On the uniqueness of correspondence under orthographic and perspective projections

a rXiv:083.3395v6[mat h.RT]13J ul28GENERALIZED HARISH-CHANDRA DESCENT AND APPLICATIONS TOGELF AND PAIRS AVRAHAM AIZENBUD AND DMITRY GOUREVITCH with appendix A by Avraham Aizenbud,Dmitry Gourevitch and Eitan Sayag Abstract.In the first part of the paper we generalize a descent technique due to Harish-Chandra to the case of a reductive group acting on a smooth affine variety both defined over arbitrary local field F of characteristic zero.Our main tool is Luna slice theorem.In the second part of the paper we apply this technique to symmetric pairs.In particular we prove that the pair (GL n (C ),GL n (R ))is a Gelfand pair.We also prove that any conjugation invariant distribution on GL n (F )is invariant with respect to transposition.For non-archimedean F the later is a classical theorem of Gelfand and Kazhdan.We use the techniques developed here in our subsequent work [AG3]where we prove an archimedean analog of the theorem on uniqueness of linear periods by H.Jacquet and S.Rallis.Contents 1.Introduction 21.1.Main results 21.2.Related works on this topic 31.3.Structure of the paper 3Acknowledgements 42.Preliminaries and notations 52.1.Preliminaries on algebraic geometry over local fields 52.2.Vector systems 72.3.Preliminaries on distributions 83.Generalized Harish-Chandra descent 93.1.Generalized Harish-Chandra descent 93.2.A stronger version 104.Distributions versus Schwartz distributions 125.Applications of Fourier transform and Weil representation 135.1.Preliminaries 135.2.Applications 146.Tame actions147.Symmetric pairs167.1.Preliminaries and notations167.2.Descendants of symmetric pairs187.3.Tame symmetric pairs187.4.Regular symmetric pairs207.5.Conjectures217.6.The pairs (G ×G,∆G )and (G E/F ,G )are tame 222A VRAHAM AIZENBUD AND DMITRY GOUREVITCH8.Applications to Gelfand pairs228.1.Preliminaries on Gelfand pairs and distributional criteria228.2.Applications to Gelfand pairs23Appendix A.Localization principle24 Appendix B.Algebraic geometry over localfields25B.1.Implicit function theorems25B.2.Luna slice theorem25Appendix C.Schwartz distributions on Nash manifolds26C.1.Preliminaries and notations26C.2.Submersion principle27C.3.Frobenius reciprocity28C.4.K-invariant distributions compactly supported modulo K.29Appendix D.Proof of archimedean homogeneity theorem30 Appendix E.Diagram32 References331.IntroductionHarish-Chandra developed a technique based on Jordan decomposition that allows to reduce cer-tain statements on conjugation invariant distributions on a reductive group to the set of unipotent elements,provided that the statement is known for all(strict)Levi subgroups(see e.g.[HCh]).In this paper we generalize part of this technique to the setting of a reductive group acting on a smooth affine algebraic variety,using Luna slice theorem.Our technique is oriented towards proving Gelfand property for pairs of reductive groups.Our approach is uniform for all localfields of characteristic zero-both archimedean and non-archimedean.1.1.Main results.The core of this paper is Theorem3.1.1:Theorem.Let a reductive group G act on a smooth affine variety X,both defined over a local field F of characteristic zero.Letχbe a character of G(F).Suppose that for any x∈X(F)with closed orbit there are no non-zero distributions on the normal space to the orbit G(F)x at x which are equivariant with respect to the stabilizer of x with the characterχ.Then there are no non-zero(G(F),χ)-equivariant distributions on X(F).Using this theorem we obtain its stronger version(Corollary3.2.2).This stronger version is based on an inductive argument which shows that it is enough to prove that there are no non-zero equivariant distributions on the normal space to the orbit G(F)x at x under the assumption that all such distributions are supported in a certain closed subset which is an analog of the cone of nilpotent elements.Then we apply this stronger version to problems of the following type.Let a reductive group G acts on a smooth affine variety X,andτbe an involution of X which normalizes the action of G. We want to check whether any G(F)-invariant distribution on X(F)is alsoτ-invariant.Evidently, there is the following necessary condition onτ:(*)Any closed orbit in X(F)isτ-invariant.In some cases this condition is also sufficient.In these cases we call the action of G on X tame.GENERALIZED HARISH-CHANDRA DESCENT3 The property of being tame is weaker than the property called”density”in[RR].However,it is sufficient for the purpose of proving Gelfand property for pairs of reductive groups.In section6we give criteria for tameness of actions.In particular,we have introduced the notion of”special”action.This notion can be used in order to show that certain actions are tame(see Theorem6.0.5and Proposition7.3.5).Also,in many cases one can verify that an action is special using purely algebraic-geometric means.Then we restrict our attention to the case of symmetric pairs.There we introduce a notion of regular symmetric pair(see Definition7.4.2),which also helps to prove Gelfand ly, we prove Theorem7.4.5.Theorem.Let G be a reductive group defined over a localfield F andθbe an involution of G. Let H:=Gθand letσbe the anti-involution defined byσ(g):=θ(g−1).Consider the symmetric pair(G,H).Suppose that all its”descendants”(including itself,see Definition7.2.2)are regular.Suppose also that any closed H(F)-double coset in G(F)isσ-invariant.Then every H(F)double invariant distribution on G(F)isσ-invariant.In particular,the pair (G,H)is a Gelfand pair(see section8).Also,we formulate an algebraic-geometric criterion for regularity of a pair(Proposition7.3.7).Using our technique we prove(in section7.6)that the pair(G(E),G(F))is tame for any reductive group G over F and a quadraticfield extension E/F.This means that the two-sided action of G(F)×G(F)on G(E)is tame.This implies that the pair(GL n(E),GL n(F))is a Gelfand pair.In the non-archimedean case this was proven in[Fli].Also we prove that the adjoint action of a reductive group on itself is tame.This is a general-ization of a classical theorem by Gelfand and Kazhdan,see[GK].In our subsequent work[AG3]we use the results of this paper to prove that the pair (GL n+k,GL n×GL k)is a Gelfand pair by proving that it is regular.In the non-archimedean case this was proven in[JR]and our proof follows their lines.In general,we conjecture that any symmetric pair is regular.This would imply van Dijk conjecture:Conjecture(van Dijk).Any symmetric pair(G,H)over C such that G/H is connected is a Gelfand pair.1.2.Related works on this topic.This paper was inspired by the paper[JR]by Jacquet and Rallis where they prove that the pair (GL n+k(F),GL n(F)×GL k(F))is a Gelfand pair for non-archimedean localfield F of characteristic zero.Our aim was to see to what extent their techniques generalize.Another generalization of Harish-Chandra descent using Luna slice theorem has been done in the non-archimedean case in[RR].In that paper Rader and Rallis investigated spherical characters of H-distinguished representations of G for symmetric pairs(G,H)and checked the validity of what they call”density principle”for rank one symmetric pairs.They found out that usually it holds, but also found counterexamples.In[vD],van-Dijk investigated rank one symmetric pairs in the archimedean case and gave the full answer to the question which of them are Gelfand pairs.In[BvD],van-Dijk and Bosman studied the non-archimedean case and gave the answer for the same question for most rank one symmetric pairs.We hope that the second part of our paper will enhance the understanding of this question for symmetric pairs of higher rank.1.3.Structure of the paper.4A VRAHAM AIZENBUD AND DMITRY GOUREVITCHIn section2we introduce notation that allows us to speak uniformly about spaces of points of smooth algebraic varieties over archimedean and non-archimedean localfields,and equivariant distributions on those spaces.In subsection2.1we formulate a version of Luna slice theorem for points over localfields(Theo-rem2.1.16).In subsection2.3we formulate theorems on equivariant distributions and equivariant Schwartz distributions.In section3we formulate and prove the generalized Harish-Chandra descent theorem and its stronger version.Section4is relevant only to the archimedean case.In that section we prove that in cases that we consider if there are no equivariant Schwartz distributions then there are no equivariant distributions at all.Schwartz distributions are discussed in Appendix C.In section5we formulate homogeneity theorem that helps us to check the conditions of the generalized Harish-Chandra descent theorem.In the non-archimedean case this theorem had been proved earlier(see e.g.[JR],[RS2]or[AGRS]).We provide the proof for the archimedean case in Appendix D.In section6we introduce the notion of tame actions and provide tameness criteria.In section7we apply our tools to symmetric pairs.In subsection7.3we provide criteria for tameness of a symmetric pair.In subsection7.4we introduce the notion of regular symmetric pair and prove Theorem7.4.5that we mentioned above.In subsection7.5we discuss conjectures about regularity and Gelfand property of symmetric pairs.In subsection7.6we prove that certain symmetric pairs are tame.In section8we give preliminaries on Gelfand pairs an their connections to invariant distributions. We also prove that the pair(GL n(E),GL n(F))is Gelfand pair for any quadratic extension E/F.In Appendix A we formulate and prove a version of Bernstein’s localization principle(Theorem 4.0.1).This is relevant only for archimedean F since for l-spaces a more general version of this principle had been proven in[Ber].This appendix is used in section4.In[AGS2]we formulated localization principle in the setting of differential geometry.Currently we do not have a proof of this principle in such general setting.In Appendix A we present a proof in the case of a reductive group G acting on a smooth affine variety X.This generality is wide enough for all applications we had up to now,including the one in[AGS2].We start Appendix B from discussing different versions of the inverse function theorem for local fields.Then we prove a version of Luna slice theorem for points over localfields(Theorem2.1.16). For archimedean F it was done by Luna himself in[Lun2].Appendices C and D are relevant only to the archimedean case.In Appendix C we discuss Schwartz distributions on Nash manifolds.We prove for them Frobe-nius reciprocity and construct a pullback of a Schwartz distribution under Nash submersion.Also we prove that K invariant distributions which are(Nashly)compactly supported modulo K are Schwartz distributions.In Appendix D we prove the archimedean version of the homogeneity theorem discussed in section5.In Appendix E we present a diagram that illustrates the interrelations of various properties of symmetric pairs.Acknowledgements.We would like to thank our teacher Joseph Bernstein for our mathemat-ical education.We also thank Vladimir Berkovich,Joseph Bernstein,Gerrit van Dijk,Stephen Gel-bart,Maria Gorelik,David Kazhdan,Erez Lapid,Shifra Reif,Eitan Sayag,David Soudry,Yakov Varshavsky and Oksana Yakimova for fruitful discussions,and Sun Biny-ong for useful remarks.GENERALIZED HARISH-CHANDRA DESCENT5 Finally we thank Anna Gourevitch for the graphical design of Appendix E.2.Preliminaries and notations•From now and till the end of the paper wefix a localfield F of characteristic zero.All the algebraic varieties and algebraic groups that we will consider will be defined over F.•For a group G acting on a set X and an element x∈X we denote by G x the stabilizer of x.•By a reductive group we mean an algebraic reductive group.We treat an algebraic variety X defined over F as algebraic variety overF,F).On X we will consider only the Zariski topology.On X(F)we consider only the analytic(Hausdorff)topology.We treatfinite dimensional linear spaces defined over F as algebraic varieties.Usually we will use letters X,Y,Z,∆to denote algebraic varieties and letters G,H to denote algebraic groups.We will usually use letters V,W,U,K,M,N,C,O,S,T to denote analytic spaces and in particular F points of algebraic varieties and the letter K to denote analytic groups.Also we will use letters L,V,W to denote vector spaces of all kinds.Definition2.0.1.Let an algebraic group G act on an algebraic variety X.A pair consisting of an algebraic variety Y and a G-invariant morphismπ:X→Y is called the quotient of X by the action of G if for any pair(π′,Y′),there exists a unique morphismφ:Y→Y′such that π′=φ◦π.Clearly,if such pair exists it is unique up to canonical isomorphism.We will denote it by(πX,X/G).Theorem2.0.2.Let a reductive group G act on an affine variety X.Then the quotient X/G exists,and everyfiber of the quotient mapπX contains a unique closed orbit.Proof.In[Dre]it is proven that the variety Spec O(X)G satisfies the universal condition of X/G. Clearly,this variety is defined over F and hence we can take X/G:=Spec O(X)G.2.1.Preliminaries on algebraic geometry over localfields.2.1.1.Analytic manifolds.In this paper we will consider distributions over l-spaces,smooth manifolds and Nash manifolds. l-spaces are locally compact totally disconnected topological spaces and Nash manifolds are semi-algebraic smooth manifolds.For basic preliminaries on l-spaces and distributions over them we refer the reader to[BZ], section1.For preliminaries on Nash manifolds and Schwartz functions and distributions over them see Appendix C and[AG1].In this paper we will consider only separated Nash manifolds.We will now give notations which will allow a uniform exposition of archimedean and non-archimedean cases.We will use the notion of analytic manifold over a localfield(see e.g.