Quantum gravity and singularities

a r X i v :p h y s i c s /0405111v 1 [p h y s i c s .g e n -p h ] 20 M a y 2004Quantum gravity and “singularities”Anastasios Mallios Dedicated to the memory of the late Professor Klaus Floret Abstract The paper concerns the ?ctitious entanglement of the so-called “singulari-ties”in problems,pertaining to quantum gravity,due,in point of fact,to the way we try to employ,in that context,di?erential geometry,the latter being associated,in e?ect,by far,classically (:smooth manifolds),on the basis of an erroneous correspondence between what we may call/understand,as “physical space”and the “cartesian-newtonian”one.1.The two issues in the title of this article are only,seemingly (!),di?er-ent ,while,as we shall see,they are,in e?ect,in a very concrete sense,quite tautosemous.So,when we try to quantize gravity ,we are inevitably confronted (cf.e.g.(3.14)below)with the so-called,thus far,“singularities”,that is,with the emerging “in?nities”etc (referred,of course,always to our (!)calculations),some-thing,that certainly remind us of the characteristic,in that context,relevant remarks of P.A.M.Dirac,already from 1975,see thus [7:p.36],in that;(1.1)“...sensible mathematics involves neglecting a quantity when it turns

out to be small not neglecting it just because it is in?nitely great and

you do not want it”.

or even P.A.M.Dirac [8:p.85],that:

(1.1′)“Some day a new quantum mechanics,a relativistic one ,will be discov-

ered,in which we will not have ...in?nities at all [(!)].”

1

2Anastasios Mallios

[Emphasis above is ours;this will also be,in principle,the case when referring to quotations,throughout the present work,unless otherwise is stated].Accordingly, by coming back to our subject,

(1.2)the?rst item,as in the title of this paper,is,in point of fact,reducible to the second one,or,equivalently,the famous problem of the quantization of gravity is virtually subject to that one of the so-called“singularities”.

In this connection,we can still refer to the criticism of R.P.Feynman thereof[10:p. 166],pertaining thus to the use of the notion of“continuum”in the quantum deep, in that:

(1.3)“...the theory that space is continuous is wrong,because we get.... in?nities and other di?culties...[while]the simple ideas of geometry, extended down into in?nitely small...are wrong!”

Now,the problem here lies essentially with our blocked intension/endevour to as-sociate our(technical)theory(:“geometry”)with physis(:natural laws).In this regard,see also our previous relevant comments in A.Mallios[19],or even in[22]. On the other hand,within the above vein of ideas,we can also quote here C.J.Isham [15:p.393],when remarking that;

(1.4)“...at the Plank-length scale,classical di?erential geometry is simply incompatible with quantum theory....[hence]one will not be able to use di?erential geometry in the true quantum gravity theory...”

Here again,as it was similarly the case with(1.3),the problem is not with the (classical)di?erential geometry itself,even at the Plank-length scale(!),when ma-thematically speaking(there are no,in e?ect,such inherent restrictions on the Rie-mannian/Lorentzian,or on any other,whatsoever,type of“metric-geometry”,by its very de?nition,pertaining to the“metric”,read,to the“space”,we use),but only with the extent to which we wish,in that context,to apply the framework(!)of that classical theory,as a model,for a mathematical-physical theory,to describe thus a physical situation;viz.the quantum domain,alias the physical laws governing that particular(physical)r′e gime.So it is worthwhile to point out here,once more,that,

Quantum gravity and“singularities”3

(1.5)the manner we try to apply,so far,the classical di?erential geometry (CDG)always refers to its standard framework,viz.to the theory of di?erential(i.e.,smooth,or even C∞-)manifolds,and not(!),to its in-herent(“leibnizian”,so to say)mechanism,as the latter aspect has been just exhibited,by the“abstract di?erential geometry”(ADG);the same still a?ords,as we shall see,a quite di?erent perspective from that one of the classical case,concerning thus potential applicabilities of ADG, provided we have also suitably chosen,so to say,our“di?erentiable func-tions”(:“generalized arithmetics”,in the latter context;cf.,for instance, (3.10)in the sequel).

The above diversi?cations from standard aspects,so far,of the same matter,will become progressively clearer,through the subsequent discussion.

Now,continuing further,within the previous point of view,as it concerns the applicability of the notion of the“continuum”(:space-time)in problems of quantum gravity we can still quote here A.Einstein himself,who,since1916,already,has declared,indeed,in a pretty caustic manner,that;

(1.6)“...continuum space-time...should be banned from theory as a supple-mentary construction not justi?ed by the essence of the problem?a construction which corresponds to nothing real[(!)]”.

See,for instance,J.Stachel[33:p.280].So we have actually been warned,already, either directly(Einstein),or indirectly(Feynman,Isham)for the inappropriateness of combining classical di?erential geometry with quantum theory(!).

On the other hand,R.Geroch(1968),trying to explain the situation one has with the“singularities”in general relativity,he further notes that;

(1.7)“...general relativity di?ers from[other?eld theories]in one important respect:...one has[in those theories]a background(Minkowskian)me-tric to which the?eld quantities can be referred,[while]in general rela-tivity the“background metric”is the very?eld whose singularities one wishes to describe”.

See R.Geroch[12].Furthermore,we have a recent similar criticism to(1.7),as above,by J.Baez[1:p.v;Preface],as it actually concerns quantization of gravity, by remarking that:

4Anastasios Mallios

(1.8)“A fundamental problem with quantum gravity...is that in...general relativity there is no background geometry to work with:the geometry of spacetime itself becomes a dynamical variable.”

Consequently,as an upshot of the preceding discussion,we do e?ectuate that;

(1.9)the fact that in general relativity one is compelled,by the very essence of the theory,to consider the“geometry”itself,as a“dynamical variable”, is a fundamental issue(problem)in quantizing gravity,the same being also intimately connected with the so-called“singularities”of the theory.

2.ADG,as a potential response.?It is now our goal,by the subsequent account,to show that the aforesaid,throughout the preceding discussion,obstacles, which appear when trying to cope with problems of quantum gravity,within the standard set-up(:di?erential–smooth–manifolds)of the classical di?erential geom-etry,do not appear,at all,when looking at the matter,within the context of ADG (:abstract di?erential geometry),according to the very de?nition of the latter:In-deed,it is thus a basic moral of the same point of view(ADG),that;

(2.1)to perform“di?erential geometry”,no“space”is virtually required(in the usual sense of the standard theory(CDG),viz.a smooth mani-fold),provided that one is equipped with a“basic di?erential”,?,alias “dx”,along with the appropriate“di?erential-geometric mechanism”, that might be a?orded thereby.

Thus,it is still a basic upshot of the very context of ADG(see also(2.1),along with (3.13)in the sequel)that the problem(see e.g.(1.8),as well as,(1.9),as above)of

(2.2)making the“geometry”into a“dynamical variable”is simply begging the question!

