below 某类对象EX弹性模量excitation 激励exec (execute)执行existing目前的exothermic reaction 放热反应expansion膨胀因子expertise 经验explicitly 明白地，明确地explicit solver显式解算器explicit(explic)显式的explicit creep 显式蠕变exponential指数函数/指数曲线Exponential distribution 指数分布exponentiate 取幕export导出extend 扩展extension 广延external外部的extra附加extract 提取extrapolate外推/外插extreme极端的extrude 压出。

基于各向异性高斯核图像边缘和角点检测

1. Edge detection via the combition of the isotropic and anisotropic Gaussian kernels. A noise-robustness edge detection algorithm is presented, which fuses small-scale isotropic Gaussian kernel and large-scale anisotropic Gaussian kernels (ANGKs) to attain the edge map strength (EMS). The main merit of the algorithm is that it has high edge resolution while it is robust to noise. From the ANGks, anisotropic directional derivatives (ANDDs) are derived to capture the locally directional intensity variation of an image, and then the EMS is obtained using the ANGKs. The noise robustness is highly depended on the scale while the edge resolution is depended on the ratio of the scale to the anisotropic factor. Moreover, the image smoothing results by the ANGKs reveal edge stretch effect. The fusion of large-scale ANGKs and small-scale isotropic Gaussian kernel generates the fused ESM, which is noise robust, high edge resolution and little edge stretch. Then the fused edge map is embedded into the framework of the Canny edge detector, thus a new noise-robustness edge detection algorithm is achieved, which included two modifications: contrast equalization and noise-dependent lower thresholds. At last, the proposed edge detector is examined using the empirical Receiver Operating Characteristic (ROC) Curves of the tested images and the Pratt‟s Figure of Merit (FOM).

半导体一些术语的中英文对照

半导体一些术语的中英文对照半导体一些术语的中英文对照离子注入机ion implanterLSS理论Lindhand Scharff and Schiott theory 又称“林汉德-斯卡夫-斯高特理论”。

沟道效应channeling effect射程分布range distribution深度分布depth distribution投影射程projected range阻止距离stopping distance阻止本领stopping power标准阻止截面standard stopping cross section退火annealing激活能activation energy等温退火isothermal annealing激光退火laser annealing应力感生缺陷stress-induced defect择优取向preferred orientation制版工艺mask-making technology图形畸变pattern distortion初缩first minification精缩final minification母版master mask铬版chromium plate干版dry plate乳胶版emulsion plate透明版see-through plate高分辨率版high resolution plate, HRP超微粒干版plate for ultra-microminiaturization 掩模mask掩模对准mask alignment对准精度alignment precision光刻胶photoresist又称“光致抗蚀剂”。

负性光刻胶negative photoresist正性光刻胶positive photoresist无机光刻胶inorganic resist多层光刻胶multilevel resist电子束光刻胶electron beam resistX射线光刻胶X-ray resist刷洗scrubbing甩胶spinning涂胶photoresist coating后烘postbaking光刻photolithographyX射线光刻X-ray lithography电子束光刻electron beam lithography离子束光刻ion beam lithography深紫外光刻deep-UV lithography光刻机mask aligner投影光刻机projection mask aligner曝光exposure接触式曝光法contact exposure method接近式曝光法proximity exposure method光学投影曝光法optical projection exposure method 电子束曝光系统electron beam exposure system分步重复系统step-and-repeat system显影development线宽linewidth去胶stripping of photoresist氧化去胶removing of photoresist by oxidation等离子［体］去胶removing of photoresist by plasma 刻蚀etching干法刻蚀dry etching反应离子刻蚀reactive ion etching, RIE各向同性刻蚀isotropic etching各向异性刻蚀anisotropic etching反应溅射刻蚀reactive sputter etching离子铣ion beam milling又称“离子磨削”。

organizedby

Manifolds with nonnegative sectional curvatureorganized byKristopher Tapp and Wolfgang ZillerWorkshop SummaryIn the past few years,the study of Riemannian manifolds with nonnegative and positive curvature has been reinvigorated by breakthroughs and by new connections to other topics, including Ricciﬂow and Alexandrov Geometry.Our workshop brought together experts and newcomers to theﬁeld,including5graduate students and researchers representing a diverse range of sub-specialties.Our goal was to discuss future directions for theﬁeld and to initiate progress solving signiﬁcant open problems.We had on average two talks every morning.These talks were primarily surveys em-phasizing open problems and possible future directions for continued progress.The talks were roughly divided between the following three sub-topics,which we identiﬁed as key to continued progress in theﬁeld:(1)Riemannian submersions and group actions in nonnegative curvature(2)Alexandrov Geometry and Collapse(3)RicciﬂowOn Monday afternoon,all participants gathered to list and discuss open problems related to nonnegative curvature.The lively discussion lasted almost3hours,and resulted in a preliminary list of about30open problems.Many of these problems prompted interesting discussions.Participants continued to add problems to this list during subsequent days of the workshop.The list will continue to evolve,and has the potential to become a useful resource for future researchers in theﬁeld.On Tuesday afternoon,we divided the workshop participants into three groups,cor-responding to the three sub-topics enumerated above.This subdivision remained roughly constant through the remainder of the week,with a few participants choosing toﬂoat be-tween the groups.The groups learned of each other’s activities informally each evening during happy hour,and more formally through group reports on Friday afternoon.Theﬁrst group explored Riemannian submersions and group actions.This group was the largest,and its members decided to further subdivide.They began by brainstorming pos-sible problems to attack in smaller subgroups.Before splitting up,they scheduled mini-talks to explain some recent unpublished work.These mini-talks helped participants(especially newcomers to theﬁeld)decide which subgroup they felt best equipped to join.One sub-group formed to begin classifying the Riemannian submersions from a compact Lie group with a bi-invariant metric.This subgroup quickly discovered an interesting non-homogeneous example,which contradicts the naive conjecture that all such submersions are bi-quotient submersions.This subgroup then spent most of the remaining time considering the case of totally geodesicﬁbers.This collaboration will likely lead to a paper in the coming months.A second subgroup formed to bound the dimension of a torus acting freely on a manifold12with nonnegative curvature,and to consider related problems.The third(largest)subgroup investigated cohomogeneity-one manifolds with nonnegative curvature.They discussed this topic from several angles.They considered cohomogeneity one manifolds with a totally ge-odesic principle orbit,and came to believe that the classiﬁcation of such spaces is within reach.A collaboration on this problem will continue and probably lead to the complete solution in the near future.They also considered obstructions to metrics of nonnegative curvature and smoothness conditions for cohomogeneity one actions.Finally,some of the members of this subgroup considered topological aspect of cohomogeneity-one manifolds,in-cluding topological invariants of known and candidate examples and the problem ofﬁnding cohomogeneity-one manifolds which are topologically interesting,and for which the prob-lem of constructing new metrics with nonnegative curvature or obstructions should thus be investigated.The second group explored Alexandrov geometry and collapse.This group began by generating a list of about20interesting open problems.They then chose three of these problems to explore in more depth.Theﬁrst of these problems was to extend(the dual version of)Wilking’s connectivity lemma to Alexandrov spaces.The group mapped out a proposal involving Morse functions to solve this problem.This work will hopefully lead to a collaborative solution in the near future.The second problem was to discover topological properties of an Alexandrov space which sits at the top of aﬁnite tower ofﬁber bundles. The third problem was the conjecture that all manifolds with almost nonnegative curvature are rationally elliptic.The group discussed a rough strategy for how a proof by induction on dimension might go.One important step in such a proof would be to show that any Alexandrov space which collapses to a point also admits nontrivial collapse.In exploring this issue,the group constructed an essentially complete proof that the torus does not collapse to an interval.The third group studied Ricciﬂow.Recent progress in the applications of the Ricci ﬂow to manifolds with positive curvature operator,positive isotropic curvature and manifolds with1/4pinching were discussed in the morning survey talks.The Ricciﬂow subsection gave a simple proof of Tachibana’s theorem that an Einstein metric with positive curvature operator is a space form.Two of the participants generalized recent work by Boehm and Wilking on even dimensional manifolds with small Weyl tensor to the odd dimensional case and gave a simple proof of the algebraic part needed in the proof of the weakly1/4pinching theorem.Furthermore,existence and stability of singularity models and the nonexistence of noncompact3dimensional shrinkers was discussed.They also discussed the problem of ruling out noncompact gradient shrinking solitons with positive curvature operator,or more generally classify the gradient shrinking solitons with certain positivity of the curvature.The participants were almost unanimous in feeling that the workshop was successful. One participant stated that“all conferences should be structured this way”.Of course one should add that this is only possible with a narrowly focused research area.The afternoon group-work varied between brainstorming ideas for solving very diﬃcult open problems and solving easier problems.Work at either extreme of this spectrum was felt to be productive and meaningful.Often the groups continued working past the5:00beginning of happy hour (even the most beer-loving of the groups)which demonstrates the energy that the group members felt.We expect that new collaborations will develop as a result of the workshop. Further,participants are returning home with new ideas that could shape the long term development of theﬁeld in less tangible ways.3 We are very thankful for the generous support of the AIM.We appreciate the guidance and hard work of the AIM staﬀin helping us conduct a successful workshop.。