[Ser],Part II,Chapter III).When we say”analytic manifold”we mean analytic manifold over some localfield.Note that an analytic manifold over a non-archimedeanfield is in particular an l-space and analytic manifold over an archimedeanfield is in particular a smooth manifold.Definition2.1.1.A B-analytic manifold is either an analytic manifold over a non-archimedean localfield,or a Nash manifold.Remark 2.1.2.If X is a smooth algebraic variety,then X(F)is a B-analytic manifold and (T x X)(F)=T x(X(F)).6A VRAHAM AIZENBUD AND DMITRY GOUREVITCHNotation2.1.3.Let M be an analytic manifold and S be an analytic submanifold.We denote by N M S:=(T M|Y)/T S the normal bundle to S in M.The conormal bundle is defined by CN M S:= (N M S)∗.Denote by Sym k(CN M S)the k-th symmetric power of the conormal bundle.For a point y∈S we denote by N M S,y the normal space to S in M at the point y and by CN M S,y the conormal space.2.1.2.G-orbits on X and G(F)-orbits on X(F).Lemma2.1.4.Let G be an algebraic group.Let H⊂G be a closed subgroup.Then G(F)/H(F) is open and closed in(G/H)(F).For proof see Appendix B.1.Corollary2.1.5.Let an algebraic group G act on an algebraic variety X.Let x∈X(F).ThenN X Gx,x(F)∼=N X(F).G(F)x,xProposition2.1.6.Let an algebraic group G act on an algebraic variety X.Suppose that S⊂X(F)is non-empty closed G(F)-invariant subset.Then S contains a closed orbit.Proof.The proof is by Noetherian induction on X.Choose x∈S.Consider Z:=G(F)y∋x}.Lemma2.1.12.Let V be an algebraicfinite dimensional representation over F of a reductive group G.ThenΓ(V)=S0.GENERALIZED HARISH-CHANDRA DESCENT7 This lemma follows from fact A on page108in[RR]for non-archimedean F and Theorem5.2 on page459in[Brk].Proposition 2.1.13.Let a reductive group G act on an affine variety X.Let x,z∈X(F) be G-semisimple elements with different orbits.Then there exist disjoint G(F)-invariant open neighborhoods U x of x and U z of z.For proof of this proposition see[Lun2]for archimedean F and[RR],fact B on page109for non-archimedean F.Corollary2.1.14.Let a reductive group G act on an affine variety X.Let an element x∈X(F) be G-semisimple.Then the set S x is closed.Proof.Let y∈G(F)y contains a closed orbit G(F)z.If G(F)z=G(F)xthen y∈S x.Otherwise,choose disjoint open G-invariant neighborhoods U z of z and U x of x.Since z∈S x,this means that U z intersects S x. Let t∈U z∩S x.Since U z is G(F)-invariant,G(F)t⊂U z.By the definition of S x,x∈U z.Hence U z intersects U x-contradiction!2.1.3.Analytic Luna slice.Definition2.1.15.Let a reductive group G act on an affine variety X.Letπ:X(F)→X/G(F) be the standard projection.An open subset U⊂X(F)is called saturated if there exists an open subset V⊂X/G(F)such that U=π−1(V).We will use the following corollary from Luna slice theorem(for proof see Appendix B.2): Theorem2.1.16.Let a reductive group G act on a smooth affine variety X.Let x∈X(F)be G-semisimple.Then there exist(i)an open G(F)-invariant B-analytic neighborhood U of G(F)x in X(F)with a G-equivariant B-analytic retract p:U→G(F)x and(ii)a G x-equivariant B-analytic embeddingψ:p−1(x)֒→N X Gx,x(F)with open saturated image such thatψ(x)=0.Definition 2.1.17.In the notations of the previous theorem,denote S:=p−1(x)and N:= N X Gx,x(F).We call the quintet(U,p,ψ,S,N)an analytic Luna slice at x.Corollary2.1.18.In the notations of the previous theorem,let y∈p−1(x).Denote z:=ψ(y). Then(i)(G(F)x)z=G(F)y(ii)N X(F)G(F)y,y ∼=N NG(F)x z,zas G(F)y-spaces(iii)y is G-semisimple if and only if z is G x-semisimple.2.2.Vector systems.In this subsection we introduce the term”vector system”.This term allows to formulate statements in wider generality.However,often this generality is not necessary and therefore the reader can skip this subsection and ignore vector systems during thefirst reading.Definition 2.2.1.For an analytic manifold M we define the notions of vector system and B-vector system over it.For a smooth manifold M,a vector system over M is a pair(E,B)where B is a smooth locally trivialfibration over M and E is a smooth vector bundle over B.For a Nash manifold M,a B-vector system over M is a pair(E,B)where B is a Nashfibration over M and E is a Nash vector bundle over B.8A VRAHAM AIZENBUD AND DMITRY GOUREVITCHFor an l-space M,a vector system over M(or a B-vector system over M)is an l-sheaf,that is locally constant sheaf,over M.Definition2.2.2.Let V be a vector system over a point pt.Let M be an analytic manifold.A constant vector system withfiber V is the pullback of V with respect to the map M→pt.We denote it by V M.2.3.Preliminaries on distributions.Definition2.3.1.Let M be an analytic manifold over F.We define C∞c(M)in the following way.If F is non-archimedean,C∞c(M)is the space of locally constant compactly supported complex valued functions on M.We consider no topology on it.If F is archimedean,C∞c(M)is the space of smooth compactly supported complex valued func-tions on M.We consider the standard topology on it.For any analytic manifold M,we define the space of distributions D(M)by D(M):=C∞c(M)∗. We consider the weak topology on it.Definition2.3.2.Let M be a B-analytic manifold.We define S(M)in the following way.If M is an analytic manifold over non-archimedeanfield,S(M):=C∞c(M).If M is a Nash manifold,S(M)is the space of Schwartz functions on M.Schwartz functions are smooth functions that decrease rapidly together with all their derivatives.For precise definition see[AG1].We consider S(M)as a Fr´e chet space.For any B-analytic manifold M,we define the space of Schwartz distributions S∗(M)by S∗(M):=S(M)∗.Definition2.3.3.Let M be an analytic manifold and let N⊂M be a closed subset.We denoteD M(N):={ξ∈D(M)|Supp(ξ)⊂N}.For locally closed subset N⊂M we denote D M(N):=DM\(GENERALIZED HARISH-CHANDRA DESCENT9 For the proof of the last two theorems see e.g.[AGS1],section7.2.Theorem2.3.8(Frobenius reciprocity).Let an analytic group K act on an analytic manifold M. Let N be a K-transitive analytic manifold.Letφ:M→N be a K-equivariant map.Let z∈N be a point and M z:=φ−1(z)be itsfiber.Let K z be the stabilizer of z in K.Let∆K and∆Kzbe the modular characters of K and K z.Let E be a K-equivariant vector system over M.Then(i)there exists a canonical isomorphismF r:D(M z,E|Mz ⊗∆K|Kz·∆−1Kz)K z∼=D(M,E)K.In particular,F r commutes with restrictions to open sets.(ii)For B-analytic manifolds F r maps S∗(M z,E|Mz ⊗∆K|Kz·∆−1K z)K z to S∗(M,E)K.For proof of(i)see[Ber]1.5and[BZ]2.21-2.36for the case of l-spaces and theorem4.2.3in [AGS1]or[Bar]for smooth manifolds.For proof of(ii)see Appendix C.We will also use the following straightforward proposition.Proposition2.3.9.LetΩi⊂K i be analytic groups acting on analytic manifolds M i for i=1...n. Let E i→M i be K i-equivariant vector systems.Suppose that D(M i,E i)Ωi=D(M i,E i)K i for all i. ThenD( M i,⊠E i)QΩi=D( M i,⊠E i)Q K i,where⊠denotes the external product.Moreover,ifΩi,K i,M i and E i are B-analytic then the same statement holds for Schwartz distributions.For proof see e.g.[AGS1],proof of Proposition3.1.5.3.Generalized Harish-Chandra descent3.1.Generalized Harish-Chandra descent.In this subsection we will prove the following theorem.Theorem3.1.1.Let a reductive group G act on a smooth affine variety X.Letχbe a character of G(F).Suppose that for any G-semisimple x∈X(F)we haveD(N X Gx,x(F))G(F)x,χ=0.ThenD(X(F))G(F),χ=0.Remark3.1.2.In fact,the converse is also true.We will not prove it since we will not use it.For the proof of this theorem we will need the following lemmaLemma3.1.3.Let a reductive group G act on a smooth affine variety X.Letχbe a character of G(F).Let U⊂X(F)be an open saturated subset.Suppose that D(X(F))G(F),χ=0.Then D(U)G(F),χ=0.Proof.Consider the quotient X/G.It is an affine algebraic variety.Embed it to an affine space A n. This defines a mapπ:X(F)→F n.Let V⊂X/G(F)be an open subset such that U=π−1(V). There exists an open subset V′⊂F n such that V′∩X/G(F)=V.Letξ∈D(U)G(F),χ.Suppose thatξis non-zero.Let x∈Suppξand let y:=π(x).Let g∈C∞c(V′)be such that g(y)=1.Considerξ′∈D(X(F))defined byξ′(f):=ξ(f·(g◦π)). Clearly,x∈Supp(ξ′)andξ′∈D(X(F))G(F),χ.Contradiction.10A VRAHAM AIZENBUD AND DMITRY GOUREVITCHProof of the theorem.Choose a G-semisimple x∈X(F).Let(U x,p x,ψx,S x,N x)be an analytic Luna slice at x..Thenξ′∈D(U x)G(F),χ.By Frobenius reciprocity it corresponds toξ′′∈Letξ′=ξ|UxD(S x)G x(F),χ.The distributionξ′′corresponds to a distributionξ′′′∈D(ψx(S x))G x(F),χ.However,by the previous lemma the assumption implies that D(ψx(S x))G x(F),χ=0.Hence ξ′=0.Let S⊂X(F)be the set of all G-semisimple points.Let U= x∈S U x.We saw thatξ|U=0. On the other hand,U includes all the closed orbits,and hence by Proposition2.1.7U=X.The following generalization of this theorem is proven in the same way.Theorem3.1.4.Let a reductive group G act on a smooth affine variety X.Let K⊂G(F)be an open subgroup and letχbe a character of K.Suppose that for any G-semisimple x∈X(F)we haveD(N X Gx,x(F))K x,χ=0.ThenD(X(F))K,χ=0.Now we would like to formulate a slightly more general version of this theorem concerning K-equivariant vector systems.Duringfirst reading of this paper one can skip to the next subsection. Definition3.1.5.Let a reductive group G act on a smooth affine variety X.Let K⊂G(F)be an open subgroup.Let E be a K-equivariant vector system on X(F).Let x∈X(F)be G-semisimple. Let E′be a K x-equivariant vector system on N X Gx,x(F).We say that E and E′are compatible if there exists an analytic Luna slice(U,p,ψ,S,N)such that E|S=ψ∗(E′).Note that if E and E′are constant with the samefiber then they are compatible.The following theorem is proven in the same way as Theorem3.1.1.Theorem3.1.6.Let a reductive group G act on a smooth affine variety X.Let K⊂G(F)be an open subgroup and let E be a K-equivariant vector system on X(F).Suppose that for any G-semisimple x∈X(F)there exists a K-equivariant vector system E′on N X Gx,x(F),compatible with E such thatD(N X Gx,x(F),E′)K x=0.ThenD(X(F),E)K=0.If E and E′are B-vector systems and K is open B-analytic subgroup1then the theorem holds also for Schwartz ly,if S∗(N X Gx,x(F),E′)K x=0for any x then S∗(X(F),E)K=0, and the proof is the same.3.2.A stronger version.In this section we give a way to validate the conditions of theorems3.1.1,3.1.4and3.1.6by induction.The goal of this section is to prove the following theorem.Theorem3.2.1.Let a reductive group G act on a smooth affine variety X.Let K⊂G(F)be an open subgroup and letχbe a character of K.Suppose that for any G-semisimple x∈X(F)such thatD(R(N X Gx,x))K x,χ=0D (Q (N X Gx,x ))K x ,χ=0.Then for any for any G -semisimple x ∈X (F )we haveD (N X Gx,x (F ))K x ,χ=0.This theorem together with Theorem 3.1.4give the following corollary.Corollary 3.2.2.Let a reductive group G act on a smooth affine variety X .Let K ⊂G (F )be an open subgroup and let χbe a character of K .Suppose that for any G -semisimple x ∈X (F )such thatD (R (N X Gx,x ))K x ,χ=0we haveD (Q (N X Gx,x ))K x ,χ=0.Then D (X (F ))K,χ=0.From now till the end of the section we fix G ,X ,K and χ.Let us introduce several definitions and notations.Notation 3.2.3.Denote•T ⊂X (F )the set of all G -semisimple points.•For x,y ∈T we say that x >y if G x G y .•T 0:={x ∈T |D (Q (N X Gx,x ))K x ,χ=0}.Note that if x ∈T 0then D (N X Gx,x (F ))K x ,χ=0.Proof of Theorem 3.2.1.We have to show that T =T 0.Assume the contrary.Note that every chain in T with respect to our ordering has a minimum.Hence by Zorn’s lemma every non-empty set in T has a minimal element.Let x be a minimal element of T −T 0.To get acontradiction,it is enough to show that D (R (N X Gx,x ))K x ,χ=0.Denote R :=R (N X Gx,x ).By Theorem 3.1.4,it is enough to show that for any y ∈R we haveD (N R G (F )x y,y )(K x )y ,χ=0.Let (U,p,ψ,S,N )be an analytic Luna slice at x .We can assume that y ∈ψ(S )since ψ(S )is open,includes 0,and we can replace y by λy for any λ∈F ×.Let z ∈S be such that ψ(z )=y .By corollary 2.1.18,(G (F )x )y =G (F )z and N R G (F )x y,y ∼=N X Gz,z (F ).Hence (K x )y =K z and thereforeD (N R G (F )x y,y )(K x )y ,χ∼=D (N X Gz,z (F ))K z ,χ.However z <x and hence z ∈T 0which means D (N X Gz,z (F ))K z ,χ=0.Remark 3.2.4.As before,Theorem 3.2.1and Corollary 3.2.5hold also for Schwartz distributions,and the proof is the same.Again,we can formulate a more general version of Corollary 3.2.2concerning vector systems.During first reading of this paper one can skip to the next subsection.Theorem 3.2.5.Let a reductive group G act on a smooth affine variety X .Let K ⊂G (F )be an open subgroup and let E be a K -equivariant vector system on X (F ).Suppose that for any G -semisimple x ∈X (F )such that(*)any K x ×F ×-equivariant vector system E ′on R (N X Gx,x )compatible with E satisfies D (R (N X Gx,x),E ′)K x =0(where the action of F ×is the homothety action),(**)there exists a K x×F×-equivariant vector system E′on Q(N X Gx,x)compatible with E such thatD(Q(N X Gx,x),E′)K x=0.Then D(X(F),E)K=0.The proof is the same as the proof of Theorem3.2.1using the following lemma that follows from the definitions.Lemma3.2.6.Let a reductive group G act on a smooth affine variety X.Let K⊂G(F)be an open subgroup and let E be a K-equivariant vector system on X(F).Let x∈X(F)be G-semisimple. Let(U,p,ψ,S,N)be an analytic Luna slice at x.Let E′be a K x-equivariant vector system on N compatible with E.Let y∈S be G-semisimple.Let z:=ψ(y).Let E′′be a(K x)z-equivariant vector system on N N Gx z,zcompatible with E′.Considerthe isomorphism N N Gx z,z(F)∼=N X Gy,y(F)and let E′′′be the corresponding K y-equivariant vectorsystem on N X Gy,y(F).Then E′′′is compatible with E.Again,if E and E′are B-vector systems then the theorem holds also for Schwartz distributions.4.Distributions versus Schwartz distributionsThe tools developed in the previous section enabled us to prove the following version of local-ization principle.Theorem4.0.1(Localization principle).Let a reductive group G act on a smooth affine variety X.Let Y be an algebraic variety andφ:X→Y be an algebraic G-invariant map.Letχbe a character of G(F).Suppose that for any y∈Y(F)we have D X(F)(φ(F)−1(y))G(F),χ=0.Then D(X(F))G(F),χ=0.For proof see Appendix A.In this section we use this theorem to show that if there are no G(F)-equivariant Schwartz distributions on X(F)then there are no G(F)-equivariant distributions on X(F).Theorem4.0.2.Let a reductive group G act on a smooth affine variety X.Let V be afinite dimensional continuous representation of G(F)over C.Suppose that S∗(X(F),V)G(F)=0.Then D(X(F),V)G(F)=0.For the proof we will need the following definition and theorem.Definition4.0.3.(i)Let a topological group K act on a topological space M.We call a closed K-invariant subset C⊂M compact modulo K if there exists a compact subset C′⊂M such that C⊂KC′.(ii)Let a Nash group K act on a Nash manifold M.We call a closed K-invariant subset C⊂M Nashly compact modulo K if there exist a compact subset C′⊂M and semi-algebraic closed subset Z⊂M such that C⊂Z⊂KC′.Remark4.0.4.Let a reductive group G act on a smooth affine variety X.Let K:=G(F)and M:=X(F).Then it is easy to see that the notions of compact modulo K and Nashly compact modulo K coincide.Theorem4.0.5.Let a Nash group K act on a Nash manifold M.Let E be a K-equivariant Nash bundle over M.Letξ∈D(M,E)K such that Supp(ξ)is Nashly compact modulo K.Then ξ∈S∗(M,E)K.。