In this regard see also A.Mallios-I.Raptis[23],[24],[25].On the other hand,con-tinuing further,within the preceding vein of ideas,by turning back again to the situation,that is connected with the so-called“Plank-length scale”(cf.(1.4)above, along with the ensuing comments therein),we can still remark,yet,here too,by virtue of the same character of ADG(see also,for instance,A.Mallios[19:(9.34), and comments following it]),that,

Quantum gravity and“singularities”5

(2.3)the(physical)“geometry”,one has in the“Plank-length scale”does not actually di?er,in principle,as well as,in substance(nature),from that one,we have,anywhere else(“physis is united”(!),we suppose); yet,nature,viz.the physical“geometry”,still,in other words,what we perceive,as such,is not,at all,our own.Indeed,the latter term (:“geometry”)concerns,in point of fact,simply our own technical(!) (:mathematical)device(in e?ect,“cartesian”(!),thus far)to describe (:to model)the former.

In this regard,cf.also here A.Mallios[22].Indeed,as it was already hinted at in (2.3),we should still remark herewith,that:

(2.4)what we usually understand as(mean by a),“physical geometry”?we are thus trapped still,by our own mathematical conception of it,in that context?is,in point of fact,the“cartesian”one,either globally(e.g. a?ne space),or even locally(thus,manifold,e.g.the so-called“space-time”).

In this connection,see also A.Mallios[19:(8.5)].Consequently,

(2.5)what we actually perceive(:de?ne),as“space”,is that one,which,in e?ect,may be called“cartesian”(or even,“newtonian”)one,hence,not, in anyway,the real“physical”one,which we may still name“euclidean”(see also loc.cit.,as above).So the latter is,in point of fact,simply,

(2.5.1)that,what constitutes it(!),

loc.cit.;(1.1),(1.4);viz.,in other words,that,what we can still call,

(2.5.2)“les objets g′e om′e triques”,

in the sense of Leibniz(the same Ref.;(2.1)).

On the other hand,as already explained elsewhere(cf.,for instance,the same Ref., as above),ADG is exactly referred to these“objets g′e om′e triques”,`a la Leibniz,as before,the same being also,of course,the“varying objects”.Thus,ADG appears too to be in accord with the point of view of general relativity,let alone,without

6Anastasios Mallios

any need to resort to a particular“background geometry(space)”,to work with(!) (see,for example,(1.8)in the preceding).Indeed,

(2.6)the(di?erential-)geometric mechanism,in the formalism of ADG,does not depend on(emanate from)any“background space”,in the sense that the latter term is,at least,understood in the classical theory(CDG).

In this regard,see also A.Mallios[19:in particular,(6.1),or even(9.8),therein], yet,A.Mallios[22].We further elaborate on(explain)our previous comments,as in(2.6)above,in a more technical manner,straightforwardly,by the subsequent Section.

3.ADG,technically speaking.?As already hinted at,just before,we come now,by the subsequent discussion,to sustain the aspect,this being also another fundamental issue of the whole formalism of ADG,that:

(3.1)within the setting of ADG,the“geometry”itself,in the sense that this notion is really understood,in that context(cf.,for example,(3.2),as well as,(3.3)in the sequel)is already,by its very de?nition(ibid.), a“dynamical variable”,therefore,by itself,of a relativistic nature(!), while,as we shall also see,by the ensuing discussion,it still appears, as such,thus far,concerning our relevant equations,as well(cf.thus Section5in the sequel).

Thus,to start with,by referring to“geometry”,within the framework of ADG, one virtually means the construction of a“geometrical calculus”,just,quite in the sense of Leibniz(see,for example,A.Mallios[19:beginning of Section1]),referring thus exclusively to interrelations of what one considers,herewith,as“geometrical objects”,hence,for the case in focus,of the“vector sheaves”involved;therefore,by employing physical terms,the aforesaid“calculus”(being,as it is actually de?ned(!), of a“di?erential-geometric”character,in the classical sense of the latter term) refers directly to the“elementary particles”(alias,“?elds”)themselves(see also(3.2) below).So,up to this point,we virtually consider the following“identi?cations”:

(3.2)“geometric object”←→vector sheaf ←→elementary particle

←→“?eld”.

Quantum gravity and“singularities”7

Indeed,as we shall see,right below(cf.thus(3.3)in the sequel),the above will be appropriately supplemented,when further applying physical terminology.In this regard,see also A.Mallios[22],for a fuller account of the nowadays notion of “geometry”,yet,in perspective with physics.

On the other hand,the term“interrelation”,as applied in the foregoing,means, by its very de?nition,a morphism between the respective sheaves(alias,a“sheaf-morphism”,the most important of all,when,in particular,referring to a“di?erential-geometric”syllabus,within the relevant setup,being,what we call,an A-connection (cf.(3.5)in the sequel).In point of fact,this particular morphism appears,as we shall see(cf.thus(5.1),or even(5.19)in the sequel),within the pertinent equa-tions,in the form,as we say,of an“invariant morphism”,something,of course,of paramount physical importance(cf.,for instance,“gauge principle”);yet,technically speaking,in the form of an“A-invariant morphism”,which,for the case at issue, is the respective curvature(:“?eld strength”)of the A-connection concerned(loc. cit.).

Thus,to put the above into a better perspective,explaining also,at the same time,the previously applied terminology,we come?rst,as already promised,for that matter,to the following amendment(:supplement)of our previous schematic version of the inherent situation herewith,as described in the preceding,at?rst sight,by(3.2).So one gets at the following associations(:identi?cations),in view of(3.2),this being,in e?ect,a more intrinsic(yet,in technical terms)aspect of the matter.That is,one has;

(3.3)“geometric object”,`a la Leibniz,←→elementary particle←→“?eld”←→Yang-Mills?eld,viz.,a pair,

(E,D).

(3.3.1)

We explain,right away,the above employed terminology,term by term.Thus,we have:

i)“geometric object”(`a la Leibniz).?We have already mentioned elsewhere(see A.Mallios[19:Section1])that G.W.von Leibniz,just at his time,demanded a“geometrical calculus”(“calcul g′e om′e trique”,see,for instance,N.Bourbaki[4: Chap.I;p.161(Note historique),ft.1])to be found,which should act directly on

8Anastasios Mallios

the“geometrical objects”,without the intervention of coordinates,that is,in other words,of any“location of the objects in the“space””;of course,concerning the latter function,one certainly needs thereon a“reference point”(alias,an“origin”(!)). However,this“?xation”,in our case,“accompanies”,in e?ect,as we shall see(cf. thus,for instance,iv)below),viz.“varies”with the(geometrical)object at issue (:vector sheaf,cf.(3.2),hence,by de?nition,a reference point–“space”–A,adjusted thus to the object,under consideration).Of course,the latter issue is of an extremely important signi?cance,pertaining to a“relativistic perspective”of the whole matter (cf.also(3.1),as above).

