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UTHEP-469,LBNL-52989,BUHEP-03-13Chiral perturbation theory at O (a 2)for lattice QCD Oliver B¨a r a ?,Gautam Rupak b ?and Noam Shoresh c ?a Institute of Physics,University of Tsukuba,Tsukuba,Ibaraki 305-8571,Japan b Lawrence Berkeley National Laboratory,Berkeley,CA 94720,U.S.A.c Department of Physics,Boston University,Boston,MA 02215,U.S.A.Abstract We construct the chiral e?ective Lagrangian for two lattice theories:one with Wilson fermions and the other with Wilson sea fermions and Ginsparg-Wilson valence fermions.For each of these theories we construct the Symanzik action through O (a 2).The chiral Lagrangian is then derived,including terms of O (a 2),which have not been calculated before.We ?nd that there are only few new terms at this order.Corrections to existing coe?cients in the continuum chiral Lagrangian are proportional to a 2,and appear in the Lagrangian at O (a 2p 2)or higher.Similarly,O(4)symmetry breaking terms enter the Symanzik action at O (a 2),but contribute to the chiral Lagrangian at O (a 2p 4)or higher.We calculate the light meson masses in chiral perturbation theory for both lattice theories.At next-to-leading order,we ?nd that there are no O (a 2)corrections to the valence-valence meson mass in the mixed theory due to the enhanced chiral symmetry of the valence sector.PACS numbers:11.15.Ha,12.39.Fe,12.38.Gc Keywords:Lattice QCD,discretization e?ects,chiral perturbation theory,partially quenched theory
I.INTRODUCTION
Chiral perturbation theory(χPT)[1,2]plays an important role in the analysis of current lattice QCD data.Simulations with the quark masses as light as realized in nature are not feasible on present day computers.Instead one simulates with heavier quark masses and performs a chiral extrapolation to the physical quark masses using the analytic predictions ofχPT.To perform the chiral extrapolation one must?rst take the continuum limit of the lattice data,sinceχPT describes continuum QCD and is not valid for non-zero lattice spacing.However,it is common practice not to perform the continuum extrapolation and nevertheless?t the lattice data to continuumχPT,assuming that the lattice artifacts are small.
A strategy to reduce this systematic uncertainty was proposed in Refs.[3,4,5,6,7](a di?erent approach was taken in Refs.[8,9]in the strong coupling limit).There it was shown how the discretization e?ects stemming from a non-zero lattice spacing can be included inχPT.The basic idea is that lattice QCD is,close to the continuum limit,described by Symanzik’s e?ective theory,which is QCD with additional higher dimensional terms [10,11,12,13,14].The derivation ofχPT from QCD can then be extended to this e?ective theory with additional symmetry breaking parameters.The result is a chiral expansion in which the leading dependence on the lattice spacing is explicit.This idea was numerically examined in Ref.[15]for a theory with two dynamical sea quarks on a coarse lattice using the results of Ref.[7].The characteristic chiral log behavior in the pseudo scalar meson mass and decay constant was observed.
A similar approach was taken in Ref.[16]for analyzing lattice theories with two types of lattice fermions–Wilson fermions for the sea quarks and Ginsparg-Wilson fermions for the valence quarks.The latter can be implemented using domain wall[17,18,19], overlap[20,21,22,23,24],perfect action[25,26],and chirally improved fermions[27,28]. There are several advantages in using di?erent lattice fermions in numerical simulations. Since massless Ginsparg-Wilson fermions exhibit an exact chiral symmetry even at non-zero lattice spacing[29],it is possible to simulate such valence fermions with masses much smaller than the valence quark masses accessible using Wilson fermions[30,31].This allows a wider numerical sampling of points in the chiral regime of QCD.In addition,the valence sector exhibits all the bene?ts stemming from the Ginsparg-Wilson relation[32]such as the absence
of additive mass renormalization,of operator mixing amongst di?erent chiral multiplets and of lattice artifacts linear in the lattice spacing a[23,24,25,26,33,34].
In this paper we extend the results of both Refs.[7]and[16]by calculating the chiral La-grangian including the O(a2)lattice e?ects.There are various reasons for doing this.First, the lattice spacings in current unquenched simulations are not very small,so that neglect-ing the O(a2)contributions might not be justi?ed.Second,the use of non-perturbatively improved Wilson fermions in lattice simulations is becoming more common.The leading corrections for these fermions are of O(a2)and hence need to be computed in order to know how the continuum limit is approached.
At O(a2)many operators enter the Symanzik action and need to be taken into account for constructing the chiral Lagrangian.4-fermion operators appear for the?rst time and operators that explicitly break Euclidean rotational symmetry are encountered.Neverthe-less,the number of new operators in the chiral Lagrangian is rather small(3for the Wilson action and4for the mixed fermion theory).This is important in practical applications,since every new operator comes with an undetermined low-energy constant.These constants en-ter the analytic expressions for physical observables and too many free parameters limit the predictability of the chiral extrapolations.
The paper is organized as follows:χPT for the Wilson action is discussed in section II, including the partially quenched case in subsection II E.The mixed theory with Wilson sea and Ginsparg-Wilson valence quarks is treated in section III.In section IV we discuss the chiral power counting and compute the pseudo scalar meson mass including the new O(a2) contributions for both cases.We end with some general comments in section V.
II.WILSON ACTION
In this section we formulate the chiral e?ective theory for the Wilson lattice action.First the Wilson action and its symmetries are brie?y reviewed,then the local Symanzik action through O(a2)is presented.Based on the symmetry properties of the Symanzik action we construct the chiral e?ective theory.Finally,we consider the extension to the partially quenched case.
We consider an in?nite hypercubic lattice with lattice spacing a.The quark and anti quark?elds are represented byψand
ψ(D W+m0)ψ(x),(1)
1
D W=
(1±γ5)the left-and right-handed fermion
2
?elds are de?ned by
ψL,R=P?ψ,ψP±.(3) Under a transformation L?R∈G these chiral components transform according to
ψL→LψL,ψL L?,(4)
ψR→RψR,ψR R?.
Only if the mass and the Wilson terms are absent is G a symmetry group of the theory.If all the quark masses are non-zero but equal the vector subgroup with L=R is a symmetry of the action S W.
To complete the de?nition of the lattice theory one should also de?ne a gauge action S Y M.However,the precise choice of the gauge action is irrelevant for the purpose of our analysis,so we leave it unspeci?ed.
The Symanzik action for the Wilson lattice action,up to and including O(a2),has been calculated?rst in Ref.[13].The analysis to O(a)has been later elaborated on in Ref.[14]. We restate these results in a slightly di?erent form.We list all operators in the action through O(a2)that are allowed by the symmetries,organized in powers of a(again,we only focus on the fermion action).We use the notation
S S=a?1S?1+S0+aS1+a2S2...,(5)
S k= i c(k+4)i O(k+4)i,
where O(n)i are operators of dimension n constructed from the quark and gauge?elds and their derivatives.The constants c(n)i are unknown coe?cients.
Some allowed operators in the Symanzik action are obtained by multiplying lower dimen-sional operators with powers of the quark mass m0.Such operators(except for the mass term itself)renormalize lower dimensional operators.For example,a tr(m0)
ψψ.(6) S0:O(4)1=ψψ.(7)
By construction,S0is the continuum QCD action.
S1:O(5)1=ψiσμνFμνψ.(8)
S2,bilinears:O(6)1=ψDμ/DDμψ,(9)
O(6)2=ψγμDμDμDμψ.
S2,4-quark operators:O(6)5=(ψt aψ)2,(10)
O(6)6=(ψt aγ5ψ)2,
O(6)7=(ψt aγμψ)2,
O(6)8=(ψt aγμγ5ψ)2,
O(6)9=(ψt aσμνψ)2,
where t a are the SU(N c)generators.This list of4-quark operators is slightly di?erent from the one in Ref.[13].Sheikholeslami and Wohlert’s list contains operators with?avor group generators.However,both lists are equivalent and are related by Fierz identities(see appendix A).Our choice of operators is guided by the fact that for the study of chiral transformation properties it is more convenient to consider4-quark operators with trivial ?avor structure.
Equations of motion can be used to reduce the number of operators in the action.This also involves changing the de?nitions of the continuum?elds that are matched to the lattice ones[14,36].The way an operator in the Symanzik action a?ects the chiral Lagrangian is essentially determined by its transformation properties under chiral rotations.By using equations of motion one can replace an operator only with operators in the same repre-sentation of the?avor symmetry group.Hence,the form of the chiral Lagrangian remains unchanged whether equations of motion are used or not.In the context of on-shell improve-ment,equations of motion have been used to reduce the number of operators at O(a)[13,14]. Only the Pauli term O(5)2is left at this order,and can be subsequently canceled by adding the clover term to the lattice action.For the formulation of the chiral e?ective theory we choose not to take this step.