As a high school student, I have always been fascinated by the art of correspondence, and one of the most enchanting forms of this is through postcards. My journey with postcards began when I was in the seventh grade, and it has since become a cherished hobby that has enriched my life in many ways.It started with a simple postcard I received from my cousin who was traveling abroad. The vibrant colors and the exotic scenery depicted on the card captivated me. It was not just a piece of paper it was a window to another world, a snippet of someones experience shared with me. This sparked my interest in collecting postcards and eventually led me to start sending my own.The first postcard I ever sent was to my best friend who had moved to another city for school. I wanted to share a piece of my hometown with her, so I chose a postcard that featured a beautiful sunset over the local park where we used to play. On the back, I wrote a heartfelt message, expressing my longing for her company and the memories we shared. It was a small gesture, but it felt incredibly meaningful to me.Over time, I began to appreciate the nuances of postcards. Each one tells a story, whether its the story of the place it represents or the story of the person who sent it. I started to pay attention to the detailsthe choice of words, the handwriting, the stamps, and even the postmarks. These elements all contribute to the charm and uniqueness of each postcard.One of the most memorable postcards Ive received was from a pen pal inJapan. The card depicted a serene scene of cherry blossoms in full bloom, and the message was written in both English and Japanese, with a translation provided. It was a beautiful blend of cultures and a testament to the power of communication across borders.Sending postcards has also taught me about patience and anticipation. Theres something magical about waiting for a postcard to arrive, not knowing when it will get there. The unpredictability adds an element of surprise and excitement to the experience.Moreover, postcards have become a way for me to connect with people from different walks of life. Ive exchanged postcards with fellow students, elderly neighbors, and even strangers Ive met during my travels. Each exchange has been a learning experience, offering me a glimpse into someone elses world and fostering a sense of global community.In an age where digital communication is prevalent, the tangible nature of postcards holds a special place in my heart. Theres a warmth and personal touch to receiving a physical postcard that an email or a text message cannot replicate. Its a reminder that despite the vast distances and differences among us, we can still reach out and touch each others lives in meaningful ways.In conclusion, my experience with postcards has been a journey of discovery and connection. It has allowed me to express myself creatively, to connect with others across the globe, and to appreciate the beauty of simple, heartfelt communication. As I continue to grow and explore theworld, I look forward to filling my collection with more postcards, each one a treasured memory and a testament to the power of human connection.。