ii)elementary particle.?Now,being primarily interested herewith in potential applications of the present point of view to quantum gravity,as also the title of this article indicates,it is natural,in principle,to associate(:in point of fact,to identify) the“geometric objects”,as above,with the“elementary particles”;in other words, the geometrical objects,yet,in the sense of Leibniz,which still,for that matter,?ll up the“space”.In this connection,see also A.Mallios(loc.cit.),as well as,[20: (7.2),and subsequent remarks therein].

iii)“?eld”.?It is certainly natural to associate the“ultimate constituents of the matter”(:elementary particles)with the notion of a“?eld”,which is also considered (see,for instance,A.Einstein[9:p.140]),as an“independent,not further reducible fundamental concept”,the same correspondence,as above,being still rooted on the classical“duality”/identi?cation.Now,by further employing mathematical termi-nology,we come to the?nal association/identi?cation,as indicated in(3.3)above, that is,to the fundamental notion,concerning,in e?ect,the whole account of ADG, namely,that one of a

iv)Yang-Mills?eld,(E,D).?Now,the terminology we apply herewith is quite technical,concerning actually,the intrinsic formalism of ADG,for which we refer to A.Mallios[16][17],or even to[21].So,for convenience,we recall that the pair

(3.4)

(E,D),

as in(3.3.1)above,consists of a vector sheaf E on an(arbitrary,in principle)topo-logical space X,that is,of a locally free A-module on X,of?nite rank n∈N, relative to an algebra sheaf A on X,along with a given A-connection D on E;now,

Quantum gravity and“singularities”9 the latter is,by de?nition,a sheaf morphism,

D:E?→E?A?1,

(3.5)

which is C-linear(here the(constant)sheaf C of the complexes is,by assumption, contained in A,see thus(3.10)below),that also satis?es the pertinent herewith “Leibniz condition”:viz.one has the relation,

D(α·s)=α·D(s)+s??(α),

(3.6)

for any(continuous)local sectionsα∈A(U)and s∈E(U),with U an open subset of X,such that

(A,?,?1)

(3.7)

is a given di?erential triad on X.Furthermore,?1stands here for an A-module on X(that occasionally might be too a vector sheaf on X),while

?:A?→?1

(3.8)

is also a morphism,having analogous properties to D,as above(we call it,the standard,or even,the basic A-connection of A);so the corresponding here with Leibniz condition for?is now reduced to the relation,

?(α·β)=α·?(β)+β·?(α),

(3.9)

valid,for anyα,βin A(U),with U?X,as in the preceding.Yet,A is,by hypothesis,a unital and commutative C-algebra sheaf on X,such that one has (:canonical injection),

C?

?→εA.

(3.10)

On the other hand,in the special case that the rank of E,as before,equals1,viz. when one has,

rk A E≡rk E=1,

(3.11)

10Anastasios Mallios

then E is called,in particular,a line sheaf on X,that is also denoted by L,while the corresponding pair,as in(3.4),by

(L,D),

(3.12)

that is still named a Maxwell?eld on X.In this connection,we further note that the electromagnetic?eld is,of course(!),a Maxwell?eld,in the previous sense,that was also our primary motivation to the above employed terminology;however,see also Yu.I.Manin[29],or even[30],as well as,A.Mallios[21:Chapt.III].In this context,we also remark that,in general,bosons are characterized(:identi?ed with) Maxwell?elds,while fermions are similarly associated with Yang-Mills?elds,that is,with pairs(E,D),as in(3.4)above,for which one has rk E=n>1.However, for a fuller,as well as,a more precise account thereon,we still refer to A.Mallios [20],or even to[21:Chapt.II].

Thus,after the above brief technical account,we are next going to show,by the subsequent Section,that;

(3.13)based on the above interpretation of the notion of a“?eld”,and,still in conjunction with the very formalism of ADG,we are,in e?ect,able to look at“the?eld itself,as a dynamical variable”,a fact that,of course, we were always intensively looking for,thus far,when,in particular, confronted with problems of the“quantum deep”,

following thus,in that context,the slogan that,

(3.14)“the?eld itself is(to be considered,as it actually is(!),for that matter, as)a dynamical variable”.

So the application here of ADG a?ords the above possibility,as in(3.13),while also, let alone,that

(3.15)(viz.,apart from having the situation,as described by(3.14))we are not,moreover,compelled to resort to any background“space”(alias,“geometry”),“to work with”(cf.thus the relevant comments of J.Baez, as in(1.8)in the preceding).

On the other hand,the situation,as described,by the latter part of(3.15),was virtu-ally the case(loc.cit.)in the standard theory(CDG),when referring,in particular,

Quantum gravity and“singularities”11

to the quantization of the other forces of nature,alas(!),except gravity(:general relativity).

Accordingly,the shortage of an analogous situation with that one,as this was described by(3.15),when,in particular,referring to general relativity,within the classical framework(:CDG),while being especially confronted,in that context,with problems of the quantum deep(let alone with those,pertaining to(3.14),as above, (viz.with“in?nities”(!)),seems to be,thus far,a“fundamental culprit”of the whole issue.

On the other hand,based here,simply,on our experience from ADG,the fol-lowing comments being,in point of fact,the main moral,thereby,one comes to the conclusion that:

(3.16)the aforementioned shortage of the classical theory(CDG),as this,in particular,concerns quantization of general relativity,seems to arrive, as a result,thus far,of our insistence on having,

(3.16.1)

the classical“dx”(hence,the whole standard di?erential-

geometric mechanism thereof,yet,another conclusion here of

ADG(!)),only from a“local presence”of the classical carte-

sian R n,while,on the other hand,we still insist to retain,as

well,as a whole(!),that particular local presence of the same

“space”;

that is,the(smooth)manifold concept itself,concerning our calcula-tions,(!),something,indeed,of a paramount inconvenience,pertaining to the aforesaid context.

We are going now,through the ensuing discussion in the following Section4,to illuminate,as well as,further support the preceding,by referring directly to the nature and the type too of fundamental di?erential equations of the classical theory, that the latter acquire,when perceived from the point of view of the abstract setup, as above.

4.Di?erential equations,within the setting of ADG.?Looking at the particular type of“di?erential equations”,that one can formulate,within the above

12Anastasios Mallios

abstract framework,as this is advocated by ADG,we are able,in principle,to remark here,yet,on the ground of a similar rationale,as before,that:

(4.1)evolution may be perceived,as an“algebraic automorphism”(cf.,for instance,Feynman);that is,as something of a relational character(cf. also Sorkin),which,in turn,can still supply an“analytic expression”. So one can associate to it“numbers”(occasionally,in the most general sense of the term;here one can think,for instance,of something reminding“(Gel’fand-)duality”,thus,e.g.,even of a“generalized”(!)

spectrum of an appropriate operator,cf.,for instance,in that connection, Z.Daouldji-Malamou[5]or even[6],therefore,?nally,through“di?er-entiation”(Hamilton–Schr¨o dinger),providing thus,yet,algebraically(!), the“time operator”(Heisenberg–Prigogine–K¨a hler–Hiley).