We distinguish two types of operators in the Symanzik action:those that break chiral symmetry and those that do not.Among the?rst type,the largest symmetry breaking e?ect comes from the term a?1O(3)1.This term can be absorbed into O(4)1by introducing the renormalized quark mass
m=m q Z m,m q=m0?m c,(11)
where the critical quark mass m c～1/a can be de?ned as the value of m0where the pion becomes massless.Similarly,the e?ect of some higher dimensional operators can be absorbed in the de?nition of the renormalized quark mass by including a factor1+b1am q+b2(am q)2
on the right hand side of eq.(11)[14,36].
At O(a)all operators break chiral symmetry.At O(a2)there are six symmetry breaking operators:O(6)5,O(6)6,O(6)9,O(6)10,O(6)11and O(6)14.Fermionic operators that do not break the chiral symmetry?rst appear at O(a2).Purely gluonic operators(which we have not listed above)also belong to the second type of operators as they are trivially invariant under chiral transformations.They too enter at O(a2).
The operator O(6)4deserves special attention.While respecting the chiral symmetries it is not invariant under O(4)rotations.This means that it does a?ect the structure of the chiral Lagrangian by inducing O(4)symmetry breaking terms in it.The analysis leading to the Symanzik action reveals that such terms must be at least of O(a2).
C.Spurion analysis
At O(a0),the Symanzik action is QCD-like.For small a and m we assume the lattice theory to exhibit the same spontaneous symmetry breaking pattern SU(N f)L?SU(N f)R→SU(N f)V as continuum QCD.Consequently,the low-energy physics is dominated by Nambu-Goldstone bosons which acquire small masses due to the soft explicit symmetry breaking by the small quark masses and discretization e?ects.The low-energy chiral e?ective?eld theory is written in terms of these light bosons.
To construct the chiral Lagrangian we follow the standard procedure of spurion analysis. We write a term in the Symanzik Lagrangian as C0O where O contains the?elds and their derivatives,and C0is the remaining constant factor.For symmetry breaking terms,O changes under a chiral transformation of the fermionic?elds,O→O′.We then promote C0 to the status of a spurion C,with the transformation C→C′such that CO=C′O′.The chiral e?ective theory is constructed from the Nambu-Goldstone?elds and the spurions with the requirement that the action is invariant under chiral transformations if the spurions are transformed as well.Once the terms in the chiral Lagrangian are obtained,each spurion is set to its original constant value C=C0.This procedure guarantees that the chiral e?ective theory explicitly breaks chiral symmetry in the same manner as the underlying theory de?ned by the Symanzik action,and reproduces the same Ward identities.
It might appear that one needs many spurion?elds to accommodate all the symmetry breaking operators in the Symanzik action.However,this is not the case.Two spurions
which transform in the same way will lead to the same terms in the chiral Lagrangian, and so it is enough to consider only one of them.This is discussed in appendix B.Since we organize the chiral perturbation theory as an expansion in m and a,we do distinguish between spurions that transform the same way but have di?erent m or a dependence.
In the following we list the representative spurions.Shown are the transformation rules for the di?erent spurions under chiral transformations,and the constant values to which the spurions are assigned in the end.
O(a0):M→LMR?,M?→RM?L?,(12)
).
M0=M?0=m=diag(m1,...,m N
f
This makes the mass termψR M?ψL invariant under the chiral transformations of eq.(4).
O(a):A→LAR?,A?→RA?L?,(13)
A0=A?0=aI,
where I is the?avor identity matrix.
O(a2):B≡B1?B2→LB1R??LB2R?,B?≡B?1?B?2→RB?1L??RB?2L?,(14) C≡C1?C2→RC1L??LC2R?,C?≡C?1?C?2→LC?1R??RC?2L?,
B0=B?0=C0=C?0=a2I?I.
These spurions are introduced to make the symmetry breaking4-quark operators invariant and therefore carry four?avor indices(see Ref.[37]and references therein).Consider,for example,the operator
(ψψ)=(ψLψR)+(ψLψR)+(ψRψL)+(ψRψL).(15) The?rst term on the right hand side can be made invariant with the spurion B as can be seen from
BψLψR=B ijkl(ψL)k(ψR)l=ψL B2ψR.(16) Similarly,all the other symmetry breaking4-quark operators can be made invariant using the spurions B,C and their hermitian conjugates.At O(a2)one also needs to consider the squares of the O(a)spurions.Note,however,that A2,A?A,(A?)2and AA?transform exactly like B,C,B?and C?,respectively,and therefore need not be treated separately.
D.Chiral Lagrangian
The chiral Lagrangian is expanded in powers of p2,m and a.Generalizing the stan-dard chiral power counting,the leading order Lagrangian contains the terms of O(p2,m,a), while the terms of O(p4,p2m,p2a,m2,ma,a2)are of next-to-leading order.In terms of the dimensionless expansion parameters m/Λχand aΛχ,whereΛχ≈1GeV is the typical chiral symmetry breaking scale,this power counting assumes that the size of the chiral symmetry breaking due to the mass and the discretization e?ects are of comparable size.1 For the Wilson action,all next-to-leading order terms have already been computed in Ref.[7],except for the O(a2)terms.We are now in the position to calculate these contri-butions,which are the ones associated with the spurions B and C.We?nd the following three new terms(and their hermitian conjugates)
B1Σ? B2Σ? →a2 Σ? 2,(17)
B1Σ?B2Σ? →a2 Σ?Σ? ,(18)
C1Σ C2Σ? →a2 Σ Σ? .(19) HereΣ=exp(2iΠ/f),withΠbeing the matrix of Nambu-Goldstone?elds.Σtransforms under the chiral transformations in eq.(4)asΣ→LΣR?.The angled brackets are traces over?avor indices,and the arrows indicate assigning B=B0,C=C0,according to eq.(14).
So far we only considered the operators in the Symanzik action that explicitly break chiral symmetry.Operators that do not break chiral symmetry also contribute at O(a2). These operators do not add any new terms to the chiral Lagrangian,but simply modify the coe?cients in front of already existing operators.At leading order,for example,the kinetic term is f2
1A more detailed discussion of the power counting scheme is given in section IV
hypercubic symmetry,an operator must carry at least four space-time indices.In the chiral Lagrangian,these are provided by the partial derivative?μ,hence the operator is at least of O(p4).Adding the fact that it is also an O(a2)e?ect,we see that the leading O(4) symmetry breaking terms in the chiral Lagrangian are of O(p4a2)(an example is the operator a2 μ ?μ?μΣ?μ?μΣ? ).Hence,up to the order considered here,O(4)breaking terms can be excluded from the analysis.
Finally we can write down the terms of O(a2)which enter the next-to-leading order chiral Lagrangian.In terms of the two parameters
?m≡2B0m=2B0diag(m1,...,m N
f
),?a≡2W0a,(20) which have been introduced in Ref.[16],2these terms are
L a2 =??a2W′6 Σ?+Σ 2??a2W′7 Σ??Σ 2??a2W′8 Σ?Σ?+ΣΣ .(21) The coe?cients W′i are new unknown low-energy constants.Putting it all together,also quoting the terms in the Lagrangian of O(a)from Ref.[7],3we?nd
Lχ=f2
4 ?mΣ?+Σ?m
??a f2
2Unlike in Ref.[16],here we de?ne?a without the factor of c SW.The coe?cient c SW is not kept explicit as we do not use equations of motion,and S1contains ac(5)1O(5)1besides the Pauli term ac SW O(5)2.
3There are some typos in the Lagrangian[eq.(2.10)]in Ref.[7].These are corrected in eq.(22).
E.Partially quenched QCD
Partially quenched QCD is formally represented by an action with sea,valence and ghost quarks[38].We collect the quark?elds inΨ=(ψS,ψV),whereψS describes the sea quarks, andψV contains both the anticommuting valence quarks and commuting ghost?elds.The same is done for the anti quark?elds.The mass matrix is given by m=diag(m S,m′V), with m S being the N f×N f mass matrix for the sea quarks and m′V=diag(m V,m V)is the 2N V×2N V mass matrix for the valence quarks and valence ghosts.
We consider partially quenched lattice QCD with Wilson’s fermion action eq.(1)for all three types of?elds.The discrete symmetries and the color gauge symmetry is as in the unquenched case.The group of chiral?avor transformations,however,is di?erent.If all the masses and the Wilson parameter r are set to zero,the action is invariant under transformations in the graded group4
G P Q=SU(N f+N V|N V)L?SU(N f+N V|N V)R.(23)
Based on the symmetries of the lattice theory the Symanzik action for partially quenched lattice QCD is obtained as before.The result is easily quoted:One can simply replaceψand
Ψbecause the only two-and four-quark operators that are invariant under the extended, graded?avor group are still
4See Ref.[39]for a more honest discussion of the symmetry group of partially quenched QCD.