九天国际教育GCSE真题November 2016 (v3) QP - Paper 3 CIE Chemistry IGCSE

READ THESE INSTRUCTIONS FIRSTWrite your Centre number, candidate number and name on all the work you hand in.Write in dark blue or black pen.You may use an HB pencil for any diagrams or graphs.Do not use staples, paper clips, glue or correction fluid.DO NOT WRITE IN ANY BARCODES.Answer all questions.Electronic calculators may be used.A copy of the Periodic Table is printed on page 16.You may lose marks if you do not show your working or if you do not use appropriate units.At the end of the examination, fasten all your work securely together.The number of marks is given in brackets [ ] at the end of each question or part question.CHEMISTRY0620/33Paper 3 Theory (Core)October/November 20161 hour 15 minutesCandidates answer on the Question Paper.No Additional Materials are required.Cambridge International ExaminationsCambridge International General Certificate of Secondary EducationThis document consists of 16 printed pages.[Turn overIB16 11_0620_33/2RP © UCLES 2016The syllabus is approved for use in England, Wales and Northern Ireland as a Cambridge International Level 1/Level 2 Certificate.0620/33/O/N/16© UCLES 20161 T he diagram shows part of the Periodic Table.LiC N O F Ne Si Ge C l Ar TiCrBr Kr SnI XeCu ZnHA nswer the following questions using only the elements in the diagram. E ach element may be used once, more than once or not at all. (a) W hich element(i) f orms 21% of the air,....................................................................................................................................... [1] (ii) r eacts with water to form a solution which turns litmus paper from red to blue, (1)(iii) f orms ions of type X 3+ which when tested with aqueous sodium hydroxide produce a greenprecipitate, (1)(iv) i s a red-brown liquid at room temperature and pressure,....................................................................................................................................... [1] (v) i s a noble gas with only three complete electron shells? (1)(b)T he table gives some information about the properties of four metals.metaldensityin g / cm3relativestrengthresistanceto corrosionrelative electricalconductivitymeltingpoint / °Cchromium7.28very good81857copper8.930good601283iron7.921poor101535titanium 4.523very good21660W hich one of these metals is most suitable for making the frame of an aircraft?E xplain your answer using information from the table............................................................................................................................................................................................................................................................................................................................................................................................................................................................. (3)[Total: 8]0620/33/O/N/16© UCLES 2016[Turn over2A scientist analysed the substances present in a 1 dm3 sample of river water in an agricultural area.sample.T he table shows the mass of each ion dissolved in the 1 dm3 Array(a) (i)W hich negative ion has the highest concentration, in g / dm3, in this sample of water? (1)(ii)G ive the name of the ion with the formula SO2–.4 (1)(iii)C alculate the mass of sodium ions in 1 dm3 of this river water. (1)(b)D escribe a test for nitrate ions.test .................................................................................................................................................................................................................................................................................................result ..........................................................................................................................................[3]© UCLES 20160620/33/O/N/160620/33/O/N/16© UCLES 2016[Turn over(c) T he sample of river water also contains insoluble materials such as clay and the remains ofdead animals and plants.(i) W hat method could be used to separate insoluble materials from river water?....................................................................................................................................... [1] (ii) S ome of the remains of dead animals and plants contain food materials.Which two of the following substances are constituents of food?T ick two boxes. alkane carbohydrate graphiteprotein[1](iii) P articles of clay suspended in river water show Brownian motion.D escribe the movement of these particles. (1)(d) M ost of the nitrate ions in river water come from fertilisers.(i) E xplain why farmers use fertilisers.............................................................................................................................................. ....................................................................................................................................... [2] (ii) A mmonium nitrate is a fertiliser.A mmonium nitrate reacts with calcium hydroxide. ammonium nitrate + calcium hydroxide → calcium nitrate + ammonia + waterExplain why adding calcium hydroxide to the soil at the same time as nitrate fertilisers results in loss of nitrogen from the soil.............................................................................................................................................. (2)[Total: 13]3E thanol can be manufactured by fermentation and from ethene.(a)D escribe the manufacture of ethanol by fermentation and from ethene.I n your answer include•t he essential conditions required for each reaction,•o ne or more relevant word equations..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... (5)he table shows some properties of different alcohols.(b)T(i)D educe the state of methanol at room temperature.E xplain your answer.............................................................................................................................................. (2)(ii)P redict the boiling point of pentanol. (1)(iii)D escribe how the relative viscosity changes with the number of carbon atoms in the alcohol. (1)© UCLES 20160620/33/O/N/16(c) (i)D raw the structure of ethanol. Show all of the atoms and all of the bonds.[2](ii)G ive one major use of ethanol. (1)[Total: 12]© UCLES 2016[Turn over0620/33/O/N/160620/33/O/N/16© UCLES 20164Jelly is a mixture of water and protein chains.waterprotein chains(a) A crystal of blue dye was placed on top of some jelly.A fter 30 minutes some of the blue colour could be seen in the jelly.A fter 1 day the blue colour had spread out further into the jelly.at the startafter 30 minutes after 1 daycrystal ofU se the kinetic particle model of matter to explain these observations. .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... (3)(b) T he diagram shows the colour changes of the indicator bromocresol green at different pHvalues.green increasing pHblueyellowP redict the colour of bromocresol green in pure water,.............................................................................................................................in a strongly acidic solution. .......................................................................................................[2]0620/33/O/N/16© UCLES 2016[Turn overT he concentration of an alkali can be found by titrating it with an acid using the apparatusshown.(i) S tate the names of the pieces of glassware labelled A and B .A .........................................................................................................................................B .........................................................................................................................................[2](ii) D escribe how you would carry out a titration using the apparatus shown.............................................................................................................................................. ............................................................................................................................................. ............................................................................................................................................. ............................................................................................................................................. ............................................................................................................................................. (3)[Total: 10]5L ime (calcium oxide) is made by heating limestone (calcium carbonate).CaCO3(s) CaO(s) + CO2(g)(a) (i)I s this reaction exothermic or endothermic?E xplain your answer.............................................................................................................................................. (1)(ii)T he reaction is reversible.W hat information in the equation shows that this reaction is reversible? (1)(b)The diagram shows a furnace for making lime.(i)O n the diagram, write•t he letter C to show where the waste gases exit the furnace,•the letter L to show where the lime is removed from the furnace.[2](ii)S uggest a reason for adding coke (carbon) to the furnace. (1)(c)E xplain why farmers use lime to treat acidic soils..................................................................................................................................................... (2)0620/33/O/N/16© UCLES 2016(d)L imestone is used to manufacture cement. The limestone is mixed with clay and heated to1500 °C. It is then mixed with calcium sulfate and crushed.(i)D escribe the test for sulfate ions.test ......................................................................................................................................result ...................................................................................................................................[2](ii)C oncrete is a mixture of cement, silicates and water. Part of the structure of a silicate is shown.D educe the formula for this silicate. (1)(e)C oncrete contains small amounts of calcium oxide.T his can react with rainwater to form calcium hydroxide.(i)C alcium hydroxide is strongly alkaline.W hat is the most likely pH of a strongly alkaline solution?D raw a ring around the correct answer.pH 2 pH 6 pH 7 pH 12[1](ii)T he calcium hydroxide on the surface of a piece of concrete reacts with carbon dioxide in the air.C omplete the chemical equation for this reaction.Ca(OH)2 + CO2→ CaCO3+ ...............[1](iii)L imewater is an aqueous solution of calcium hydroxide. A teacher left an open beaker of limewater in the laboratory.A fter a week, the solution in the beaker was pH 7 and a white precipitate was observed.U se the information in (e)(i) and (e)(ii) to help you explain these observations.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................. (3)[Total: 15]6T he Periodic Table is a method of classifying elements.(a)(i)I n what order are the elements arranged in the Periodic Table? (1)(ii)H ow does the character of the elements change from left to right across a period? (1)(iii)D escribe two trends in the properties of the elements going down Group I........................................................................................................................................................................................................................................................................................... (2)(b)T he halogens are a group of elements with diatomic molecules.(i)C hlorine reacts with an aqueous solution of sodium iodide.C l2 + 2Na I→I2+ 2NaC lW hat colour change would be observed in the solution?from ............................................................. to (2)(ii)A statine, At2, is a halogen.S uggest why astatine does not react with aqueous potassium iodide. (1)(c)C hlorine reacts with hydrogen to form hydrogen chloride.(i)C omplete the chemical equation for this reaction.C l2 + ............... → ......HC l[2](ii)D raw a diagram to show the electronic structure of a molecule of hydrogen chloride.S how only the outer shell electrons.[2](iii) H ydrochloric acid reacts with lithium hydroxide.C omplete the word equation for this reaction.hydrochloric acid+lithium hydroxide→+[2][Total: 13]Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonableeffort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity.To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at after the live examination series.Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.7 T he diagram shows the changes of state when phosphorus is cooled slowly to room temperature.(a) G ive the names of the changes of state labelled A and B .A ................................................................................................................................................B ................................................................................................................................................[2] (b) D escribe the arrangement and motion of the particles in solid phosphorus.arrangement .............................................................................................................................. motion ........................................................................................................................................[2] (c) I s phosphorus(V ) oxide an acidic oxide or basic oxide?E xplain your answer. (1)(d) P hosphorus sulfide is a covalent molecule.PPP S S SP P redict two properties of phosphorus sulfide. .................................................................................................................................................... . (2)(e) M any metal ores contain sulfides.W hen zinc sulfide is heated in air the following reaction takes place.zinc sulfide + oxygen → zinc oxide + sulfur dioxideExplain why this reaction may be harmful to the environment. .................................................................................................................................................... (2)[Total: 9]G r o u pT h e P e r i o d i c T a b l e o f E l e m e n t s1Hh y d r o g e n12H eh e l i u m4I I I I I II V V V I V I I V I I I3L il i t h i u m 74B eb e r y l l i u m 9a t o m ic n u m b e ra t o m i c s y mb o l K e yn a m er e l a t i v e a t o m i c m a s s 11N as o d i u m 2312M gm a g n e s i u m2419Kp o t a s s i u m 3920C ac a l c i u m 4037R br u b i d i u m 8538S rs t r o n t i u m 8855C sc a e s i u m 13356B ab a r i u m 13787F rf r a n c i u m –88R ar a d i u m –5B b o r o n 1113A l a l u m i n i u m 2731G a g a l l i u m7049I ni n d i u m11581T lt h a l l i u m2046C c a r b o n 1214S i s i l i c o n2832G eg e r m a n i u m7350S nt i n11982P bl e a d20722T i t i t a n i u m 4840Z r z i r c o n i u m 9172H f h a f n i u m 178104R f r u t h e r f o r d i u m –23V v a n a d i u m5141N b n i o b i u m9373T a t a n t a l u m181105D b d u b n i u m–24C r c h r o m i u m 5242M o m o l y b d e n u m 9674W t u n g s t e n 184106S g s e a b o r g i u m –25M n m a n g a n e s e 5543T c t e c h n e t i u m –75R e r h e n i u m 186107B h b o h r i u m –26F e i r o n 5644R u r u t h e n i u m 10176O so s m i u m 190108H s h a s s i u m –27C o c o b a l t 5945R h r h o d i u m 10377I r i r i d i u m 192109M t m e i t n e r i u m –28N i n i c k e l 5946P d p a l l a d i u m 10678P t p l a t i n u m 195110D s d a r m s t a d t i u m–29C u c o p p e r 6447A g s i l v e r 10879A ug o l d197111R gr o e n t g e n i u m–30Z n z i n c 6548C dc ad m i u m11280H gm e r c u r y201112C nc o p e r n i c i u m–114F lf l e r o v i u m–116L vl i v e r m o r i u m–7N n i t r o g e n1415Pp h o s p h o r u s3133A sa r s e n i c7551S ba n t i m o n y12283B ib i s m u t h2098Oo x y g e n1616Ss u l f u r3234S es e l e n i u m7952T et e l l u r i u m12884P op o l o n i u m–9Ff l u o r i n e1917C lc h l o r i n e35.535B rb r o m i n e8053Ii o d i n e12785A ta s t a t i n e–10N en e o n2018A ra r g o n4036K rk r y p t o n8454X ex e n o n13186R nr a d o n–21S c s c a n d i u m 4539Y y t t r i u m 8957–71l a n t h a n o i d s89–103a c t i n o i d s57L al a n t h a n u m 13989A cl a n t h a n o i d sa c t i n o i d sT h e v o l u m e o f o n e m o l e o f a n y g a s i s 24 d m 3 a t r o o m t e m p e r a t u r e a n d p r e s s u r e (r .t .p .).a c t i n i u m –58C e c e r i u m 14090T h t h o r i u m 23259P r p r a s e o d y m i u m14191P a p r o t a c t i n i u m23160N d n e o d y m i u m 14492U u r a n i u m 23861P m p r o m e t h i u m –93N p n e p t u n i u m–62S m s a m a r i u m15094P up l u t o n i u m–63E ue u r o p i u m15295A ma m e r i c i u m–64G dg a d o l i n i u m15796C mc u r i u m–65T bt e r b i u m15997B kb e r k e l i u m–66D yd y s p r o s i u m16398C fc a l i f o r n i um–67H oh o l m i u m16599E se i n s t e i n i u m–68E re r b i u m167100F mf e r m i u m–69T mt h u l i u m169101M dm e n d e l e v i u m–70Y by t t e r b i u m173102N on o b e l i u m–71L ul u t e t i u m175103L rl a w r e n c i u m–。