In this connection,see also B.J.Hiley[14],as well as,A.Mallios[19:(3.27)].Hence, one thus arrives within the preceding framework,at the conclusion,that the

(4.2)(“di?erential”)equations acquire thus a“dynamical character”,more akin to“second quantization”,in point of fact,to the“?eld”itself,under consideration,and not merely to the vector states in the carrier space of a particular representation of CCR(:?rst quantization);in this context, the latter simply entails,in e?ect,the“carrier space”,thus,in turn,the supporting“space-time manifold”(alias,“continuum”),whose presence, however,creates again,as already noted in the foregoing,?nally,an, indeed,fundamental problem for the whole set-up.

Consequently,as a really instrumental outcome of the preceding,one thus realizes that:

(4.3)based on ADG,we are able to refer to the equations of quantum?eld theory,directly,in terms of the?elds themselves;therefore,without the intervention of any“background space”,which would provide,according to the classical pattern(CDG),the“di?erential-geometric”apparatus, employed in that framework.

Quantum gravity and“singularities”13

The above constitutes,in e?ect,as already noted before,the quintessence,indeed,of the whole potential applicability of“ADG formalism”in problems of quantum?eld theory;let alone,of course,the fact that one is able,another upshot,as well,of the general theory of ADG,as it was also pointed out in the preceding,to incorporate (classical)“singularities”(:in?nities,and the like)in(the(local)sections of)the structure sheaf A.

Therefore,equivalently,by referring to our previous last comments,one thus concludes that;

(4.4)the“ADG formalism”can read over(or even,see through)“singulari-ties”,in the standard(:CDG)sense of the latter term.

Yet,in other words,one can say,by still applying a language,akin to quantum?eld theory(cf.,for instance,R.Haag[13:p.326]),that:

(4.5)the“ADG formalism retains the information,one can(locally(!))get, even through(or else,(locally)supplied(!)by)“singularities”.

In this connection,see also,for example,A.Mallios-I.Raptis[26],concerning an appropriate relevant formulation of the well-known“Finkelstein(coordinate)singu-larities”[11].Yet,see A.Mallios[19:(0.6)and subsequent comments therein],for an early account of the same matter.

In toto,by summarizing the preceding,we can thus,?nally,say that:

(4.6)the“di?erential equations”,that one obtains,within the framework of ADG,pertaining,thus directly to the?eld itself,by virtue of the(as-sumed)correspondences(3.3),hence,being,so to say,in character,“sec-ond quantized ones”,

(4.6.1)

represent,in point of fact,the very quantized equation(s)of

the?eld(viz.of the elementary particle),in focus.

In this connection,we can further remark that,the manner of understanding the physis of elementary particles(thus,of the“fundamental entities”),by virtue of the above correspondences(:identi?cations),as in(3.3)in the foregoing,may still be construed,as being in accord with recent tendencies of taking into account“dynamic individuation of fundamental entities”(see J.Stachel[34]).Now,in our case,as

14Anastasios Mallios

above,this can,very likely,be assigned to the notion of the pair,

(4.7)

(E,D),

as the latter has been applied,throughout the preceding discussion(yet,cf.(3.3), herewith),along with the concomitant invariance of the whole theory(:ADG), under the action of the group

(4.8)

A ut E

(group sheaf of A-automorphisms of E).Yet,the latter notion might be perceived, in point of fact,within our present abstract perspective,actually supplied by the ADG formalism,still(Klein)as a(“variant”(!))“space-time”(see also A.Mallios [19:(3.23)and(3.26)]).

Next,we specialize the preceding,straightforwardly,by the subsequent Section, through concrete fundamental instances of the classical theory:

5.Concrete examples.?As already said,our aim in this?nal Section of the present article is to illuminate the preceding account,by referring,in particular,to fundamental examples of the standard situation,thus far:Thus,we start with the following Subsection.

5.(a).Einstein’s equation(in vacuo).?The(“di?erential”)equation,re-ferred to in the title of this Subsection,has actually,just,the same form with the homonymous one,as in the classical case(:CDG),however,now,quite a di?erent meaning(!).We thus change point of view,as well as,the respective formalism,the latter being now,that one of the abstract di?erential geometry(:ADG),in conjunc-tion with our perspective,in that context,as exhibited by(3.3)in the preceding. So the said equation has also herewith the familiar form,

(5.1)

R ic(E)=0,

which thus in our case is the Einstein’s equation(in vacuo).Now,concerning the technical part of the previous relation(5.1),we still refer to A.Mallios[18],or even,for a full account thereof,to the forthcoming2-volume detailed treatment

Quantum gravity and“singularities”15

in A.Mallios[22:Chapt.IX;Section3].For convenience,however,of the ensuing discussion,herewith,we do recall,in brief,the following items about(5.1);that is, one thus sets:

Ric(E)=tr(R(·,s)t)≡tr(R(D E)(·,s)t):E(U)?→A(U),

(5.2)

where s,t are local(continuous)sections of the Yang-Mills?eld concerned,

(5.3)

(E,D≡D E),

in such a manner that the A(U)-morphism,as in(5.2)above,stands here for a“local instance”(viz.,by restriction to a local gauge U?X of E)of the so-called Ricci operator of

(5.4)

E≡(E,D E≡D),

such that one further de?nes;

(5.5)

R ic(E)≡(Ric U(E)≡Ric(E)),

with U running over a given local frame of E,the last relation yielding thus the ?rst member of(the equation)(5.1),as an A-morphism(as it actually entails any (“di?erential”)equation,whatsoever,cf.also(5.10)in the sequel)of the A-modules (in fact,vector sheaves,see thus below)concerned,locally identi?ed,through(5.2).

Now,the A-module E,as brie?y indicated by(5.4)above,that is involved here-with,is,in point of fact,a“Lorentz vector sheaf”(loc.cit.,Chapt.IX;(2.14),or even Note3.1therein)on a given topological space X,common base space,by def-inition,of all the A-modules(sheaves)appeared throughout.Furthermore,within this same context,one assumes an appropriate“di?erential triad”on X,

(5.6)

(A,?,?1)

(see also Section3in the preceding for the relevant terminology applied here),while we still suppose that,in particular,one has;

(5.7)

?1=E?,

16Anastasios Mallios

the second member of(5.7)standing for the“dual”vector sheaf of E(ibid.Chapt. IX;Section3,see,in particular,De?nition3.1,along with the subsequent Scholium 3.1therein).

On the other hand,by further looking at(5.1),as above,we also remark that any?eld,that is(see thus(3.2),or even(3.3.1)in the foregoing),a pair,as in(5.3) (but,see also(5.4),for an abbreviated form,of common usage too),that satis?es (5.1),thus,in other words,a“solution of Einstein’s equation”,appears in the latter equation,by itself,or even,precisely speaking,via its“?eld strength”(:curvature),

R(D E)≡R(D)≡R,

(5.8)

cf.thus(5.2)above.