III.MIXED ACTION
In this section we consider a lattice theory with Wilson sea quarks and Ginsparg-Wilson valence quarks.As before we?rst construct Symanzik’s e?ective action through O(a2).We then derive the chiral Lagrangian for this theory.
https://www.doczj.com/doc/2c15146054.html,ttice action
The use of di?erent lattice fermions for sea and valence quarks is a generalization of partially quenched lattice QCD.Theoretically it too is formulated by an action with sea and valence quarks and valence ghosts.However,in addition to allowing di?erent quark masses (m S=m V),the Dirac operator in the sea sector is chosen to be di?erent from the one for the valence quarks and ghosts.For this reason we will refer to this type of lattice theories as“mixed action”theories.
The mixed action theory with Wilson sea quarks and Ginsparg-Wilson valence quarks is de?ned in Ref.[16].We refer the reader to this reference for details and notation.Here we just quote that the?avor symmetry group of the mixed lattice action is
G M=G Sea?G V al,(24)
G Sea=SU(N f)L?SU(N f)R,
G V al=SU(N V|N V)L?SU(N V|N V)R.
The quark mass term in the mixed action breaks both G Sea and G V al.However,in the massless case G V al becomes an exact symmetry[29]while G Sea is still broken by the Wilson term.Because of the di?erent Dirac operators there is no symmetry transformation that mixes the valence and sea sectors,in contrast to the partially quenched case[cf.eq.(23)].
B.Symanzik action
The Symanzik action for the mixed theory can be derived using the results of the previous section.It is convenient to separately discuss three types of terms-those that contain only sea quark?elds,those that contain only valence?elds,and those that contain both.
For the?rst type of terms the analogy with the previous section is evident:the relevant symmetry group is G Sea=G,and the explicit symmetry breaking structure is the same.
Thus,all bilinear operators O(n)i(ψ)and4-quark operators O(n)i(ψ,ψ),listed in subsection II B,appear in Symanzik’s action,onceψis replaced byψS.5
The construction of the purely valence terms is also analogous to the one for the Wilson action in subsection II B.However,there are stricter symmetry constraints for Ginsparg-Wilson quarks and ghosts because the Ginsparg-Wilson action possesses an exact chiral symmetry when the quark mass is set to zero.All operators without any insertions of the quark mass must therefore be chirally invariant.In particular,all dimension-3and dimension-5operators are forbidden.Several dimension-6operators at O(a2)are also ex-cluded.Only the bilinears of eq.(9)and the4-quark operators O(6)i(ψV,ψV),i=7,8,12, and13,of eq.(10),are G V al invariant and are therefore allowed.
For terms of the third type,note that the symmetry group G M forbids bilinears that mix valence and sea quarks.Thus,the only terms containing both sea and valence?elds are 4-quark operators which are products of two bilinears-one from each sector.Again,only the four terms O(6)i(ψS,ψV),i=7,8,12,and13,are allowed.All the others break the chiral symmetry in the valence sector when m V=0.
From these considerations it follows that the Symanzik action for the mixed lattice action up to and including O(a2)contains the following terms:
S?1:O(3)1(ψS),(25) S0:O(4)i(ψS),O(4)i(ψV),i=1,2.(26) S1:O(5)i(ψS),i=1,2.(27) S2,bilinears:O(6)i(ψS),O(6)i(ψV),i=1?4.(28)
S2,4-quark operators:O(6)i(ψS,ψS),i=5?14,(29)
O(6)i(ψV,ψV),O(6)i(ψS,ψV),i=7,8,12,13.
As before we have not listed operators that are obtained by multiplying lower dimensional operators with powers of the bare mass.
ψ1?Jψ1
C.Spurion analysis
S0,the leading term in the Symanzik action,6is just the continuum action of partially quenched QCD.In the m→0limit it is invariant under the?avor symmetry group G P Q in eq.(23),which is larger than G M[eq.(24)],the symmetry group of the underlying lattice action.For a su?ciently small a(and m),S0determines the spontaneous symmetry breaking pattern and the symmetry properties of the Nambu-Goldstone particles in the theory.It follows that the mixed theory contains the same set of light particles as partially quenched QCD.
For the construction of the chiral e?ective theory we introduce spurion?elds that make the entire Symanzik action invariant under G P Q.Notice that all the operators proportional to a and a2break G P Q,the?avor symmetry of the leading term.This is obvious for operators which appear with sea quark?elds only,like the dimension5operators.However,even if an operator appears“symmetrically”inψS andψV,as in eq.(28),it still breaks G P Q.To illustrate this point let us consider any of the bilinear terms,suppressing allγmatrices and color group generators.Any bilinear that is invariant under all rotations of G P Q must have the?avor structureψSψS+ψSψS and
6As before,we assume that the sea quark mass renormalization e?ectively accounts for S?1–see eq.(11).
valence quarks and ghosts).
O(a0):M→LMR?,M?→RM?L?,(31)
M0=M?0=m=diag(m S,m′V).
O(a):A→LAR?,A?→RA?L?,(32)
A0=A?0=aP S.
The last spurion arises from the sea sector symmetry breaking terms at O(a).
All the quark bilinears at O(a2)couple?elds with the same chirality.Since there are bilinears for both sea and valence?elds we obtain the following spurions:
O(a2),bilinears:B→LBL?,C→RCR?,(33)
B0,C0∈{a2P S,a2P V}.
We can distinguish two types of4-quark operators.The?rst type is made of bilinears which couple?elds of the same chirality only.These operators appear with only sea or valence?elds as well as in the“mixed”form O(ψS,ψV).The remaining4-quark operators, which couple?elds with opposite chirality,appear only with sea quarks.We therefore introduce the following spurions:
O(a2),4-quark operators:
D≡D1?D2→LD1L??LD2L?,E≡E1?E2→RE1R??RE2R?,
F≡F1?F2→LF1L??RF2R?,G≡G1?G2→RG1R??LG2L?,
D0,E0,F0,G0∈{a2P S?P S,a2P S?P V,a2P V?P V},(34)
H≡H1?H2→LH1R??LH2R?, J≡J1?J?2→LJ1R??RJ?2L?,H?≡H?1?H?2→RH?1L??RH?2L?, J?≡J?1?J2→RJ?1L??LJ2R?,
H0=H?0=J0=J?0=a2P S?P S.(35) Squaring the spurions of O(a)does not lead to any new spurions.
D.Chiral Lagrangian
The chiral Lagrangian for the mixed action theory including the cut-o?e?ects linear in a is derived in Ref.[16].Terms of O(a2)are constructed from the spurions in eqs.(33-35).It
is easily checked that the spurions B,C,D and E lead necessarily to operators higher than O(a2)[at least O(p2a2,ma2)],so we can ignore them.From the other spurions we obtain the following independent operators(and their hermitian conjugates):
F1ΣF2Σ? →a2 τ3Στ3Σ? ,(36)
H1Σ?H2Σ? →a2 P SΣ?P SΣ? ,(37)
H1Σ? H2Σ? →a2 P SΣ? P SΣ? ,(38)
J1Σ? J?2Σ →a2 P SΣ? P SΣ .(39)
For eq.(36)we use the fact that P S=1
2
(I?τ3),withτ3=diag(I S,?I V). When assigning F1,2=(I±τ3)and expanding,the?eldsΣandΣ?are next to each other and cancel whenever the identity matrix is inserted,so the only non-trivial operator is the one shown in eq.(36).
We conclude that for the mixed action theory with Wilson sea and Ginsparg-Wilson valence quarks the terms of O(a2)in the chiral Lagrangian are
L a2 =??a2W M τ3Στ3Σ? (40)??a2W′6 P SΣ?+ΣP S 2??a2W′7 P SΣ??ΣP S 2??a2W′8 P SΣ?P SΣ?+ΣP SΣP S . The parameters?m and?a are de?ned as in the unquenched case in eq.(20).Note that the projector P S in the last three terms implies that these operators involve only the sea-sea block ofΣ.
The?nal result,including the terms from Ref.[16],reads
Lχ=f2
4 ?mΣ?+Σ?m
??a f2
The chiral Lagrangian for the mixed action theory at O(a2)has4terms while there are only3terms at this order in the chiral Lagrangian for the Wilson action.The reason that the mixed theory has an additional operator(and consequently an additional unknown low-energy constant multiplying it)is its reduced symmetry group,G M in eq.(24),compared to G P Q in eq.(23).The use of di?erent Dirac operators for sea and valence quarks for-bids transformations between the sea and valence sectors,and allows the additional term τ3Στ3Σ? in eq.(40).
