The Variables and Invariants in the Evolution of Logic Optical Lithography Process
f(R) Theories Of Gravity
a r X i v :0805.1726v 2 [g r -q c ] 24 M a y 2008f (R )theories of gravityThomas P.SotiriouCenter for Fundamental Physics,University of Maryland,College Park,MD 20742-4111,USA ∗Valerio FaraoniPhysics Department,Bishop’s University,2600College St.,Sherbrooke,Qu`e bec,Canada J1M 1Z7†Modified gravity theories have received increased attention lately due to combined motivation coming from high-energy physics,cosmology and astrophysics.Among numerous alternatives to Einstein’s theory of gravity,theories which include higher order curvature invariants,and specifically the particular class of f (R )theories,have a long history.In the last five years there has been a new stimulus for their study,leading to a number of interesting results.We review here f (R )theories of gravity in an attempt to comprehensively present their most important aspects and cover the largest possible portion of the relevant literature.All known formalisms are presented —metric,Palatini and metric-affine —and the following topics are discussed:motivation;actions,field equations and theoretical aspects;equivalence with other theories;cosmological aspects and constraints;viability criteria;astrophysical applications.ContentsI.Introduction1A.Historical1B.Contemporary Motivation 2C.f (R )theories as toy theories 3II.Actions and field equations4A.Metric formalism 5B.Palatini formalism6C.Metric-affine formalism81.Preliminaries 92.Field Equations10III.Equivalence with Brans–Dicke theory andclassification of theories 11A.Metric formalism 12B.Palatini formalism 13C.Classification13D.Why f (R )gravity then?14IV.Cosmological evolution and constraints15A.Background evolution151.Metric f (R )gravity 152.Palatini f (R )gravity 17B.Cosmological eras18C.Dynamics of cosmological perturbations19V.Other standard viability criteria 20A.Weak-field limit 201.The scalar degree of freedom 202.Weak-field limit in the metric formalism 223.Weak-field limit in the Palatini formalism 25B.Stability issues 271.Ricci stability in the metric formalism 272.Gauge-invariant stability of de Sitter space in themetric formalism 293.Ricci stability in the Palatini formalism 304.Ghost fields 30C.The Cauchy problem 312and,consequently,thefield equations with no apparent theoretical or experimental motivation is not very ap-pealing.However,the motivation was soon to come. Beginning in the1960’s,there appeared indications that complicating the gravitational action might indeed have its merits.GR is not renormalizable and,therefore, can not be conventionally quantized.In1962,Utiyama and De Witt showed that renormalization at one-loop de-mands that the Einstein–Hilbert action be supplemented by higher order curvature terms(Utiyama and DeWitt, 1962).Later on,Stelle showed that higher order ac-tions are indeed renormalizable(but not unitary)(Stelle, 1977).More recent results show that when quantum corrections or string theory are taken into account, the effective low energy gravitational action admits higher order curvature invariants(Birrell and Davies, 1982;Buchbinder et al.,1992;Vilkovisky,1992).Such considerations stimulated the interest of the scientific community in higher-order theories of gravity, i.e.,modifications of the Einstein–Hilbert action in order to include higher-order curvature invariants with respect to the Ricci scalar[see(Schmidt,2007)for a historical review and a list of references to early work].However, the relevance of such terms in the action was considered to be restricted to very strong gravity regimes and they were expected to be strongly suppressed by small couplings,as one would expect when simple effective field theory considerations are taken into account. Therefore,corrections to GR were considered to be important only at scales close to the Planck scale and, consequently,in the early universe or near black hole sin-gularities—and indeed there are relevant studies,such as the well-known curvature-driven inflation scenario (Starobinsky,1980)and attempts to avoid cosmo-logical and black hole singularities(Brandenberger, 1992,1993,1995;Brandenberger et al.,1993; Mukhanov and Brandenberger,1992;Shahid-Saless, 1990;Trodden et al.,1993).However,it was not expected that such corrections could affect the gravita-tional phenomenology at low energies,and consequently large scales such as,for instance,the late universe.B.Contemporary MotivationMore recently,new evidence coming from astrophysics and cosmology has revealed a quite unexpected picture of the universe.Our latest datasets coming from different sources,such as the Cosmic Microwave Background Radi-ation(CMBR)and supernovae surveys,seem to indicate that the energy budget of the universe is the following: 4%ordinary baryonic matter,20%dark matter and76% dark energy(Astier et al.,2006;Eisenstein et al.,2005; Riess et al.,2004;Spergel et al.,2007).The term dark matter refers to an unkown form of matter,which has the clustering properties of ordinary matter but has not yet been detected in the laboratory.The term dark en-ergy is reserved for an unknown form of energy which not only has not been detected directly,but also does not cluster as ordinary matter does.More rigorously, one could use the various energy conditions(Wald,1984) to distinguish dark matter and dark energy:Ordinary matter and dark matter satisfy the Strong Energy Condi-tion,whereas Dark Energy does not.Additionally,dark energy seems to resemble in high detail a cosmological constant.Due to its dominance over matter(ordinary and dark)at present times,the expansion of the universe seems to be an accelerated one,contrary to past expec-tations.1Note that this late time speed-up comes to be added to an early time accelerated epoch as predicted by the infla-tionary paradigm(Guth,1981;Kolb and Turner,1992; Linde,1990).The inflationary epoch is needed to ad-dress the so-called horizon,flatness and monopole prob-lems(Kolb and Turner,1992;Linde,1990;Misner,1968; Weinberg,1972),as well as to provide the mechanism that generates primordial inhomogeneities acting as seeds for the formation of large scale structures(Mukhanov, 2003).Recall also that,in between these two periods of acceleration,there should be a period of decelerated expansion,so that the more conventional cosmological eras of radiation domination and matter domination can take place.Indeed,there are stringent observational bounds on the abundances of light elements,such as deu-terium,helium and lithium,which require that Big Bang Nucleosynthesis(BBN),the production of nuclei other than hydrogen,takes place during radiation domination (Burles et al.,2001;Carroll and Kaplinghat,2002).On the other hand,a matter-dominated era is required for structure formation to take place.Puzzling observations do not stop here.Dark mat-ter does not only make its appearance in cosmological data but also in astrophysical observations.The“missing mass”question had already been posed in1933for galaxy clusters(Zwicky,1933)and in1959for individual galax-ies(Kahn and Woltjer,1959)and a satisfactoryfinal an-swer has been pending ever since(Bosma,1978;Ellis, 2002;Moore,2001;Persic et al.,1996;Rubin and Ford, 1970;Rubin et al.,1980).One,therefore,has to admit that our current picture of the evolution and the matter/energy content of the universe is at least surprising and definitely calls for an explanation.The simplest model which adequatelyfits the data creating this picture is the so called concordancea=−4πG3 model orΛCDM(Λ-Cold Dark Matter),supplemented bysome inflationary scenario,usually based on