Yet,in this regard,we should further remark that,based on the preceding(see thus(3.3.1),or even(5.4)),and“dynamically speaking”,so to say,we also assume, throughout the present discussion,the basic correspondence(:identi?cation),

?eld←→D E≡D,

(5.9)

as it concerns,in e?ect,a given pair(:a Yang-Mills?eld),as in(5.4).However, an A-connection D,as above,is,by its very de?nition(cf.(3.5)in the preceding), only,a C-linear morphism,therefore,not a“tensor”,thus,technically speaking,not an A-morphism of the A-modules concerned(ibid.),as it actually is,its curvature (:?eld strength),R(D)≡R,hence,the appearance of the latter in the corresponding equations,describing the?eld at issue:Indeed,something that we already hinted at in the preceding(cf.thus the pertinent comments,following(5.5)above),we should explicitly point out herewith the fundamental principle,in e?ect,that,

(5.10)the(“di?erential”)equations,describing a?eld(yet,otherwise,by ob-viously“abusing language”(!),herewith,the“equations of a?eld”),are to be formulated,via“tensors”;hence,in other words,in terms always of A-morphisms,the equation itself entailing thus an A-morphism,as well(expressing,by its very substance,for that matter,a physical law, that one,determined by the“?eld”,in focus).

In this connection,we can still say that the above may also be construed,as another upshot of the same“principle of general covariance”;in this regard,see also,for

Quantum gravity and“singularities”17

instance,D.Bleecker[3:p.50,Section3.3,in particular,p.52,Theorem3.3.6].Yet, within the same vein of ideas,we can further remark,in point of fact,that;

(5.11)when considering the(“di?erential”)equations,describing?elds(:na-tural laws),as being A-morphisms(see(5.10)above),this may be con-strued,in e?ect,as another,just,technical(!),equivalent expression of the same“principle of general covariance”.

On the other hand,by specializing our previous considerations in the preceding Section4(cf.,for instance,(4.6)therein)to the case of the equation(5.1),thus,by further looking at the particular issues,involved in the same equation,one realizes that;

(5.12)Einstein’s equation(in vacuo)refers to the?eld itself,that is,e o i p s o to the respective quantum(hence,for the case at issue,to the“graviton”). Therefore,it is,by its very formulation,already a quantized equation, and,as a matter of fact,a“second quantized”one,therefore,an equation, within the setting of quantum?eld theory.

Indeed,the sheaf-theoretic character of the framework,within which that equation has been formulated,provides also its relativistic perspective,being thus,at the same time,as already remarked(see comments following(5.5)),a covariant one,as well.

On the other hand,by further continuing our concrete specialization of the pre-ceding(see thus our general comments on“quantizing gravity”in Section1,or even (2.6)above)to the particular case,considered by the present Subsection,we can still remark that,what is to be viewed,herewith,as of a particular signi?cance,especially pertaining to problems of“quantum gravity”,being also in complete diversi?cation with the manner we apply,in that context,the classical theory(:CDG),see,for instance,(1.5),(1.9),or even(2.6)in the preceding,is the following fact,already mentioned,generally speaking,in the https://www.wendangku.net/doc/4d18175029.html,ly,one can still remark here that:

18Anastasios Mallios

(5.13)the whole formalism of ADG,according to its very de?nition,

(5.13.1)is entirely“space-independent”,

in the usual sense of this term.That is,by applying herewith classical parlance(loc.cit.),

(5.13.2)one does not need any“background geometry to work with”,

while,at the same time,this same“geometry”,within the present ab-stract set-up(:ADG),being,in point of fact,represented by(alias, emanated from)the same“structure sheaf of coe?cients”,A(viz.our “generalized arithmetics”),

(5.13.3)is actually still varied with us(!),as well,

according to the very de?nition of the objects,that are entangled in the equation,for instance,(5.1),as above.For,

(5.13.4)“everything[there]boils down[locally(!)]to A”.

Yet,as a result of the preceding,we can still say that;

(5.14)re?ecting,within the framework of ADG,we realize that the“observer”(viz.“we”,to the extent that this is expressed,trough our“arithme-tics”A)becomes a“dynamical variable”,as well,acquiring thus too,a “relativistic character”.

Thus,the following claim has here its relative position,being also in accord with previous similar considerations;that is,one can say that

(5.15)“all is dynamical”(!)

The above might also be related with J.Stachel’s,quite recently stated,principle of dynamic individuation of the fundamental entities”,see thus[34:p.32].

Now,by still commenting on our previous remark in(5.13.4),as above,we further note that our last indication(“locally”)therein reminds us,of course,of a quite recent comment of R.Haag[13:p.326]in that,

(5.16)“all information characterizing the[quantum?eld]theory is strictly lo-cal”,

Quantum gravity and“singularities”19 this being also,

(5.17)“the central message”of nowadays Quantum Field Theory(loc.cit.).

In this connection,it is further worth mentioning here that the same author,as above (ibid.),refers to the aforesaid situation,about today QFT,while advocating a sheaf-theoretic approach to that theory,as being thus more akin to the“local character”of the latter(one thus considers here,neighborhoods),in contradistinction with the“point-character”of?ber bundle theory,(e.g.vector bundles),that has been employed,so far.On the other hand,by

(5.18)applying ADG,one also gets,via its overall sheaf-theoretic character,at a“synthesis of the knowledge gained in...di?erent approaches”,

a fact,that was also in perspective,by the aforesaid author(loc.cit.).Thus,in the case of ADG,one has,for instance,the following synthesis:

(5.19)

ADG←→CDG

(5.19.1)

in a new perspective,since no Calculus is employed,at all(!),plus

sheaf theory,

(5.19.2)

in conjunction with sheaf cohomology.

We terminate the present discussion,by still pointing out,within the preceding framework,another,indeed,de?nitive aspect of the formalism of ADG,that we have also hinted at in the foregoing,within the abstract setting of our general commentary herein;namely,the

(5.20)possibility of working,within the framework of ADG,by employing func-tions,in point of fact,local sections of A,that may have/incorporate a large,in e?ect,the biggest,thus far(!),amount of“singularities”,in the classical sense of this term,as if(a signi?cant,indeed,advantage of the aforesaid mechanism)the latter classical anomalies of the standard theory(CDG)were not present,at all(!)

Thus,to say it,once more,emphatically,

20Anastasios Mallios

(5.21)the formalism of ADG can,indeed“absorb”the“singularities”of the classical theory,

by appropriately chosen A(see also(4.4),(4.5)in the preceding).In this connection cf.A.Mallios-E.E.Rosinger[27],[28],for an early account of the subject,as well as,the recent work in A.Mallios-I.Raptis[26];yet,cf.A.Mallios[19],along with A.Mallios[21:Chapt.IX;Subsection5.(b)],concerning a more general perspective thereon;yet,the latter can actually be viewed,as the outcome of some recent categorical considerations,pertaining to the formalism of ADG,as presented in the relevant work of M.Papatrianta?llou[31],[32].