The presence of more terms in the Lagrangian does not entail that chiral expressions for all observables in the mixed theory depend on more free parameters than inχPT for the Wilson action.By de?nition,the correlation functions measured in numerical simulations involve operators that are made of valence quarks only,and the enhanced chiral symmetry of the Ginsparg-Wilson?elds plays an important role in that sector.The chiral symmetry leads to constraints on operators in the Symanzik action that contain valence?elds,and ultimately it restricts and simpli?es the form of chiral expressions for valence quark observables.This can already be seen by considering the terms in L[a2]with coe?cients W′6,W′7and W′8[see eq.(40)].These terms depend only on the sea-sea block ofΣ.This entails that all the multi-pion interaction vertices obtained from these terms necessarily contain some mesons with at least a single sea quark in them.Consequently,these terms cannot contribute at tree level to any expectation value of operators made entirely out of valence?elds.This is easily understood:the W′terms arise from the breaking of chiral symmetry in the sea sector by the Wilson term,and this breaking is communicated to the valence sector through loop e?ects only.A more concrete demonstration of this point is provided by the calculation of the pseudo scalar valence-valence meson mass in the next section.
IV.APPLICATION
We conclude our analysis of the chiral e?ective theories for the Wilson action and the mixed action theory with an explicit calculation of the light meson masses.Before presenting the calculations,however,a discussion of the chiral power counting is appropriate.
A.Power counting
χPT reproduces low-momentum correlation functions of the underlying theory,provided that the typical momentum p and the mass of the Nambu-Goldstone boson M NGB are su?ciently small,p?Λχand M NGB?Λχ.The standard convention is to consider p and M NGB as formally of the same order,and take a single expansion parameter?～M2NGB/Λ2χ～p2/Λ2χ.Thus,a typical next-to-leading order(1-loop)expression for a correlation function inχPT has the structure
C=C LO+C NLO+···,(42)
C LO=O(?)=O M2NGBΛ2χ ,
C NLO=O(?2)=O M4NGBΛ4χ,M2NGB p2
fermions.If,in addition,the lattice spacing in a simulation is large such that a2Λ2χ～m/Λχ, an expansion in two parameters may be required,and the leading order contributions are O(p2/Λ2χ,m/Λχ,a2Λ2χ).
B.Pseudo scalar meson masses
We now turn to the calculation of the pseudo scalar meson masses.As in Refs.[7]and [16],we only consider mesons with di?erent valence?avor indices(A=B).For the partially quenched Wilson action we?nd
1
M2AB=( m Val+ a)+
( m Val+ a)[N f(L4 m Sea+W4 a)+L5 m Val+W5 a]
f2
8N f
+
[L8 m2Val+W8 m Val a+W′8 a2]+O(?3),(43)
f2
where N f is the number of sea quark?avors.
Next,we consider the mixed action theory.As explained in section III,the terms in the Lagrangian at O(a2)with coe?cients W′6,W′7and W′8[see eq.(40)]cannot contribute at tree level to any expectation value of operators made entirely out of valence?elds.In particular,at next-to-leading order they do not enter the self energy of mesons made out of valence quarks,and thus do not contribute to their mass.In addition,the leading order meson mass(M2AB)LO in the mixed action theory is independent of a,which means that contributions at O(?2)of the type a(M2NGB)LO(from the terms proportional to W4and W5 )are only linear in a.Thus,the only possible source for a2corrections to the mass comes from the term in L[a2]with coe?cient W M.However,an explicit calculation shows that this contribution vanishes and there are no O(a2)corrections to the pseudo scalar meson mass.The expression for M AB is therefore the same as in Ref.[16],which we quote here for
completeness:7
1
M2AB= m Val+
f2 m Val[(L5?2L8) m Val+N f(L4?2L6) m Sea+N f(W4?W6) a]+O(?3).
The fact that there are no O(a2)contributions at next-to-leading order is not as surprising as one might think at?rst.Only the valence quark mass term breaks the chiral symmetry for Ginsparg-Wilson fermions.Hence the pseudo scalar meson mass is proportional to the quark mass and vanishes in the limit m V al→0.It follows that any lattice contribution to M2AB is suppressed by at least one factor of m V al,and the largest lattice correction quadratic in the lattice spacing is of O(m V al a2).Note that this higher order term becomes the leading discretization e?ect in the meson mass if an O(a)improved Wilson action is used for the sea quarks.This example illustrates the bene?cial properties of Ginsparg-Wilson fermions which are preserved even in the presence of a“non-Ginsparg-Wilson”sea sector.
V.SUMMARY