some scalarfield called inflaton.Besides not explaining the originof the inflaton or the nature of dark matter by itself,theΛCDM model is burdened with the well known cosmolog-ical constant problems(Carroll,2001a;Weinberg,1989):the magnitude problem,according to which the observedvalue of the cosmological constant is extravagantly smallto be attributed to the vacuum energy of matterfields,and the coincidence problem,which can be summed upin the question:since there is just an extremely shortperiod of time in the evolution of the universe in whichthe energy density of the cosmological constant is com-parable with that of matter,why is this happening todaythat we are present to observe it?These problems make theΛCDM model more of anempiricalfit to the data whose theoretical motivationcan be regarded as quite poor.Consequently,therehave been several attempts to either directly motivatethe presence of a cosmological constant or to proposedynamical alternatives to dark energy.Unfortunately,none of these attempts are problem-free.For instance,the so-called anthropic reasoning for the magnitude ofΛ(Barrow and Tipler,1986;Carter,1974),even whenplaced into thefirmer grounds through the idea of the“anthropic or string landscape”(Susskind,2003),stillmakes many physicists feel uncomfortable due to itsprobabilistic nature.On the other hand,simple sce-narios for dynamical dark energy,such as quintessence(Bahcall et al.,1999;Caldwell et al.,1998;Carroll,1998;Ostriker and Steinhardt,1995;Peebles and Ratra,1988;Ratra and Peebles,1988;Wang et al.,2000;Wetterich,1988)do not seem to be as well motivated theoreticallyas one would desire.2Another perspective towards resolving the issues de-scribed above,which might appear as more radical to some,is the following:gravity is by far the dominant in-teraction at cosmological scales and,therefore,it is the force governing the evolution of the universe.Could it be that our description of the gravitational interaction at the relevant scales is not sufficiently adequate and stands at the root of all or some of these problems?Should we con-sider modifying our theory of gravitation and if so,would this help in avoiding dark components and answering the cosmological and astrophysical riddles?It is rather pointless to argue whether such a perspec-tive would be better or worse than any of the other so-lutions already proposed.It is definitely a different way to address the same problems and,as long as these prob-lems do notfind a plausible,well accepted and simple,2κ d4x√2κ d4x√4 as a series expansion,i.e.,f(R)=...+α2R−2Λ+R+R2β3+...,(4)where theαi andβi coefficients have the appropriate dimensions,we see that the action includes a number of phenomenologically interesting terms.In brief,f(R) theories make excellent candidates for toy-theories—tools from which one gains some insight in such gravity mod-ifications.Second,there are serious reasons to believe that f(R)theories are unique among higher-order grav-ity theories,in the sense that they seem to be the only ones which can avoid the long known and fatal Ostro-gradski instability(Woodard,2007).The second question calling for an answer is related to a possible loophole that one may have already spotted in the motivation presented:How can high-energy modifi-cations of the gravitational action have anything to do with late-time cosmological phenomenology?Wouldn’t effectivefield theory considerations require that the co-efficients in eq.(4)be such,as to make any corrections to the standard Einstein–Hilbert term important only near the Planck scale?Conservatively thinking,the answer would be posi-tive.However,one also has to stress two other serious factors:first,there is a large ambiguity on how grav-ity really works at small scales or high energies.Indeed there are certain results already in the literature claiming that terms responsible for late time gravitational phe-nomenology might be predicted by some more funda-mental theory,such as string theory[see,for instance, (Nojiri and Odintsov,2003b)].On the other hand,one should not forget that the observationally measured value of the cosmological constant corresponds to some energy scale.Effectivefield theory or any other high-energy the-ory consideration has thus far failed to predict or explain it.Yet,it stands as an experimental fact and putting the number in the right context can be crucial in ex-plaining its value.Therefore,in any phenomenological approach,its seems inevitable that some parameter will appear to be unnaturally small atfirst(the mass of a scalar,a coefficient of some expansions,etc.according to the approach).The real question is whether this initial “unnaturalness”still has room to be explained.In other words,in all sincerity,the motivation for in-frared modifications of gravity in general and f(R)grav-ity in particular is,to some extent,hand-waving.How-ever,the importance of the issues leading to this motiva-tion and our inability tofind other,more straightforward and maybe better motivated,successful ways to address them combined with the significant room for specula-tion which our quantum gravity candidates leave,have triggered a,probably reasonable,increase of interest in modified gravity.To conclude,when all of the above is taken into ac-count,f(R)gravity should neither be over-nor under-estimated.It is an interesting and relatively simple alter-native to GR,from the study of which some useful con-clusions have been derived already.However,it is still a toy-theory,as already mentioned;an easy-to-handle deviation from Einstein’s theory mostly to be used in or-der to understand the principles and limitations of mod-ified gravity.Similar considerations apply to modifying gravity in general:we are probably far from concluding whether it is the answer to our problems at the moment. However,in some sense,such an approach is bound to be fruitful since,even if it only leads to the conclusion that GR is the only correct theory of gravitation,it will still have helped us to both understand GR better and secure our faith in it.II.ACTIONS AND FIELD EQUATIONSAs can be found in many textbooks—see,for ex-ample(Misner et al.,1973;Wald,1984)—there are actually two variational principles that one can apply to the Einstein–Hilbert action in order to derive Ein-stein’s equations:the standard metric variation and a less standard variation dubbed Palatini variation[even though it was Einstein and not Palatini who introduced it(Ferraris et al.,1982)].In the latter the metric and the connection are assumed to be independent variables and one varies the action with respect to both of them(we will see how this variation leads to Einstein’s equations shortly),under the important assumption that the mat-ter action does not depend on the connection.The choice of the variational principle is usually referred to as a for-malism,so one can use the terms metric(or second order) formalism and Palatini(orfirst order)formalism.How-ever,even though both variational principles lead to the samefield equation for an action whose Lagrangian is lin-ear in R,this is no longer true for a more general action. Therefore,it is intuitive that there will be two version of f(R)gravity,according to which variational principle or formalism is used.Indeed this is the case:f(R)gravity in the metric formalism is called metric f(R)gravity and f(R)gravity in the Palatini formalism is called Palatini f(R)gravity(Buchdahl,1970).Finally,there is actually even a third ver-sion of f(R)gravity:metric-affine f(R)gravity (Sotiriou and Liberati,2007a,b).This comes about if one uses the Palatini variation but abandons the