5.(b).Yang-Mills equations.?As it was the case in the preceding with Einstein’s equation(in vacuo),see equa.(5.1)above,the equations in the title of the present Subsection,still retain,within the abstract set-up,employed herewith, the familiar form,they have,in the standard setting of the classical theory(CDG). Thus,the aforesaid equations preserve,here too,their classical form,within,of course,the appropriate now formalism adapted to ADG.That is,one gets at the relations;

δE nd E(R)=0,

(5.22)

or even,equivalently,

?E nd E(R)=0,

(5.23)

which thus constitute,within the abstract setting of ADG,the corresponding Yang-Mills equations.

Now,concerning the relevant technical details,connected with the previous equa-tions,we still refer to A.Mallios[16],or even,for a complete account thereof,to A.Mallios[21:Chapt.VIII].For convenience,of course,we simply recall(loc.cit.), that E here stands for a Yang-Mills?eld,as in(5.3)above(cf.also,for instance, (5.4)),whose?eld strength is R(cf.(5.8)).Thus,here too,one realizes that;

(5.24)a complete analogous rationale,as that one,we have exhibited above, pertaining to Einstein’s equation(in vacuum),is also in force,referring now to the Yang-Mills equations,as in(5.22)/(5.23).

地球物理勘查名词术语

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快速减载系统操作手册

ABB

ABB Copyright ? ABB. 2010

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1. 1.1 1.2 ABB – Asea Brown Boveri – AC800M - Advant 800 Advant Controller 800 series AI – Analog Input CB – Circuit breaker DCS – Distributed Control System DI – Digital Input DO – Digital Output DI DO AI AO HSI – Human System Interface Loadbusbar – MW – Mega Watt MVar – Mega Var OS – Operator Station P – Active Power Q – Reactive Power SR – Spinning Reserve

android layout_gravity 和 android gravity 的区别

1.gravity 这个英文单词是重心的意思，在这里就表示停靠位 置的意思。 android:layout_gravity 和android:gravity 的区别 从名字上可以看到，android:gravity是对元素本身说的，元素本身的文本显示在什么地方靠着换个属性设置，不过不设置默认是在左侧的。 android:layout_gravity是相对与它的父元素说的，说明元素显示在父元素的什么位置。 比如说button：android:layout_gravity 表示按钮在界面上的位置。android:gravity表示button上的字在button上的位置。 可选值 这两个属性可选的值有：top、bottom、left、right、center_vertical、fill_vertical、center_horizontal、fill_horizontal、center、fill、clip_vertical。

而且这些属性是可以多选的，用“|”分开。 默认这个的值是：Gravity.LEFT 对这些属性的描述： 出自： https://www.wendangku.net/doc/4d18175029.html,/guide/topics/resources/drawable-res ource.html https://www.wendangku.net/doc/4d18175029.html,/reference/android/graphics/drawable /ClipDrawable.html Value Description top Put the object at the top of its container, not changing its size. 将对象放在其容器的顶部，不改变其大小. bottom Put the object at the bottom of its container, not changing its size. 将对象放在其容器的底部，不改变其大小.

快速装车站系统手册样本

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(精校版)沪教牛津版初中英语单词表

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沪教版七年级上单词表 Unit 1 German adj. 德国的 blog n. 博客 grammar n。语法 sound n. 声音complete v。完成 hobby n. 爱好 country n. 国家 age n。年龄 dream n.梦想 everyone pron. 人人；所有人 Germany n. 德国mountain n。山；山脉elder adj. 年长的friendly adj。友爱的；友好的 engineer n.工程师 world n. 世界 Japan n。日本 flat n. 公寓 yourself pron.你自己 US n. 美国close to （在空间、时间上） 接近 go to school 去上学 （be) good at 擅长 make friends with 与..。... 交朋友 all over 遍及 ’d like to = would like to 愿意 Unit2 daily adj。每日的；日常 的 article n。文章 never adv. 从不 table tennis n.兵乓球 ride v. 骑；驾驶 usually adv。通常地 so conj. 因此;所以 seldom adv.不常；很少 Geography n. 地理 break n. 休息 bell n。钟；铃 ring v。（使）发出钟声， 响起铃声 end v。结束；终止 band n。乐队 practice n. 练习 together adv。在一起 market n。集市；市场 guitar n。吉他 grade n. 年级 junior high school 初级 中学 on foot步行 take part in 参加 have a good time 过得愉快 go to bed 去睡觉 get up 起床 Unit3 Earth n. 地球 quiz n。知识竞赛；小测 试 pattern n。模式;形式 protect v.保护 report n。报告 part n. 部分

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。 中国太平洋保险（集团）股份有限公司Xxxxxxxxxxx项目 系统安装部署手册 V1.0 项目经理： 通讯地址： 电话： 传真： 电子邮件：

文档信息 1引言 (3) 1.1编写目的 (3) 1.2系统背景 (3) 1.3定义 (3) 1.4参考资料 (3) 2硬件环境部署 (3) 2.1硬件拓扑图 (3) 2.2硬件配置说明 (3) 3软件环境部署 (3) 3.1软件环境清单 (3) 3.2软件环境部署顺序 (3) 3.3操作系统安装 (4) 3.4数据库安装 (4) 3.5应用级服务器安装 (4) 3.6其他支撑系统安装 (4) 4应用系统安装与配置 (4) 4.1应用系统结构图 (4) 4.2安装准备 (4) 4.3安装步骤 (4) 4.4系统配置 (5) 5系统初始化与确认 (5) 5.1系统初始化 (5) 5.2系统部署确认 (5) 6回退到老系统 (5) 6.1配置回退 (5) 6.2应用回退 (5) 6.3系统回退 (5) 6.4数据库回退 (5) 7出错处理 (5) 7.1出错信息 (5) 7.2补救措施 (5) 7.3系统维护设计......................................................... 错误！未定义书签。

1 引言 1.1 编写目的 [说明编写系统安装部署手册的目的] 1.2 系统背景 [ a ． 说明本系统是一个全新系统还是在老系统上的升级； b ． 列出本系统的使用单位/部门、使用人员及数量。] 1.3 定义 [列出本文件中用到的专门术语的定义和缩写词的原词组。] 1.4 参考资料 [列出安装部署过程要用到的参考资料，如： a ． 本项目的完整技术方案； b ． 系统运维手册； c ． 其他与安装部署过程有关的材料，如：工具软件的安装手册] 2 硬件环境部署 2.1 硬件拓扑图 [列出本系统的硬件拓扑结构，如服务器、网络、客户端等。] 2.2 硬件配置说明 [列出每一台硬件设备的详细配置，如品牌、型号、CPU 数量、内存容量、硬盘容量、网卡、带宽、IP 址址、使用、应部署哪些软件等等] 3 软件环境部署 3.1 软件清单 [列出需要用到哪些软件，包括操作系统软件、数据库软件、应用服务器软件和其他支撑系统软件等，要列明每个软件的全称、版本号、适用操作系统、LICENSE 数量等] 3.2 软件环境部署顺序 [列出每一台硬件上的软件安装顺序，如果不同硬件间的软件安装顺序存有依赖关系，也要在备注中列出，