Gravity on Conformal Superspace

a r X i v :g r -q c /0311034v 1 11 N o v 2003Gravity on Conformal SuperspaceBryan KelleherThesis submitted in fulﬁllment of the requirements of the degree of Doctor of Philosophy from the Department of PhysicsUniversity College CorkNational University of Ireland,CorkSupervisor :Prof.Niall ´OMurchadha June 2003To my familyAcknowledgementsThere are many people I wish to thank.Firstly,Niall,go raibh m´ıle,m´ıle maith agat.It has been both a pleasure and a privilege.Thanks to the entire physics department for everything over the years-great times,a superb atmosphere and lifelong st -but most deﬁnitely not least-thanks to my parents,my brothers and sister,my wife to be Gill and my extended family and friends.I could not(and more than likely would not)have done it without you.AbstractThe conﬁguration space of general relativity is superspace-the space of all Riemannian 3-metrics modulo diﬀeomorphisms.However,it has been argued that the conﬁguration space for gravity should be conformal superspace-the space of all Riemannian3-metrics modulo diﬀeomorphisms and conformal transformations.Taking this conformal nature seriously leads to a new theory of gravity which although very similar to general relativity has some very diﬀerent features particularly in cosmology and quantisation.It should reproduce the standard tests of general relativity.The cosmology is studied in some detail.The theory is incredibly restrictive and as a result admits an extremely limited number of possible solutions.The problems of the standard cosmology are addressed and most remarkably the cosmological constant problem is resolved in a natural way. The theory also has several attractive features with regard to quantisation particularly regarding the problem of time.Contents1Introduction11.1Introduction (1)1.2General Relativity (2)1.3(3+1)-Decomposition (2)1.4York’s Approach (5)1.4.1Gauge Fixing in GR (6)1.5Lagrangian and Hamiltonian Formulations (7)1.5.1The Lagrangian (7)1.5.2Constraints and Evolution Equations (9)1.5.3The Hamiltonian (11)1.6Jacobi Action (11)1.7Conformally Related Solutions (12)1.8Topological Considerations (14)1.8.1Integral Inconsistencies (14)1.9Other Results (18)1.10Problem (18)2A New Hope202.1The Need For A Change (20)2.1.1Resolving The problem(s) (20)2.2The Hamiltonian Formulation (24)2.3Jacobi Action (26)2.4Conformally Related Solutions (27)2.4.1What ofξc? (28)2.5Topological Considerations (30)2.5.1Integral Inconsistencies(Slight Return) (30)2.5.2New Constraints (31)2.6The Hamiltonian Formulation (32)2.7The Volume (34)2.8Jacobi Action (34)2.9Comparison with GR (35)2.10Time (36)2.11Light Cones (37)2.12Matter in General Relativity (38)2.12.1Cosmological Constant (38)2.12.2Electromagnetism (39)2.12.3Dust (40)2.13Matter and Conformal Gravity (40)2.13.1Cosmological Constant (41)2.13.2Electromagnetism (41)2.13.3Dust (42)3Four Dimensions!443.1Introduction (44)3.2BOM Conformal Gravity (44)3.2.1The Action (45)3.2.2Dimensional Properties of Conformal Transformations (45)3.2.3Varying with respect to gαβ (46)3.2.4Varying with respect toφ (47)3.2.5A note on the action (47)3.2.6(3+1)-Decomposition (47)3.3Conformally Related Solutions (51)3.3.1Topological Considerations (51)3.4New Conformal Gravity (54)3.4.1Non-Compact (55)3.4.2Compact Manifold (56)3.5Special Case (61)3.6The Solar System (61)3.7Comment (62)4Cosmology634.1Introduction (63)4.2Cosmology In General Relativity (63)4.2.1Open Universe (63)4.2.2Flat Universe (64)4.2.3Closed Universe (64)4.3Cosmology in the Conformal Theory (66)4.3.1Open Universe (66)4.3.2Flat Universe (67)4.3.3The Closed Universe (68)4.4Cosmological Parameters (71)4.4.1Hubble Parameter (72)4.4.2Deceleration Parameter (72)4.5Problems of the Standard Cosmology (72)4.5.1The Cosmological Constant Problem (73)4.5.2The Flatness Problem (73)4.5.3The Horizon Problem (74)4.6Some Numbers (76)4.7A Non-Standard Cosmology:Anisotropy (78)4.7.1The Kasner Universe (78)4.7.2Eﬀective Anisotropic Energy Density (79)4.8Discussion (80)5Discussion81Chapter1Introduction1.1IntroductionAs formulated by Einstein,the natural arena for gravity as represented by general rela-tivity(GR)is spacetime.We have a purely4-dimensional structure and the4-geometry reigns.(The invention of GR was a truly monumental achievement and no oﬀence is intended by any attempt here to suggest an alternative theory.)Dirac[1]and Arnowitt, Deser and Misner(ADM)[2]reformulated the theory in canonical form which is more in-keeping with other areas of modern physics.This formulation led to Wheeler’s identi-ﬁcation of the conﬁguration space as superspace and GR as the theory of the evolution of the3-geometry which led to the coining(again by Wheeler)of geometrodynamics.To get superspace oneﬁrst considers Riem the space of all Riemannian3-geometries.Super-space is then Riem modulo diﬀeomorphisms,that is,we identify all3-geometries related by diﬀeomorphisms.York[3]went further and identiﬁed the conformal3-geometry with the dynamical degrees of freedom of the gravitationalﬁeld.The correct conﬁguration space for gravity should not be superspace but rather conformal superspace-superspace modulo conformal trans-formations.Barbour and´O Murchadha(BOM)[4]went further again and formulated a theory with conformal superspace at the very core.We’ll begin with a brief review of GR as found from the Einstein-Hilbert action and the ADM formulation.We’ll then discuss the York approach and the original BOM the-ory.All of this will serve as a warm up(albeit,a necessary warm up)to the real focus ofthis work.1.2General RelativityAlthough Einstein developed GR using beautiful physical reasoning and principles it is the Hilbert derivation from an action principle which is more instructive to us.(We will however refer to the action as the Einstein-Hilbert action as it was Einstein’s work which inspired Hilbert toﬁnd the action to begin with.)The Einstein-Hilbert action of general relativity is well known.It has the formS=1−(4)g(4)R d4x(1.1)where gαβis the4-metric and(4)R is the four dimensional Ricci scalar.The action is varied with respect to gαβand the resulting equations are the(vacuum)Einstein equationsGαβ= Rαβ−1δgαβ(1.4) 1.3(3+1)-DecompositionBefore we consider the new theory it will be instructive to recall the ADM treatment of general relativity as much of this will carry straight over to the new theory.The idea in the ADM treatment is that a thin-sandwich4-geometry is constructed from two3-geometries separated by the proper time dτ.The4-metric found from the ADM construction is(4)g00(4)g0k(4)g i0(4)g ik=(N s N s−N2)N kN i g ik(1.5)N=N(t,x,y,z)is the lapse function given bydτ=N(t,x,y,z)dt(1.6) and N i=N i(t,x,y,z)are the shift functions given byx i2(x m)=x i1−N i(t,x,y,z)dt(1.7) where x i2is the position on the“later”hypersurface corresponding to the position x i1on the“earlier”hypersurface.The indices in the shift are raised and lowered by the3-metric g ij.The reciprocal4-metric is(4)g00(4)g0k(4)g i0(4)g ik=−1/N2N k/N2N i/N2g ik−N i N k/N2(1.8)The volume element has the formg dt d3x(1.9)This construction of the four metric also automatically determines the components of the unit timelike normal vector n−(4)g(4)R d4x(1.12) Using the Gauss-Codazzi relations we get(4)R=R−(trK)2+K ab Kab−2Aα;α(1.13) where Aαis given by(as earlier)Aα= nαtrK+aα (1.14)nαis the unit timelike normal andaα=nα;βnβ(1.15) is the four-acceleration of an observer travelling along nN.Substituting into the action givesS= N√2£n.In the coordinates we are using here the extrinsic curvature takes the formK ab=−1∂t−N a:b−N b;a(1.18)The action is varied with respect to∂g abg g ab trK−K ab (1.19) and varied with respect to N and N a to give the initial value equationsH=0and H a=0(1.20) respectively,whereH=√2(trπ)2 −√∂t =δH∂t =−δH1.4York’s ApproachThe Hamiltonian and momentum constraints correspond to the00and0a components of Einstein’s equations(1.3).They are equivalently initial-value constraints.We need to be able toﬁnd initial data which satisfy these.One method was proposed by Baerlein, Sharp and Wheeler(BSW)[5].This is known as the thin-sandwich conjecture.First the pair{g ab,∂g ab√g ab trπ(1.31)3Now,if the CMC condition holds then the momentum constraint reduces to▽bσab=0(1.32)Now the tracefree part is transverse-traceless(TT).This property is invariant under the conformal transformationg ab−→ω4g ab(1.33)σab−→ω−4σab(1.34) It is important here that trπnow transforms in a diﬀerent way to the tracefree partσab. For the momentum constraint to be conformally invariant we need to deﬁnetrp=trπg−→trp(1.35)That is,trp transforms as a conformal scalar.Since there is a well known method toﬁnd a TT tensor we canﬁnd the pair{g ab,σab}easily.The Hamiltonian constraint transforms to becomeσabσabφ−7−13trp(1.37) is often interpreted as a notion of time,the York time,due to the properties of trp noted above.1.4.1Gauge Fixing in GRIt is important to notice the diﬀerence between a single use of the CMC condition toﬁnd initial data and subsequent use of the condition when the data is propagated.This is byno means guaranteed.As noted earlier,once the initial data has been speciﬁed the lapse and shift are freely speciﬁable.To maintain the CMC slicing during the evolution it is necessary to choose the lapse in a particular ing the evolution equations we get∂trp4+▽c trπg N c(1.38) To ensure CMC slicing we need to set▽c trπ=0and∂trp∂t =2NR−2▽2N+N(trp)2∂t=0.Thus∂trpg(R−(trK)2+K ab K ab)dtd3x(1.42)To ﬁnd the conformal action we simply transform the Lagrangian under the transforma-tiong ab −→ψ4g ab(1.43)We need to deﬁne how the lapse and shift are transformed under such a transformation.In a later chapter we will see that this theory can be found using a 4-dimensional action whereg αβ−→ψ4g αβ(1.44)and under this we would haveN −→ψ2N(1.45)andN i −→ψ4N i(1.46)Let’s adopt these as our transformation rules.Under such a transformationR −→ψ−4R −8▽2ψ2N∂g ab∂t−→ψ4∂g ab∂t(KN )ab −→ψ4(KN )ab +4ψ3g ab N c ▽c ψ(1.50)ThusK ab −→ψ2B ab =−ψ2∂t−(KN )ab −θg ab(1.51)whereθ=−4∂t −N c ▽c ψ(1.52)The Lagrangian is thusL =N√ψ+B ab B ab −(trB )2(1.53)Noteψ ˙ψ−ψ,i N i .We can alsoﬁnd a coordinateindependent form for B.This isB=−1(ψ4g)(1.54)This is analogous to the expressionK=−1(g)(1.55)for the extrinsic curvature K in general relativity.1.5.2Constraints and Evolution EquationsWe can perform the usual variations toﬁnd the constraints of the theory.Let’s vary with respect to Nﬁrst.This gives us,R−8▽2ψψ −▽2 Nψ3 =0(1.60)where we have used the other constraints to simplify.The constraints may appear more familiar if we write them in terms of the canonical momentum rather than B ab.Weﬁnd the canonical momentum,πab by varying the action with respect to∂g abgψ4 g ab trB−B ab (1.61) Then using equation(1.58)we getπab=−√The constraints are then,πabπab−gψ8 R−8▽2ψψ −▽2 Nψ3 =0(1.66)Equation(1.63)corresponds to the Hamiltonian constraint of General Relativity.Equa-tion(1.64)is the usual momentum constraint of general relativity which represents diﬀeo-morphism invariance.Equation(1.65)is new and represents conformal invariance.Our initial data consists of a pair(g ab,πab)which must satisfy equations(1.64)and(1.65). These are the initial value equations.Equation(1.63)is used toﬁnd the“conformal ﬁeld”ψonce we have speciﬁed the initial data.Equation(1.66)is a lapse-ﬁxing equation which is used to determine N throughout.We must check if these constraints are propa-gated under evolution.The evolution equations are found in the usual way.They are∂g ab2ψ−4πab+(KN)ab−θg ab(1.67) and∂πabgNψ4 R ab−g ab R−8▽2ψ2ψ−4πacπb c+√gψ3 ▽a▽bψ+3g ab▽2ψ+4√g▽(a(Nψ3)▽b)ψ+▽c N cπab −πbc▽c N a−πac▽c N b−θπab (1.68)It can be veriﬁed that these equations do indeed preserve the constraints.We can see how similar the results are to those in York’s approach.The Hamiltonian constraint has become the Lichnerowicz equation.The momentum is TT.Also,the lapseﬁxing equation is the gauge requirement of GR to preserve the trπ=0constraint.Of course,those equations are all secondary in GR whereas here they have arisen directly through a variational procedure!1.5.3The HamiltonianNow that we have found the momentum it is straightforward to ﬁnd the Hamiltonian.As usual we haveH =πab∂g abgψ4R −8▽2ψ2(trπ)2√2(trπ)2−√ψ−2N a ▽b πab +θtrπd 3x(1.71)Recalling the constraints we see that yet again,as found by Dirac and ADM,the Hamil-tonian is a sum of the constraints with Lagrange multipliers.1.6Jacobi ActionBaerlein,Sharp and Wheeler [5]constructed a Jacobi Action for general relativity.Their action was,S =+g√T GR d 3x(1.72)whereT GR =g ac g bd −g ab g cd∂g ab∂t−(KN )cd(1.73)Variation with respect to∂g abgR∂t−(KN )cd(1.74)This expression is squared to give the Hamiltonian constraint.The variation with respect to N a gives the momentum constraint.The evolution equations are found in the usual way.The equations found with the Jacobi action are those of general relativity if we identify 2N and R .We want to construct the analogous case in conformal gravity.Letus return to our (3+1)Lagrangian,L =N√ψ−(trB )2+B ab B ab(1.75)We can write this asL =√ψ+1∂t−(KN )ab −θg ab.We now extremise with respect to N .This gives us,N =+2βab βab −(trβ)21ψ −1dλ √R −8▽2ψT d 3x(1.78)where T =βab βab −(trβ)2 .This is the conformal gravity version of the BSW action (1.72).We can do all the usual variations here:N a ,˙ψand ψ.These give the momentum constraint,the conformal constraint and the lapse-ﬁxing equation respectively.Becauseof the independent variations of ˙ψand ψ,it turns out that we may vary with respect to θand ψto get the conformal constraint and the lapse-ﬁxing equation respectively.When we ﬁnd the canonical momentum πab we can “square”it to give the “Hamiltonian constraint.”Actually,this is precisely the BOM action found by starting with the BSW action and conformalising it under conformal transformations of the 3-metricg ab −→ψ4g ab(1.79)The Jacobi action is manifestly 3-dimensional and its conﬁguration space is naturally conformal superspace -the space of all 3-D Riemannian metrics modulo diﬀeomorphisms and conformal rescalings.1.7Conformally Related SolutionsIn conformal superspace conformally related metrics are equivalent.Thus conformally related solutions of the theory must be physically equivalent and so it is crucial that we have a natural way to relate such solutions.Suppose we have one set of initial data (g ab ,πab ).These must satisfy the constraints (1.64)and (1.65).We solve the Hamiltonian constraint (1.63)for our “conformal ﬁeld”ψ.Suppose now we start with a diﬀerent pair (h ab ,p ab )where h ab =α4g ab and ρab =α−4πab .Our new initial data is conformallyrelated to the original set of initial data.This is allowed as“transverse-traceless”-ness is conformally invariant and so our initial data constraints are satisﬁed.All we must do is solve the new Hamiltonian constraint for our new conformalﬁeldχsay.This constraint is nowρabρab=hχ8 R h−8▽2hχ.That is,ψis automati-αcally transformed when our initial data is transformed.Now,ψ4χ4h ab==Ng−1∂t=−√gN∂t−√2πacπb c−▽c N aπbc−▽c N bπac+▽c N cπab (1.88)These are exactly those of general relativity on a maximal slice.Thus,solutions of general relativity in maximal slicing gauge are also solutions here.There are of course solutions of general relativity which do not have a maximal slicing and these are not solutions of the conformal theory.1.8Topological ConsiderationsSo far we have not considered any implications which the topology of the manifold may have.In an asymptoticallyﬂat case we have no problems with the theory as it stands. This is not the case however in a topology which is compact without boundary.1.8.1Integral InconsistenciesRecall the lapse-ﬁxing equation of the theory in the physical representation(removing the “hats”for simplicity),NR−▽2N=0(1.89) Let’s integrate this equation:√g▽2N d3x=0(1.90)The second term integrates to zero and so we just have√gNR d3x− √gNR d3x(1.93) is positive deﬁnite.The second integral is− √gN▽c N dΣc(1.95)whereΣc is the boundary on which N=0.Since N is decreasing on the boundary we have that this term is positive deﬁnite.This means however that we have a vanishing sum of two positive deﬁnite quantities.This is a contradiction.Thus we must have N≡0. We get frozen dynamics.(This is not the case with a manifold which is asymptotically ﬂat so the earlier analysis works in that case.)Frozen dynamics also arises in general relativity if one imposes aﬁxed trπ=0gauge condition.However,this is a problem of the gauge rather than a problem of the theory as with conformal gravity.(See[6]for a treatment of this problem.)The easiest way to resolve this problem involves a slight change to the action.We intro-duce a volume term.The inspiration for this term comes from the Yamabe theorem.The action isS= N√V2/3 R−8▽2ψgψ6d3x(1.97) The power of2gψ4V4/3 R−8▽2ψψ −▽2 Nψ3 =Cψ5(1.102)The term C is given byC= N√V R−8▽2ψV(ψ)4/3(1.104)▽bπab=0(1.105) trπ=0(1.106) NR−▽2N=C(1.107)where C is nowC=1g√T: A is the average of A given by the usual notion of averageA = √ √gNR d3x− ▽2N d3x− √gNR d3x− √:Although we have only used the physical representation in our integral tests it can be veriﬁed easily that everything also works out in the general representation.Of course,in EVERY situation,this must be true.We are losing nothing by working in the physical representation.We should consider the evolution equations again now that we have changed the action. The evolution equations become∂g ab2V(ψ)2∂t =−N√V2/3 R ab−g abR−8▽2ψ√gψgψ3gg ab g3√V2/3(1.114)where C is as in(1.103).As usual we can write these in the physical representation.In this form the evolution equations are∂g ab2V2∂t =−N√V2/3 R ab−g ab R−2NV4/3gπacπb c +√V2/3 ▽a▽b N−g ab▽2N+▽c N cπab −πbc▽c N a−πac▽c N b−2gg ab CThe lapse-ﬁxing equation isNR−▽2N=C(1.121) These are precisely the constraints and gaugeﬁxing conditions for propagated maximal slicing in GR.The evolution equations are∂g ab2 πab+(KN)ab(1.122) and∂ πab g R ab−g ab R −2N g πac πb c+√3√V2/3(1.123)which are identical to those in GR apart from the global C term in the equation forπab. We can easilyﬁnd the Hamiltonian and the Jacobi action for the new form.They are H= N V2/3gψ4 πabπab−1gψ4ψ d3x(1.124) andS= dλ √R−8▽2ψT3d3x(1.125) Note again the homogeneity throughout inψ.1.9Other ResultsThere has been work on other aspects of this theory not described here.It is unnecessary from the point of view of this work while,of course,being valuable in itself with a number of worthwhile results most notably on the constraint algebra and the Hamilton-Jacobi theory.The interested reader canﬁnd this in[7].1.10ProblemAlthough the theory has emerged beautifully and easily form very natural principles we canﬁnd at least one major problem immediately.Consider the volume of a hypersurface VV= √Taking the time derivative of this we get∂V2√∂td3x(1.127)This becomes∂V∂t=0(1.129) and the volume of the universe is static.This rules out expansion and thus the stan-dard cosmological solution is lost.In particular,the red-shift,an experimental fact,is unexplained.This is a serious shortcoming.All is not lost however...Chapter2A New Hope2.1The Need For A ChangeDespite all the promising features of the theory there is at least one major drawback.We canﬁnd the time derivative of the volume quite easily and get that it is proportional to trπand thus is zero.That is,the volume does not change and so the theory predicts a static universe and we cannot have expansion.This is quite a serious problem as the pre-diction of expansion in GR is considered to be one of the theory’s greatest achievements. We are left with the following options:(a)Abandon the theory;(b)Find a new explanation of the red-shift(among other things);(c)Amend the theory to recover expansion.Theﬁrst option seems quite drastic and the second,while certainly the most dramatic, also seems to be the most diﬃcult.Thus,let’s check what we canﬁnd behind door(c).2.1.1Resolving The problem(s)Any change to the theory needs to be made at the level of the Lagrangian and so we’ll return to our earlier expression for LL=N√ψ+B ab B ab−(trB)2 (2.1)but naively change the form of B ab toB ab=−1∂t−(KN)ab−▽cξc g ab(2.2)Let’s vary the action with respect toξc.We getδL=N√gψ4 B ab−2trBg ab −1gψ4trB▽cδξc(2.3)Integrating by parts givesδL=2√3g ab trB(2.8) We shall retain the new form of B ab as deﬁned above in(2.2)all the same.The Lagrangian now readsL=N√ψ+S ab S ab−2gψ4 R−8▽2ψ3ψn(trB)2 (2.10)Before we continue,one interesting point about S ab is the following.We haveS ab =B ab −12N∂g ab3g abg cd ∂g cd2N∂g ab2Ng ab ▽c ξc −1∂t−g cd (KN )cd−32N∂g ab3g abg cd∂g cd3g ab trK (2.15)That is,S abis the tracefree part of the extrinsic curvature and is independent of anyconformal ﬁelds.Let us ﬁnd πab .This is done as usual by varying with respect to∂g abgψ42S ab δS ab −4gψ4S ab δB ab −13ψn trBg ab δB ab=2N√3ψn trBg abδB ab=−√3ψn S ab trBδ∂g abgψ4S ab +2gψn +4g ab trB(2.17)Splitting πab into its trace and tracefree parts will further clear things up.We’ll label the split asπab =σab +1gψ4S ab (2.19)and the trace is given bytrπ=2ψn+4trB(2.20) Note that our value of n is undeﬁned as yet.The constraints are found by varying with respect toξc,ψ,N and N a.The confor-mal constraint and the lapse-ﬁxing equation are given by varying with respect toξc and ψrespectively.These give▽c trπ=0(2.21) andNψ3 R−7▽2ψ4=0(2.22) respectively.From the variation with respect to N we getS ab S ab−2ψ =0(2.23) which in terms of the momentum isσabσab−1ψ =0(2.24) andﬁnally,from the variation with respect to N a we get▽bπab=0(2.25) We require conformal invariance in our constraints.Under what conditions is the momen-tum constraint(2.25)invariant?The tracefree part of the momentum,σab,has a natural weight of−4(from the original theory).That isσab−→ω−4σab(2.26) If trπ=0then we have conformal invariance.If not however,we require various further conditions.We need▽bσab=0(2.27)▽c trπ=0(2.28)and thattrp=trπg−→trp(2.29)under a conformal transformation.In our theory we have theﬁrst two conditions emerg-ing directly and naturally from the variation.Thus we simply deﬁne trp to transform as a conformal scalar as required.With this done our momentum constraint is conformally invariant.23Transforming the constraint(2.24)givesσabσab−1ψ =0(2.30) and so we must have n=−12for conformal invariance.The constraint then becomesσabσab−1ψ =0(2.31)(Note:This is exactly the Lichnerowicz equation from GR.However,we have found it directly from a variational procedure.)Thus we have determined the unique value of n and our constraints areσabσab−1ψ =0(2.32)▽bπab=0(2.33)▽c trπ=0(2.34) Nψ3 R−7▽2ψ4=0(2.35) Let’s proceed to the Hamiltonian formulation.2.2The Hamiltonian FormulationThe earlier expression forπab can be inverted to get∂g ab∂t =2Ngψ4 σab−1√6(trπ)2ψ12−gψ8 R−8▽2ψAs a consistency check let’sﬁnd∂g ab∂t =2Ngψ4 σab−1∂t=−N√ψ−2N gψ4 πacπb c−16πab trπψ12+√gψ3 ▽a▽bψ+3g ab▽2ψ+4g ab√g▽(a(Nψ3)▽b)ψ+▽c πab N c −πbc▽c N a−πac▽c N b− πab−1∂tquite easily.(Of course,we need the evolution equations to propagate all of the constraints.We will deal with the others later.)Weﬁnd that∂trp∂t =∂πab3∂g ab trπWorking through the details gives us∂σabgψ4 R ab−1ψ −2N gψ4σacσb c+√3g ab▽2(Nψ3)+N√3g ab▽2ψ(2.42)+4g ab√g▽(a(Nψ3)▽b)ψ+▽c σab N c −σbc▽c N a−σac▽c N b−σab▽cξc+Nψ8gσab trπ2.3Jacobi ActionWe can alsoﬁnd the Jacobi action of this theory.Recall the(3+1)Lagrangian,L=N√ψ+S ab S ab−2gψ4 N R−8▽2ψ4N ΣabΣab−21ψ−12(trβ)2 1ψ −13dλ √R−8▽2ψT d3x(2.46) where T= ΣabΣab−22.4Conformally Related SolutionsWe can do almost exactly the same thing here as we did in the section with the same name in Chapter1.Suppose we start with initial data{g ab,σab,trp}obeying the initial data conditions(2.33)and(2.34).We then solve(2.32)forψSuppose instead that we start with the conformally related initial data{h ab,ρab,trp}= {α4g ab,α−4σab,trp}.These automatically satisfy the initial data conditions by the con-formal invariance.We now solve the Hamiltonian constraint for the conformal“ﬁeld”χ, say.Just like before it can be shown thatχ=ψ6(trπ)2− g R=0(2.49)▽b πab=0(2.50)▽c trπ=0(2.51)N R− ▽2 N+( trp)26(trπ)2−gR=0(2.53)▽bπab=0(2.54)▽c trπ=0(2.55) Evolution of the CMC condition givesNR−▽2N+(trp)22.4.1What ofξc?Precious little has been revealed about whatξc may be or even how it transforms.This needs to be addressed.First let’s recall that we demanded thattrB−→ω−8trB(2.57)under a conformal transformation.This will be enough to reveal the transformation properties ofξc.Taking the trace gives ustrB=−1∂t−g ab(KN)ab−3▽cξc(2.58)Under a conformal transformation we getω−8trB=−1∂t+12˙ω2ω2N g ab∂g ab2ω2N▽cξc+3ω3N ˙ω−ω,c N c=ω−2trB+3ω ˙ω−ω,c N c(2.59)Thus,3ω ˙ω−ω,c N c=−1ω ˙ω−ω,c N c−2N∂t =2 N g σab−1and∂ σab g R ab−1g ▽a ▽b N−13ω(˙ψ−ψ,c N c)−2Nψ(˙ψ−ψ,c N c)+2N3trB(1−ψ−6)(2.66)whereθis as in the original theory.Thus,the exact form ofξc is determined.We needed ▽cξc to be zero in the physical representation for constraint propagation and so we should check that this is the case with our newly found expression for▽cξc.We can check this easily.In the physical representationθ=0andψ=1.Thus,we do have that ▽c ξc is zero.It is vital to note that this is strictly a POST-VARIATION identiﬁcation.If we use this form forξc in the action we will run into problems,not least an inﬁnite sequence in the variation of trB with respect toξc.(This is because we would have trB deﬁned in terms of trB itself.)We see thatξc is intimately related with howψchanges from slice to slice.Our constraints in the physical representation areσabσab−1。