assumption that the matter action is independent of the connection.Clearly,metric affine f(R)gravity is the most general of these theories and reduces to metric or Palatini f(R)gravity if further assumptions are made. In this section we will present the actions andfield equations of all three versions of f(R)gravity and point out their difference.We will also clarify the physical meaning behind the assumptions that discriminate them.For an introduction to metric f(R)gravity see also(Nojiri and Odintsov,2007a),for a shorter review of metric and Palatini f(R)gravity see (Capozziello and Francaviglia,2008)and for an ex-5 tensive analysis of all versions of f(R)gravity and otheralternative theories of gravity see(Sotiriou,2007b).A.Metric formalismBeginning from the action(3)and adding a matterterm S M,the total action for f(R)gravity takes the formS met=1−g f(R)+S M(gµν,ψ),(5)whereψcollectively denotes the matterfields.Variation with respect to the metric gives,after some manipula-tions and modulo surface termsf′(R)Rµν−1√δgµν,(7) a prime denotes differentiation with respect to the ar-gument,∇µis the covariant derivative associated withthe Levi-Civita connection of the metric,and ≡∇µ∇µ.Metric f(R)gravity wasfirst rigorously studied in(Buchdahl,1970).3It has to be stressed that there is a mathematical jump in deriving eq.(6)from the action(5)having to do with the surface terms that appear in the variation:as in the case of the Einstein–Hilbert action,the surface terms do not vanish just byfixing the metric on the boundary. For the Einstein–Hilbert action,however,these terms gather into a total variation of a quantity.Therefore, it is possible to add a total divergence to the action in order to“heal”it and arrive to a well-defined variational principle(this is the well known Gibbons–Hawking–York surface term(Gibbons and Hawking,1977;York,1972)). Unfortunately,the surface terms in the variation of the action(3)do not consist of a total variation of some quantity(the reader is urged to calculate the variation in order to verify this fact)and it is not possible to“heal”the action by just subtracting some surface term before performing the variation.The way out comes from the fact that the action in-cludes higher order derivatives of the metric and,there-fore,it should be possible tofix more degrees of freedom on the boundary than those of the metric itself.There is no unique prescription for such afixing in the literature so far.Note also that the choice offixing is not void of phys-ical meaning,since it will be relevant for the Hamiltonian formulation of the theory.However,thefield equations (6)would be unaffected by thefixing chosen and from a6 2006a).4Finally,let us note that it is possible to write thefieldequations in the form of Einstein equations with an ef-fective stress-energy tensor composed of curvature termsmoved to the right hand side.This approach is question-able in principle(the theory is not Einstein’s theory andit is artificial to force upon it an interpretation in termsof Einstein equations)but,in practice,it has been provedto be useful in scalar-tensor gravity.Specifically,eq.(6)can be written asGµν≡Rµν−1f′(R)+gµν[f(R)−Rf′(R)]f′(R)(10)orGµν=κκ f(R)−Rf′(R)4Energy-momentum complexes in the spherically symmetric case have been computed in(Multamaki et al.,2008).the independent connection.Note that the metric is not needed to obtain the latter from the former.For clarity of notation,we denote the Ricci tensor constructed with this independent connection as Rµνand the correspond-ing Ricci scalar5is R=gµνRµν.The action now takes the formS pal=1−g f(R)+S M(gµν,ψ).(13)GR will come about,as we will see shortly,when f(R)= R.Note that the matter action S M is assumed to de-pend only on the metric and the matterfields and not on the independent connection.This assumption is crucial for the derivation of Einstein’s equations from the linear version of the action(13)and is the main feature of the Palatini formalism.It has already been mentioned that this assumption has consequences for the physical meaning of the independent connection(Sotiriou,2006b,d;Sotiriou and Liberati, 2007b).Let us elaborate on this:recall that an affine connection usually defines parallel transport and the co-variant derivative.On the other hand,the matter action S M is supposed to be a generally covariant scalar which includes derivatives of the matterfields.Therefore,these derivatives ought to be covariant derivatives for a general matterfield.Exceptions exist,such as a scalarfield,for which a covariant and a partial derivative coincide,and the electromagneticfield,for which one can write a co-variant action without the use of the covariant derivative [it is the exterior derivative that is actually needed,see next section and(Sotiriou and Liberati,2007b)].How-ever,S M should include all possiblefields.Therefore, assuming that S M is independent of the connection can imply one of two things(Sotiriou,2006d):either we are restricting ourselves to specificfields,or we are implic-itly assuming that it is the Levi-Civita connection of the metric that actually defines parallel transport.Since the first option is implausibly limiting for a gravitational the-ory,we are left with the conclusion that the independent connectionΓλµνdoes not define parallel transport or the covariant derivative and the geometry is actually pseudo-Riemannian.The covariant derivative is actually defined by the Levi-Civita connection of the metric{λµν}.This also implies that Palatini f(R)gravity is a met-ric theory in the sense that it satisfies the metric pos-tulates(Will,1981).Let us clarify this:matter is mini-mally coupled to the metric and not coupled to any other fields.Once again,as in GR or metric f(R)gravity, one could use diffeomorphism invariance to show that the stress energy tensor is conserved by the covariant derivative defined with the Levi-Civita connection of the metric,i.e.,∇µTµν=0(but¯∇µTµν=0).This can7also be shown by using the field equations,which we will present shortly,in order to calculate the divergence of T µνwith respect to the Levi-Civita connection of the metric and show that it vanishes(Barracoet al.,1999;Koivisto,2006a).6Clearly then,Palatini f (R)gravityis ametrictheoryaccording to the definition of (Will,1981)(not to be confusedwiththeterm“metric”in“metricf(R)gravity”,which simply refers to the fact that one only varies the action with respect to the metric).Conven-tionally thinking,as a consequence of the covariant con-servation of the matter energy-momentum tensor,test particles should follow geodesics of the metric in Palatini f (R )gravity.This can be seen by considering a dust fluid with T µν=ρu µu νand projecting the conserva-tion equation ∇βT µβ=0onto the fluid four-velocity u β.Similarly,theories that satisfy the metric postulates are supposed to satisfy the Einstein Equivalence Principle as well (Will,1981).Unfortunately,things are more compli-cated here and,therefore,we set this issue aside for the moment.We will return to it and attempt to fully clarify it in Secs.VI.B and VI.C.2.For now,let us proceed with our discussion of the field equations.Varying the action (13)independently with respect to the metric and the connection,respectively,and using the formulaδR µν=¯∇λδΓλµν−¯∇νδΓλµλ.(14)yieldsf ′(R )R (µν)−1−gf ′(R )g µν+¯∇σ√−gf ′(R )g σµ=0,(17)which implies that we can bring the field equations intothe more economical formf ′(R )R (µν)−1−gf ′(R )g µν=0,(19)It is now easy to see how the Palatini formalism leads to GR when f (R )=R ;in this case f ′(R )=1and−h h µν=√7See,however,(Sotiriou,2007b)for further analysis of the f (R )action and how it can be derived from first principles in the two formalisms.8This calculation holds in four dimensions.When the num-ber of dimensions D is different from 4then,instead of us-ing eq.