In the vast expanse of the universe, Earth stands as a unique haven for life, teeming with a multitude of species, both human and animal. The relationship between humans and animals has been a subject of fascination and study for centuries, with countless stories and scientific research exploring the depths of this bond.The bond between humans and animals is not only a result of necessity but also of mutual affection. Pets, for instance, have been an integral part of human society for thousands of years. Dogs, cats, and other domesticated animals provide companionship, emotional support, and even physical protection. They are often considered as family members, sharing in the joys and sorrows of their human counterparts.In addition to pets, humans have also formed relationships with animals in the wild. Conservation efforts and wildlife rehabilitation programs have shown that humans can play a crucial role in protecting and preserving animal species. Through these initiatives, humans have learned to coexist with animals, respecting their habitats and understanding their behaviors.However, the relationship between humans and animals is not always harmonious. The encroachment of human activities on natural habitats has led to the displacement and endangerment of many species. Deforestation, pollution, and climate change are just a few of the factors that have contributed to the decline of wildlife populations.To address these challenges, it is essential for humans to take proactive steps to protect the environment and the animals that inhabit it. This includes implementing sustainable practices, conserving resources, and promoting biodiversity. Education and awareness campaigns can also play a significant role in fostering a greater understanding and appreciation for the natural world.Moreover, advancements in technology have opened up new avenues for exploring the relationship between humans and animals. Remote monitoring and tracking devices allow researchers to study animal behavior in their natural habitats without disturbing them. Virtual reality and augmented reality technologies offer immersive experiences that can help people better understand and empathize with the lives of animals.In conclusion, the relationship between humans and animals is a complex and multifaceted one. It is a testament to the interconnectedness of all life on Earth and the importance of coexistence. By recognizing and respecting the intrinsic value of all species, humans can work towards a more harmonious and sustainable future for both themselves and the animals with whom they share this planet.。

考研英语阅读难点及应对之历年长句解析An invisible border divides those arguing for computers in the classroom / on the behalf of students' career prospects and those arguing for computers in the classroom for broader reasons of radical educational reform.(1999年真题) 【分析】句子主干是由divides...and...(使……对立)搭配连接而成。

两个those后面的分词短语作定语修饰those。

句子主干为：An invisible border divides those and those.【翻译】一种看不到的鸿沟，把那些主张为了学生有更好的职业前景而把电脑引入课堂的人和那些主张为了获得彻底教育革新更广泛理由而把电脑引入课堂的人的观点对立起来。

Very few writers on the subject have explored this distinction——indeed, contradiction——Which goes to the heart of what is wrong with the campaign / to put computers in the classroom.(1999年真题)【分析】句子主干为：Very few writers have explored this distinction. “on the subject”介词短语作状语修饰“very few writers”,“which goes to the heart of what is wrong with the campaign to put computers in the classroom”是定语从句修饰“this distinction”的。