In the previous sections we presented the chiral Lagrangian for two lattice theories:one with Wilson fermions and the other with Wilson sea fermions and Ginsparg-Wilson valence fermions.One consequence of the analysis is that corrections to the low-energy constants of continuumχPT(coming from symmetry-conserving discretization e?ects)are of O(a2). Since the coe?cients in the chiral Lagrangian themselves multiply terms of O(p2)(B0and f)and O(p4)(Gasser-Leutwyler coe?cients),such e?ects can only be detected by measuring observables at the accuracy of O(a2p2)and O(a2p4),respectively.Another important dis-cretization e?ect that enters the Symanzik action at O(a2)is the breaking of O(4)rotational invariance.An O(4)breaking term in the chiral Lagrangian,however,must contain at least four derivatives,so it is a higher order term as well(at least O(a2p4)).
The main purpose of constructing chiral e?ective theories for lattice actions is to capture discretization e?ects analytically and to guide the chiral extrapolations of numerical lattice data.This is achieved by the explicit a dependence of observables that can be calculated in these e?ective theories.In particular,the chiral Lagrangian is su?cient for the determina-
《计算机绘图A 》 学号： 第一次作业 姓名：梁陈荣 主观题（共22小题） 19.写出下面AutoCAD 常用工具栏上引出线所指按钮的含义或 功能。 答：用户界面 （1）构造线 （2）样条曲线 （3）图块定义 （4）图案填充 （5）复制 （6）比例缩放 （7）打碎 （8）打印预览 （9）特性匹配 （10）返回上一次显示画面 20.写出AutoCAD 中图层工具条上引出线所指按钮及图标的含 义。 把对象的图层 设置为当前层 答：图层对话框、图层打开图标、图层解冻图标、图层开锁图标、图层颜色 图标 21. AutoCAD 点的定位，可以用键盘敲入点的极坐标，其形式为100<45 （试举例说明），其中前者表示点到原点的距离，后者表示点到原点 连线与水平的夹角。 22.AutoCAD 中许多命令的名字很长，为了节省敲键时间，AutoCAD 给 一些命令规定了别名，它们通常是原始命令的头一个或几个字母。 23．AutoCAD 命令中的字母大小写是 等价的 。一条命令敲完后要敲 空格 键或 回车 键结束键入。 24．AutoCAD 中点的相对坐标是相对于 前一点的 ， 相对坐 标在使用时要在坐标数字前键入一个 @ 符号。 （5） （1） （6） （7） （8） （9） （10） （3） （2） （4）
25．AutoCAD中有些命令的名字前面加了一个连线符“-”，这样的命令表示其选项和参数是在命令行操作的。 26．AutoCAD中有些命令可以透明地执行，即在别的命令执行过程中执行它。透明使用这样的命令时要在命令名前键入一个单引号（’）号。 27．AutoCAD点的定位，可以用键盘敲入点的绝对坐标，绝对坐标是以原点为基准进行度量的。也可以用键盘敲入点的相对坐标，相对坐标用 @ 符号表示，它后面的数字是相对于前一点的。 28．AutoCAD中角度的输入，在缺省状态下是自正X方向逆时针 时针度量的，通常用度表示。 29．对图形进行编辑时，需要从图上选取目标构成选择集。其中窗口方式用一个矩形方框选择，它表示矩形方框内的实体是被选中的目标，与方框边界相交的实体不是被选中的对象。交叉窗口方式也是用一个矩形方框选择，它表示与矩形方框相交的图形 是选中的对象。 30．夹点是布局在实体上的控制点，这些点以小方格的形式显示出来，当夹点出现后可以直接对实体进行拉伸、平移、旋转、镜像等操作。 31．AutoCAD中图层可以想像成没有厚度的透明图片。通常把一幅图的不同颜色线型和内容的图形画在不同的图层上，而完整的图形则是所有图层的叠加。各个图层具有统一的坐标
剑桥国际英语教程 《剑桥国际英语教程》是专为非英语国家的学习者编写的大型英语教程，是国际上最受欢迎，最有影响的英语教材之一。教程的内容包括听、说、读、写四种技能。同时进行语言训练和词汇扩展，尤其强调听说技能和英语交际能力的培养。该书主要使用美国英语，但是其内容并不局限于某个国家、地区或文化，而是反应了英语作为国际交流用语的丰富性和多元性。该套教材的主要产品包括学生用书（附赠词汇手册）、教师用书、练习册、音带或CD、录像教材、DVD和CD-ROM等。 《剑桥国际英语教程》由两个系列丛书组成，共分六级。New Interchange系列包括4个级别。入门级，1级，2级，3级。入门级：针对没有英语基础的“真正”初学者，讲解基础语法结构、词汇和语言功能。1级：针对有初级英语水平的学习者，旨在进一步培养语法、词汇和语用技能，使学习者达到初级偏高水平。2级：针对有初级偏高英语水平的学习者，旨在进一步培养语法、词汇和语用技能，使学习者达到中级水平。3级：针对有初中级英语水平的学习者，培养学习者用比较流利、精确的英语进行交际的能力。本教材通过大量富于启发性、挑战性的练习活动，使学生能够进一步巩固和发展他们用英语交际的能力。除应用型技能练习外，教材还设计了各种更高级的语篇理解能力练习。听力练习包括：听叙述、听广告、听讨论、听采访。阅读活动包括：跨文化交际主题，生活方式主题，不同的价值观主题等取材真实的阅读篇章。完成以上4级的学习之后，学习者可
以达到中级偏高水平。Passages系列包括两个系别。NNew Interchange的后续高级教程。 新版《剑桥国际英语教程》（第3版）（Irnterchange Third Edition）是《剑桥国际英语教程》（New Interchange）的全面修订版。作为世界上最受欢迎，最有影响的英语教程之一，《剑桥国际英语教程》推动了中国传统语言学习模式的革新，加快了我国英语教学的国际化进程。新版总结了上一版在全球的课堂实践经验，为读者奉献了更时尚的内容、更丰富的语法训练和更多的听说实践机会。 本套教材的主要产品包括学生用书（附赠词汇手册）、教师用书、练习册、音带或CD、录像教材、DVD和CD-ROM等。另外，学生用书和练习册分两个版本——全一册和A、B分册。便于广大师生根据需要选择。录像教材可以作为视听说培训教材单独使用。 本教程共分4级：本书是《剑桥国际英语教程第》3级。
剑桥国际英语教程 本人是个学生，依照自己学习英语的经历发表一些浅见。请各位不要将以下内容做商业用途。 误区： 1、在我看来，简单地评判一本英语教材是否优秀，可以观察以下几点：1. 有无CEFR欧洲标准评级，2. 是否是近10年出版3. 是否是英美出版社出版（或英美引进） 1：CEFR是最重要的欧洲各语言标准，分为A1-C2六个等级，若不清楚可以上网查一下。教材若没有明确指出，则教材要么落伍20年以上，要么就不符合常规的CEFR大纲的英语教学，对于一般学生来说不选为妙。2：CEFR标准和考试从上个世纪开始实施，相关教材出版过很多，教材内容也不断更新，当今能继续出版的多是更先进的教材，不建议买10年以前的教材，最好是近5年的书。一本脱节的教材因为所谓的“经典”而一直用下去是错误的(新概念1/2的几乎每篇课文我都背过，这个问题是毋庸置疑的)。3：我从未见过母语为中文的作家出过任何优秀的英语教材（不包括参考书），这其实很好理解，你见过哪个中文学生从小学习外国人出的语文书吗？且英美是当今教育实力最强的两个国家，优秀教材数不胜数，没有什么理由选非英美甚至国内的教材。下面会抽一些国内知名垃圾教材说明。
当然以上标准也有例外情况，但我见过的大多数教材都符合以上要求。 2、国内知名教材的简单评价：国内最出名的教材无非以下几种：初高中教科书、新概念、赖世雄美语、走遍美国等，很显然，按照上面的标准，它们都过于落后了。这些教材与近五年新出的英语教材的差距都在二十年以上，是骡子是马直接买本下面推荐的优秀教材比对就能看得一清二楚（当然第一次见到这种教材可能不适应），我没必要具体说明。 3、对我国英语教育的吐槽：国内多数学校，包括许多重点中学（外国语除外）仍是使用落后的教科书，如旧版人教英语教科书本世纪初出版，质量本就一般，且竟过了近15-20年才再版，而我大致浏览了新版，发现并没有先进多少，依旧落后。唯一值得欣慰的是少部分大学教材直接引进了英美先进教材，但大概学的大学生也少之又少。而课外的培训机构仍大多迷信几十年前的极其落伍的教材如新概念，那些先进教材则因为教授所赚的利润不大而被遗弃，导致如今竟仍有人觉得新概念是宝。而国内多数中大学生的英语能力如何呢？中考2000词汇量，A2水平，高考4000词汇量，B1水平，四六级6000词汇量，B2水平。且中高考对听说写的要求不高或只以应试为准，实际远没有达到以上级别。这可是在连续学了至少10年英语的背景上！英语难吗？总比中文简单，我见过有些小孩小学就过了B2甚至C1的