(22),the conformal metric h µνshould be introduced as h µν≡[f ′(R ]2/(D −2)g µνin order for eq.(23)to still hold.8Then,eq.(19)becomesthedefinitionofthe Levi-Civita connectionofhµνandcan besolved algebraicallyto giveΓλµν=hλσ(∂µhνσ+∂νhµσ−∂σhµν),(24)or,equivalently,in terms of gµν,Γλµν=121f′(R) ∇µ∇ν−12(f′(R))2(∇µf′(R))(∇µf′(R))+3f′Tµν−1f′ (28)+1212gµν(∇f′)2 .Notice that,assuming that we know the root of eq.(20),R=R(T),we have completely eliminated the indepen-dent connection from this equation.Therefore,we havesuccessfully reduced the number offield equations to oneand at the same time both sides of eq.(28)depend onlyon the metric and the matterfields.In a sense,the the-ory has been brought to the form of GR with a modifiedsource.We can now straightforwardly deduce the following:•When f(R)=R,the theory reduces to GR,asdiscussed previously.•For matterfields with T=0,due to eq.(21),R andconsequently f(R)and f′(R)are constants and thetheory reduces to GR with a cosmological constantand a modified coupling constant G/f′.If we de-note the value of R when T=0as R0,then thevalue of the cosmological constant is1f′(R0) =R02κ d4x√9Note that,apart from special cases such as a perfectfluid,Tµνand consequently T already includefirst derivatives of the matterfields,given that the matter action has such a dependence.Thisimplies that the right hand side of eq.(28)will include at leastsecond derivatives of the matterfields,and possibly up to thirdderivatives.9 1.PreliminariesBefore going further and derivingfield equations fromthis action certain issues need to be clarified.First,sincenow the matter action depends on the connection,weshould define a quantity representing the variation of S Mwith respect to the connection,which mimics the defi-nition of the stress-energy tensor.We call this quantitythe hypermomentum and is defined as(Hehl and Kerling,1978)∆µνλ≡−2−gδS M4Qνµν,(33)and the Cartan torsion tensorSλµν≡Γλ[µν],(34) which is the antisymmetric part of the connection.By allowing a non-vanishing Cartan torsion tensor we are allowing the theory to naturally include torsion.Even though this brings complications,it has been considered by some to be an advantage for a gravity theory since some matterfields,such as Diracfields,can be cou-pled to it in a way which might be considered more natural(Hehl et al.,1995):one might expect that at some intermediate or high energy regime,the spin of particles might interact with the geometry(in the same sense that macroscopic angular momentum interacts with geometry)and torsion can naturally arise.Theories with torsion have a long history,probably starting with the Einstein–Cartan(–Sciama–Kibble)theory(Cartan, 1922,1923,1924;Hehl et al.,1976;Kibble,1961;Sciama, 1964).In this theory,as well as in other theories with an independent connection,some part of the connection is still related to the metric(e.g.,the non-metricity is set to zero).In our case,the connection is left completely unconstrained and is to be determined by thefield equa-tions.Metric-affine gravity with the linear version of the action(30)was initially proposed in(Hehl and Kerling, 1978)and the generalization to f(R)actions was consid-ered in(Sotiriou and Liberati,2007a,b).Unfortunately,leaving the connection completely un-constrained comes with a complication.Let us consider the projective transformationΓλµν→Γλµν+δλµξν,(35) whereξνis an arbitrary covariant vectorfield.One can easily show that the Ricci tensor will correspondingly transform likeRµν→Rµν−2∂[µξν].(36) However,given that the metric is symmetric,this implies that the curvature scalar does not changeR→R,(37) i.e.,R is invariant under projective transformations. Hence the Einstein–Hilbert action or any other action built from a function of R,such as the one used here, is projective invariant in metric-affine gravity.However, the matter action is not generically projective invariant and this would be the cause of an inconsistency in the field equations.One could try to avoid this problem by generalizing the gravitational action in order to break projective in-variance.This can be done in several ways,such as allowing for the metric to be non-symmetric as well, adding higher order curvature invariants or terms in-cluding the Cartan torsion tensor[see(Sotiriou,2007b; Sotiriou and Liberati,2007b)for a more detailed discus-sion].However,if one wants to stay within the framework of f(R)gravity,which is our subject here,then there is only one way to cure this problem:to somehow constrain the connection.In fact,it is evident from eq.(35)that, if the connection were symmetric,projective invariance would be broken.However,one does not have to take such a drastic measure.To understand this issue further,we should re-examine the meaning of projective invariance.This is very similar to gauge invariance in electromagnetism(EM).It tells us that the correspondingfield,in this case the connections Γλµν,can be determined from thefield equations up to a projective transformation[eq.(35)].Breaking this invari-ance can therefore come byfixing some degrees of free-dom of thefield,similarly to gaugefixing.The number of degrees of freedom which we need tofix is obviously the number of the components of the four-vector used for the transformation,i.e.,simply four.In practice,this means that we should start by assuming that the con-nection is not the most general which one can construct, but satisfies some constraints.Since the degrees of freedom that we need tofix are four and seem to be related to the non-symmetric part of the connection,the most obvious prescription is to demand that Sµ=Sσσµbe equal to zero,which wasfirst suggested in(Sandberg,1975)for a linear ac-tion and shown to work also for an f(R)action in。
高二英语数学建模方法单选题20题
高二英语数学建模方法单选题20题1.In the process of mathematical modeling, the factor that determines the outcome is called_____.A.independent variableB.dependent variableC.control variableD.extraneous variable答案：B。
本题考查数学建模中的基本术语。
独立变量（independent variable）是指在实验或研究中被研究者主动操纵的变量；因变量dependent variable）是指随着独立变量的变化而变化的变量，在数学建模中决定结果的因素通常是因变量；控制变量（control variable）是指在实验中保持不变的变量；无关变量（extraneous variable）是指与研究目的无关，但可能会影响研究结果的变量。
2.The statement “The value of y depends on the value of x” can be represented by a mathematical model where y is the_____.A.independent variableB.dependent variableC.control variableD.extraneous variable答案：B。
在“y 的值取决于x 的值”这句话中，y 是随着x 的变化而变化的变量，所以y 是因变量。
3.In a mathematical model, the variable that is held constant toobserve the effect on other variables is_____.A.independent variableB.dependent variableC.control variableD.extraneous variable答案：C。
连续介质力学基本理论物理量的不变性表示
A 2 CA ACA 2 I1 ( A) ACA I 3 ( A)C tr( AC) A 2 [I 1 ( A) tr( AC) tr( A 2 C)]A I 3 ( A) tr( C)I A 2 CA 2 I 2 ( A) ACA I 3 ( A)( AC CA )
兑关锁 Dui Guansuo
Beijing Jiaotong University
Institute of Engineering Mechanics
非线性连续介质力学中通常其物理量可表示为两种不同形 式, 一种是Hill在应变主轴标架思想基础上系统发展起来的主轴 表示。但主轴法是在主标架给出的，不仅需要求解特征值而且 需要计算相应的特征向量，主轴法将耗费很大的计算量。为了 避免计算特征值及其特征向量, 物理量还可以表示为与坐标无关 的不变性表示形式, 不变性表示理论及其在连续介质力学中的应 用吸引了众多学者的研究兴趣。 例如
Trusdell, Noll, Coleman, CC Wang, C-s Man
Hill, Ogden 国内教材
兑关锁 Dui Guansuo
Beijing Jiaotong University
Institute of Engineering Mechanics
郭仲衡
兑关锁 Dui Guansuo
In particularly, for n=3
A 3 I1A 2 I 2 A I 3I 0
The number of useful results can be extracted from the Cayley– Hamilton theorem.
兑关锁 Dui Guansuo
Beijing Jiaotong University
advised by
Diploma Thesis
Thomas Wies
September 2004