集成电路专业英语词汇

Abrupt junction 突变结Accelerated test ing 加速实验Acceptor 受主Acceptor atom 受主原子Accumulati on 积累、堆积Accumulat ing con tact 积累接触Accumulatio n regi积累区Accumulati on layer 积累层onActive regio n 有源区Active comp onent 有源元Active device 有源器件Activati on 激活Activati on en ergy 激活能Active regi on 有源（放大）区Admitta nee 导纳Allowed band 允带Alloy-j unction device 合金结器件Alumi nu m(Alumi nium) 铝Alumi num -oxide 铝氧化物Alu minum passivati on 铝钝化Ambipolar 双极的Ambie nt temperature 环境温度Amorphous 无定形的，非晶体的Amplifier 功放扩音器放大器An alogue(A nalog) comparator 模拟比较器An gstrom 埃Ann eal 退火An isotropic 各向异性的Anode 阳极Arse nic (AS) 砷Auger 俄歇Auger process 俄歇过程Avala nche 雪崩Avala nche breakdow n 雪崩击穿Avala nche excitatio雪崩激发nBackgro und carrier 本底载流子Backgro und dop ing 本底掺杂Backward 反向Backward bias 反向偏置Ballast ing resistor 整流电阻Ball bond 球形键合Band 能带Band gap 能带间隙Barrier 势垒Barrier layer 势垒层Barrier width 势垒宽度Base 基极Base con tact 基区接触Base stretchi ng 基区扩展效应Base tran sit time 基区渡越时间Base tran sport efficie ncy 基区输运系数Base-width modulatio n 基区宽度调制Basis vector 甘/r 基人Bias 偏置Bilateral switch 双向开关Binary code 二进制代码Binary compo und semic on ductor 二元化合物半导体Bipolar 双极性的Bipolar Ju nction Tran sistor (BJT) 双极晶体管Bloch 布洛赫Block ing band 阻挡能带Block ing con tact 阻挡接触Body - cen tered 体心立方Body-ce ntred cubic structure 体立心结构Boltzma nn 波尔兹曼Bo nd 键、键合Bonding electro n 价电子Bonding pad 键合点Bootstrap circuit 自举电路Bootstrapped emitter follower 自举射极跟随器Boro n 硼Borosilicate glass 硼硅玻璃Boun dary con diti on 边界条件Bound electr on 束缚电子Breadboard 模拟板、实验板Break dow n —R 击穿Break over 转折Brilloui n 布里渊Brillo uin zone 布里渊区Built-in 内建的Build-i n electric field 内建电场Bulk 体/体内Bulk absorpti on 体吸收Bulk gen erati on 体产生Bulk recomb in ati on 体复合Bur n - in 老化Burn out 烧毁Buried cha nnel 埋沟Buried diffusi on region隐埋扩散区CanCapture cross secti on 外壳俘获截面Capacita neeCapture carrier电容俘获载流子Carrier 载流子、载波Carry bit 进位位Carry-i n bit 进位输入Carry-out bit 进位输出Cascade 级联Case 官冗Cathode 阴极Cen ter 中心Ceramic 陶瓷（的）Channel 沟道Channel breakdow n 沟道击穿Channel curre nt 沟道电流Channel dop ing 沟道掺杂Channel shorte ning 沟道缩短Channel width 沟道宽度Characteristic impeda nee 特征阻抗Charge 电何、充电Charge-compe nsati on effects 电荷补偿效应Charge con servati on 电荷守恒Charge n eutrality con diti on 电中性条件Charge drive/exchange/sharing/transfer/storage 电荷驱动/ 交换/ 共享/ 转移/ 存储Chemmical etch ing 化学腐蚀法Chemmically-Mecha ni cally Polish (CMP) Chip yield 芯片成品率Clampi ng diode 箝位二极管Clock rate 时钟频率Clock flip-flop 时钟触发器Close-loop gain 闭环增益Collisio n 碰撞Common-base/collector/emitter connection Common-gate/dra in/source connection Common-m ode gain 共模增益Common-m ode rejectio n ratio (CMRR)Chemically-Polish 化学抛光化学机械抛光Chip 芯片Clamped 箝位Cleavage pla ne 解理面Clock gen erator 时钟发生器Close-packed structure 密堆积结构Collector 集电极Compe nsated OP-AMP 补偿运放共基极/集电极/发射极连接共栅/漏/源连接Common-m ode in put 共模输入Compleme ntary Metal-Oxide-Semico nductor Field-Effect-Tra nsistor(CMOS) 互补金属氧化物半导体场效应晶体管Compleme ntary error fun ctio n 余误差函数Computer-aided design (CAD)/test(CAT)/manufacture(CAM) 计算机辅助设计/ 测试/制化合物半导体Con ducta nee 电导导带（底）Conduction level/state 导带态Con ductor 导体Con figurati on 组态Con pled Con figurati on Devices等能面Con sta nt-source diffusion 恒定源扩散电流密度Curvature截止Current drift/dirve/sharingCurrent Sense Custom in tegrated circuit Czochralshicrystal Czochralski tech nique电流取样定制集成电路直立单晶切克劳斯基技术CurvatureCyli ndrical(Cz法直拉晶体J)悬挂键空载时间德布洛意分贝深受主能级深度杂质能级Dark curre ntDebye len gthDecderateDecodeDeep donor levelDeep trap暗电流德拜长度减速译码深施主能级深陷阱Con tactContinu ity equati onCon tact pote ntial 接触连续性方程接触电势Con tam in ati onCon tact holeContinuity con ditionContra dop ingCon verter反掺杂Con trolled 转换器Con veyerCopper in terc onnection system 铜互连系统CoupingCovale nt Critical共阶的Crossover临界的Crossu nder治污接触孔连续性条件受控的传输器耦合跨交穿交Crucible 坩埚Crystal defect/face/orie ntation/lattice 晶体缺陷/晶面/晶向/晶Compatibility 兼容性Compe nsated impurities 补偿杂质Compleme ntary Darlington circuitCompe nsati on 补偿Compe nsated semic on ductor互补达林顿电路补偿半导体Compo und Semic on ductorCon duct ion band (edge)Con ductivityCon lomb结构组态Con sta nts 电导率库仑物理常数Con sta nt en ergy surface 格Current den sityCut off曲率电流漂移/驱动/共享弯曲柱面的Dan gli ng bonds Dead timeDe.broglieDecibel (dB)Deep acceptor level Deep impurity levelDefeat Dege nerate semic on ductorDegradati on 退化缺陷简并半导体 Dege neracy Degree Celsius(ce ntigrade) /Kelvin 简并度摄氏/开氏温度Delay Den sity of states 延迟态密度Den sity Depleti on 密度耗尽Depleti on approximati on 耗尽近似 Depleti on con tact 耗尽接触Depleti on depth Depleti on layer Depleti on regi onDepositi onprocessDie Dielectric Differe nee-mode input 耗尽深度耗尽层耗尽区淀积工艺芯片(复数dice ) 介电的差模输入 Depleti on effect Depletio n MOS Deposited film Desig n rules Diode Dielectric isolatio n Differe ntial amplifier耗尽效应耗尽MOS 淀积薄膜设计规则二极管介质隔离差分放大器 Differen tial capacitanee Diffusio n Diffusi on con sta nt微分电容扩散扩散常数 Diffusi on capacita nce/barrier/curre nt/furnace Digital circuit 数字电路 Dipole layer 偶极层 Diffused jun ction 扩散结 Diffusi on coefficie nt 扩散系数 Diffusivity 扩散率扩散电容/势垒/电流/炉偶极畴直接耦合Dipole doma in Direct-coupli ngDirect-gap semic on ductor 直接带隙半导体 Direct tran sition直接跃迁Discharge放电 Discrete component分立元件Dissipati on Distributed capacita nee Displaceme nt 位移 Domai n 畴施主耗尽掺杂半导体双扩散MOS. 耗散分布电容 Distributi onDistributed model Dislocati on Donor Donor exhausti on Doped semic on ductor Double-diffusive MOS(DMOS) Drift Drift mobility Dry/wet oxidati on Duty cycle Dyn amicsDopa ntDopi ng concen tratio n分布分布模型位错施主掺杂剂掺杂浓度漂移 Drift field 迁移率干/湿法氧化Dry etch ing Dose 工作周期 Dual-in-I ine package 动态Dyn amiccharacteristics漂移电场干法腐蚀剂量(DIP )双列直插式封装动态属性Early effect Effective massElectric Erase Programmable Read Only Memory(E2PROM)储器Electrode电极 Wlectromin ggratim电迁移 Electro n affinity电子亲和势Electronic -grade电子能Electr on-beam photo-resist exposure 光致抗蚀剂的电子束曝光Electr on gas电子气 Electro n-grade water电子级纯水 Electr on trapp ing cen ter 电子俘获中心Electro n Volt (eV) 电子伏 Electrostatic 静电的Eleme nt兀素/兀件/配件 Eleme ntal semic on ductor 兀素半导体 Ellipse椭圆Ellipsoid椭球Emitter 发射极 Emitter-coupled logic 发射极耦合逻辑 Emitter-coupled pair 发射极耦合对Emitter follower 射随器Empty band空带Emitter crowdi ng effect 发射极集边（拥挤）效应En dura nee test =life test 寿命测试 En ergy state能态Energy momentum diagram 能量-动量(E-K) 图 Enhancement mode 增强型模式En ha nceme nt MOS 增强性MOS En tefic （低）共溶的En viro nmen tal test 环境测试 Epitaxial 外延的 Epitaxial layer 外延层 Epitaxial slice 外延片 Expitaxy外延Equivale nt curcuit等效电路Equilibrium majority /mi nority carriers 平衡多数/少数载流子Erasable Programmable ROM (EPROM) 可搽取（编程）存储器Error function compleme nt余误差函数Etch 刻蚀 Etcha nt刻蚀剂Etch ing mask 抗蚀剂掩模 Excess carrier过剩载流子 Excitati on en ergy 激发能 Excited state 激发态 Excito n激子 Extrapolati on外推法Extri nsic 非本征的Extri nsic semic on ductor杂质半导体Face - cen tered 面心立方 Fall time下降时间Fan-in扇入 Fan-out扇出 Fast recovery 快恢复 Fast surface states 快界面态 Feedback反馈Fermi level 费米能级Fermi-Dirac Distributi on 费米-狄拉克分布 Femi pote ntial费米势 Fick equati on 非克万程（扩散） Field effect tran sistor场效应晶体管 Field oxide 场氧化层 Filled band满带 Film薄膜Flash memory闪烁存储器Dyn amic impeda nee动态阻抗厄利效应 Early failure 有效质量Ein ste in relation( ship) 早期失效爱因斯坦关系一次性电可擦除只读存Flat band平带Flat pack扁平封装Flicker no ise 闪烁（变）噪声 Flip-flop toggle触发器翻转 Floati ng gate 浮栅 Fluoride etch 氟化氢刻蚀 Forbidde n band 禁带 Forward bias 正向偏置 Freque ncy response 频率响应 Fun cti on函数 Gai n增益 Gallium-Arse nide(GaAs) 砷化钾 Gamy ray r 射线 Gate 门、栅、控制极 Gate oxide 栅氧化层 Gauss ( ian ) 咼斯Forward blocking/conducting 正向阻断 / 导通 Freque ncy deviati on no ise 频率漂移噪声 Gaussia n distributi on profile 高斯掺杂分布 Gen erati on-recomb in ati on 产生-复合 Geometries Graded Graded junction Gradie nt Guard ring Gunn - effect 几何尺寸缓变的缓变结梯度保护环狄氏效应Germa nium(Ge) Graded (gradual) cha nnel Grain Grow n jun ctio n Gummel-Poom model锗缓变沟道晶粒生长结葛谋-潘模型Harde neddevice Heat sink Heavy saturati onHeteroj un ctio n 辐射加固器件散热器、热沉重掺杂异质结 Heteroj un ctio n Bipolar Tran sistor High field property 高场特性 High-performa nee MOS.( H-MOS) Horiz on tal epitaxial reactorHeat of formati on Heavy/light hole band Hell - effectHeteroj unctionstructure(HBT )异质结双极型晶体高性能MOS. 卧式外延反应器 Hormalized Hot carrior Hybrid in tegrati on 混合集成形成热重/轻空穴带霍尔效应异质结结构归一化热载流子 Image - force Impeda nee Impla ntati on dose Impurity In creme ntal resista nee In dium tin oxide (ITO) In frared 镜象力阻抗注入剂量杂质 In put offset voltage In sulated Gate FET(IGFET) 电阻增量（微分电阻）铟锡氧化物红外的输入失调电压绝缘栅FETImpact ioni zati on Imperfect structure Impla nted ion Impurity scatteri ng In-con tact mask In duced cha nnelInjectio nIn sulatorIn tegrated inject ionlogic碰撞电离不完整结构注入离子杂志散射接触式掩模感应沟道注入绝缘体集成注入逻辑In tegrati on In terc onn ecti on time delay集成、积分互连延时In terc onn ecti on In terdigitated structure互连交互式结构In terfaee 界面 Intern ati onal system of unionsIn terfere nee国际单位制 In ternally seatteringMegeto - resista nee磁阻 Mesa台面MESFET-Metal Semico nductor 金属半导体FETMetallizatio n 金属化 Microeleetr onic tech nique微电子技术 Microeleetr onics 微电子学 Mille n in dices密勒指数 Minority carrier少数载流子Misfit失配Inversion 反型 In verter 倒相器Ion 离子 Ion beam离子束Ion etch ing 离子刻蚀 Ion impla ntatio n 离子注入 loni zati on 电离 loni zati on en ergy 电离能 Irradiati on 辐照Isolati on land隔离岛Isotropic各向同性Junction FET(JFET) 结型场效应管 Junction isolati on 结隔离 Junction spaci ng结间距Junction side-wall结侧壁Latch up 闭锁 Lateral 横向的 Lattice晶格Layout版图In verse operati on 晶格结合力/晶胞/晶格/晶格常熟Lattice bin di ng/cell/c on sta nt/defect/distorti on干涉谷间散射Intrin sie In terpolati on 内插法In tri nsic semic on ductor 本征半导体本征的反向工作/晶格缺陷/晶格畸变 Leakage curre nt (泄)漏电流 Life time寿命共价键 Lin ked bond Liquid — phase epitaxial growth tech niqueLithography光刻 Load line or Variable 负载线纵向的洛沦兹Level shift ing lin earity Liquid Nitroge n液相外延生长技术电平移动线性度液氮Lon gitudinal Lore ntz Light Emitti ngDiode(LED)Locati ng and Wiring Logic swi ng Lumped model发光二极管布局布线逻辑摆幅集总模型 Majority carrier Mask level Mass - action law 多数载流子掩模序号质量守恒定律匹配平均自由程 Mask 掩膜板，掩模组光刻板 Matchi ng Mean free pathMea n time before failure (MTBF) Mask set Master-slave D flip-flopMaxwell麦克斯韦Mean dered emitter junction平均工作时间主从D 触发器梳状发射极结Mismatchi ng 失配Mobile ions 可动离子Mobility 迁移率Module 模块Modulate 调制Molecular crystal 分子晶体Mon olithic IC 单片IC MOSFET金属氧化物半导体场效应晶体管Mos. Tran sistor(MOST )MOS. 晶体管Multiplicati on 倍增Modulator 调制Multi-chip IC 多芯片ICMulti-chip module(MCM) 多芯片模块Multiplicatio n coefficie nt 倍增因子Naked chip 未封装的芯片（裸片）Negative feedback 负反馈Negative resista nee 负阻Nest ing 套刻Negative-temperature-coefficie nt 负温度系数Noise margin 噪声容限Non equilibrium 非平衡No nrolatile 非挥发（易失）性Normally off/on 常闭/开Numerical an alysis 数值分析Occupied band 满带Officie nay 功率Offset 偏移、失调On sta ndby 待命状态Ohmic con tact 欧姆接触Open circuit 开路Operati ng point 工作点Operat ing bias 工作偏置Operatio nal amplifier (OPAMP) 运算放大器Optical phot on -phot on 光子Optical que nching 光猝火Optical tran siti on 光跃迁Optical-coupled isolator 光耦合隔离器Organic semic on ductor 有机半导体Orien tati on 晶向、定向Outli ne 外形Out-of-co ntact mask 非接触式掩模Output characteristic 输出特性Output voltage swi ng 输出电压摆幅Overcompe nsati on 过补偿Over-curre nt protectio n 过流保护Over shoot 过冲Over-voltage protect ion 过压保护Overlap 交迭Overload 过载Oscillator 振荡器Oxide 氧化物Oxidati on 氧化Oxide passivati on 氧化层钝化Package 封装Pad 压焊点Parameter 参数Parasitic effect 寄生效应Parasitic oscillati on 寄生振荡Pass in ati on 钝化Passive comp onent 无源元件Passive device 无源器件Passive surface 钝化界面Parasitic tran sistor 寄生晶体管Peak-po int voltage 峰点电压Peak voltage 峰值电压Perma nen t-storage circuit : 永久存储电路Period 周期Periodic table 周期表Permeable - base 可渗透基区Phase-lock loop 锁相环Phase drift 相移Phonon spectra 声子谱Photo con duct ion 光电导Photo diode 光电二极管Photoelectric cell 光电池Photoelectric effect 光电效应Photoe nic devices 光子器件Photolithographic process 光;(光敏)抗腐蚀剂 Pin夹断Pinning of Fermi levelPlanar process 平面工艺 Planar tran sistor 平面晶体管 Plasma等离子体 Plezoelectric effect 压电效应Poiss on equati on 泊松方程 Point con tact 点接触Polarity极性Polycrystal多晶Polymer semic on ductor 聚合物半导体Poly-silic on 多晶硅 Pote ntial （电）势Pote ntial barrier 势垒Pote ntial well 势阱 Power dissipati on 功耗 Power tran sistor 功率晶体管 Preamplifier 前置放大器Primary flat 主平面Prin cipal axes 主轴Prin t-circuit board(PCB)印制电路板Probability 几率Probe探针Process工艺Propagati on delay 传输延时 Pseudopote ntial method 膺势发 Punch through 穿通Pulse triggeri ng/modulat ing脉冲触发/调制PulseWiden Modulator(PWM) 脉冲宽度调制Pun chthrough 穿通 Push-pull stage 推挽级Quality factor品质因子Quan tizatio n量子化Qua ntum量子 Quan tum efficie ncy量子效应 Quan tum mecha nics 量子力学Quasi -Fermi—level 准费米能级 Quartz 石英Radiati on conductivity辐射电导率 Radiatio n damage辐射损伤Radiati on flux den sity 辐射通量密度 Radiation harde ning 辐射加固 Radiati on protect ion 辐射保护Radiative - recomb in ati on 辐照复合Radioactive 放射性 Reach through穿通 Reactive sputteri ng source 反应溅射源 Read diode里德二极管 Recomb in ati on 复合 Recovery diode 恢复二极管 Reciprocal lattice 倒核子Recovery time 恢复时间 Rectifier 整流器（管） Rectify ing con tact 整流接触 Referenee 基准点基准参考点Refractive in dex 折射率 Register 寄存器 Registrati on对准Regulate 控制调整 Relaxati on lifetime驰豫时间 Reliability 可靠性 Resonance 谐振 Resista nee 电阻 Resistor 电阻器 Resistivity 电阻率 Regulator稳压管（器） Relaxatio n驰豫Resonant freque ncy共射频率 (photo) resist Pin ch off管脚费米能级的钉扎（效应）Resp onse timeReverse bias 响应时间反向偏置Reverse 反向的Sampli ng circuit Satellite valley 取样电路卫星谷Sapphire 监宝石(Al2O3 )Saturated curre nt range电流饱和区Saturati on regi onScaled dow n Schockley diode Schottky barrier Schrodi ngen Secon dary flat Seed crystalSelectivity 饱和区按比例缩小肖克莱二极管肖特基势垒薛定厄次平面籽晶选择性 Saturati on Scatteri ng Schottky Schottky con tactScrib ing grid 饱和的散射肖特基肖特基接触划片格 Segregati onSelf alig ned 分凝自对准的Self diffusi on Semic on ductor-c on trolled rectifier Serial自扩散 Settle time Shield Shot no ise Sidewall capacita nee Silica glass Silic on carbide Silicon Nitride(Si3N4)串行/串联建立时间屏蔽散粒噪声边墙电容石英玻璃碳化硅氮化硅 Semic on ductor可控硅 Sen dsitivitySeries in ducta nee Sheet resista nee Short circuit Shu ntSig nalSilicon 半导体灵敏度串联电感薄层电阻短路分流信号硅Silico n dioxide (SiO2) Silic on On In sulator 二氧化硅绝缘硅Siliver whiskers Sin gle crystal Ski n effect Sn eak path Solar battery/cell Solid Solubility Source Space charge Speed-power product Spi n Spontan eous emissi onSputter Static characteristic 银须单晶趋肤效应潜行通路太阳能电池固溶度源极空间电荷速度功耗乘积自旋自发发射溅射静态特性 Simple cubic Si nk Snap time Sulethreshold Solid circuit Sonband Source follower Specific heat(PT) Spherical Split简立方沉急变时间亚阈的固体电路 -H-P子带源随器执八、、球面的分裂Spreadi ng resista nee 扩展电阻Stacki ng fault层错Stimulated emissio n 受激发射 Stimulated recomb in ation 受激复合 Storage time 存储时间Stress 应力Straggle偏差Sublimati on 升华Substrate 衬底Substituti onal 替位式的Superlattice 超晶格Supply 电源Surface 表面Surge capacity 浪涌能力Subscript 下标Switch ing time 开关时间Switch 开关Taili ng 扩展Termi nal 终端Tensor 张量Ten sorial 张量的Thermal activati on 热激发Thermal con ductivity 热导率Thermal equilibrium 热平衡Thermal Oxidati on 热氧化Thermal resista nee 热阻Thermal sink 热沉Thermal velocity 热运动Thermoelectricpovoer 温差电动势率Thick-film tech nique 厚膜技术Thin-film hybrid IC 薄膜混合集成电路Thi n-Film Tran sistor(TFT) 薄膜晶体Threshlod 阈值Thyistor 晶闸管Transcon ducta nee 跨导Tran sfer characteristic 转移特性Tran sfer electron 转移电子Tran sfer function 传输函数Tran sie nt 瞬态的Tran sistor agi ng(stress) 晶体管老化Tran sit time 渡越时间Tran siti on 跃迁Tran siti on-metal silica 过度金属硅化物Tran siti on probability 跃迁几率Tran siti on regi on 过渡区Tran sport 输运Tran sverse 横向的Trap 陷阱Trapp ing 俘获Trapped charge 陷阱电荷Trian gle gen erator 三角波发生器Triboelectricity 摩擦电Trigger 触发Trim 调配调整Triple diffusi on 三重扩散Truth table 真值表Tolerahce 容差Tunn el( ing) 隧道（穿）Tunnel curre nt 隧道电流Turn over 转折Turn - off time 关断时间Ultraviolet 紫外的Unijun cti on 单结的Un ipolar 单极的Unit cell 原（元）胞Uni ty-ga in frequency单位增益频率Un ilateral-switch 单向开关Vacancy 空位Vacuum 真空Vale nce(value) band 价带Value band edge 价带顶Vale nee bond 价键Vapour phase 汽相Varactor 变容管Varistor 变阻器Vibratio n 振动Voltage 电压Wafer 晶片Wave equati on 波动方程Wave guide 波导Wave nu mber 波数Wave-particle duality Wire rout ing波粒二相性布线Wear-outWork function烧毁功函数Worst-case device 最坏情况器件Yield 成品率Zener breakdow n 齐纳击穿Zone melti ng 区熔法。