1.重点词汇及短语 first name名字；last name姓；nice美好的；meet 见面；sorry 对不起；again 再，又；spell 拼写；classmate同班同学；single 单身的；married 已婚的；female 女性；male 男性；Mr. 先生；Miss 小姐；Ms. 女士，小姐；Mrs. 夫人；great 极好的，棒的；not bad. 不错；excuse me 对不起，打扰一下. over there在那边 good morning/afternoon/evening 早上/下午/晚上好 See you later. / See you tomorrow. 明天见。/ 一会儿见。 Good night. 晚安；Thank you. / Thanks谢谢 Work phone工作电话；Home phone 家庭电话；Cell phone 手机； 2.重点句型 My name is … . / Our names are … . / Y our name is … . His name is … . / Her name is … . / Their names are … . I’m … . / He is … . / She is … . / We are … . / They are … . It’s nice to meet you. 很高兴见到你。 Nice to meet you, too. 也很高兴见到你。 What’s your last name again? 再问下你的姓是什么？ It’s Carson. 是Carson How do you spell your name? 你的名字怎么拼？ How are you? 你好吗？ I’m just fine / I’m ok. / Good. / Great. 我很好。/ Not bad. 不错。So-so. 一般。How about you? 你呢？ This is your book. 这是你的书 This is Jennifer. 这位是Jennifer。 Y ou are in my class, right? 你在我们班，对吗？ Are you Steven Carson? 你是Steven Carson吗？ See you later. 一会见。/ See you tomorrow. 明天见。 Good night. 晚安。 Have a nice day. 过得愉快。 Have a good evening. 有个美好的晚上。 3.语法点总结 1) 代词主格与be动词的搭配形容词性物主代词 缩写完整形式(后面须加名词使用) I’m I am my you’re you are your he’s he is his she’s she is her it’s it is its we’re we are our they’re they are their 2) 元音字母：Aa Ee Ii Oo Uu
2016年秋西南大学网教[0921]《计算机绘图》 新版作业及参考答案 1、线性标注按钮是（），半径标注按钮是（），直径标注按钮是（），角度标注按钮是（），形位公差框格标注按钮是（）？(B) A、BCDE B、ABCDH C、ABCGH D、ABCDF 2、要设置多重引线样式，应点击按钮（），在“多重引线样式管理器”中进行设置。要进行多重引线标注，应点击按钮（）。(C) A. B B. AC C. A D. BA 3、设置尺寸标注样式要点击“标注”工具栏中的按钮（D ），打开“标注样式管理器”进行设置。(D) 4、在命令行输入（），可激活单行文本命令；单击命令按钮（），可激活多行文本命令；（）已书写好的文本，可自动进入编辑状态，进行相应修改。(C) A. BC B. DH C. ADF D. EF 5、旋转命令按钮是（），倒直角命令按钮是（），修剪命令按钮是（）打断命令按钮是（），倒圆角命令按钮是（），延伸命令按钮是（）缩放命令按钮是（）(A)
A. CJFIKGD B. DFGIABC C. ADBCEFG D. BEDFGIA 6、移动命令按钮是（），复制命令按钮是（），偏移命令按钮（），镜像命令按钮是（），阵列命令按钮是（）。(D) A. BCDE B. CDEF C. DEFA D. FBDCE 7、矩形命令按钮是（），圆命令按钮是（），图案填充命令按钮是（）。(B) A. BC B. ABE C. DF D. EF 8、直线命令按钮是（），多段线命令按钮是（），正多边形命令按钮是（）。(B) A. BC B. ACD C. ACE D. ADC 9、“正交模式”的按钮是（），“极轴追踪”的按钮是（），“对象捕捉”的按钮是（），“对象捕捉追踪”的按钮是（）。(B) A. BCD B. CGBI
Third Edition (第3版) interchange 剑桥国际英语教程 Jack C. Richards 词汇手册入门级 Unit 1 it’s nice to meet you afternoon n. 下午 again ad. 又，再 am (=’m) v. 是，在 and conj. 和，与 are (=’re) v. 是，在 book n. 书 (English / math) class n. （英语/ 数学）课 1 classmate n. 同班同学 country n. 国家；国土 evening n. 傍晚 famous a. 著名的 female n. 女子 first a. 第一的 first / last name 名字/ 姓 he pro. 他 her a. 她的 2 his a. 他的 I pron. 我 in (my class) prep. 在（我的班）里 is (=’s) v. 是，在 it pron. 它 last a. 最后的 male n. 男子 married a. 已婚的 3 math n. 数学 Miss n. (未婚)女士,小姐 morning n. 早晨 Mr. n. 先生 Mrs. n. 夫人，太太 Ms. n. 女士 my a. 我的 name n. 名字 nickname n. 绰号 4 night n. 夜晚 no ad. 不 oh int. 噢 our a. 我们的 phone n. 电话 phone number 电话号码 popular a. 受欢迎的 she pron. 她 single a. 单身的 5 teacher n. 教师 (over) there ad. （在）那里 too (=also) 也 yes 是 you 你，你们 your 你的，你们的 Unit2 What’s this? address 地址 address book 地址簿 bag 袋子，包 behind 在……后面 board 写字板 book bag 书包 box 盒子 7 briefcase 公文包 （English）book （英语）书 café咖啡厅，快餐店 camera 照相机 car 汽车 cassette player 磁带放音机 CD player CD机 8 cell phone （=cellphone）手机 chair 椅子 chopstick 筷子 clock 钟，时钟 desk 书桌 （English）dictionary （英文）词 典 DVD player DVD机 earring 耳环 9 encyclopedia 百科全书 eraser 橡皮 exercise 练习，训练 glasses 眼镜 gone 丢失的 great 好的，伟大的 10 I bet 我认为…… in front of 在……前面 interesting 有趣的 key 钥匙 location 地点，场所 notebook 笔记本 now 好了（用于停顿或引起别人 注意） 11 on 在……前面 open 打开 pen 钢笔 pencil 铅笔
计算机绘图作业及答案 计算机绘图作业答案 第一章总论 作业一：思考题及答案(选做)： 1．计算机绘图系统由几部分组成，它们分别包括那些基本内容 答：计算机绘图系统由硬件和软件两部分组成。 硬件包括主机、大容量外存贮器、图形输入和图形输出设备，其中图形显示器、打印机、绘图机和键盘为微机绘图所需外部设备的基本配置。 软件包括系统软件(如：Windows系统)、支撑软件、应用软件(如：AutoCAD)。 2．利用AutoCAD绘图的优势有那些 答：效率高、精度高、易修改、出图质量好。 第二章 AutoCAD二维绘图基础 作业一： 一．基础知识填空题及答案(必做) 1．AutoCAD 2004的整个绘图与编辑过程都是用一系列命令完成的，这些命令一般都是通过(下拉菜单)、 (屏幕菜单)、 (工具栏)、 (命令行输入)等方式执行的。 2．AutoCAD2004的菜单栏可分为(标准菜单)和(右键快捷菜单)两大类。 3．将鼠标放置在工具栏边界上，按住鼠标左键不放，可以将该工具栏拖放到屏幕上的任意位置。工具栏位于屏幕中间区域时，称为(浮动工具栏)，工具栏位于屏幕边界时会自动调整其形状成初始大小，此时称为(固定工具栏)。 4. 缺省设置下的绘图区窗口没有任何边界，是一个无限大的区域，这个缺省设置下的绘图区窗口叫做AutoCAD2004的(模型空间Model)。 5. 在绘图区的下部有3个小标签Model、Layoutl、Layout2，它们分别用于在(模型空间)和(图纸空间)之间进行切换。
6. 绘图界限就是标明用户的工作区域和图纸的边界，以防止用户绘制的图形超出该边界，绘图界限的设置一般有(单击菜单栏[格式]／[图形界限]命令)和(在命令行中输入[1imits]命令)两种方式。 7. 在使用AutoCAD 2004绘图时，通常用坐标来精确定位点的位置，默认的坐标系是(世界坐标系(WCS))，但在绘图过程中，经常要修改坐标系的原点和方向，AutoCAD 2004为我们提供了可变的(用户坐标系(UCS))以方便用户绘图。 8. 在精确的定位点的位置时，常采用键盘直接输入点的坐标值，一般常用的坐标输入方式有(绝对坐标)、 (相对坐标)和(极坐标)3种。 9. AutoCAD 2002提供了([放弃])命令和([重做])命令，我们能够使用它 们来取消已经做出的错误操作，以提高我们的绘图质量和绘图速度。另外，通过使用([Esc])键，AutoCAD 2004还可以轻松地取消正在执行中的错误命令，重新回到发出命令的初始状态。 10．AutoCAD 2004提供了11种画弧的方法，在执行“起点、圆心、端点方式”画弧时，在命令行提示下，如果不拾取端点而是输入“A”或“L”，就用弧的中心角或弦长来确定弧。如果输入正的中心角时，系统按(逆时针)方向画弧；如果输入负的中心角时，按(顺时针)方向画弧。 11．绘制正多边形时，先输入边数，再选择按边(Edge)或是按中心(Center of Polygon)。如果按中心绘制正多边形，有指定(外接圆半径)和(内切圆半径)两种画法。 二．基础知识选择题及答案(选做) 1．在绘图过程中，用户可以随时删除一些不用的图层，在下列选项中选取不能被删除的图层。 (ABCD)