advised by Prof. Dr. Andreas Podelski
Universit¨ at des Saarlandes Fachbereich 6.2 - Informatik
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . .
. . . .
. . . .
. . . .
Байду номын сангаас
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
英汉对照计量经济学术语
英汉对照计量经济学术语计量经济学术语A校正R2（Adjusted R-Squared）：多元回归分析中拟合优度的量度，在估计误差的⽅差时对添加的解释变量⽤⼀个⾃由度来调整。
对⽴假设（Alternative Hypothesis）：检验虚拟假设时的相对假设。
AR（1）序列相关（AR(1) Serial Correlation）：时间序列回归模型中的误差遵循AR（1）模型。
渐近置信区间（Asymptotic Confidence Interval）：⼤样本容量下近似成⽴的置信区间。
渐近正态性（Asymptotic Normality）：适当正态化后样本分布收敛到标准正态分布的估计量。
渐近性质（Asymptotic Properties）：当样本容量⽆限增长时适⽤的估计量和检验统计量性质。
渐近标准误（Asymptotic Standard Error）：⼤样本下⽣效的标准误。
渐近t 统计量（Asymptotic t Statistic）：⼤样本下近似服从标准正态分布的t 统计量。
渐近⽅差（Asymptotic Variance）：为了获得渐近标准正态分布，我们必须⽤以除估计量的平⽅值。
渐近有效（Asymptotically Efficient）：对于服从渐近正态分布的⼀致性估计量，有最⼩渐近⽅差的估计量。
渐近不相关（Asymptotically Uncorrelated）：时间序列过程中，随着两个时点上的随机变量的时间间隔增加，它们之间的相关趋于零。
衰减偏误（Attenuation Bias）：总是朝向零的估计量偏误，因⽽有衰减偏误的估计量的期望值⼩于参数的绝对值。
⾃回归条件异⽅差性（Autoregressive Conditional Heteroskedasticity, ARCH）：动态异⽅差性模型，即给定过去信息，误差项的⽅差线性依赖于过去的误差的平⽅。
⼀阶⾃回归过程[AR（1）]（Autoregressive Process of Order One [AR(1)]）：⼀个时间序列模型，其当前值线性依赖于最近的值加上⼀个⽆法预测的扰动。
第三章平行结构(Parallelism)(精)
第三章平行结構(Parallelism)一、起源平行结構起源於拉丁文的Parallelu s，意謂兩個事物想法或概念極為類似或雷同，以相同的結構平行呈現，利用對稱連接詞（and, or, but, notonly…but also, either…or, neither…nor）來銜接成一個合句。
連接詞所銜接的句型必須一致，(例如：動詞片語and 動詞片語；分詞片語or 分詞片語)。
目的在顯示出兩件(或多件)事物間的相似性，類似中文修辭裡的"對偶" 或"排比". 也藉者句型的重複性來強調某些概念或事物的重要性或顯著性。
E. P.J. Corbett 曾經詳細的描述平行结構如下：Parallelism is one of the basic principles of grammar andrhetoric. The principle demands that equivalent things be setforth in co-ordinate grammatical structures. So nouns must beyoked with nouns, prepositional phrases with prepositionalphrases, adverb clauses with adverb clauses. When thisprinciple is ignored, not only is the grammar of co-ordinationviolated, but also the rhetoric of coherence is wrenched.Students must be made to realize that violations of parallelismare serious, not only because they impair communication butalso because they reflect disorderly thinking. Whenever yousee a coordinated conjunction in one of your sentences, youshould check to make sure that the elements joined by theconjunction are of the same grammatical kind. (428-9)平行結構的理論基礎在於反映大自然對稱的特質。
Chiral and axial anomalies in the framework of generalized Hamiltonian BFV-quantization
(hereafter we use the following notations: α, β, µ, ν, ρ, σ = 0, 1, 2, 3; i,j,k,l = 1, 2, 3; ηµν = diag (− + ++); γ µ γ ν + γ ν γ µ = 2η µν 1, γ 5 ≡ iγ 0 γ 1 γ 2 γ 3 ; γ 5 γ 5 = 1; γ µ γ 5 + γ 5 γ µ = 0; ε0123 = 1, ta –anti-Hermitian normalized basis in the Lee algebra of symmetry group, f abc – corresponding totally anti-symmetrical 1 ab δ . Aa ν and F b µν — structural constants: ta , tb = f abc tc , Sp ta , tb = − 2
massive theory. As a result the both symmetries are described by the BRST operator equations of the similar structure. Then we are proposing an explicitly covariant regularization scheme for the operator constrained Hamiltonian formalism and finding the Schwinger terms in the involution relations. These Schwinger terms induce the breakdown both of the BRST charge nilpotency and its conservation law, thereby the anomaly becomes of the BRST form. The mode of operation does not have a specific character for the both chiral and axial anomalies in its key points and the proposed procedure may serve as a possible tool in a study of other anomalies in d > 2. Let us mention that the Schwinger terms turn out to be consistent with the covariant anomaly divergence of the corresponding current but they break down Wess-Zumino consistency condition for the operator BRST-algebra. It is worth noting that the contradiction between covariance and Wess-Zumino condition is the usual phenomenon for an anomalous theory itself (see [12]) and for the BRST-anomaly in particular. So, in an anomalous theory one usually has a difficult choice between the covariance and the Jacoby identity. We prefer the covariance for the following reason. We believe that the physically sensible theory should provide gauge anomalies to be mutually cancelled and thus it must have the nilpotent and conserved BRST charge (as far as an anomalous theory does not meet today any satisfactory physical interpretation). This mutual cancellation should be actual for an arbitrary coordinate system; thus we seemingly have to describe covariantly each of the anomalies, while the consistency condition could be provided for the total BRST charge as far as it becomes nilpotent in each coordinate system. Thus we may treat the considered theory as a part of a more wide one, where the proposed description of the anomaly could be applied for the anomaly cancelation covariant control. Let us begin with a suitable (for our goal) outline of the symmetries in theory of the massless fermions chiraly coupled to nonabelian massless vector field. The theory is described by the action S= ¯ µ ∇µ ψ − 1 F aµν F a µν . d4 x iψγ 4 (1.1)
On the central critical value of the triple product L-function
2
ቤተ መጻሕፍቲ ባይዱ
¨ S. BOCHERER AND R. SCHULZE-PILLOT
difficult (for the first step they have already been carried out in [2]). The actual computation of the integral is elementary and needs only standard results from the theory of newforms. It yields a Dirichlet series (2.41) whose Euler product decomposition is then computed in Section 3. The cases that p divides one, two or all three of the levels Ni or is coprime to N must all be treated separately, which makes the discussion somewhat lengthy. However, the actual computation in each of these cases is again fairly straightforward. In Section 4 we show that the Euler factors defined in Section 3 are the “right ones” by proving the functional equation. In order to exhibit the central critical value as a square (up to elementary factors) we follow a similar strategy as [12]: the Eisenstein series E at s = 0 is expressed as a linear combination of genus theta series of quaternary positive definite integral quadratic forms. At most one of these genera (depending on the levels Ni and the eigenvalues of the fi under the Atkin-Lehner involutions) contributes to the integral. Eichler’s correspondence between cusp forms for Γ0 (N ) and automorphic forms on definite quaternion algebras allows then to express this contribution as an (explicitly computable) square of an element of the coefficient field of the fi ; this element arises as value of a trilinear form on a space of automorphic forms on the quaternion algebra and may be interpreted as value of a height pairing similar to [11]. It may be of interest to compare the advantages of the different methods applied to this problem. Although the adelic method makes it easier to obtain general results, the explicit computations needed here appear to become somewhat simpler in the classical context. In particular, by making use of the theory of newforms and of orthogonality relations for the theta series involved from [2] we can use the same Eisenstein series E independent of the fi . This is of advantage since the pullback formalism is especially simple for this type of Eisenstein series and leads to the remarkably simple computations in Sectiions 2 and 3. Most of this article was written while both authors were guests of the MSRI in Berkeley during its special year on automorphic forms. We wish to thank the MSRI for its hospitality and financial support. R. Schulze-Pillot was also supported by Deutsche Forschungsgemeinschaft during a visit of one month at MSRI and was a guest of the Max-Planck-Institut f¨ ur Mathematik in Bonn in the final stage of the preparation of this manuscript.