Improving your technical writing skillsVersion 4.125 September 2003Norman FentonComputer Science DepartmentQueen Mary (University of London)London E1 4NSnorman@/~norman/Tel: 020 7882 7860AbstractThis document describes the basic principles of good writing. It is primarily targeted at students and researchers writing technical and business reports, but the principles are relevant to any form of writing, including letters and memos. Therefore, the document contains valuable lessons for anybody wishing to improve their writing skills. The ideas described here are, apart from fairly minor exceptions, not original. They are drawn from a range of excellent books and have also been influenced by various outstanding authors I have worked with. Thus, the approach represents a kind of modern consensus. This approach is very different to the style that was promoted by the traditional English schools’ system, which encouraged students to write in an unnecessarily complex and formal way. The approach described here emphasises simplicity (‘plain English’) and informality. For example, it encourages shorter sentences and use of the simplest words and phrases possible. It explains how you can achieve simplicity by using the active rather than the passive style, personal rather than impersonal style, and by avoiding noun constructs in favour of verbs. Crucially, this approach leads to better reports because they are much easier to read and understand.Document change historyVersion 1.0, 11 September 2000: Derived from Norman Fenton’s ‘Good Writing’ web pages. Version 2.0, 21 September 2001. Minor changes including addition of student project guidelines.Version 2.1, 20 September 2002. Minor corrections made.Version 3.0, 14 September 2003. Major revision.Version 4.0, 23 September 2003. Restructuring and editing.Version 4.1, 25 September 2003. Various typos fixed and polemic removed.Table of contents1.INTRODUCTION (4)2.BEFORE YOU START WRITING (5)ING PLAIN ENGLISH: STYLE (6)3.1S ENTENCE AND PARAGRAPH LENGTH (6)3.2B ULLET POINTS AND ENUMERATED LISTS (7)3.3U SING THE SIMPLEST WORDS AND EXPRESSIONS POSSIBLE (8)3.3.1Replace difficult words and phrases with simpler alternatives (9)3.3.2Avoid stock phrases (9)3.3.3Avoid legal words and pomposity (10)3.3.4Avoid jargon (10)3.4A VOIDING UNNECESSARY WORDS AND REPETITION (10)3.5U SING VERBS INSTEAD OF NOUNS (12)3.6U SING ACTIVE RATHER THAN PASSIVE STYLE (13)3.7U SING PERSONAL RATHER THAN IMPERSONAL STYLE (13)3.8E XPLAIN NEW IDEAS CLEARLY (15)3.9U SE CONSISTENT NAMING OF THE SAME ‘THINGS’ (15)3.10P AINLESS POLITICAL CORRECTNESS (16)3.11S UMMARY (17)ING PLAIN ENGLISH: THE MECHANICS (18)4.1A VOIDING COMMON VOCABULARY AND SPELLING ERRORS (18)4.2A BBREVIATIONS (19)4.3P UNCTUATION (19)4.3.1Capital letters (20)4.3.2Apostrophes (20)4.3.3Commas (21)4.3.4Exclamation marks (21)4.4S UMMARY (22)5.BASIC STRUCTURE FOR REPORTS (23)5.1W HAT EVERY REPORT SHOULD CONTAIN (23)5.2G ENERAL LAYOUT (24)5.3S ECTIONS AND SECTION NUMBERING (24)5.4T HE CRUCIAL ROLE OF ‘INTRODUCTIONS’ AND SUMMARIES (25)5.5F IGURES AND TABLES (26)5.6 A STRUCTURE FOR STUDENT PROJECT REPORTS (27)5.7S UMMARY AND CHECKLIST FOR WHEN YOU FINISH WRITING (28)6.ABSTRACTS AND EXECUTIVE SUMMARIES (29)7.WRITING THAT INCLUDES MATHEMATICS (31)8.SUMMARY AND CONCLUSIONS (32)9.REFERENCES (33)1. IntroductionCompare the following two sentences that provide instructions to a set of employees (this Example is given in [Roy 2000]):1. It is of considerable importance to ensure that under no circumstances shouldanyone fail to deactivate the overhead luminescent function at its local activationpoint on their departure to their place of residence, most notably immediatelypreceding the two day period at the termination of the standard working week.2. Always turn the lights out when you go home, especially on a Friday.The meaning of both sentences is, of course, equivalent. Which one was easier to read and understand? The objective of this document is to show people how to write as in the second sentence rather than the first. If you actually prefer the first, then there is little point in you reading the rest of this document. But please do not expect to win too many friends (or marks) from any writing that you produce.Unfortunately, the great shame for anybody having to read lots of reports in their everyday life is that the schools’ system continues to produce students who feel they ought to write more like in the first sentence than the second. Hence, the unnecessarily complex and formal style is still common. This document shows you that there is a better way to write, using simple, plain English.One of the good things about technical writing is that you really can learn to improve. You should not believe people who say that being a good writer is a natural ability that you either have or do not have. We are talking here about presenting technical or business reports and not about writing novels. I speak from some experience in this respect, because in the last ten years I have learned these ideas and applied them to become a better writer. When I was writing my first book in 1989 an outstanding technical editor highlighted the many problems with my writing. I was guilty of many of the examples of bad practice that I will highlight throughout this document. You too can improve your writing significantly if you are aware of what these bad practices are and how to avoid them.The document contains the following main sections:• Before you start writing (Section 2): This is a simple checklist that stresses the importance of knowing your objective and audience.• Using plain English: style (Section 3). This is the heart of the document because it explains how to write in the simplest and most effective way.• Using plain English: the mechanics (Section 4). This covers vocabulary, spelling, and punctuation.• Basic structure for reports (Section 5). This section explains how to organise your report into sections and how to lay it out.• Abstracts and executive summaries (Section 6). This explains the difference between informative and descriptive abstracts. It tells you why you should always use informative abstracts and how to write them.• Writing that includes mathematics (Section 7). This contains some simple rules you should follow if your writing includes mathematical symbols or formulas.2. Before you start writingBefore you start producing your word-processed report you must make sure you do the following:• Decide what the objective of the report is. This is critical. If you fail to do this you will almost certainly produce something that is unsatisfactory. Every report should have a single clear objective. Make the objective as specific as possible.• Write down the objective. Ideally, this should be in one sentence. For example, the objective of this document is “to help students write well structured, easy-to-understand technical reports”. The objective should then be stated at the beginning of the report. If you cannot write down the objective in one sentence, then you are not yet ready to start any writing.• Always have in mind a specific reader. You should assume that the reader is intelligent but uninformed. It may be useful to state up front what the reader profile is. For example, the target readers for this document are primarily students and researchers with a good working knowledge of English. The document is not suitable for children under 13, or people who have yet to write documents in English. It is ideal for people who have written technical or business documents and wish to improve their writing skills.• Decide what information you need to include. You should use the objective as your reference and list the areas you need to cover. Once you have collected the information make a note of each main point and then sort them into logical groups. Ultimately you have to make sure that every sentence makes a contribution to the objective. If material you write does not make a contribution to the objective remove it – if it is good you may even be able to reuse it in a different report with a different objective.• Have access to a good dictionary. Before using a word that ‘sounds good’, but whose meaning you are not sure of, check it in the dictionary. Do the same for any word you are not sure how to spell.• Identify someone who can provide feedback. Make sure you identify a friend, relative or colleague who can read at least one draft of your report before you submit it formally. Do not worry if the person does not understand the technical area – they can at least check the structure and style and it may even force you to write in the plain English style advocated here.The following checklist should be applied before you give even an early draft of your document out for review:• Check that the structure conforms to all the rules described in this document.• Run the document through a spelling checker.• Read it through carefully, trying to put yourself in the shoes of your potential readers.3. Using plain English: styleWhen you are producing a technical or business report you want it to ‘get results’. If you are a student this can mean literally getting a good grade. More generally we mean that you want to convince the reader that what you have to say is sensible so that they act accordingly. If the report is a proposal then you want the reader to accept your recommendations. If the report describes a piece of research then you want the reader to understand what you did and why it was important and valid. Trying to be ‘clever’ and ‘cryptic’ in the way you write will confuse and annoy your readers and have the opposite effect to what you wanted. In all cases you are more likely to get results if you present your ideas and information in the simplest possible way. This section describes how to do this.The section is structured as follows:• Sections 3.1 and 3.2 describe structural techniques for making your writing easier to understand. Specifically:o Sentence and paragraph length: keeping them short is the simplest first step to improved writing.o Bullet points and lists: using these makes things clearer and less cluttered.• Sections 3.3 and 3.4 describe techniques for using fewer words. Specifically: o Using the simplest words and expressions available: this section also describes words and expressions to avoid.o Avoiding unnecessary words: this is about removing redundancy.• Sections 3.5 to 3.7 describe techniques for avoiding common causes of poorly structured sentences. Specifically:o Using verbs instead of nounso Using active rather than passive styleo Using personal rather than impersonal style• Section 3.8 describes how to explain new ideas clearly.• Section 3.9 explains the importance of naming things consistently.• Section 3.10 gives some rules on how to achieve political correctness in your writing without adding complexity.3.1 Sentence and paragraph lengthContrary to what you may have learnt in school, there is nothing clever about writing long, complex sentences. For technical writing it is simply wrong. You must get used to the idea of writing sentences that are reasonably short and simple. In many cases shorter sentences can be achieved by sticking to the following principles:1. A sentence should contain a single unit of information. Therefore, avoid compoundsentences wherever possible. In particular, be on the lookout for words like and, or and while which are often used unnecessarily to build a compound sentence.2. Check your sentences for faulty construction. Incorrect use of commas (see Section4.3 for how to use commas correctly) is a common cause of poorly constructed andexcessively long sentences.Example (this example fixes some other problems also that are dealt with below) Bad: “Time division multiplexed systems are basically much simpler, thecombination and separation of channels being affected by timing circuitsrather than by filters and inter-channel interference is less dependent onsystem non-linearities, due to the fact that only one channel is using thecommon communication medium at any instant.”Good: “Systems multiplexed by time division are basically much simpler.The channels are combined and separated by timing circuits, not byfilters. Interference between channels depends less on non-linear featuresof the system, because only one channel is using the commoncommunication medium at any time.”3. Use parentheses sparingly. Most uses are due to laziness and can be avoided bybreaking up the sentence. Never use nested parentheses if you want to retain your reader.Learning about some of the principles described below, especially using active rather than passive constructs, will go a long way toward helping you shorten your sentences.Just as it is bad to write long sentences it is also bad to write long paragraphs. A paragraph should contain a single coherent idea. You should always keep paragraphs to less than half a page. On the other hand, successive paragraphs that are very short may also be difficult to read. Such an approach is often the result of poorly structured thinking. If you need to write a sequence of sentences that each express a different idea then it is usually best to use bullet points or enumerated lists to do so. We consider these next.3.2 Bullet points and enumerated listsIf the sentences in a paragraph need to be written in sequence then this suggests that there is something that relates them and that they form some kind of a list. The idea that relates them should be used to introduce the list. For example, the following paragraph is a mess because the writer is trying to make what is clearly a list into one paragraph:Getting to university on time for a 9.00am lecture involves following a number of steps. First of all you have to set your alarm – you will need to do this before you go to bed the previous night. When the alarm goes off you will need to get out of bed.You should next take a shower and then get yourself dressed. After getting dressed you should have some breakfast. After breakfast you have to walk to the tube station, and then buy a ticket when you get there. Once you have your ticket you can catch the next train to Stepney Green. When the train arrives at Stepney Green you should get off and then finally walk to the University.The following is much simpler and clearer:To get to university on time for a 9.00am lecture:1. Set alarm before going to bed the previous night2. Get out of bed when the alarm goes off3. Take a shower4. Get dressed5. Have some breakfast6. Walk to the tube station7. Buy ticket8. Catch next train to Stepney Green9. Get out at Stepney Green10. Walk to the UniversityThe simple rule of thumb is: if what you are describing is a list then you should always display it as a list.The above is an example of an enumerated list. The items need to be shown in numbered order. If there is no specific ordering of the items in the list then you should use bullet points instead. For example consider the following paragraph:Good software engineering is based on a number of key principles. One such principle is getting a good understanding of the customer requirements (possibly by prototyping). It is also important to deliver in regular increments, involving the customer/user as much as possible. Another principle it that it is necessary to do testing throughout, with unit testing being especially crucial. In addition to the previous principles, you need to be able to maintain good communication within the project team (and also with the customer).The paragraph is much better when rewritten using bullet points:Good software engineering is based on the following key principles:• Get a good understanding of the customer requirements (possibly byprototyping).• Deliver in regular increments (involve the customer/user as much aspossible).• Do testing throughout, (unit testing is especially crucial).• Maintain good communication within the project team (and also with the customer).There are numerous examples throughout this report of bullet points and enumerated lists. You should never be sparing in your use of such lists. Also, note the following rule for punctuation in lists:If all the list items are very short, by which we normally mean less than one line long, then there is no need for any punctuation. Otherwise use a full stop at the end of each list item.3.3 Using the simplest words and expressions possibleOn a recent trip to Brussels by Eurostar the train manager made the following announcement: “Do not hesitate to contact us in the event that you are in need if assistance at this time”. What she meant was: “Please contact us if you need help now”,but she clearly did not use the simplest words and expressions possible. While this maybe acceptable verbally, it is not acceptable in writing.The golden rules on words and expressions to avoid are:• Replace difficult words and phrases with simpler alternatives;• Avoid stock phrases;• Avoid legal words and pomposity;• Avoid jargon.We will deal with each of these in turn.3.3.1 Replace difficult words and phrases with simpler alternativesTable 1 lists a number of words and expressions that should generally be avoided in favour of the simple alternative.Table 1 Words and expressions to avoidWord/expression to avoid SimplealternativeWord/expressiontoavoidSimplealternativeutilise use endeavourtry facilitate help terminate end,stop at this time now transmit sendin respect of about demonstrate showcommence start initiate begin terminate end, stop assist, assistance helpascertain findout necessitate needin the event of if in excess of more thanin consequence so dwelling houseenquire askAlso, unless you are talking about building maintenance or computer graphics, never use theverb ‘render’ as in:The testing strategy rendered it impossible to find all the faults.The ‘correct’ version of the above sentence is:The testing strategy made it impossible to find all the faults.In other words, if you mean ‘make’ then just write ‘make’ not ‘render’.3.3.2 Avoid stock phrasesStock phrase like those shown in Table 2 should be avoided in favour of the simpler alternative. Such phrases are cumbersome and pompous.Table 2 Stock phrases to avoidBAD GOODThere is a reasonable expectation that ... Probably …Owing to the situation that … Because, since …Should a situation arise where … If …Taking into consideration such factors as … Considering …Prior to the occasion when … Before …At this precise moment in time … Now …Do not hesitate to … Please …I am in receipt of … I have …3.3.3 Avoid legal words and pomposityLawyers seem to have a language of their own. This is primarily to ensure that their documents are so difficult to understand that only other lawyers can read them. This ensures more work and money for lawyers because it forces ordinary people to pay lawyers for work they could do themselves. For some strange reason ordinary people often think they are being very clever by using legal words and expressions in their own writing. Do not fall into this trap. Avoid legal words like the following:forthwith hereof Of the (4th) inst. thereofhenceforth hereto thereat whereat hereat herewith therein whereonAlso avoid nonsensical legal references like the following:“The said software compiler…”which should be changed to“The software compiler…”and:“The aforementioned people have agreed …”which should be changed to“A and B have agreed…”3.3.4 Avoid jargonExpressions like MS/DOS, Poisson distribution, and distributor cap are examples of jargon.In general, jargon refers to descriptions of specific things within a specialised field. The descriptions are often shorthand or abbreviations. If you are certain that every reader of your report understands the specialist field then it can be acceptable to use jargon. For example, if your only potential readers are computer specialists then it is probably OK to refer toMS/DOS without the need to explain what MS/DOS is or stands for. The same applies to Poisson distribution if your readers are all statisticians or distributor cap if your readers are car mechanics. In all other cases (which is almost always) jargon should be avoided. If you cannot avoid it by using alternative expressions then you should define the term the first time you use it and/or provide a glossary where it is defined.3.4 Avoiding unnecessary words and repetitionMany sentences contain unnecessary words that repeat an idea already expressed in another word. This wastes space and blunts the message. In many cases unnecessary words are causedby ‘abstract’ words like nature, position, character, condition and situation as the following examples show:BADGOOD The product is not of a satisfactory natureThe product is unsatisfactory The product is not of a satisfactory characterThe product is unsatisfactory After specification we are in a position tobegin detailed designAfter specification we can begin detailed design We are now in the situation of being able tobegin detailed designWe can now begin detailed designIn general, you should therefore use such abstract words sparingly, if at all.Often writers use several words for ideas that can be expressed in one. This leads to unnecessarily complex sentences and genuine redundancy as the following examples show:WITH REDUNDANCYWITHOUT REDUNDANCY The printer is located adjacent to the computerThe printer is adjacent to the computer The printer is located in the immediate vicinity of the computerThe printer is near the computer The user can visibly see the image movingThe user can see the image moving He wore a shirt that was blue in colour He wore a blue shirt The input is suitably processed The input is processed This is done by means of inserting an artificial faultThis is done by inserting an artificial fault The reason for the increase in number of faults found was due to an increase in testingThe increase in number of faults found was due to an increase in testing It is likely that problems will arise with regards to the completion of thespecification phaseYou will probably have problemscompleting the specification phase Within a comparatively short period we will be able to finish the designSoon we will be able to finish the designAnother common cause of redundant words is when people use so-called modifying words. For example, the word suitable in the sentence “John left the building in suitable haste” is a modifying word. It is redundant because the sentence “John left the building in haste” has exactly the same meaning. Similarly, the other form of a modifying word – the one ending in ‘y’ as in suitably – is also usually redundant. For example, “John was suitably impressed” says nothing more than “John was impressed”. Other examples are:BADGOOD absolute nonsense nonsense absolutely critical critical considerable difficulty difficulty considerably difficult difficultModifying words can be fine when used with a concrete reference, as in the example “Jane set John a suitable task” but in many cases they are not and so are best avoided: Here are the most common modifying words to avoid:appreciable excessive sufficientapproximate fair suitablecomparative negligible unduedefinite reasonableutter evident relativeFinally, one of the simplest ways to shorten and simplify your reports is to remove repetition. Poorly structured reports are often characterised by the same idea being described in different places. The only ‘allowable’ repetition is in introductions and summaries, as we shall see in Section 5.4. You can avoid repetition by checking through your report and jotting down a list of the key ideas as they appear. Where the same idea appears more than once, you have to decide once and for all the place where it should best go and then delete and/or merge the text accordingly.3.5 Using verbs instead of nounsLook at the following sentence:“Half the team were involved in the development of system Y”.This sentence contains a classic example of a common cause of poor writing style. The sentence is using an abstract noun ‘development’ instead of the verb ‘develop’ from which it is derived. The simpler and more natural version of the sentence is:“Half the team were involved in developing system Y”.Turning verbs into abstract nouns always results in longer sentences than necessary, so you should avoid doing it. The following examples show the improvements you can achieve by getting rid of nouns in favour of verbs:BAD GOODHe used to help in the specification of newsoftwareHe used to help specify new softwareAcid rain accounts for the destruction of ancientstone-workAcid rain destroys ancient stone-work When you take into consideration … When you consider …Clicking the icon causes the execution of the program The program executes when the icon is clickedMeasurement of static software properties was performed by the tool The tool measured static software propertiesThe analysis of the software was performed byFredFred analysed the software The testing of the software was carried out by Jane Jane tested the softwareIt was reported by Jones that method x facilitated the utilisation of inspection techniques by the testing team Jones reported that method x helped the testing team use inspection techniquesThe last example is a particular favourite of mine (the bad version appeared in a published paper) since it manages to breach just about every principle of good writing style. It uses a noun construct instead of a verb and it includes two of the forbidden words (facilitated, utilisation). However, one of the worst features of this sentence is that it says “It was reported by Jones” instead of simply “Jones reported”. This is a classic example of use of passive rather active constructs. We deal with this in the next section.3.6 Using active rather than passive styleConsider the following two sentences:1. Joe tested the software2. The software was tested by JoeBoth sentences provide identical information. The first is said to be in the active style and the second is said to be passive style. In certain situations it can make sense to use the less natural passive style. For example, if you really want to stress that a thing was acted on, then it is reasonable to use the passive style as in “the city was destroyed by constant bombing”. However, many writers routinely use the passive style simply because they believe it is more ‘formal’ and ‘acceptable’. It is not. Using the passive style is the most common reason for poorly structured sentences and it always leads to longer sentences than are necessary. Unless you have a very good reason for the change in emphasis, you should always write in the active style.The following examples show the improvements of switching from passive to active: BAD GOODThe report was written by Bloggs, and was found to be excellent Bloggs wrote the report, and it was excellentThe values were measured automatically by the control system The control system measured the values automaticallyIt was reported by the manager that the project was in trouble The manager reported that the project was in troubleThe precise mechanism responsible for this antagonism cannot be elucidated We do not know what causes this antagonismThe stability of the process is enhanced by co-operation Co-operation improves the stability of the process3.7 Using personal rather than impersonal styleSaying“My results have shown…”is an example of a sentence using the personal (also called first person) style. This contrasts with:“The author’s results have shown…”which is an example of the impersonal (also called third person) style.。