宇宙的未来（英文版）

Future of the UniverseLucyna Kedziora-ChudczerPrinciple of general relativitymodels which predicted cosmological redshiftwith a flat curvature (boundary case betweenWhich model is correct?,and the critical density rCRIT=70km/s/Mpcis 7 atoms per cubic metrePositive spherical curvature, closed universe -> will collapseFlat space, open infinite universe -> decelerates to restNegative curvature, open infinite universe -> expands forever= 1 is an unstable critical point for the geometry of the Universe.Flat universe model is favoured,but how was the Universefine-tuned to be so flat?Answer: inflationCan W be measured by direct observations?Hubble diagramEnergy density of vacuum Dark matter evident from: dynamics ofNot enough for the flat UniverseExperiments like WMAP, BOOMERANGand MAXIMA measure the fluctuationsin the CMB on the scale of 1deg.Such result is expected for the flatUniverse (W~1).What causes this acceleration?- Repulsive force- Not an ordinary matter- It contributes negative energypressuremay have a variable speed – was slower in the past.Fluctuations of Cosmic Microwave BackgroundWeak lensing mass census - WmThe expansion/contraction may be cyclic. With each cycle the Universe gains energy.Photons emitted by the Sun today will become gradually redshifted because of expansion, losing energy.As the Universe contracts, the photons are blueshifted, gaining energy. They eventually become blueshifted, until they are more energetic than they were at the time of emission.So during contraction, the Universe is hotter than it was at the corresponding time during contraction.TrapeziumEvolution of the Sun Changes of our EarthFate of the EarthThe Ultimate fate of the Sun。