计量经济学(英文)重点知识点考试必备
第一章1.Econometrics（计量经济学）:the social science in which the tools of economic theory，mathematics, and statistical inference are applied to the analysis of economic phenomena.the result of a certain outlook on the role of economics, consists of the application of mathematical statistics to economic data to lend empirical support to the models constructed by mathematical economics and to obtain numerical results.2.Econometric analysis proceeds along the following lines计量经济学分析步骤1)Creating a statement of theory or hypothesis。
建立一个理论假说2)Collecting data.收集数据3)Specifying the mathematical model of theory.设定数学模型4)Specifying the statistical，or econometric, model of theory。
设立统计或经济计量模型5)Estimating the parameters of the chosen econometric model.估计经济计量模型参数6)Checking for model adequacy ：Model specification testing。
核查模型的适用性：模型设定检验7)Testing the hypothesis derived from the model。
拓扑理论
The topological approach to perceptual organizationLin ChenKey Laboratory of Cognitive Science,Graduate School and Institute of Biophysics,Chinese Academy of Sciences,Beijing,ChinaTo address the fundamental question of``what are the primitives of visual per-ception'',a theory of topological structure and functional hierarchy in visual perception has been proposed.This holds that the global nature of perceptual organization can be described in terms of topological invariants,global topological perception is prior to the perception of other featural properties,and the primitives of visual form perception are geometric invariants at different levels of structural stability.I n Part Iof this paper,Iwill illustrate why and how the topological approach to perceptual organization has been advanced.In Part II,I will provide empirical evidence supporting the early topological perception,while answering some commonly considered counteraccounts.In Part III,to complete the theory,I will apply the mathematics of tolerance spaces to describe global properties in discrete sets.In Part IV,I will further present experimental data to demonstrate the global-to-local functional hierarchy in form perception,which is stratified with respect to structural stability defined by Klein's Erlangen Program.Finally,in Part V,Iwill discuss relations of the global-to-local topological model to other theories:The topological approach reformulates both classical Gestalt holism and Gibson's direct perception of invariance,while providing a challenge to com-putational approaches to vision based on the local-to-global assumption.INTRODUCTIONA great divide:Local-to-global vs.global-to-localAs a Chinese proverb says:Everything is difficult at its very beginning. Historically,major schools of vision diverge in their answers to the question of ``Where visual processing begins?''(Pomerantz,1981)or``What are the pri-mitives of visual perception?''(Chen,1982).The question is so fundamental and also so controversial as to serve as a watershed,a Great Divide,separatingPlease address all correspondence to:Lin Chen,Key Laboratory of Cognitive Science,Graduate School and Institute of Biophysics,Chinese Academy of Sciences,15Datun Road,Beijing100101, China.Email:chenl@Supported by National Nature Science Foundation of China Grant(697900800);Ministry of Science and Technology of China Grant(1998030503);Chinese Academy of Sciences Grants (KGCX2-SW-101,KJCX1-07).This work was partly done during author's sabbaticals,at Institute of Medical Psychology,University of Munich,and at National Institute of Mental Health./journals/pp/13506285.html DOI:10.1080/13506280444000256554CHENtwo most basic and sharply contrasting lines of thinking in the study of perception.Early feature analysis:Local-to-global.On one side of the Great Divide,the early feature-analytic viewpoint holds that perceptual processing is from local to global:Objects are initially decomposed into separable properties and components,and only in subsequent processes are objects recognized,on the basis of the extracted features.The computational approach to vision by Marr (1982),representative of``early feature-analysis''viewpoint,claims that the primitives of visual-information representation are simple components of forms and their local geometric properties,such as,typically,line segments with slopes. Early holistic registration:Global-to-local.On the other side of the Great Divide,the viewpoint of early holistic registration claims that perceptual processing is from global to local:Wholes are coded prior to perceptual analysis of their separable properties or parts,as indicated by the conception of perceptual organization in Gestalt psychology.As we will see in the following discussion,with respect to the fundamental question of``Where to begin'',the core contribution of the Gestalt idea goes far beyond the notion that``Whole is more than the simple sum of it parts'';rather it is that``Holistic registration is prior to local analysis''.The idea of early feature analysis has gained wide acceptance,and dominates much of the current study of visual cognition.Intuitively,it seems to be natural and reasonable that visual processing begins with analysing simple components and their local geometric properties,typically as line segments with slopes,as they are readily to be considered physically simple and computationally easy. An underlying idea of Marr's computational system of vision was,in Marr's own words,``In the early stages of the analysis of an image,the representations used depend more on what it is possible to compute from an image than on what is ultimately desirable.''(Marr,1978).Nevertheless,a starting point of the present paper is that physically or computationally simple doesn't necessarily mean psychologically simple or perceptually primitive;therefore,the question of which variables are perceptual primitives is not a question answered primarily by logical reasoning or analysis of computational complexity but rather by empirical findings.Topological structure and functional hierarchyinform perceptionTo address the fundamental question of what are the primitives of visual percep-tion,based on a fairly large set of data on perceptual organization reviewed here,a theory of``early topological perception''has been proposed.This holds that:GLOBAL TO LOCAL TOPOLOGICAL PERCEPTION555A primitive and general function of the visual system is the perception of topo-logical properties.The time dependence of perceiving form properties is system-atically related to their structural stability under change,in a manner similar to Klein's hierarchy of geometries;in particular,topological perception(based on physical connectivity)is prior to the perception of other geometrical properties.The invariants at different geometrical levels are the primitives of visual form perception.These include,in a descending order of stability(from global to local), topological,projective,affine,and Euclidean geometrical invariants.The topological approach is based on one core idea and includes two main aspects.The core idea is that perceptual organization should be understood in the perspective of transformation and perception of invar-iance over transformation.The first aspect emphasizes the topological struc-ture in form perception,namely,that the global nature of perceptual organization can be described in terms of topological invariants.The sec-ond aspect further highlights early topological perception,namely,that topo-logical perception is prior to perception of local featural properties.The ``prior''has two strict meanings:First,it implies that global organizations, determined by topology,are the basis that perception of local geometrical properties depends on;and second,topological perception(based on physi-cal connectivity)occurs earlier than the perception of local geometrical properties.The hypothesis of early topological perception led to a major finding that the relative perceptual salience of different geometric properties is remarkably consistent with the hierarchy of geometries according to Klein's Erlangen Program(see Part II and IV),which stratifies geometries in terms of their relative stability over transformations.Based on the finding,a functional hier-archy in form perception has been established as a formal and systematic definition of``global-to-local''relations:A property is considered more global (or stable)the more general the transformation group is,under which this property remains invariant;relative to geometrical transformations,the topolo-gical transformation is the most general and hence topological properties are the most global.The framework of the topological structure and functional hierarchy high-lights a fundamental empirical prediction,namely a time dependence of per-ceiving form properties,in which visual processing is from global to local:The more global a form invariant is the earlier its perception occurs,with topological perception being the most global and occurring earliest.The framework further highlights a series of novel empirical predictions about long-standing issues related to the study of perceptual organization,and many Gestalt-type phe-nomena in form perception may be explained in this unified manner.They include the following examples:556CHEN.With respect to the relationship between different organizational factors, proximity is the most fundamental organizational factor(even in comparison with uniform connectivity;Palmer&Rock,1994)(see Part III),and there is a time course of processing different organizational principles:Proximity precedes similarity,and topological similarity precedes similarity of local geometric properties(see Part VI)..Early topological perception predicts the visual sensitivity to distinction made in topology.For example,two stimuli that are topologically different are more discriminable under a near-threshold condition than are other pairs of forms that are topologically equivalent despite their difference in local features(see Part II)..With respect to the question of whether motion perception precedes form perception or vice versa,topological discrimination should occur earlier than and determine motion perception(see Part II)..Configural superiority effects(Pomerantz,1981)demonstrated by configural relations between line segments,such as the``triangle±arrow pair'',may simply demonstrate the superiority effect for perception of holes over indi-vidual line segments(see Part IV)..With respect to``global precedence''(Navon,1977),according to the functional hierarchy,the performance advantage for compound letters is quite straightforward:Global precedence reflects the primacy of proximity in perceptual organization(see Part III)..If topological properties are primitive,illusory conjunctions(Treisman& Gelade,1980)of topological properties,such as holes,should sometimes take place(see Part II)..With respect to the definition of perceptual