[英语作文]My penpal我的笔友Title: My PenpalI have always found the idea of having a penpal to be quite enchanting—the thought of exchanging letters with someone from a different corner of the world, sharing stories and perspectives, truly excites me. My penpal, whose name is [Penpal's Name], has been a source of joy and a wonderful connection to a culture that is distinctly different from my own.[Penpal's Name] hails from [Penpal's Country or City], and our correspondence began when we were both participating in an international pen-pal program at school. What initially drew us together was our shared love for literature and our curiosity about each other's lives. Over the months, we have exchanged countless letters, each one filled with tales of our daily adventures, thoughts on books we've read, and dreams for the future.Through [Penpal's Name]'s words, I have traveled vicariously to the far-off lands of [Describe Penpal's Homeland], gaining insights into not just the geography, but the rich cultural tapestry thatdefines [Penpal's Country or City]. It's fascinating to learn about local traditions, festivals, and everyday norms that are so different from what I experience in my daily life.Our conversations are never dull; they are a constant reminder of the diversity of our global village. We discuss everything from the latest technology trends to our favorite recipes, and there's always something new to discover about each other's worlds.One of the most meaningful aspects of our friendship is how it has bridged the gap between distant lands. Despite the geographical distance, we have formed a bond that transcends borders. In a world where digital communication dominates, the patience and care involved in waiting for a handwritten letter to arrive, then sitting down to craft a thoughtful response, feels like a precious ritual.My penpal has become much more than just a friend from afar; [Penpal's Name] is a window to another world, a mentor in cultural exchange, and a confidant in shared dreams and aspirations. Our ongoing correspondence continues to unfold like a treasure hunt, each letter revealing more about the world and ourselves.In conclusion, I cherish my relationship with [Penpal's Name] deeply. The uniqueness of our penpal friendship has taught me patience, open-mindedness, and the beauty of maintaining connections across continents. It's a testament to the power of simple acts of reaching out and the enriching experiences that can blossom from them.。