Resolution

http://www.lsv.ens−cachan.fr/Publis/Research Report LSV−02−8, Lab. Spécification et Vérification, CNRS & ENS de Cachan, France, Jul. 2002A Note on the Completeness of Certain Reﬁnements ofResolutionJean Goubault-LarrecqLSV/CNRS UMR8643,ENS Cachan61,av.du pr´e sident-Wilson94235Cachan Cedex,FranceAbstract.Resolution and many reﬁnements thereof have now been known fora long pleteness is usually proved by semantic means(e.g.,semantictrees,Bachmair-Ganzinger forcing),or by syntactic tricks(Bledsoe’s excess lit-eral technique).The purpose of this paper is to note that there is a completelyproof-theoretic way of proving completeness for several reﬁnements of resolu-tion,resembling Gentzen’s method of cut-elimination.While this technique hasa number of shortcomings compared to the semantic arguments cited above,it isvaluable in that the completeness proofs for different reﬁnements are the same.We have found this proof technique to be effective in teaching the ins and outs ofreﬁnements of resolution to masters level students.This can also be used to ex-tract propositional proofs in one resolution format from resolution proofs in someother format automatically;in theﬁrst-order case,the same technique allows oneto extract ordered resolution or hyperresolution proofs from proofs in any otherresolution format mechanically.1IntroductionContext and related work.Resolution[10]has several reﬁnements,among which hy-perresolution,semantic resolution,ordered strategies,and combinations thereof,mostof which being complete[2].The classical proofs of their completeness is different ineach case,and sometimes feel ad hoc.Amongst these techniques,weﬁnd Kowalski andHayes’semantic tree technique[9],which works nicely for ordered resolution,hyper-resolution and semantic resolution;Bachmair and Ganzinger’s forcing technique[1],which is often used for ordered resolution with selection and provides an explicit modelconstruction in case no refutation can be found.These semantic methods have severaladvantages,including the fact that it is easy to show that several deletion strategies(tautology elimination,subsumption)as well as additional rules(e.g.,splitting[5]orcondensation[8])can be used without destroying the completeness of the base calcu-lus.One syntactic method of proving completeness is Boyer’s excess literal technique,which applies notably to Boyer’s locking and to semantic resolution,including hyper-resolution:see[2].It is then usually somewhat awkward to teach students a course in resolution tech-niques,as reﬁnements look like a hodge-podge of ad hoc tricks,with speciﬁc semanticor syntactic completeness arguments.The purpose of this paper is threefold:–First,to show one unifying intuition behind several reﬁnements of resolution:res-olution,at least in the propositional case,is Gentzen’s cut rule,and there are many ways that a proof using only cuts can be rearranged by permuting and distribut-ing cuts across each other.Normal proofs will then be of speciﬁc forms,allowing one to reduce non-determinism in proof search.This is very much in the spirit of Gentzen’s Hauptsatz for sequent calculi[13].While permuting and distributing cuts is easy,showing that this process terminates is slightly more challenging.(Note that weak termination would be enough,but it is not much harder to show strong termination:all rewrites terminate.)–Second,to provide a simple argument by which termination is ensured,hence from which completeness follows.This simple argument(condition(5)below)is enough to retrieve some of the most well-known reﬁnements of resolution.–Third,to provide effective translations between resolution formats.For example, our argument allows one to rewrite any ordered resolution refutation into a positive hyperresolution refutation from the same set of clauses.This can be used to provide human-readable proofs from machine-generated proofs;positive hyperresolution derivations,notably,tend to be more readable than ordered resolution proofs.On the other hand,it is not the purpose of this paper to show that resolution reﬁne-ments are still complete in the presence of such or such deletion strategy,for which semantic trees or Bachmair and Ganzinger’s technique are preferable.This is mostly an orthogonal concern.For example,it is still possible to show that subsumed clauses and tautologies can be eliminated,when they can,by syntactic methods[7].Neither is it the purpose of this paper to introduce new reﬁnements of resolution,or to introduce a uni-versal completeness proof.In particular,it seems that certain reﬁnements of resolution, e.g.,ordered resolution with free selection of negative literals,are not easily amenable to the technique described here.Outline.Because we can always rest on lifting arguments,we mostly deal with propo-sitional resolution in this paper—that is,until Section5.Weﬁx notations and recall the resolution principle in Section2.We then introduce our proof transformation rules in Section3,and give sufﬁcient conditions for them to terminate,thus implying complete-ness.Section4illustrates a number of known reﬁnements that can be shown complete by this technique.Although the stress is put on the propositional case in this paper,we deal with theﬁrst-order case in Section5;this is more difﬁcult to tackle without going through liftingﬁrst.This effort pays up:our technique provides an effective translation from anyﬁrst-order resolution refutation to ordered,or to hyperresolution refutations. We conclude in Section6.2ResolutionLet usﬁx a vocabulary of atoms,,...;literals are either positive atoms or negative atoms.Clauses,,...,areﬁnite sets of literals,seen as disjunctions.2On propositional formulas,which is the case we deal with except in Section5,the resolution rule is nothing else than Gentzen’s Cut rule:where comma denotes union,and disjoint union in premises(in particular,, ).We write the empty clause,the negation of literal,i.e.,,.In the Cut rule above,is called the cut formula.A resolution proof of a clause from the set of clauses is anyﬁnite tree of resolution inferences(instance of Cut) whose leaves(at the top)are clauses in and whose root(at the bottom)is.A refutation from is a resolution proof of from.The completeness of resolution,i.e.,that there is a refutation from whenever is inconsistent,can be established by semantic means,or by appealing to the syntactic device of cut elimination:let denote Gentzen’s sequent calculus for classical logic augmented with non-logical axioms taken from(clauses being read as sequents), then eliminating cuts from any-proof of the empty sequent yields one where the only rule is(Cut)[6,7].That resolution is complete will be assumed in the sequel. To show that some reﬁnement of resolution is complete,we only need to rewrite any given refutation of into a refutation of that obeys the constraints of the reﬁnement. We shall do this by using rewrite rules(–below)that express all possible ways of permuting one cut past another.(1)(2)The,,,marks have been added for future reference.For example,we shall say that the-cuts are those marked with(the topmost cuts in each rule,on the right).Recall that we assume that in premises such as,the literals and are distinct and not in.These rules do not terminate in general.However we shallﬁnd a series of condi-tions that ensure termination in the next section,and demonstrate that several known reﬁnements of resolution obey these conditions.33Completeness via Selection FunctionsLet us specify a reﬁnement of resolution by means of a selection function mapping each clause to a subset of literals that we are allowed to take as cut formulas in the (Cut)rule.For example,if returns the set of all-maximal literals for some ordering,and compares literals by comparing the underlying atoms,then we get ordered resolution. We shall give more examples of functions in Section4.We ask the selection function to obey the following axioms.First,selects from literals in the clause:(3) Then,should select the unique literal from each unit(one-literal)clause:(4) for every literal.That is,it is not allowed to select nothing from a unit clause.Before we introduce the last condition,deﬁne the-cut rule as the restriction of the(Cut)rule where and.A resolution proof is a -resolution proof if and only if it only uses-cuts.We then require the following condition:In(1),(2),if the-cut is a-cut,and the-cut is not,then the-cuts are-cuts.(5)If is a non--resolution refutation from,there must be a lowest instance of (Cut)that is not a-cut.This lowest non--cut cannot be the last one.Indeed,because for every unit clause,anyﬁnal instance of(Cut)in a refutation,which derives and therefore must be a cut between two unit clauses,is a-cut.So there must be a-cut below the lowest non--cut.That is,contains the following conﬁguration:(6)non--cut-cutwhich we call a redex.Condition(5)says that(1)and(2)rewrite redexes to conﬁgu-rations where the-cut was permuted upwards:all topmost cuts in right-hand sides are required to be-cuts.Since the non--cut in the redex(6)is an instance of(Cut),the clause must equal.Then either contains but not(or by symmetry does not but does),or both and contain:these yield the left-hand sides of rules(1)and(2) respectively.We claim that under the assumptions(3),(4),(5),these rewrite rules(1)and(2) restricted to apply to redexes(6)terminate.It will follow that we can always transform any resolution refutation into a-resolution refutation;in particular,-resolution will be complete.4To establish termination,deﬁne an interpretation of proofs asﬁrst-order terms built on one constant and two binary function symbols and,representing-cuts and non--cuts respectively.Recall that is the set of clauses that we start from:-cutnon--cutWe assume and to be commutative,i.e.,to avoid ambiguity in this translation.(We might also untie the knot by imposing that be,say,the translation of the premise where the cut formula is positive,but this would unnecessarily duplicate the cases to handle.)Translating(1)and(2)through yields the following rewrite rules:(7)(8) where ranges over.That is,by rules(1)or(2)impliesby rules(7)or(8);this is an easy check,using condition(5).Lemma1(Termination).The rewrite system(7),(8)terminates.Proof.Let be the set of terms that are terminating,i.e.such that every rewrite starting from isﬁnite.Let the contexts be terms with one hole,denoted( where ranges over terms);denotes with the hole replaced by the term .Similarly,let be the context obtained by replacing the hole of by the context .Deﬁne-contexts inductively by:is an-context,and if is an-context and,then is an-context.Finally,say that a term is reducible if and only if for every-context;the set of reducible terms is written .Observe that:(a).This is because is an-context.Note also:(b)if and,then.Indeed,for ev-ery-context,is an-context by construction and,so .We also have:(c)if and then.Indeed,for any -context,,so the one-step reduct is in,too.We claim that:(d)if,then.To this end,let be any-context,and let us show that.Toﬁx ideas,write as,and let us show the claim by induction on ordered lexicographically,where terms,,,...,are com-pared via the relation—which is well-founded on and on5.Then look at one-step reducts from.Some of them are obtained by contracting a redex in,in,or in some,:then by(c)and the induction hypothesis the obtained one-step reduct is in;another is obtained by contracting the redex,provided.If this reduces by(8),the reduct is where. Note that and are in by(b).So,if,we may con-clude by the induction hypothesis(with decreased by,and replaced by);if ,then is in becauseis an-context(in particular is in by(c)and(a))and. Similarly,every one-step-reduct obtained by contracting by(7)is in :if,is in by induction hypothesis,using the fact that and by(b);if,is in since is an-context(using)and. Since every one-step reduct of is in,is in,too.Since is arbitrary,is reducible.It is easy to see that.Indeed,for every,is indeed in(an easy induction on).It follows that every term is reducible,by structural induction on.We have just dealt with the base case,and the inductive cases are dealt with by(d)when ,by(b)and(a)when.Since every term is reducible,by(a) every term is in.Most standard methods in rewriting fail to prove Lemma1.In particular,the recur-sive path ordering[3]cannot deal with rules(7)or(8)when.In fact,this rewrite system is not simply terminating,and being included in a recursive path ordering im-plies simple termination.Recall that a rewrite system is simply terminating if and only if plus the simpliﬁcation rules,is terminating.A counter-example is:by(8)by simpliﬁcation Condition(5)is also maximal in that any liberalization leads to non-termination; in fact even weak termination(existence of normal forms)fails with any liberalized form of condition(5).Allowing some-cuts to be non--cuts while the-cut is a-cut would mean creating redexes on the right-hand side,leading immediately to non-terminating behavior.More subtly,allowing some-cuts as well as the-cut to be non--cuts,which might seem a benign extension,also leads to non-termination. Consider the case of rule(1)for example,then allowing the latter would enable the fol-lowing non-terminating behavior.We have elided the actual clauses,which are unim-portant.Firstnon--cutnon--cut -cut rewrites to-cutnon--cut-cut-cut6by(2),then to-cut non--cutnon--cut-cutif we allow for the indicated liberalization of rule(1).The latter derivation is then again a-cut under a non--cut under a non--cut,which allows us to start this cycle of reductions all over again.Having proved Lemma1,we are now done:Theorem1.Every resolution refutation from can be effectively transformed into a -resolution refutation from,provided(3),(4),(5)hold.Corollary1.Every reﬁnement of resolution based on a selection function satisfying (3),(4),(5)is complete.4ApplicationsWeﬁrst show that Corollary1allows us to justify some standard reﬁnements of resolu-tion.As announced in the introduction,we won’t deal with every known reﬁnement of resolution.In particular,free selection functions[1],even in the propositional case that we are now considering,do not seem toﬁt well in this framework.4.1Ordered resolution.Let be a strict ordering on atoms.Ordered resolution is the case where is the set of all literals such that is maximal in:i.e.,there is no in,with a possibly different sign,such that.Clearly,(3)and(4)hold.For(5),assume that,,and is a -cut in(1):In other words,(a)is maximal in,and(b)no atom in is greater than in.If is not a-cut,then there must be an atom greater than in.It cannot be in by(b),so we must have.So is maximal in, otherwise there would be a greater atom in:then,contradicting (b).must also be maximal in,otherwise there would be a greater atom in ,so,contradicting(b)again.7The argument for rule(2)is similar:Since is a-cut,(a)is maximal in,and(b)is maximal in. If some-cut is not a-cut,say the left one by symmetry,then is not maximal in ,so by(b).Then is maximal in both premises of.Therefore (5)holds.In particular by Corollary1ordered resolution is complete.There was in fact an easier syntactic proof of termination here.Modify the trans-lation so that:where for each atom there is a new binary commutative function symbol.Then the termination of the rewrite system on proofs of Section3in this case can be shown by using a multiset path ordering[3]with the precedence iff.This applies since there are onlyﬁnitely many atoms in any given refutation,therefore is well-founded.4.2Positive hyperresolution.Let be the set of negative atoms in if any,otherwise.A-cut between two parent clauses and is then such that only contains positive atoms,otherwise cannot be selected.Conversely,every positive hyperres-olution step deﬁned as(Cut)where one premise is a positive clause(a clause containing only positive atoms)is a-cut with this deﬁnition of.Again,conditions(3)and(4)are clear.If the left-hand side of(1)(or(2))is a redex, we claim that must be a positive clause.Indeed,since the-cut is a positive hyperresolution step,the only other possibility is that,resp.,is a positive clause;then one of the premises of the-cut must be positive,so the-cut would be a-cut,contradicting the fact that we have got a redex.So is a positive clause,hence the-cut is a positive hyperresolution step,i.e.,a-cut.Similarly for (2).Since is a positive clause in any case,condition(5)holds.Therefore positive hyperresolution is complete.In fact,we can spell out the rewrite rules(1),(2)in this case as follows:(9)8(10) where it should be clearer that the-cuts must be positive hyperresolution steps.Here is positive,however we shall use the same rules in Section4.4without this restriction.The alternate form of hyperresolution where macro-steps with premises are used,consisting of a non-positive clause(the nucleus)and positive clauses(the elec-trons)[2]which are resolved in steps to yield a new positive clause is complete,too. It is enough to notice that in a positive hyperresolution refutation,as deﬁned above,if some-cut has a non-positive conclusion,then going down the refutation we eventu-ally reach a positive clause:at the latest,is a positive clause.Hence we can extract a refutation consisting entirely of macro-steps from any-resolution refutation.We let the interested reader check that the only role of in the argument of Sec-tion3is to show that the last instance of(Cut)in a refutation is a-cut.In positive hyperresolution we may generalize:any resolution derivation of any positive clause must end in a positive hyperresolution step(a-cut).We can then replay Section3: every resolution derivation of can be effectively transformed into a positive hy-perresolution derivation of.In particular,we get the well-known fact that positive hyperresolution derives exactly the same positive clauses as unconstrained resolution. 4.3Negative hyperresolution,semantic resolution.Negative hyperresolution is obtained similarly by letting be the set of all positive atoms in if any,otherwise.In general,-resolution,a.k.a.semantic resolution,where is a set of atoms,that is,a Herbrand interpretation,is obtained by letting be the set of all literals that are true in if any,otherwise.Just as in the case of positive hyperresolution,in rules(1)and(2)must be restricted to be false in.We do not need to use Corollary1here,though:completeness of-resolution fol-lows from that of positive hyperresolution by renaming every atom to in clauses whenever,and noticing that such a renaming preserves the existence of refuta-tions.4.4Semi-ordered hyperresolutionWhile imposing an ordering constraint on both premises of positive hyperresolution steps destroys completeness,it is well-known that imposing that be maximal only in the positive clause leads to a complete reﬁnement of resolution.This is called semi-ordered hyperresolution in[7]to distinguish it from the aforementioned incom-plete ordered reﬁnement of hyperresolution.This is obtained by letting be the set of all negative atoms in if any,otherwise is the set of maximal(positive) atoms in.9Curiously,Corollary1does not apply directly.The reason is that condition(5)is not satisﬁed.Indeed,note that-cuts are cuts where one premise is positive and the cut formula is maximal.Then in rule(1)it might be the case that is a positive clause with and maximal in(whence is a-cut),but is also positive with but is not maximal in(whence is not a-cut);in this case there is no reason why should be a-cut:neither nor is positive,in particular.It might be that there is still a way of showing that(1)and(2)terminate in this case,too.However,an easier way of converting any resolution refutation into a semi-ordered positive hyperresolution refutation is to proceed in two steps.First convert into an ordered refutation as in Section4.1,then convert into a positive hyperreso-lution refutation as in Section4.2,using rules(9)and(10).Then apply the following lemma.Call an instance of(Cut)-ordered if and only the cut formula is maximal in the one premise where is positive.A resolution derivation is-ordered if every step in it is.It is clear that ordered resolution derivations are-ordered,while positive hyperres-olution derivations that are-ordered are exactly the semi-ordered positive hyperres-olution derivations.For convenience,assume that is total here:if is maximal in then is greater than or equal to all atoms in.Lemma2.If by rules(9)or(10)and is-ordered,then is-ordered. Proof.The-cuts are clearly-ordered.It remains to show that the-cuts are,too. If is a positive literal in(9),then by assumption;since, in particular,so the-cut is-ordered.If is a negative literal in(9)then by assumption,so again the-cut is-ordered.In the case of rule(10),by symmetry we may assume that;then by assumption, so since;so the-cut is-ordered.It follows that the end refutation obtained in the two-step process above is both a positive hyperresolution refutation and-ordered,so it is a semi-ordered positive hyperresolution refutation.Again,similar arguments show that semi-ordered negative hyperresolution or semi-ordered-resolution are complete,and give an effective procedure to transform any resolution refutation in one of the required format.5The First-Order CaseWhile this is not completely immediate,the rewrite rules(1)and(2)generalize to the ﬁrst-order case,where the resolution rule reads:provided is the most general uniﬁer of.For any substitution ,we write the result of applying the substitution to the atom,and we use10similar notations for substituting in clauses.As is traditional,we leave implicit the fact that the parent clauses and areﬁrst renamed so that they have no free variable in common.For short,write for the set.Such sets will always be assumed to be non-empty,i.e.,.Let denote the most general uni-ﬁer of.Extend this to literals and to several equality signs:is the most general common substitution(if any) that instantiate to the same literal,and...,andto the same literal.Let,the domain of,be the set.We shall always assume that most general uniﬁers are idempotent,i.e.,for every,is not free in. If is idempotent,letting be the set of all equations,,then ;furthermore,if and are idempotent and,then(we write the substitution mapping every variable to ),and if two sets of equations and have the same unquantiﬁed equational consequences,then.In particular if then.The rewrite rules(1)and(2)change as follows in theﬁrst-order case.The left-hand side of rule(1)now reads:(11)Now,because clauses are renamed apart,.Also,.It follows in particular thatexists,that exists and that.We can then rewrite the redex above to:(12)where.Since,the conclusion of(12)is the same as that of(11).The case of rule(2)is slightly more complicated.The left-hand side is:(13)11where denotes the union of the sets and.Recall that both and are assumed not empty.Again,and.Let now be a renaming(one-to-one,mapping variables to variables)substitution so that as no free variable in common with.Then is a uniﬁer of ,so exists;is a uniﬁer of,soexists.Then Note that and have disjoint domains.So the union makes sense,and we claim that it uniﬁes.Indeed, uniﬁes and(because is their common most general uniﬁer),and also and.In particular,uniﬁes,and is therefore an instance of.(For any atom,clause or substitution,an instance of is any atom,clause or substitution, for any substitution.We then say that is more general than if and only if is an instance of.)We may then generate the derivation:(14) Now because all clauses are renamed apart,the bottom clause in(14)is alsoThen,and we have already noticed that is an instance of this.In general,is not the same as,however,contrarily to the case for rule(1).For example,take be the atom,take to be, to be,to be.Let us say that is,and assume that every two variables with different names are distinct.Then,and therefore also ,map,,,to the same term.On the other hand,only equates with and with.With the rules(11)(12)and(13)(14)we can now deﬁne a transformation on derivations as follows:if and only if either is of the form(11)and is(12), or is of the form(13)and is(14),or is(15)12and,where derives a clause that is more general than—say, and—,and is:(16) Note indeed that uniﬁes,hence the substitution mapping every variable free in to and every variable free in to uniﬁes.So exists and is more general than.Since,is also more general than.It follows that the relation is well-deﬁned and rewrites derivations of clauses into derivations of more general clauses.In particular,it rewrites refutations into refutations.As in Section3,let be a selection function,and call aﬁrst-order resolution step a-step if and only if all literals resolved upon(and in(15)for example)are selected in their respective clauses.The last difﬁculty that awaits us in adapting the arguments of Section3is that the resolution step of(15)might be a-step while that of(16)fails to be,or conversely.It turns out that converting a non--step into a-step is benign,while the converse leads to non-termination.The former requires us to add the following rewrite rule to(7),(8):(17)Then the termination Lemma1extends smoothly:Lemma3.The rewrite system(7),(8),(17)is terminating.Proof.As for Lemma1.The only additional case is in claim(d),where inthe subterm rewrites to,with, and,.Then the contractum is,where is in and is clearly an-context, so that.At this point,the technique of Section3applies almost without modiﬁcation.We only have to replace condition(5)by the condition that,whenever the bottom resolution step in(11)is a-step and the top step is not,then the top resolution step in(12)is a -step;and similarly,that whenever the bottom resolution step in(13)is a-step and the top step is not,then the top two resolution steps in(14)are-steps.By extension, call this condition(5)again.Then:Theorem2.Let the selection function be stable:for every literal,clause,and substitution,if then.Assume that conditions(3),(4)and(5) hold.Then every resolution derivation of from can be effectively transformed into a-resolution derivation of some clause more general than from.13Proof.If(15)is a-step,then.Letting as above be and be,it obtains,so by stability.Therefore(16) is a-step ing the obvious adaptation of the interpretation,terminates if the rewrite system(7),(8),(17)does.Then apply Lemma3.Finally,observe that-normal forms consist only of-steps,using condition(4).This requires showing that the last resolution step in a refutation is a-step,and taking bottommost non--steps,as in Section3.And indeed the last resolution step must be a resolution between clauses and,with;by condition(4),since this is a unit clause,and similarly,so by stability and, therefore the last resolution step is a-step.This can be used to extract ordered refutations from any resolution refutation: Corollary2.Let be a stable ordering,i.e.,implies for every sub-stitution.Then everyﬁrst-order resolution refutation can be effectively transformed into an ordered resolution refutation(wrt.)from the same set of clauses.Proof.Apply a similar argument as in Section4.1.Consider rule(11)(12),and as-sume that(a)every literal in is maximal in,and(b)every literal in is max-imal in.By stability,(b’)every literal in is maximal in. If the topmost resolution step in(12)is not ordered,then,because of(a),some literal in is less than some literal in.By(b’)this literal in must be less than some literal in:(c).We claim that every literal in is maximal in.Otherwise there would be a literal in that is less than some literal in.First,cannot be in:since instantiates all of to the same atom,if then by stability,contradicting.Second,cannot be in:otherwise implies by stability;since by(c)and stability, ,contradicting(b).Third,cannot be in:otherwise(by stability)(by(c)and stability), a contradiction.So every literal in is indeed maximal in,therefore the topmost resolution step in(12)is a-step.The case of rule(13)(14)is similar.Now since deﬁned as the set of maxi-mal literals in is stable in the sense of Lemma3,the result follows.It is probably more interesting to rearrange resolution refutations into hyperreso-lution proofs instead.Recall that positive hyperresolution derivations can be seen as proofs deriving new facts from old facts[2]:arguably these proofs look like standard mathematical proofs,proceeding from assumptions to theorems.The following corol-lary can then be used to improve the readability of proofs obtained by any resolution theorem prover,provided it keeps a trace of the proof obtained,by extracting a positive hyperresolution proof from it.Corollary3.Everyﬁrst-order resolution refutation can be effectively transformed into a positive hyperresolution refutation from the same set of clauses.Proof.As in Section4.2,this is the case where is a positive clause in(11),(12), (13),(14).Condition(5)is then veriﬁed.Moreover,the corresponding selection func-tion,which selects literals based on signs,is clearly stable.14。