[by:英语听力网|https://www.doczj.com/doc/2c15146054.html,|人人论坛|https://www.doczj.com/doc/2c15146054.html,|人人听力网] [00:00.00]喜欢https://www.doczj.com/doc/2c15146054.html,，就把https://www.doczj.com/doc/2c15146054.html,复制到QQ个人资料中！2.CONVERSATION [00:03.71]A.Listen and practice [00:07.03]Julia:I'm so excited!We have two weeks off!What are you going to do? [00:12.96]Nancy:I'm not sure.I guess I'll just stay home.Maybe I'll catch up on my reading.What about you?Any plans? [00:21.73]Julia:Well,my parents have rented a condominium in Florida.I'm going to take long walks along the beach every day and do lots of swimming. [00:32.18]Nancy:Sounds great! [00:33.83]Julia:Say,why don't you come with us?We have plenty of room. [00:38.19]Nancy:Do you mean it?I'd love to ! [00:41.93]3.GRAMMAR FOCUS Future with be going to and will [00:50.47]Use be going to + verb to talk about plans you've decided https://www.doczj.com/doc/2c15146054.html,e will + verb with maybe,probably,I guess,or I think to talk about possible plans before you've made a decision. [01:09.90]Where are you going to go? I'm going to go to the beach. [01:14.45]I'm not sure.Maybe I'll catch up on my reading. [01:18.71]I'm not going to take a vacation. [01:22.05]I probably won't take a vacation this year. [01:25.08]What are you going to do?I'm going to do lots of swimming. [01:30.31]I guess I'll just stay home. [01:33.36]I don't know.I think I'll go camping. [01:37.10]7.CONVERSATION [01:40.44]A.Listen and practice. [01:44.10]Lucy:Hey,Mom.I want to backpack around Europe this summer.What do you think? [01:49.30]Mom:Backpack around Europe?That sounds dangerous!You shouldn't go by yourself.You ought to go with someone. [01:58.26]Lucy:Yes,I've thought of that. [02:00.69]Mom:And you'd better talk to your father first. [02:04.42]Lucy:I already did.He thinks it's a great idea.He wants to come with me! [02:10.43]8.GRAMMAR FOCUS Modals for necessity and suggestion [02:18.21]Describing necessity [02:20.33]You have to get a passport. [02:22.92]You must get a visa for some countries. [02:26.44]You need to take money. [02:28.61]For some countries,you don't have to get any vaccinations. [02:33.16]Giving suggestions [02:35.19]You'd better talk to your father. [02:37.73]You ought to go with someone. [02:40.31]You should take warm clothes. [02:42.87]You shouldn't go by yourself. [02:45.59]9.PRONUNCIATION Ought to and have to [02:51.96]A.Listen and practice.Notice the pronunciation of ought to and have to in these
本次作业是本门课程本学期的第3次作业，注释如下： 一、单项选择题(只有一个选项正确，共23道小题) 1. AutoCAD中的（），不但可以消隐、着色，还具有体积、质量等物理属性。 (A) 线框模型 (B) 面模型 (C) 实体模型 (D) 前三种模型都可 你选择的答案： C [正确] 正确答案：C 解答参考： 2. 世界坐标系原点的位置，将（）。 (A) 始终为（0,0,0）点 (B) 由Ucs命令设定 (C) 由图形界限Limits命令设定 (D) 由视图缩放Zoom命令设定 你选择的答案： [前面作业中已经做正确] [正确] 正确答案：A 解答参考： 3. 若要使坐标系图标始终在绘图窗口的左下角显示，应选取图标Ucsicon命令的（）选项。 (A) 开ON (B) 全部A (C) 非原点N (D) 原点OR 你选择的答案： C [正确] 正确答案：C 解答参考： 4. 使用（）命令可以控制坐标系图标的显示样式、大小、可见性以及在绘图窗口中的位置。 (A) Ucs (B) 动态Ucs (C) Zoom (D) Ucsicon
你选择的答案： D [正确] 正确答案：D 解答参考： 5. 使用UCS命令的“三点”选项建立新的UCS时，其中第2点确定（）。 (A) 新原点 (B) X轴的正向 (C) Y轴的正向 (D) 以上三项都有可能 你选择的答案： [前面作业中已经做正确] [正确] 正确答案：B 解答参考： 6. 下列（）平面为世界坐标系WCS的XY坐标平面。 (A) 主视图 (B) 俯视图 (C) 左视图 (D) 右视图 你选择的答案： B [正确] 正确答案：B 解答参考： 7. 下图中左侧的坐标系图标是通过（）操作后，得到右侧的结果。 (A) 绕X轴旋转-90 (B) 绕X轴旋转90 (C) 绕Y轴旋转-90 (D) 绕Y轴旋转90 你选择的答案： [前面作业中已经做正确] [正确] 正确答案：B 解答参考： 8. “视图”工具栏上提供了（）个方向的标准视图。 (A) 4 (B) 6 (C) 8
“已知甲比乙多（少）几分之几，求乙比甲少（多）几分之几”教学设计 教学内容：已知甲比乙多（少）几分之几，求乙比甲少（多）几分之几。 教学目标：1、使学生学会掌握“已知甲比乙多（少）几分之几，求乙比甲少（多）几分之几”分数问题的解答方法，能熟练地运用“画线段图”、“举数字”、“多少+，少多-”等方法来解答。 2、进一步培养学生自主探索问题解决的能力和分析、推理和判断等思维能力，提高画线段图的能力。 教学重点：弄清单位“1”的量，能正确分析题中的数量关系，运用“画线段图”、“举数字”、“多少+，少多-”等方法来正确解答。 教学难点：画线段图分析数量关系及解题思路和解题方法。 教学准备：课件，预学、导学、练习纸。 课前游戏：玩魔方 教学过程： 一、温故知新（展示预学单） 甲数是50，乙数40。 ① 甲数比乙数多几分之几？ ②乙数比甲数少几分之几？ （50-40）÷40 = 14 （50-40）÷50 = 15 题型：求一个数比另一个数多（或少）几分之几 解题方法：（大数-小数）÷单位“1”的量 关键：找准单位“1”的量 观察——发现：当甲数比乙数多（ ），则乙数比甲数少（ ）。 师：如果不告诉甲和乙具体是多少，只告诉甲比乙多几分之几，求乙比甲少几分之几，能解决吗？我们今天就来一起研究——已知甲比乙多（少）几分之几，求乙比甲少（多）几分之几（揭示课题） 二、探究新知 （一）合作探究 若甲数比乙数多 25 ，则乙数比甲数少（ ）。 1、先让孩子在小组内合作探究 2、展学（小组作业单） （二）课件演示： 1、画线段图 乙数： 甲数： “1” 多 2 5
(1)乙数：1 甲数:1+25 = 75 (75 -1)÷75 = 27 (2)乙数：5 甲数：7 (7-5)÷7= 27 3、再画线段图（单位“1”变了） 甲数： 乙数： （三）小试牛刀 1、如果甲数比乙数少 1 4 ，那么乙数比甲数多（ ）。 ①学生自主完成（选择你喜欢的方法解答）。 ②展示学生的作业。（你是怎么想的） 2、看图填空 若梨的筐数比苹果少 2 5 ，则苹果的筐数比梨多（ ）。 （四）多少+，少多- 师：我这儿还有一种很特别的方法，想不想知道？ 观察4道题，你发现了什么？ （提示）已知条件里面的分子与答案里面的分子、已知条件里面的分母与答案里面的分母之间的关系，你能发现什么？ 方法："多少＋"，就是分子不变，分子、分母相加的和作分母； "少多－"，就是分子不变，分子、分母相减的差作分母。 三、巩固练习 1、抢答。 ①男生人数比女生少 1 6 ，那么女生人数比男生多（ ） ②小红的身高比小华高 2 3 ，那么小华的身高比小红矮（ ）。 ③如果上衣比裤子贵 3 8 ，那么裤子比上衣便宜（ ）。 ④若大豆重量比玉米轻 2 7 ，那么玉米重量比大豆重（ ）。 2、拓展（认知冲突）。 ①若甲跳绳比乙跳绳长 1 5 ，则乙跳绳比甲跳绳短 1 6 。（ ） ②若小红的年龄比小华大 1 2 岁，则小华的年龄比小红小 1 3 岁。（ ） ③若男生人数比女生多全班人数的 1 4 ，则女生人数比男生少全班人数的 1 4 。（ ） 5 2“1” 少 2 7
计算机绘图综合作业概述 摘要：经过一个学期的学习，我对CAD技术有了基本了解，并可以运用CAD2004软件绘制出基本的图形以及装配图。由于个人早知道CAD技术对于工程技术人员的重要性，所以，这一学期刚刚接触到CAD软件，也是很认真的去学习了。我相信只有掌握好CAD绘图技术，将来自己才能更好的发展。下面我将从国内外计算机二维绘图技术的发展概况、学习CAD的心得以及体会、结合装配图说明绘制步骤以及对CAD技术的总结展望四个方面来阐述本学期学习CAD绘图的所学所得。 关键字：二维软件发展优缺点心得体会装配图总结展望 一、国内外二维绘图软件技术的发展以及概况 CAD (计算机辅助设计及制造)技术产生于本世纪50年代后期发达国家的航空和军事工业中，随着计算机软硬件技术和计算机图形学技术的发展而迅速成长起来。三十几年来CAD技术和系统有了飞速的发展，CAD的应用迅速普及。在工业发达国家，CAD技术的应用已迅
速从军事工业向民用工业扩展，由大型企业向中小企业推广，由高技术领域的应用向日用家电、轻工产品的设计和制造中普及。 CAD技术经过几十年的发展，先后走过大型机、小型机、工作站、微机时代，每个时代都有当时流行的CAD软件。下面我将大概介绍国内外一些流行的软件及它们的状况。 国内：CAXA电子图版、开目Cad（子公司开发了尧创Cad与AutoCad界面相似）、清华天河（需挂在同版本AutoCad上）、机械工程师（需挂在同版本AutoCad上且已无后续开发）、中望Cad、纬衡Cad、凯图Cad、高华Cad、浩辰ICAD 、华途Cad，需挂在同版本AutoCad上）前面几种用的人较多。 国外的有中文版的大都是AutoCad、少量是TuoberCad。 对于国内的绘图软件，用的比较广泛的有CAXA电子图版、中望CAD。国外大都是AutoCad。但国内使用AutoCad的人远远多于使用CAXA电子图版、中望Cad的人。因此AutoCad是运用的最最广泛的二维绘图软件。 国外主要软件： 1.AutoCAD及MDT AutoCAD系统是美国Autodesk公司为微机开发的一个交互式绘