Density and Specific Gravity

Density and Specific Gravity (1) Density (r)of a material is defined as mass per unit volume and is usually represented in the SI system by the units g*cm-3or kg*m-3. There are a number of different methods and devices for determining the density of a substance, and there are also a number of factors that can affect the density of a sample. If the exact composition of a non-homogeneous ) can be determined using the mass material is known, its solid density(r s and density of each of the n components using the following equation: The density of materials such as agricultural grains, which consist of many small particles, can be expressed in terms of solid density,particle density or bulk density. Solid density considers the mass and volume of the solid matter only and does not include any air spaces within the mixture. Particle density describes the mass per unit volume of an individual particle (e.g. corn kernel) from the sample. Bulk density, on the other hand, considers the total mass and total volume of a large quantity of the particles. Specific gravity is a dimensionless term used to compare the densities of different materials relative to water. Specific gravity of a substance is defined as the ratio of the substance’s density to the density of wa ter at the same temperature. From this information, it can be determined that if the specific gravity of a material is less than 1, it is less dense than water, and if the specific gravity is greater than 1, it is more dense than water. 1. Stroshine and Hamann. 1994. Physical Properties of Agricultural Materials and Food Products. 17-18.

快速装车站(1)

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5、实现的效果 优化铁路装车作业工序，将原本需要大量人工作业的工序全部集成为自动化操作，特别是系统具备的自动打印报表等功能，方便了企业产销统计工作。自动配煤、自动装车不仅减少了人工投入，实现在作业环节中的安全保障，同时提高了装车的速度、精度，节省了装车作业时的滞留时间。 二、技术内容 1.技术方案 选用LCS-100型定量快速装车站系统替代了传统多人员参与装车系统，减少劳动力、提高安全管理水平。LCS-100型定量快速装车站主要参数如下： 装车能力：标准5500 t／h，最高可达到8000 t／h； 缓冲仓缓冲能力：0～1000 t； 称量范围：0～100 t； 称重传感器精度：0.01%； 装车精度：单车装车精度0.1%，整列车装车精度0.05%； 铁路线形：贯通式、环线、尽头式及各类站场； 铁路线间距：≥5m； 牵引方式：电力机车、内燃机车、蒸汽机车、调车绞车等； 牵引速度：0.8～2.0 km／h 适用车型：现有各类车型； 操作方式：自动、半自动、手动； 结构形式：单线单套快速定量装车站

东亚贸易gravity model应用-国际经济学

WORLD TRADE（EAST ASIA）: The Gravity Model PART1: 从更具体的省份?角度来看中韩贸易，??广东、江苏、?山东和上海稳固占据前4席位，其中??广东是中韩进出?口的第?一?大省，进出?口份额均约1/4。东部省市是中韩贸易的主体，但中?西部省市凸显?高速增?长态势。 按照gravity model，距离越?小，贸易量会越?大，从上表中可知排名第三的?山东、第六的辽宁第七的天津都是距离韩国较近的省份，故贸易量会排在较前。但是??广东、江苏、

上海、浙江的省经济贸易总量?非常?大，以?至于其影响超过了距离对东部各省与韩的贸易量的影响，使??广东各省稳居前列。 PART2: ?自1992年8?月建交以来，中韩两国间经贸往来取得了较快发展，两国双边贸易额由建交时的50亿美元增加到2012年2742．38亿美元，增长了五??十余倍。2013年，韩国是中国的第三?大贸易伙伴，?而中国已成为韩国的第?一?大贸易伙伴、第?一?大出??口?目的地和第?一?大进??口来源地。

上?方所显?示的四张图是中?日、中韩的近?几年贸易量显?示。总体来讲由于中国本?身国家经济贸易总量的增?长，中?日、中韩之间的贸易量基本都是增?长趋势，其增?长额度与速度也不容?小觑，加上本?身同为东亚的各国的距离上的优势以及?日本与韩国的GPD多年来的增?长，形成以上图的状况。图?二中2003-2013中韩贸易额占中国贸易总额微有所下降，中韩贸易额占韩国贸易总额却有明显的上升，?一?方?面是由于中国本?身国家经济贸易总量的快速?大量增?长超越了韩国的GDP增?长，另?一?方?面由于中国扩?大了贸易发展?面，贸易多边化，中韩所占中国贸易总量?比重减少。 图四所显?示，2006-2010中?日贸易同?比增?长和中国外贸同?比增?长基本趋势趋于?一致，除2009年点处，中国外贸同?比增?长数额明显?大于中?日贸易同?比增?长数额。原因还是中国贸易的国际化快速发展。 PART3:

火车装车控制使用手册

装车控制使用手册 装车站为机、电、液一体的自动装车系统，下面就如何操作作简单介绍，仅供参考。 (一)、系统启动前的准备工作 1．快速装车系统操作使用前的一般规定： 1)操作员具有高中(中技)以上文化程度； 2)操作员上岗前要经过安全和业务培训； 3)操作员要熟悉快速装车站的工艺流程； 4)操作员应具有机械、液压、电气控制及计算机的基本知识； 5)操作员要掌握给煤机、胶带运输机和装车设备的工作原理、基本结构和主要性能； 6)岗位司机应取得钢缆皮带操作工上岗资格。 2．开机前的检查项目： 1)检查胶带机电机、液力偶合器、减速器、制动器、逆止器、防堵开关、拉绳开关、跑偏开关、防打滑装置、纵向撕列保护、胶带、各类滚筒、拉紧装置、皮带清扫器、托辊等设备及部件是否完好。 2)检查给煤机各螺栓是否紧固，吊挂是否平衡稳固，电机两端防护罩是否完好，弹簧座软连接是否磨损 3)检查装车塔楼内液压站、油缸、泵、阀等是否有渗漏现象，各截止阀位置是否正确，油路是否畅通。 4)检查液压泵(1#2#)动力电是否送好，油位、油温是否正常。 3．检查工控机的通讯情况：将计算机打开，进入控制画面，若控制画面图象显示正常，则说明网络通讯正常，反之则不正常。 4．冬季，液压系统启动前应注意： 由于冬季环境温度底，导致达不到液压系统的工作温度，因此要在装车前对液压系统进行热油循环或油箱预热。 5．对液压系统进热油循环或油箱预热： 进入控制画面，在控制画面的右上角点击“热油循环”或“油箱预热”按钮，系统将自动开始加热循环(注意：此时溜槽不能停在装车位置) (二)、系统的运行操作 1．“系统启动”按钮的功能及操作： 功能：开启液压油泵。操作：按下1s 2．“系统停止”按钮的功能及操作： 功能：停止液压油泵。操作：按下1s 3．“缓冲仓全部闸门”按钮的功能及操作： 功能：控制缓冲仓全部闸门的开和闭。操作：按下，闸门开；弹起，闸门关闭4．“缓冲仓闸门开”按钮的功能及操作： 功能：控制缓冲仓补料闸门的开和闭。操作：按下，闸门开；弹起，闸门关闭5．“卸料闸门启动”按钮的功能及操作： 功能：打开卸料闸门。操作：按下1s 6．“卸料闸门停止”按钮的功能及操作： 功能：关闭卸料闸门。操作：按下ls 7．“重读”按钮的功能及操作： 功能：在定量仓处在就绪状态下，重新配煤时用。操作：按下ls 8．“卸料闸门点动”按钮的功能及操作：