a sick note英语作文大学

a sick note英语作文大学全文共3篇示例，供读者参考篇1A sick note, also known as a doctor's note or medical certificate, is a document that is issued by a medical professional to certify an individual's illness or injury. This note is often required by employers, schools, or other institutions as a proof of a person's absence due to health reasons.There are several reasons why a sick note may be required. In the case of an employee, a sick note is often needed to justify time off work and to ensure that the individual is not taking advantage of sick leave policies. For students, a sick note may be necessary to excuse their absence from school or to request special accommodations for missed assignments or exams.In order to obtain a sick note, an individual must visit a doctor or healthcare provider and explain their symptoms or condition. The doctor will then conduct a physical examination or review the individual's medical history to determine the appropriate course of action. If the doctor believes that the individual is genuinely ill or injured, they will issue a sick notedetailing the diagnosis, treatment plan, and expected time frame for recovery.It is important for individuals to be honest when requesting a sick note and to provide as much information as possible about their condition. Falsifying information or misrepresenting one's health status can have serious consequences, including disciplinary action from employers or academic institutions.In some cases, a sick note may not be necessary if the individual is only experiencing minor symptoms or if they are able to complete their work or school tasks from home. However, it is always best to consult with a healthcare provider to ensure that the proper care is received and to prevent the spread of illness to others.In conclusion, a sick note is a valuable document that helps to protect the health and well-being of individuals and those around them. By obtaining a sick note when needed and following the advice of healthcare professionals, individuals can ensure a speedy recovery and a safe return to work or school.篇2A sick note is a document issued by a doctor to a patient who is unable to attend work or school due to illness. It serves asevidence of the individual's medical condition and provides an explanation for their absence. In a university setting, a sick note may be required by professors or academic advisors to excuse a student from missed classes, exams, or assignments.The process of obtaining a sick note typically involves scheduling an appointment with a healthcare provider, such as a general practitioner or a campus health center. During the appointment, the doctor will assess the patient's symptoms, perform a physical examination, and make a diagnosis. If the doctor deems the patient too unwell to attend classes or exams, they will issue a sick note detailing the reason for the absence and the expected duration of recovery.Once the sick note is obtained, it should be submitted to the relevant faculty member or academic advisor as soon as possible. In some cases, students may be required to notify their professors of their absence and provide a copy of the sick note within a certain timeframe, such as within 24 hours of the missed class or exam. Failure to do so may result in penalties, such as a lower grade or an incomplete for the missed assignment.It is important for students to understand the university's policy regarding sick notes and absences. Some institutions may have specific requirements for the format and content of sicknotes, such as including the doctor's contact information or a signature from the healthcare provider. In addition, students should be aware of any limitations on the number of sick days allowed per semester and the procedures for requesting extensions or accommodations due to illness.In some cases, students may encounter challenges in obtaining a sick note, such as difficulty scheduling an appointment with a healthcare provider or financial constraints. In such situations, it is important for students to communicate openly and honestly with their professors or academic advisors about their circumstances. Professors may be willing to make accommodations or provide alternative options for students who are unable to obtain a sick note.Overall, a sick note is a valuable tool for students who are unable to attend classes or exams due to illness. By following the proper procedures for obtaining and submitting a sick note, students can ensure that their absences are excused and that they are able to stay on track with their academic progress. It is important for students to prioritize their health and well-being, and to seek help and support when needed.篇3A Sick NoteIn college, there are times when students may fall ill and are unable to attend classes. During these times, it is important for students to obtain a sick note from a healthcare provider to provide to their professors as proof of their illness. A sick note is a document that verifies a student's illness or medical condition and provides information on the student's ability to attend classes.There are several reasons why a sick note is necessary in college. First and foremost, a sick note helps to ensure that the student's absence from classes is legitimate and justified. Without a sick note, students may be seen as skipping classes or being irresponsible, which can lead to poor academic performance and even disciplinary action. By providing a sick note, students can demonstrate that their absence is due to a genuine health issue.Additionally, a sick note can help in the determination of whether a student is eligible for excused absences or accommodations. For example, if a student has a chronic illness that requires frequent medical appointments or treatment, a sick note can provide documentation of the student's condition andthe need for accommodations such as flexible deadlines or alternative testing arrangements.In order to obtain a sick note, students should visit a healthcare provider such as a doctor or nurse practitioner. The healthcare provider will conduct a medical evaluation to assess the student's condition and provide a diagnosis. Based on the evaluation, the healthcare provider will then write a sick note that includes information such as the student's name, date of visit, diagnosis, recommended treatment plan, and expected duration of illness.Once the sick note is obtained, students should provide a copy to their professors as soon as possible. It is important to communicate with professors about the student's illness and any accommodations that may be needed. By providing a sick note, students can demonstrate their commitment to their academic responsibilities and ensure that their professors are aware of their health status.Overall, a sick note plays a crucial role in college by providing verification of a student's illness and facilitating communication with professors about any needed accommodations. It is important for students to take the necessary steps to obtain a sick note and communicateeffectively with their professors in order to ensure their academic success during periods of illness.。

会议变态发言稿英语版范文

Good morning/afternoon/evening, everyone. It is with great pleasure and a touch of trepidation that I stand before you today to share some thoughts that might, quite frankly, be considered somewhat unconventional. However, as we are all here to explore new ideas and push the boundaries of our understanding, I believe that a bit of unconventional thinking can be quite refreshing.Firstly, let us consider the concept of normalcy. In our society, we are often conditioned to seek normalcy, to fit in, to conform. But what if normalcy is, in fact, the enemy of progress? What if the most groundbreaking ideas and innovations come from those who dare to step outside the bounds of what is considered "normal"?Take, for instance, the internet. When it was first introduced, many people considered it a fad, a passing curiosity. Today, it is anintegral part of our lives, changing the way we communicate, work, and even think. Had we sought normalcy, we might still be using telegrams and rotary phones.Now, let's talk about the environment. We are currently facing a global crisis of climate change. The "normal" approach has been to try and mitigate the effects of climate change while continuing with our current lifestyle. But what if the "normal" way is precisely what is causing the problem? What if we need to adopt a completely new, unconventional approach to save our planet?Consider the following ideas:1. Negative Emissions Technology: Instead of focusing on reducing carbon emissions, why not develop technology that actively removes carbon from the atmosphere? This might sound like science fiction, but it could be the key to reversing climate change.2. Habitat Reconstruction: Instead of trying to preserve the status quo of our natural habitats, why not reconstruct them to be more resilient and sustainable? This could involve creating new ecosystems that are better equipped to handle the changing climate.3. Global Energy Grid: Imagine a world where energy is produced and distributed in a way that is not only efficient but also equitable. A global energy grid could harness renewable energy sources from aroundthe world and distribute it to where it is needed most.Moving on to our personal lives, we often seek to achieve "normal" happiness by following the same paths as others. But what if happinessis not a destination but a journey? What if the "normal" path to happiness is actually a recipe for dissatisfaction and unfulfillment?Consider the following suggestions:1. Pursue Your Passions: Instead of trying to conform to societal expectations, why not pursue your true passions? This might lead you on a path that is unconventional, but it is far more likely to lead to a fulfilling life.2. Embrace Imperfection: In our quest for perfection, we often overlook the beauty in imperfection. Embracing our flaws and learning from them can lead to a more authentic and satisfying life.3. Practice Gratitude: Instead of constantly seeking more, why not focus on what we already have? Practicing gratitude can shift our perspective and make us appreciate the small joys in life.In conclusion, I urge each and every one of you to question the norm. To challenge the status quo. To embrace the idea that sometimes, the most revolutionary ideas come from those who dare to be different. Let us not be afraid to take risks, to think outside the box, and to pursue a path that might seem unconventional but could very well lead us to a brighter, more sustainable, and more fulfilling future.Thank you for your attention, and I look forward to the conversationsand ideas that will undoubtedly emerge from this meeting. Together, we can shape a world that is not just normal, but exceptional.Good day/afternoon/evening.。

上课把零食当作文具用英文

上课把零食当作文具用英文Paragraph 1: Snacks as stationery? Absolutely! Who needs a boring old pen when you can use a crunchy bag of chips to write your notes? It adds a whole new level of excitement to the classroom. Plus, the sound of the crunching chips can serve as a unique form of motivation to keep you focused on your work. So forget about traditional stationery, snacks are the way to go!Paragraph 2: Have you ever tried using a candy bar as an eraser? Trust me, it's a game-changer. Not only does it satisfy your sweet tooth, but it also gets rid of those pesky mistakes. Just imagine the joy of indulging in a delicious chocolate bar while simultaneously fixing your errors. It's like a double win!Paragraph 3: Need to highlight important information in your textbook? Look no further than a pack of gummy bears. These colorful and chewy treats can be used as mini highlighters. Just pick a gummy bear in your desired color,press it onto the text, and voila! You've got a fun and tasty way to mark important passages. Who knew studying could be so delicious?Paragraph 4: Pencils are so last season. Why not use pretzel sticks instead? They're not only great for snacking on during class, but they can also serve as a uniquewriting tool. Just dip the pretzel stick in ink or paint and use it to create your masterpiece. It adds a touch of creativity to your work and who doesn't love a salty snack?Paragraph 5: Forget about rulers, use a licorice lace instead! Not only can you measure things with it, but you can also enjoy a sweet treat while doing so. It's a fun and unconventional way to bring some excitement to your math or art class. Plus, licorice laces come in various colors, so you can coordinate your measurements with your mood.Paragraph 6: Lastly, who needs a textbook when you can use a bag of popcorn to learn? Simply write important information on individual pieces of popcorn and arrange them in the order of your lesson. As you eat each piece,you'll absorb the knowledge. It's a tasty and interactive way to study. Just be careful not to eat all your notes before the test!Remember, these suggestions are purely for fun and creativity. In reality, it's important to use appropriate stationery and focus on your studies. But hey, a little snack-inspired imagination never hurts!。