Respecting the beauty of diversity is a fundamental aspect of creating a harmonious and inclusive society.In our globalized world,we encounter people from various backgrounds,cultures,and beliefs.Embracing these differences not only enriches our lives but also fosters understanding and cooperation among individuals.Firstly,respecting differences means acknowledging that each person has unique experiences and perspectives.By valuing these individual contributions,we can learn from one another and gain a broader understanding of the world.For instance,in a multicultural classroom,students can share their cultural traditions and customs,leading to a more diverse and engaging learning environment.Secondly,appreciating diversity promotes creativity and innovation.When people with different skills,talents,and ideas come together,they can collaborate to solve complex problems and create new solutions.In the workplace,a diverse team can bring fresh perspectives to a project,leading to more effective and efficient outcomes. Moreover,respecting differences helps to break down stereotypes and prejudices.When we take the time to understand and appreciate the unique qualities of others,we are less likely to make assumptions based on superficial characteristics.This can lead to stronger relationships and a more inclusive society.However,respecting diversity also comes with challenges.It requires openmindedness and a willingness to step outside of our comfort zones.It means being willing to listen to different viewpoints and to engage in respectful dialogue.It also involves educating ourselves about other cultures and traditions to better understand and appreciate their value.In conclusion,respecting the beauty of diversity is essential for building a more tolerant and harmonious world.By embracing and celebrating our differences,we can create a society that is rich in culture,ideas,and opportunities.It is up to each of us to take the initiative to learn about and appreciate the unique qualities of others,and to promote a culture of inclusivity and respect.。

a r X i v :m a t h /07175v1[mat h.GR]24J an27ON THE UNIQUENESS OF LOOPS M (G,2)PETR VOJT ˇECHOVSK ´Y Abstract.Let G be a finite group and C 2the cyclic group of order 2.Con-sider the 8multiplicative operations (x,y )→(x i y j )k ,where i ,j ,k ∈{−1,1}.Define a new multiplication on G ×C 2by assigning one of the above 8mul-tiplications to each quarter (G ×{i })×(G ×{j }),for i ,j ∈C 2.When G is nonabelian then exactly four assignments yield Moufang loops that are not associative;all (anti)isomorphic,known as loops M (G,2).Keywords:Moufang loops,loops M (G,2),inverse property loops MSC2000:20N05 1.Introduction Because of the specialized topic of this paper,we assume that the reader is familiar with the theory of Bol and Moufang loops (cf.[5]).Chein introduced the following construction in [1]to obtain Moufang loops from groups:Let G be a finite group and let x ;x ∈G }be a set of new elements.Define multiplication ∗on G ∪y =x ∗y =x ∗Work partially supported by Grant Agency of Charles University,grant number 269/2001/B-MAT/MFF.12PETR VOJTˇECHOVSK´Y2.NotationLet us introduce a notation that will better serve our purposes.Consider the permutationsι,σ,τof G×G defined by(x,y)ι=(x,y),(x,y)σ=(y,x),and (x,y)τ=(y−1,x).Sinceσ2=τ4=ιandστσ=τ−1,the group A generated byσandτis isomorphic to D8,the dihedral group of order8.The elementsψof A are described byψ(x,y)(y,x)(y−1,x)(x−1,y−1)(y,x−1)(x−1,y)(y−1,x−1)(x,y−1). We like to think of these elements as multiplications in G,and often identifyψ∈A with the mapψ∆:G×G→G,where(x,y)∆=xy.For instance,the permutation στdetermines the multiplication x∗y=x−1y.Note thatσ∆=ι∆when G is abelian,and that A∆=ι∆when G is an elementary abelian2-group.To avoid trivialities,we assume throughout the paper that G is not an elementary abelian2-group,and that|G|>1.It is natural to split the multiplication table of M(G,2)into four quarters G×G, G×G×G and G,as in∗G.Then Chein’s construction(1)can be represented by the matrix (2)M c= ισστ3τ .For example,we can see from M c that(x,y)στ3=LOOPS M(G,2)3 The equation y=1∗y holds for every y∈G if and only if y=(1,y)α,which happens if and only ifα∈{ι,σ,τ3,στ}.Similarly,the equation y=y∗1holds for every y∈G if and only ifα∈{ι,σ,τ,στ3}.Altogether,y=y∗1=1∗y holds for every y∈G if and only ifα∈{ι,σ}.Following the same strategy,y holds for every y∈G if and only ifβ∈{ι,σ,τ3,στ},and y∗1holds for every y∈G if and only ifγ∈{ι,σ,τ,στ3}.Once M is a loop,it must have two-sided inverses:REVISION:IN THE ORIGINAL VERSION I CLAIMED THAT IF M IS A LOOP THAT IS BOL THEN IT MUST BE MOUFANG.IT’S NOT TRUE. Lemma2.If M is a loop then it is an inverse property loop.Proof.Assume that x∗y=1for some x,y∈G∪G,by Lemma1.We therefore want to show that(x,y)ε=1 implies(y,x)ε=1for everyε∈A and x,y∈G.Pickε∈A.Then(x,y)ε=(x i y j)k for some i,j,k∈{−1,1}.Assume that (x,y)ε=1.Then x i y j=1and y j x i=1.If i=j,we conclude from the latter equality that y i x j=1,and thus(y,x)ε=1.The inverse of the former equality yields y−j x−i=1.If i=−j,we immediately have y i x j=1,and thus(y,x)ε=1.Hence M is an inverse property loop.Given M as in(3),letM op= σασγσβσδ .Lemma3.The quasigroup M op is opposite to M.Proof.Denote by◦the multiplication in M op.Thenx◦y=(x,y)σα=(y,x)α=y∗x,x◦(x,y)σγ=y∗x,(x,y)σβ=x,y=(x,y)σδ=(y,x)δ=x,for every x,y∈G.Let us assume from now on that G is nonabelian.Then the identity xy=yx and any other identity that reduces to xy=yx do not hold in G,of course.We will come across the identity xxy=yxx.Note that this identity holds in G if and only if the center of G is of index2in G.We would like to know when M is a Bol(and hence Moufang)loop.Assume from now on that M is a loop.Recall that the opposite of a Moufang loop is again Moufang.We can therefore combine Lemmas1,3and assume that the loop M satisfiesα=ι.Since every Moufang loop is diassociative,we are going to have a look at such loops:Lemma 4.If G is nonabelian and M is a diassociative loop withα=ιthen (β,γ,δ)is one of the eight triples(4)(ι,ι,ι),(τ3,ι,στ),(σ,σ,σ),(στ,σ,τ3), (τ3,τ,τ2),(ι,τ,στ3),(σ,στ3,τ),(στ,στ3,στ2).4PETR VOJTˇECHOVSK´YProof.The identities(x)∗y=x∗y),x)=(x hold in M,for every x,y∈G.They translate into(5)(x,x)δy=(x,(x,y)γ)δ,(6)(x,(y,x)β)δ=((x,y)γ,x)δ,respectively.We arefirst going to check which pairs(γ,δ)satisfy(5).Assume thatγ=ι.Then(5)becomes(x,x)δy=(x,xy)δ.Denote this identity by I(δ).Then I(ι)is xxy=xxy(true),I(σ)is xxy=xyx(false),I(τ)is y=y−1 (false),I(τ2)is x−2y=x−1y−1x−1(false),I(τ3)is y=xyx−1(false),I(στ)is y=y(true),I(στ2)is x−2y=y−1x−1x−1(false),and I(στ3)is y=xy−1x−1 (false).Assume thatγ=σ.Then(5)becomes(x,x)δy=(x,yx)δ.Verify that this identity holds only ifδ=σorδ=τ3.(The caseδ=σleads to the identity xxy=yxx mentioned before this Lemma.)Whenγ=τ,(5)holds only ifδ=τ2orδ=στ3.Whenγ=στ3,(5)holds only ifδ=τorδ=στ2.Altogether,(5)can be satisfied only when(γ,δ)is one of the8pairs(ι,ι), (ι,στ),(σ,σ),(σ,τ3),(τ,τ2),(τ,στ3),(στ3,τ),(στ3,στ2).All these pairs will now be tested on(6).Straightforward calculation shows that(6)can be satisfied only when(β,γ,δ)is one of the8triples listed in(4).The Moufang identity((xy)x)z=x(y(xz))will help us eliminate4out of the8 possibilities in(4).We have((x∗y∗(x∗z))in M,and thus (7)(((x,y)β,x)γ,z)γ=(x,(y,xz)γ)β.The pairs(β,γ)=(σ,σ),(τ3,ι),(ι,τ),(στ,στ3)do not satisfy(7).For instance, (β,γ)=(σ,σ)turns(7)into zxyx=xzyx,i.e.,zx=xz.The four remaining triples from(4)yield Moufang loops,as we are going to show.The quadruple(α,β,γ,δ)=(ι,ι,ι,ι)=Gιcorresponds to the direct product of G and the two-element cyclic group.The quadruple(ι,σ,στ3,τ)=M c is the Chein Moufang loop M(G,2)that is associative if and only if G is abelian,by[1].(We can also verify this directly.)Set Gτ=(ι,τ3,τ,τ2)and Mσ=(ι,στ,σ,τ3).We claim that Gιis isomorphic to Gτ,and M c is isomorphic to Mσ.Lemma5.Define T:A4→A4byM= αβγδ → ατ3βγττ2δ =MT.If((x,y)β∆)−1=(y−1,x−1)β∆and((x,y)γ∆)−1=(x−1,y)γτ∆then M is iso-morphic to MT.Proof.Consider the permutation f of G∪x)=G.With x,y∈G,we have(x∗y)f=(x,y)α∆f=(x,y)α∆=x◦y=xf◦yf,(y)f=(x,y)δ∆f=(x,y)δ∆=(x−1,y−1)τ2δ∆=yf.LOOPS M(G,2)5Using the assumption onβandγ,we also have(x∗(x,y)β∆f=(y−1,x−1)β∆=yf, and((x,y)γ∆f=(x−1,y)γτ∆=G,whereα,β,γ,δ∈A= σ,τ ,and (x,y)σ=(y,x),(x,y)τ=(y−1,x).When G is nonabelian,then L is a Moufang loop if and only if M is equal to one of the following matrices:Gι= ιιιι ,G opι= σσσσ ,Gτ= ιτ3ττ2 ,G opτ= σστστ3στ2 ,M c= ισστ3τ ,M op c= στ3ιστ ,Mσ= ιστστ3 ,M opσ= σιτστ3 .The loops X op are opposite to the loops X.The isomorphic loops Gι,Gτand their opposites are groups.The isomorphic loops M c,Mσand their opposites are Moufang loops that are not associative.References[1]Orin Chein,Moufang loops of small order,Memoirs of the American Mathe-matical Society,Volume13,Issue1,Number197(1978).[2]O.Chein,H.O.Pflugfelder,The smallest Moufang loop,Arch.Math.22(1971),573–576.[3]Aleˇs Dr´a pal and Petr Vojtˇe chovsk´y,Moufang loops that share associator andthree quarters of their multiplication tables,submitted.[4]Edgar G.Goodaire,Sean May,Maitreyi Raman,The Moufang Loops of Orderless than64,Nova Science Publishers,1999.[5]H.O.Pflugfelder,Quasigroups and Loops:Introduction,Sigma series in puremathematics7,Heldermann Verlag Berlin,1990.[6]Petr Vojtˇe chovsk´y,The smallest Moufang loop revisited,to appear in Resultsin Mathematics.E-mail address:petr@Department of Mathematics,University of Denver,2360S Gaylord St,Denver,CO 80208,USA。