object,the topological approach ties a formal definition of``object''to invariance over topological trans-formation(see Part I).From this definition,it follows that as an object is moving along and a hole appears in it,this topological change would dis-turb object continuity,while changes of shape and colour wouldn't(Wolfe, personal communication).For example,in an MOT(multiple object track-ing)test(Pylyshyn&Storm,1988;vanMarle&Scholl,2003),attentive tracking processes would be impaired by objects changing topology by getting a hole,while it does not matter if they change local features and colours..With respect to its ecological function and functional anatomy,long-range apparent motion works by abstracting form invariants,and hence is asso-ciating with form perception and activates the ventral pathway in the two visual systems(Ungerleider&Mishkin,1982).Specifically,the fMRIacti-vation should be correlated with the form stability under change(see Parts II and IV)..From the perspective of biological evolution,if topological perception is indeed a fundamental property of vision,one might expect topologicalGLOBAL TO LOCAL TOPOLOGICAL PERCEPTION557 properties to be extracted by all visual systems,including the relativelysimple ones possessed by insects,such as bees(see Part II).In summary,the framework of topological structure and functional hierarchy in form perception provides a new analysis of the fundamental question,i.e., ``What are the primitives of visual perception?'',in which primitives of visual form perception are considered to be geometric invariants(as opposed to simple components of objects,such as line segments)at different levels of structural stability.In the following,I will illustrate why and how the topological approach to perceptual organization has been advanced(Part I);provide empirical evidence supporting the topological perception,while answering some commonly con-sidered counteraccounts(Part II);complete the theory of topological perception, using the mathematics of tolerance spaces that describe global properties in discrete sets(Part III);present experimental data to demonstrate the functional hierarchy in form perception,which is stratified with respect to structural sta-bility defined by Klein's Erlangen Program(Part IV);and finally,discuss relations of early topological perception to other theories,including Gestalt psychology,Gibsonian psychology,and the computational approach(Part V).PART I:WHY AND HOWÐA TOPOLOGICALAPPROACH TO PERCEPTUAL ORGANIZATIONA paradoxical problem of``where to put the master map'' Fundamental problems faced by the early feature-analysis approach are typically embodied in a paradoxical problem of``where to put the master map''as posed by the feature-integration theory of Treisman and co-workers(e.g.,Treisman& Gelade,1980).Feature-integration theory,consistent with the early feature-analysis approach,adopts a``two-stage model'':In the first,preattentive stage, primitive features,such as colours and orientations,are abstracted effortlessly and in parallel over the entire visual field,and registered in special modules of feature maps;and in the second,attentive stage,focal attention is required to recombine the separate features to form objects.A master map of locations plays a central role in feature binding by tying the separate feature maps together.Within the master map,a focal attention mechanism selects a filled location,binding the activated features linked to that location together to form a coherent object. Problems for feature-integration theory are,however,represented by the question of where exactly the master map of locations fits into the feature integration mechanism.In Treisman's own words,``I have hedged my bets on where to put the master map of locations by publishing two versions of the figure!In one of them,the location map received the output of the feature558CHENmodules(e.g.,Treisman,1986)and in other is placed at an earlier stage of analysis(e.g.,Treisman,1985;Treisman&Gormican,1988)''(Treisman,1988, pp.203±204).To place the master map of locations at an early stage of analysis,in Treisman's own words,``implies that different dimensions are initially conjoined in a single representation before being separately analysed''(Treisman,1988,pp. 203±204).This contradicts the basic position of early feature analysis.Placing the master map later,however,contradicts some behavioural data.One of the strengths of feature-integration theory is that it draws on a number of major pieces of counterintuitive evidence,including illusory conjunction and visual search, which appear to provide strong support for early feature analysis(e.g.,Treisman &Gelade,1980;Treisman&Gormican,1988).However,it is interesting to see that problems for this theory also arise here(e.g.,for a general review,see Humphreys&Bruce,1989).For example,despite the fact that line segments are commonly considered to be basic features,there is markedly little evidence for illusory conjunction where line segments are miscombined into letter-like objects,when letter shapes and line segments forming the letter shapes are used as stimulus forms(e.g.,Duncan,1984).In contrast,there is much stronger evidence that whole letter shapes migrate across words and produce illusory conjunctions of the entire letter shapes,rather than of line segments making up the letter shapes. These findings indicate that letter shapes,as combinations of line segments, behave psychologically as holistic objects,even though line segments are commonly considered to be basic features.Apparently attention,as it relates to feature binding,is not needed for holistic object perception.This suggests that before a stage of separate featural analysis,there must be a stage of early holistic perception in which objects like letters are coded as wholes.Treisman and co-workers,in their effort to explain these counterexamples, have augmented feature-integration theory with new strategies and new mechanisms of attention,such as``guided search''(e.g.,Wolfe,1994),``map suppression'',and dividing items into different depth planes.Nevertheless,these efforts are not completely successful(e.g.,Duncan&Humphreys,1989)but rather in fact illustrate that,despite the attractions of feature-integration theory, the paradoxical problem of``Where to put the master map''stems directly from the fundamental question of``Where visual processing begins''.Perceptual organization:To reverse back theinverted(upside-down)problem of feature binding Regardless of how an object is decomposed into properties and components,the decomposed features themselves are unlikely to be sufficient for achieving object recognition.Indeed,we do not normally perceive isolated features such as brightness,colours,and orientations free from an object,leading to the con-tention that there must be a further process of feature binding.This problem ofGLOBAL TO LOCAL TOPOLOGICAL PERCEPTION559 feature binding presents a central problem for current vision research in parti-cular,and for parallel and distributed modelling of cognition in general(e.g., MuÈller,Elliott,Herman,&Mecklinger,2001).However,despite the centrality of the issue for perceptual theory,it is questionable whether any breakthrough has been made after extensive efforts. Both spatial and temporal factors have been considered as cues for binding features together.But the principles for feature binding based on either space or time are neither always obeyed nor exclusive.Feature binding and perceptual organization appear to be very similar pro-blems(Duncan,1989)in the sense that both deal with similar processes,such as ``what goes with what'',and with similar concepts,such as belongingness and assignment.It turns out that,even though the early feature-analysis viewpoint emphasizes the fundamental importance of early parallel processing,the issue of perceptual organization remains indispensable.Yet,the concepts of``perceptual organization''and``feature binding''involve very different underlying issues, with the former rooted in the idea of early holistic registration and the latter originating from an assumption of early feature analysis.Thus,with respect to the fundamental question of``Where to begin'',perceptual organization and feature binding can be considered contrary concepts,going in opposite directions.In terms of our understanding of objects in the real word,there may be little disagreement that the real features of an object,whether geometrical or physical, exist together as a coherent whole belonging to one entity in the outside world. The question of how the perceptual system represents perceptual objects as fundamental units of conscious perceptual experience,however,has either given rise to much controversy when considered,or not been considered at all.But the truth remains that real features of a real object,at a given time,originally coexist together rather than being separated;a real object is an integral stimulus,a single thing.This truth is a fundamental property of a real-world object.No one doubts the direct perception of various featural properties such as brightness,colour, line orientation,and so on.Why,then,is only this fundamental property of ``belonging together as a whole''excluded from the membership of primitives in our perceptual world?The assumption that the visual system cannot directly perceive a real integral object has not yet been proved or disproved.Indeed,the continuing challenges to the issues of feature binding suggest that this question deserves closer scrutiny.From the perspective of early holistic registration,the feature-binding pro-blem is an ill-posed question:Not just a question of getting off on a wrong foot but even a question of``standing upside down''.In this sense,the feature-binding problem might be a wrong,inverted question.Kubovy and Pomerantz (1981)commented:``the main problems facing us today are quite similar to those faced by the Gestalt psychologists in the first half of this century''.After half a century,the study of visual perception appears,in some sense,to be back560CHENto square one.This situation leads us to wonder whether the problems of feature binding are due to difficulties in posing the fundamental question of``Where to begin''.Where does the above analysis leave us?It suggests that early holistic registration may provide a way to avoid the feature-binding problem by focusing on issues of perceptual organization.In other words,we may apply the concept of perceptual organization to reverse back the inverted(upside-down)question of feature binding.The topological approachDespite its deep and rational core in the idea of early holistic registration,the notion of perceptual organization has its own problems.In particular,like other Gestalt concepts,it has suffered from a lack of proper theoretical treatment. Gestalt evidence has often been criticized for being mainly phenomenological and relying mainly on conscious experience.Consequently,explanations from theories of perceptual organization usually rely on intuitive or mentalistic concepts that are somewhat vague and elusive.What is needed is a proper formal analysis of perceptual organization that goes beyond intuitive approa-ches,and provides a theoretical basis for describing or defining precisely the core concepts related to perceptual organization,e.g.,``global''vs.