粉笔批改英语作文

粉笔批改英语作文As an English teacher, correcting essays can be a meticulous yet rewarding process. Here's how I would approach correcting English essays using chalk on a blackboard, which is a traditional method that still holds educational value:1. Preparation:- Gather all the essays to be corrected.- Ensure the blackboard is clean and has enough space for writing.2. Reading Through:- Read each essay thoroughly to understand the student's main ideas and arguments.3. Identifying Key Issues:- Look for common mistakes such as grammatical errors, spelling mistakes, and punctuation issues.- Note down structural problems like unclear thesis statements or lack of coherence.4. Correction Process:- Use colored chalk to make the corrections stand out. For example, use red for spelling and grammar, blue for structure, and green for positive feedback.- Write the corrected word or phrase clearly on the blackboard, referencing the student's name or essay number.5. Structural Feedback:- For structural feedback, draw attention to the flow of ideas and suggest improvements using arrows or bullet points on the blackboard.6. Vocabulary Enhancement:- Highlight any repetitive or basic vocabulary and suggest more sophisticated alternatives.7. Positive Reinforcement:- Point out the strengths of each essay, such as a well-argued point or a particularly well-written paragraph.8. General Comments:- Write general comments on the board that apply to the entire class, such as common mistakes to avoid or areas for improvement.9. Interactive Session:- After all essays have been corrected, conduct an interactive session where students can ask questions about their corrections.10. Follow-Up:- Assign a follow-up task where students rewrite their essays, incorporating the corrections and feedback.11. Documentation:- While the blackboard serves as an immediate teaching tool, it's important to document the corrections and feedback for future reference and to track student progress.By using the blackboard, students can see the mistakes and corrections in real-time, which can be a powerful teaching moment. It also allows for a class discussion on common errors and how to avoid them, fostering a collaborative learning environment.。

如何脱稿英文作文范文

如何脱稿英文作文范文英文回答：Effective off-the-cuff English essay writing requires a combination of preparation, practice, and presence of mind. Here are some strategies to help you master this art:1. Gather Your Thoughts: Before you start writing, takea moment to gather your thoughts and organize your ideas. Identify the main points you want to convey and their supporting evidence.2. Use Mind Maps: Visualizing your ideas through mind maps can help you connect thoughts and establish a logical flow. Start with a central topic or question, and then branch out with subtopics and supporting details.3. Practice Regularly: The more you practice writingoff the cuff, the better you will become. Set aside dedicated time each week to practice writing essays withoutany notes. Start with short, timed intervals and gradually increase the time and complexity of your essays.4. Build Your Vocabulary: Expand your vocabulary by reading widely and incorporating new words into your writing. The more words you know, the easier it will be to express yourself eloquently and concisely.5. Stay Informed: Keep up with current events and general knowledge to have a wider pool of topics to draw upon. This will give you confidence in speaking extemporaneously on various subjects.6. Use Transition Words and Phrases: Transition words, such as "moreover," "in addition," and "however," help connect ideas and ensure a smooth flow of thought. Incorporate them into your writing to make your essays more coherent.7. Speak Aloud: Practice speaking your thoughts aloud to simulate the experience of writing off the cuff. This will help you develop fluency and articulate your ideaseffectively.8. Don't Be Afraid to Make Mistakes: Mistakes are part of the learning process. Don't get discouraged if you stumble or deviate from your intended path. Embrace mistakes as opportunities to refine your thinking and improve your writing.9. Use Storytelling and Anecdotes: Weave personal anecdotes and stories into your essays to make them more engaging and relatable. This will help you connect with your audience and make a lasting impression.10. Stay Calm and Collected: When writing off the cuff, it's essential to stay calm and collected. Take deep breaths, focus on your thoughts, and let your words flow naturally.中文回答：如何脱稿写好一篇英文作文。

托福写作172句型精编版

（论证positive方面常用）1.Be supposed to – should 应该…In fact, going on a diet is supposed to be more than just eating little. It should also be about eating healthily.实际上，节食应该不只关乎吃得少，也应关乎吃得健康。

2.Hold fast to - 牢牢的把握住In his biography, the massage of perseverance and holding fast to one’s dreams in the face of adversity is very strong.他的传记非常强调坚持不懈与不放弃梦想的精神。

近义词：put a high value on/ treasure/ cherish/value/put emphasis on/ attach great importance to3.Bear in mind - 牢记Bear in mind that the next generation of cell phone is just around the corner.请牢记，新一代手机已呼之欲出。

4.There is no better way to ... than to ...…… 是做某事做好的方法没有比运动更好的减肥方法了5.Go a long way --- 大有帮助Sincere praise, encouragement and appreciation go a long way- even just saying “thank you” leaves a lasting impression.真诚的表扬，鼓励与赞赏总是非常有效地，即使只是一句谢谢也给对方留下长久的印象。

6.It’s worthwhile to ... / be worth doing sth - 值得做某事Before you vote for a certain candidate in an election, it’s worthwhile to find his/her true worth.在你为某位大选候选人投票前，值得去发现他的真正价值。

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a rX iv:mat h /311344v2[mat h.DG ]25J un24A NOTE ON NEGATIVE ISOTROPIC CUR V ATURE HARISH SESHADRI Abstract.We prove that any smooth orientable closed four-manifold admits a Riemannian metric with negative isotropic curvature in the sense of Micallef and Moore. 1.Introduction An interesting notion of curvature,that of curvature on isotropic 2-planes ,was introduced by Micallef and Moore in [5].The deﬁnition is as follows.Let (M,g )be a Riemannian manifold of dimension at least 4.Consider the complexiﬁcation T M ⊗R C .The inner products on T M and ∧2T M extend to complex bilinear maps on T M ⊗C and ∧2T M ⊗C and we denote these extensions by , in both cases.A subspace W ⊂T p M ⊗C is said to be isotropic if w,w =0for all w ∈W .Now let R :∧2T M ⊗C →∧2T M ⊗C denote the complex linear extension of the curvature operator of g .The metric g is said to have positive (resp.negative)isotropic curvature if R (v ∧w ),w >0(resp.<0)whenever {v,w }span an isotropic 2-plane.Considerations of the second variation of area of minimal surfaces lead natu-rally to isotropic curvature.Examples of manifolds with isotropic curvature of a ﬁxed sign are pointwise quarter-pinched manifolds and manifolds with positive or negative curvature operators.The relationship between the sign of isotropic curvature and the “classical curva-tures”is somewhat mysterious.It is known (cf.[6])that positive isotropic curvature does not imply positive Ricci curvature but does imply positive scalar curvature (a similar statement is true for negative isotropic curvature).As in the case of positive scalar curvature,one knows that if two manifolds have positive isotropic curvature then their connected sum admits a metric with positive isotropic curvature.How-ever,unlike the case of positive scalar curvature,surgeries in lower codimension need not preserve positive isotropic curvature.Pursuing the analogy between isotropic and scalar curvatures,one is tempted to ask whether every smooth closed manifold admits a metric with negative isotropic curvature.In this note we show that this is the case in dimension 4.Theorem 1.1.Any smooth orientable closed four-manifold admits a C ∞metric with negative isotropic curvature.Remarks :(i)A K¨a hler metric can never have negative isotropic curvature.In fact,if v and w are such that (v ∧w )1,1=0,then R (v ∧w ),w =0.There exist v and w such that (v ∧w )1,1=0and C {v,w }is an isotropic 2-plane [3].2HARISH SESHADRI(ii)In the deﬁnition of isotropic curvature,if one demands that R(v∧w),v∧w < 0for any v∧w∈∧2T M⊗C,then one says,following Sampson[8],that g has nega-tive Hermitian sectional curvature.This curvature occurs in the study of harmonic maps of K¨a hler manifolds and clearly implies negative sectional curvature.Hence it has strong topological implications while Theorem1.1states that negative isotropic curvature,in dimension4at least,has no topological implications.The proof of Theorem1.1essentially hinges on the fact that in dimension four, the negative isotropic curvature condition can be rephrased as a variational prob-lem.The functional involved will be the integral of a scalar curvature-like function σand its domain will be the space of metrics.Once this is noted the proof in-volves showing that the functional becomes negative for a suitable choice of metric g.Then standard techniques in conformal geometry(as in[2])can be applied to show that there exists a conformal deformation of g for whichσis negative.The geometric part of the proof is the construction of g.We do this by gluing in a product of a circle with a hyperbolic3-manifold ofﬁnite(but large)volume.This hyperbolic3-manifold will be a knot complement in S3and the gluing will be done without changing the smooth topology of M.Note that the proof below only gives a C2,αmetric with negative isotropic cur-vature.The density of C∞metrics in the space of C2,αmetrics(with the C2,αnorm)will then give us a C∞metric with negative isotropic curvature. Acknowledgements.I would like to thank the referee for pointing out various errors in the original draft.2.ProofProof.For the sake of exposition we divide the proof into two steps.STEP1:Let M be a closed smooth four-manifold.A simple calculation,as in [5],shows that a metric g on M has negative isotropic curvature if and only if (1)s6+|W|<0,where|W|denotes the pointwise norm of W(regarded as an element of⊗4T∗M). This is because W is a symmetric operator and hence W(α),α ≥−|W||α|2.The main property ofσg is that under conformal changes it transforms like scalar curvature.More precisely,if g→˜g=u4n−2σg−4n−1n−2△u.This follows immediately from the corresponding transformation law for the usual scalar curvature s cf.[1]and the fact that|W f2g|f2g=f−2|W g|g for any smooth positive function f.NEGATIVE ISOTROPIC CUR V ATURE3 Consider the functional on the space of C2metrics on M deﬁned byF(g)= Mσg dV g,where dV g is the volume form of the metric g.A standard argument,which we outline below,shows that if F(g)<0for some g,then there is a C2,αmetric˜g in the conformal class of g with˜σg<0.Deﬁne the operator L on C∞(M)by(n−1)L=−4n−2˜g.By(2)and(3),we see thatσg=u−(n−2)N=S3−ν(K)⊂N,whereν(k)is an open tubular neighbourhood of K.Fix an embedding ofN×S1which are isometric to aﬁxed standard metric near∂(N×S1).For large values of c we get metrics for which F<0.The actual construction is as follows:N has one cusp end and this end is iso-metric to(E:=[0,∞)×T2,g H:=dt2+e−2t g e),where T2is the2-torus and g e aﬂat metric on it.Now consider the metrics c2g H,where c∈R+on N.We note two features of these metrics.(i)The sectional curvatures of these metrics go to zero as c→∞.Also,by choosing a(c)appropriately(in fact a(c)=c log(c4HARISH SESHADRINow the scaled metric c2g H on the end can be written asc2g H:=dt2+c2e−2t.2Further,letf c(t)=φ t−a(c) ce−t,let a(c)=c log(c).If t∈[a(c),a(c)+1],then1c< 1.log(2)From this an easy computation shows that the C2distance between the metrics k c (deﬁned in(5))and dt2+g e on[a(c),a(c)+1]×T2is bounded,(6)|k c−(dt2+g e)|C2≤A,with A independent of c.The metric g c that we are interested(on M)is now given by:g c=c2g H+dθ2on N0×S1=k c+dθ2on[0,a(c)+1]×T2×S1=g is aﬁxed extension of the standard product metric dt2+g e+dθ2on a tubular neighbourhood of T2×S1∼=∂ M−(N×S1).This extension can obviously be chosen to be independent of c.Now we claim that F(g c)<0for large c.To see this,noteﬁrst that the scalar curvature integral M s g c dV g c is equal to(7) N a(c)×S1s g c dV g c+ [a(c),a(c)+1]×T2×S1s g c dV g c+ M−(N in the last integral.Now,as c→∞,the ﬁrst integral goes to−∞by(4).The second remains bounded by(6)and the third is independent of c.Hence(8) M s g c dV g c→−∞as c→∞. Similarly,the Weyl curvature integral M|W g c|dV g c is equal to(9) N a(c)×S1|W g c|dV g c+ [a(c),a(c)+1]×T2×S1|W g c|dV g c+ M−(NEGATIVE ISOTROPIC CUR V ATURE5is with respect to g c.Again,for the same reasons,the where the norm of W gcsecond and third integrals will remain bounded as c→∞.To see that theﬁrst integral is zero,recall that the metric on N a(c)×S1is the product metric and the product of a constant sectional curvature metric with S1is conformallyﬂat.Hence(10) M|W g c|dV g c remains bounded as c→∞.Combining(8)and(10)we see that F(g c)<0for large c.Remarks:(i)The proof above actually shows that on any closed4-manifold,for anyµ>0, there exists a metric g withµs g+|W g|<0.(ii)In the case of negative scalar curvature,one knows by the work of Lohkamp [4],that the space of such metrics M is contractible.One can ask if the same is true for the space of negative isotropic curvature metrics.Note that in Lohkamp’s paper,a variational approach similar to Step1is used to extend a map of S n into M to a map of B n+1(in order to prove thatπn(M)=0).(iii)The proof hinges on the fact that in dimension4,there exists an expression of the formµs g∓|W g|,withµ>0such that if this expression has a certainﬁxed sign,then the isotropic curvature also has the same sign.In order to extend the proof to dimensions n≥5,one needs to know if a similar statement is true for all n. However the following example,given in[6]shows that such a statement is not true in the positive case.The product metric on M=Σ×S2m,whereΣis a compact surface of constant curvature−1and S2m is the unit sphere of dimension2m≥4, is conformallyﬂat and has positive scalar curvature.However,since H2(M,R)=0 by Theorem2.1of[6],M does not admit a metric of positive isotropic curvature.References[1]A.L.Besse,Einstein Manifolds,Springer-Verlag,Berlin-Heidelberg,1978.[2]M.J.Gursky,Four-manifolds withδW+=0and Einstein constants of the sphere,Math.Ann.318(2000),no.3,417-431.[3]A.El Souﬁ,R.Petit,Immersions minimales et immersions pluriharmoniques entre vari´e t´e sRiemanniennes:r´e sultats de non existence et de rigidite´e,Ann.Inst.Fourier(Grenoble),50 (2000),no.1,235-256.[4]J.Lohkamp,The space of negative scalar curvature metrics,Invent.Math.110(1992),403-407.[5]M.J.Micallef and J.D.Moore,Minimal two-spheres and the topology of manifolds withpositive curvature on totally isotropic two-planes,Ann.of Math.127(1988),199-227.[6]M.J.Micallef and M.Y.Wang,Metrics with nonnegative isotropic curvature,Duke Math.Journal,72(1993),649-672.[7]R.Riley,A quadratic parabolic group,Math.Proc.Cambridge Philos.Soc.,77(1975),281-288.[8]J.H.Sampson,Application of harmonic maps to K¨a hler geometry,Contemp.Math.,77(1984),125-134.E-mail address:harish@isibang.ac.inStat-Math Unit,,Indian Statistical Institute,,Bangalore,India。