Unit 1 Please call me Beth. 语法点：A.含有be动词的特殊疑问句和一般疑问句及回答。 B. 代词主格(I you him/her/it we you they) 形容词性物主代词（my, your, his/her/its, our, your, their）。 1.人们怎么称呼你？What do people call you? 大家叫我Jim。Everyone calls me Jim. 2.-我来自巴西，他呢？I’m from Brazil. What/How about him? -他也是。So is he. 3.你的姓怎么拼写？How do you spell your last name? S-I-L-V-A. 4.我们在同一个数学班。We’re in the same math class. 5.让我们去打个招呼。Let’s go and say hello. 6.首尔是什么样子的？What’s Seoul like? 它真的很漂亮。It’s really very nice. 7.日子过的怎么样？How’s it going? 非常好。Pretty good. 8.我在去学校的路上。I’m on my way to school. / 他在回家的路上。He is on his way home. 9.你有空么？Are you free? 10.你和Beth在同一个班级么？Are you and Beth in the same class? 11.你和你的好朋友同岁吗？Are you and your best friend the same age? 12.一会见。See you later. 明天见。See you tomorrow. 13.祝你愉快。/祝你度过快乐的一天。Have a good day. *14. Name is an important part of your identity. 名字是你身份的重要组成部分。 *15. Why are some names unpopular? 为什么一些名字不受欢迎呢？ Unit 2 How do you spend your day? 语法点：A.含实意动词的一般现在时(单三) B. 介词（表示时间at/ in/ on/ around/ until/before/ after） 1.我带人们去南美的一些国家旅行。I take people on tours to countries in South America, like Peru. 2. 多有趣啊！How interesting! 3. 我有一份兼职工作。I have a part-time job. 4. 他在北京为一家打电脑公司工作。He works for a big computer company in Beijing. 5. 你什么时候上班？What time do you go to work? 6. 你晚上什么时候到家？When do you get home at night? 7. 难道你不认识我么？Don’t you recognize me ? 短语：care for 照顾answer the phone 接电话by the way 顺便说一下get up 起床go to bed睡觉leave work 下班get home到家stay up 熬夜wake up醒来 in the morning 在上午in the afternoon 在下午in the evening在晚上at night 在晚上on Sundays 在周日on weekdays 在工作日
2012—2013学年第2学期课程离线作业 课程名称：计算机绘图A第二次作业 班级（全称）：土木工程(工民建)2013-18班(专本) 姓名：王轶 学号： 13825497 西南交通大学网络教育学院知金教育东莞数字化学习中心
三、主观题 19.线性尺寸的起止符号为用中粗实线绘制的短斜线，其倾斜方向为与尺寸界线按 顺时针时针成45 角，长度宜为 2~3 mm。 20.尺寸界线的起始端应偏离被标注的端点 2~3 mm，终止端应超出尺寸线 2~3 mm。 21.图形中可见轮廓线使用粗实线表示,不可见轮廓线使用中粗虚线表示。圆的中心线、圆柱的轴线用细点划线表示。 22. AutoCAD中可以建立三种形式的三维模型：线框模型、表面模型、实体模型，其中可以被消隐、渲染的有表面模型、实体模型。 23.三维表面模型中的边界曲面（EDGESURF）是由用户指定的 4 条首尾相接的边界线来构造的一个三维网格曲面。边界线需事先画出，它们可 以是直线、样条曲线、圆弧等。用户选择的第 1 条边决定了网格 的M方向，与它相邻的边即为网格的N方向，系统变量 surftab1 控制M方向 上网格的划分数目，系统变量 surftab2 控制N方向上网格的划分数目。 24.在三维绘图中使用 UCS 命令可以移动坐标原点，改变坐标轴的方向，建立用户作图所使用的坐标系。二维绘图命令是针对 XOY 坐标面 设计的，在三维绘图中要使用 UCS 命令将 XOY 定义在物体的某个表面上，才可以对该表面使用二维绘图命令画图。 25.什么是目标捕捉？请写出目标捕捉操作中的４种捕捉类型，并分别解释它们的含义 解答： 目标捕捉是帮助用户迅速而准确地捕捉到可见图形实体上的特殊点，供作图定位使用。例如线段的中点、直线与圆的交点等等，目标捕捉本身并不产生实体，而是配合其他命令使用的。 例如目标捕捉操作中的以下4种捕捉类型： 端点：捕捉线段、圆弧、椭圆弧、多义线等实体的端点。 中点：捕捉线段、圆弧、椭圆弧、多义线、样条曲线等实体的中点。 交点：捕捉直线、圆、圆弧、椭圆、椭圆弧、多义线、样条曲线、构造线等实体之间的交点。 切点：捕捉圆、圆弧、椭圆、椭圆弧上与作图过程中最后一点的连线形成相切的离光标最近的点。 26.什么是目标捕捉追踪？什么是角度追踪？ 解答： 目标捕捉追踪是按与对象的某种特定关系来追踪，这种特定关系确定了一个事先并不知道的角度。目标捕捉追踪必须与目标捕捉模式同时工作。 角度追踪是按事先设定的角度增量来追踪点。
unitl about prep. 关于 after prep.（时间）在... 之后after noon n. 下午 again ad. 又,再一次agree v. 同意 alphabet n.字母表 am v.是 American n. 美国人 and conj. 和,以及 answer n.答案,回答 are v.是 article n. 文章 ask v.要求 回原处at prep.在... 里；在... 旁；向,朝；出席,参加；在... 时间back adv. bad adj. 坏的 basketball n.篮球box n.方框 boy n. 男孩
bye int.再见 bye-bye int. 再见 candy bar （单独包装的块状）糖果 chart n.图表 check v. 检查，打对勾circle v. 圈出 class n.课 classmate n. 同学 colored adj. 有颜色的,彩色的 com mon adj.常见的,普遍的complete v. 完成adj. 完整的connect v. 连接;结合con versati on n. 对话 cool adj. 酷的,绝妙的correct adj. 正确的v. 更正day n. 日子differe nt adj. 不同的disagree v. 不同意,反正 do v.［作为助动词，构成疑问句、否定句等］
dog n.狗 each adj.( 两个或两个以上人或物的:）每个,各 each other 互相,彼此easy adj. 容易的 e-pal n.网友 eve ning n. 夜晚 false adj. 不真实的，不正确的 female n. 女性 fine adj. 好的,健康的 first adj. 第一个的 first n ame 名字（不包括姓）focus n. 集中点，重点 friend n. 朋友 from prep. 从...；源... get v.得到 girl n. 女孩
计算机绘图A 第一次作业 三、主观题(共22道小题) 19. 答： （1）构造线（2）样条曲线（3）图块定义（4）图案填充（5）复制（6）比例缩放（7）打碎 （8）打印预览（9）特性匹配（10）返回上一次显示画面 图层对话框、图层打开图标、图层解冻图标、图层开锁图标、图层颜色图标 20.
答：图层对话框、图层打开图标、图层解冻图标、图层开锁图标、图层颜色图标 21.AutoCAD点的定位，可以用键盘敲入点的极坐标，其形式为（试举例说明），其中前者表示，后者表示。 答：100<45、点到原点的距离、点到原点连线与水平的夹角 22.AutoCAD中许多命令的名字很长，为了节省敲键时间，AutoCAD给一些命令规定 了，它们通常是原始命令的。 答：别名、头一个或几个字母 23.AutoCAD命令中的字母大小写是。一条命令敲完后要敲键或键结束键入。 答：等价的、空格、回车 24.AutoCAD中点的相对坐标是相对 于，相对坐标在使用时要在坐标数字前键入一个符号。 答：前一点的、@ 25.AutoCAD中有些命令的名字前面加了一个连线符“-”，这样的命令表示其选项和参数是在操作的。 答：命令行 26.AutoCAD中有些命令可以透明地执行，即执行它。透明使用这样 的命令时要在命令名前键入一个号. 答：在别的命令执行过程中、单引号（’） 27.AutoCAD点的定位，可以用键盘敲入点的绝对坐标，绝对坐标是 以为基准进行度量的。也可以用键盘敲入点的相对坐标，相对坐标用符号表示，它后面的数字是相对 于。 答：原点、@、前一点的
28.AutoCAD中角度的输入，在缺省状态下是自正X方向时针度量的，通常用表示。 答：逆时针、度 29.对图形进行编辑时，需要从图上选取目标构成选择集。其中窗口方式用一个矩形方框选择，它表示实体是被选中的目标，与方框边界相交的实体被选中的对象。交叉窗口方式也是用一个矩形方框选择，它表示是选中的对象。 答：矩形方框内的、不是、与矩形方框相交的图形 30.夹点是布局在实体上的控制点，这些点以的形式显示出来，当夹点出现后可以直接对实体进 行、、、等操作。 答：小方格、拉伸、平移、旋转、镜像 31.AutoCAD中图层可以想像成。通常把一幅图的画在不同的图层上，而完整的图形则是。各个图层具有的坐标系、绘图界限和显示时的缩放倍数。 答：没有厚度的透明图片、不同颜色线型和内容的图形、所有图层的叠加、统一 32.一个图层可以是不可见的。不可见的图层不 能、和 。 答：显示、编辑、输出 33.一个图层可以被锁定，锁定了的图层仍然，但不 能。 答：可见、编辑 34.一个图层可以被冻结，冻结了的图层除不 能、和 外，也 不。 答：显示、编辑、输出、参与图形重构运算 35.