THE LOG OF GRAVITY

THE LOG OF GRAVITY J.M.C.Santos Silva and Silvana Tenreyro* Abstract—Although economists have long been aware of Jensen’s in-equality,many econometric applications have neglected an important implication of it:under heteroskedasticity,the parameters of log-linearized models estimated by OLS lead to biased estimates of the true elasticities.We explain why this problem arises and propose an appropri-ate estimator.Our criticism of conventional practices and the proposed solution extend to a broad range of applications where log-linearized equations are estimated.We develop the argument using one particular illustration,the gravity equation for trade.We?nd signi?cant differences between estimates obtained with the proposed estimator and those ob-tained with the traditional method. I.Introduction E CONOMISTS have long been aware that Jensen’s in- equality implies that E(ln y) ln E(y),that is,the expected value of the logarithm of a random variable is different from the logarithm of its expected value.This basic fact,however,has been neglected in many economet-ric applications.Indeed,one important implication of Jen-sen’s inequality is that the standard practice of interpreting the parameters of log-linearized models estimated by ordi-nary least squares(OLS)as elasticities can be highly mis-leading in the presence of heteroskedasticity. Although many authors have addressed the problem of obtaining consistent estimates of the conditional mean of the dependent variable when the model is estimated in the log linear form(see,for example,Goldberger,1968;Man-ning&Mullahy,2001),we were unable to?nd any refer-ence in the literature to the potential bias of the elasticities estimated using the log linear model. In this paper we use the gravity equation for trade as a particular illustration of how the bias arises and propose an appropriate estimator.We argue that the gravity equation, and,more generally,constant-elasticity models,should be estimated in their multiplicative form and propose a simple pseudo-maximum-likelihood(PML)estimation technique. Besides being consistent in the presence of heteroskedas-ticity,this method also provides a natural way to deal with zero values of the dependent variable. Using Monte Carlo simulations,we compare the perfor-mance of our estimator with that of OLS(in the log linear speci?cation).The results are striking.In the presence of heteroskedasticity,estimates obtained using log-linearized models are severely biased,distorting the interpretation of the model.These biases might be critical for the compara-tive assessment of competing economic theories,as well as for the evaluation of the effects of different policies.In contrast,our method is robust to the different patterns of heteroskedasticity considered in the simulations. We next use the proposed method to provide new esti-mates of the gravity equation in cross-sectional https://www.wendangku.net/doc/4d18175029.html,ing standard tests,we show that heteroskedasticity is indeed a severe problem,both in the traditional gravity equation introduced by Tinbergen(1962),and in a gravity equation that takes into account multilateral resistance terms or?xed effects,as suggested by Anderson and van Wincoop(2003). We then compare the estimates obtained with the proposed PML estimator with those generated by OLS in the log linear speci?cation,using both the traditional and the?xed-effects gravity equations. Our estimation method paints a very different picture of the determinants of international trade.In the traditional gravity equation,the coef?cients on GDP are not,as gen-erally estimated,close to1.Instead,they are signi?cantly smaller,which might help reconcile the gravity equation with the observation that the trade-to-GDP ratio decreases with increasing total GDP(or,in other words,that smaller countries tend to be more open to international trade).In addition,OLS greatly exaggerates the roles of colonial ties and geographical proximity. Using the Anderson–van Wincoop(2003)gravity equa-tion,we?nd that OLS yields signi?cantly larger effects for geographical distance.The estimated elasticity obtained from the log-linearized equation is almost twice as large as that predicted by PML.OLS also predicts a large role for common colonial ties,implying that sharing a common colonial history practically doubles bilateral trade.In con-trast,the proposed PML estimator leads to a statistically and economically insigni?cant effect. The general message is that,even controlling for?xed effects,the presence of heteroskedasticity can generate strikingly different estimates when the gravity equation is log-linearized,rather than estimated in levels.In other words,Jensen’s inequality is quantitatively and qualitatively important in the estimation of gravity equations.This sug-gests that inferences drawn on log-linearized regressions can produce misleading conclusions. Despite the focus on the gravity equation,our criticism of the conventional practice and the solution we propose ex-tend to a broad range of economic applications where the equations under study are log-linearized,or,more generally, transformed by a nonlinear function.A short list of exam-ples includes the estimation of Mincerian equations for wages,production functions,and Euler equations,which are typically estimated in logarithms. Received for publication March29,2004.Revision accepted for publi- cation September13,2005. *ISEG/Universidade Te′cnica de Lisboa and CEMAPRE;and London School of Economics,CEP,and CEPR,respectively. We are grateful to two anonymous referees for their constructive comments and suggestions.We also thank Francesco Caselli,Kevin Denny,Juan Carlos Hallak,Daniel Mota,John Mullahy,Paulo Parente, Manuela Simarro,and Kim Underhill for helpful advice on previous versions of this paper.The usual disclaimer applies.Jiaying Huang provided excellent research assistance.Santos Silva gratefully acknowl- edges the partial?nancial support from Fundac?a?o para a Cie?ncia e Tecnologia,program POCTI,partially funded by FEDER.A previous version of this paper circulated as“Gravity-Defying Trade.” The Review of Economics and Statistics,November2006,88(4):641–658 ?2006by the President and Fellows of Harvard College and the Massachusetts Institute of Technology

简单解释一下android 中layout_gravity和gravity_华清远见

简单解释一下android 中layout_gravity和gravity 本篇文章的目的就是为大家简单解释android 中layout_gravity和gravity，希望看完对大家有帮助。 相信很多学习了android的人，都知道布局中存在两个很相似的属性：android :layout_gravity和android:gravity。一般的都知道， android :layout_gravity 是表示该组件在父控件中的位置 android:gravity 是表示该组件的子组件在自身中的位置 例子： 如上的代码显示效果如下： 经管很明了，但是某些时候也会出现失效的情况，如下：

常用金融术语

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