``local'', ``objects'',``grouping'',and others.Until the intuitive notions of these Gestalt-inspired concepts become properly and precisely defined,the proposed principles of perceptual organization cannot be entirely testable.Delimiting the concept of perceptual organizationTo give a precise description of the essence of perceptual organization,we first need to properly delimit the concept of perceptual organization.On the one hand,as Rock(1986)pointed out,``The concept of perceptual organization should not be defined so loosely as to be a synonym of perception'';on the other hand,this concept should not be so limited as to be unable to cover the great variety and the commonplace occurrence of perceptual organization.The fol-lowing definition of perceptual organization given by Rock(1986)is considered to define properly the very notion of perceptual organization:The meaning of organization here is the grouping of parts or regions of the field with one another,the``what goes with what''problem,and the differentiation of figure from ground.According to this definition,the study of perceptual organization is concerned with early stages of perceptual processing divorced from high-level cognition, and therefore such delimitation pitches our discussion at the right level to answer the basic question of``Where visual processing begins''.On the other hand,theGLOBAL TO LOCAL TOPOLOGICAL PERCEPTION561 concept of perceptual organization discussed in the present paper deals withgeneral processes,such as figure±ground differentiation,grouping,``what goes with what'',belonging and assignment(not particular processes,such as dif-ferentiating luminance flux,discriminating orientation,or recognizing a face), and with abstract things,such as objects,units,and wholes as well as their counterparts,such as items,elements,and parts(not concrete things,such as a line segments,a geometrical figure,a friend's face).These general processes and abstract things represent the variety and commonplace occurrence of per-ceptual organization.Figure1illustrates this.The percept of Figure1A may be described at a semantic level:Either a vase or two profiles face to face.On the other hand,it may be described in terms of the vocabulary of perceptual orga-nization:Two boundaries(units)grouped into one object(as the basis of the percept of a vase)or two boundaries(units)separated into two objects(as the basis of the percept of the two profiles face to face).It is the latter level,the level of perceptual organization,which our present research focuses on.Furthermore, as Figure1B demonstrates,perceptual organization may be perceived without semantic meaning.Here even though there is little semantic meaning involved in the stimulus,either the black parts are perceived to be an unified whole as a figure and the grey parts,another unified whole as background,or vice versa. Top-down processing of prior knowledge or expectation may influence per-ceptual organization,but it will avoid possible confusion if we consider per-ceptual organization and top-down processing of high-level cognition separately.This strengthens the rationale for defining the terminology for describing perceptual organization,emphasizing the primacy of perceptual organization.Major challenges to establishing a proper theoretical treatment on perceptual organization:Its commonplace,and its general and abstract characteristicsThe concept of perceptual organization reflects the most common fact that the phenomenal world contains objects separated from one another by space or background.Phenomena in perceptual organization are usually so common that they have not been looked on as an achievement of the perceptual system,and, thus,as something to be explained(Rock,1986).For example,tremendous efforts have been made to study how to detect line segments with orientation and location,but little attention has been paid to the question of how to perceive a line segment as a single object.While the study of face recognition has advanced considerably,the fundamental grouping question of``which eyes go with which noses,which noses with which mouths,and so forth''(Pomerantz,1981)has been almost completely ignored.One more example shows how commonplace characteristics of perceptual organization make it difficult to realize that there are problems requiringexplanation.In 1990,Rock and Palmer revealed two primary laws of perceptual organization:``Connectedness''and ``common region'',referring to the pow-erful tendency of the visual system to perceive any connected or enclosed regionas a single unit.The phenomena relating to the two laws are so common andself-evident that even classical Gestalt psychologists failed to realizethat an explanation was required for why elements that are either physically connected or enclosed by a closed curve are perceived as a single unit.As ourdiscussionABFigure 1.(A)An ambiguous figures of ``a vase vs.two faces'',showing competing organization.(B)An example of ambiguous figures,showing competing organization without involving semantic meaning.562CHENGLOBAL TO LOCAL TOPOLOGICAL PERCEPTION563 goes on,we will see that these two Gestalt laws closely approach the preciseformal(topological)description of the essence of Gestalt principles.Never-theless,they were neglected for more than a half a century!Besides the problem of being easily overlooked,one more major challenge to establishing a scientific framework for perceptual organization stems from the abstract and general characteristics of the concept.A theoretical explanation of perceptual organization,to possess explanatory power,must be built on even more general and abstract concepts than this vocabulary.The next question, therefore,is:What kinds of concept are more general and abstract than,for example,``what goes with what'',grouping,belongingness,wholes,and per-ceptual objects,and therefore,suitable for the formal analysis of organizational processes?It is not difficult to see that featural properties commonly used in the feature-analysis approach,such as orientation,distance,and size,cannot help out in dealing with the problems facing us in finding a formal explanation of perceptual organization.Topology provides a formal description of perceptual organization:Insight from invariants over shape-changing transformationsTopology has been considered a promising mathematical tool for providing a formal analysis of concepts and processing of perceptual organization(e.g., Chen,1982,2001).Topology is a branch of mathematics that aims at studying invariant properties and relationships under continuous and one-to-one trans-formations,termed topological transformations.The properties preserved under an arbitrary topological transformation are called topological properties.A topological framework of visual perception can be broad enough to encom-pass the variety of phenomena in perceptual organization,such as``what goes with what'',grouping,belonging,and parsing visual scenes into potential objects,and,on the other hand,precise enough to be free from intuitive approaches.Topology is often considered as one of the most abstract branches of mathematics.If the concepts of topology,their relevance and applicability to perceptual organization are difficult to contemplate in the abstract,an appeal to illustrative examples might be helpful.In the following,I will analyse in some depth three typical cases of perceptual organization to demonstrate why and how to advance the topological approach to perceptual organization.Question1:Figure and ground perceptionÐwhat attributes of stimuli determine the segregation of figure from background?Despite the common acceptance that figure±ground perception is fundamental and occurs at the early stage of perception,and despite the large body of empirical findings about。
- 1、下载文档前请自行甄别文档内容的完整性，平台不提供额外的编辑、内容补充、找答案等附加服务。
- 2、"仅部分预览"的文档,不可在线预览部分如存在完整性等问题,可反馈申请退款(可完整预览的文档不适用该条件!)。
- 3、如文档侵犯您的权益，请联系客服反馈,我们会尽快为您处理(人工客服工作时间：9:00-18:30)。
The Variables and Invariants in the Evolution of Logic Optical Lithography Process
Qiang Wu1,*
1 Shanghai IC R&D Center, Shanghai, China, 201210
Abstract: Photolithography has been a major enabler for the continuous shrink of the semiconductor manufacturing design rules. Throughout the years of the development of the photolithography, many new technologies have been invented and successfully implemented, such as image projection lithography, chemically amplified photoresist, phase shifting mask, optical proximity modeling and correction, etc. From 0.25 μm technology to the current 7 nm technology, the linewidth has been shrunk from 250 nm to about 20 nm, or 12.5 times. Although imaging resolution is proportional to the illumination wavelength, with the new technologies, the wavelength has only been shrunk from 248 nm to 134.7 nm (193 nm immersion in water), less than 2 times. Would it mean that the imaging performance has been continuously declining? Or we have yet fully utilized the potential of the photolithography technology? In this paper, we will present a study on the key parameters and process window performance of the image projection photolithography from 0.25 μm node to the current 7 nm node.
1. Introduction
Image projection photolithography has replaced contact/proximity printing for better resolution and defect consideration. The earliest projection exposure tools are developed by David A. Markler and Abe Offner from Perkin-Elmer company [1–2], where an all reflective 1:1 imaging system has been adopted. It has been found that the use of a ring field can cancel aberration and make good imaging. This system can offer a numerical aperture (NA) of 0.33, a ring field of 3 inch diameter, 1 mm width. The imaging resolution is 2 μm with 5.5 μm depth of focus and +/-1 μm overlay. Its illumination uniformity is +/-10% and distortion is +/- 1 μm. The system takes minimally 6 seconds for 1 wafer exposure. The advantage of all reflective design is that the alignment and exposure wavelengths can share the same optics and there is no chromatic aberration. But the limitation is numerical aperture and it cannot correct all aberration with just a few reflective surfaces. Later, partially refractive or all refractive designs that adopted double gauss, double telecentric designs have been used for more advanced design rules [3-4].