给外国人回信英语作文Dear [Name], I hope this letter finds you well. I wanted to take the time to write to you and express my appreciation for your recent correspondence. It is always a pleasure to hear from friends across the globe and to engage in meaningful conversations.First and foremost, I want to thank you for sharing your insights and perspectives on the topic we discussed. Your thoughtful observations and experiences have provided me with a deeper understanding of the cultural nuances and societal dynamics at play in your country. I am grateful for the opportunity to learn from you and to broaden my worldview.In response to your questions about my own cultural traditions and customs, I am happy to provide you with a glimpse into my way of life. In my country, [Country], we place a strong emphasis on [custom/tradition], which isdeeply rooted in our history and values. This[custom/tradition] plays a significant role in shaping our identity and fostering a sense of community among our people.I am proud to uphold these traditions and celebrate them with my loved ones.Furthermore, I am eager to learn more about your traditions and the rituals that hold special significance in your culture. I am fascinated by the diversity of practices around the world and the meaningful ways in which they bring people together. I believe that through sharing our traditions, we can develop a richer appreciation for the beauty and uniqueness of each other's heritage.In addition to cultural exchange, I am also interested in exploring potential opportunities for collaboration and partnership between our respective countries. As a firm believer in the power of international cooperation, I am keen on identifying areas where we can work together to addressshared challenges and contribute to mutual growth and development.I look forward to the possibility of furthering our dialogue and strengthening the bond between our nations. It is my hope that through open communication and a spirit of goodwill, we can build bridges that foster understanding, respect, and friendship.In closing, I extend my warmest regards to you and your loved ones. May our continued exchange serve as a source of inspiration and enrichment for both of us.。

In the realm of personal development and selfimprovement,the concept of uniqueness is a cornerstone.It is the idea that each individual possesses a set of qualities and characteristics that make them distinct from others.This uniqueness is not only a source of pride but also a driving force for personal growth and success.The Essence of UniquenessYour uniqueness stems from a combination of factors,including your experiences,values, beliefs,skills,and passions.It is the amalgamation of these elements that shapes your identity and defines who you are.Recognizing and embracing your uniqueness is essential for several reasons:1.SelfAcceptance:Understanding your uniqueness allows you to accept yourself as you are,with all your strengths and weaknesses.This acceptance is the first step towards selflove and confidence.2.Personal Growth:When you acknowledge your individuality,you are more likely to pursue personal growth in a way that aligns with your authentic self.This means setting goals that are true to your values and aspirations.3.Creativity:Your unique perspective can be a wellspring of creativity.It enables you to think outside the box and come up with innovative ideas and solutions that might not occur to others.4.Diversity:In a world that often seeks conformity,your uniqueness contributes to the diversity of thought and culture.It enriches the tapestry of society and promotes a more inclusive and tolerant environment.Cultivating UniquenessTo make the most of your uniqueness,consider the following strategies:SelfReflection:Spend time reflecting on your experiences,what you value,and what you believe in.This introspection can help you better understand your authentic self. Continuous Learning:Embrace lifelong learning to expand your knowledge and skills. This not only makes you more versatile but also allows you to contribute your unique insights to various fields.Express Yourself:Find ways to express your individuality,whether through art,writing,public speaking,or any other medium that resonates with you.Embrace Your Passions:Pursue what you are passionate about.Your enthusiasm for certain subjects or activities can set you apart and lead to meaningful contributions.Challenge Yourself:Step out of your comfort zone and take on new challenges.This can help you discover new aspects of your personality and capabilities.The Impact of Your UniquenessYour uniqueness has the power to inspire others and make a positive impact on the world. By being true to yourself,you can:Lead by Example:Show others the value of embracing their own uniqueness and the benefits it brings to personal and professional life.Influence Change:Use your unique perspective to advocate for causes that you believe in and to influence positive change in your community or industry.Create Opportunities:Your distinctive skills and ideas can open up new opportunities for you and others,leading to innovation and progress.In conclusion,recognizing and celebrating your uniqueness is not just about personal fulfillment its about contributing to a richer,more diverse,and dynamic world.By understanding and leveraging your individuality,you can achieve greater success and make a meaningful difference.。

Perseverance is a quality that is highly valued in many cultures,and it is often celebrated as a key to success in various aspects of life.Here is an essay on the importance of perseverance in English:The Value of PerseveranceIn the journey of life,we encounter numerous challenges and obstacles that test our resolve and determination.It is in these moments that the spirit of perseverance becomes invaluable.Perseverance is the unwavering commitment to continue striving for a goal despite the difficulties encountered along the way.It is a testament to ones character and resilience.The Essence of PerseverancePerseverance is not just about enduring hardship it is about embracing the process of growth that comes with overcoming challenges.It is the quiet strength that allows individuals to push through their limitations and achieve what initially seemed impossible. It is the force that drives inventors to refine their prototypes,artists to perfect their craft, and athletes to break records.Historical Examples of PerseveranceHistory is replete with examples of individuals who have demonstrated extraordinary perseverance.Thomas Edison,for instance,faced numerous failures before he successfully invented the light bulb.His famous quote,I have not failed.Ive just found 10,000ways that wont work,encapsulates the essence of perseverance.Similarly,J.K. Rowling,the author of the Harry Potter series,was rejected by multiple publishers before her work became a global phenomenon.The Role of Perseverance in Personal GrowthOn a personal level,perseverance is crucial for selfimprovement and the pursuit of ones aspirations.It is the driving force that helps us to learn from our mistakes,to adapt to new situations,and to grow stronger in the face of adversity.When we persevere,we develop a deeper understanding of our capabilities and limitations,which in turn,helps us to set realistic goals and to work towards them with renewed vigor.Cultivating PerseveranceCultivating perseverance is not an easy task,but it is a worthwhile endeavor.It requires patience,selfdiscipline,and a positive mindset.One must learn to view setbacks as opportunities for learning rather than as insurmountable barriers.Additionally,setting small,achievable goals can help to build momentum and foster a sense of accomplishment,which in turn,can bolster ones perseverance.ConclusionIn conclusion,perseverance is a vital quality that can significantly impact ones ability to achieve success and fulfillment in life.It is the bedrock of personal growth and the key to unlocking our potential.By embracing perseverance,we can navigate the complexities of life with greater confidence and resilience,ultimately shaping a more meaningful and rewarding journey.This essay highlights the significance of perseverance and encourages readers to embrace this quality in their own lives.It provides historical context and offers practical advice on how to develop perseverance,emphasizing its importance in both personal and professional realms.。

Perseverance is a quality that can lead to great achievements in life.It is the ability to continue working towards a goal despite facing obstacles and setbacks.Here are some points to consider when writing an essay on the topic of perseverance:1.Definition of Perseverance:Begin your essay by defining what perseverance means.It is the continuous effort to achieve a goal,even when faced with challenges.2.Importance of Perseverance:Explain why perseverance is important.It is a key to success in various fields such as academics,sports,business,and personal growth.3.Historical Examples:Provide examples of individuals who have demonstrated perseverance.Thomas Edisons numerous attempts to create a practical light bulb,or Abraham Lincolns multiple political defeats before becoming the President of the United States,are classic examples.4.Overcoming Failure:Discuss how perseverance helps individuals to overcome failure. It allows them to learn from their mistakes and keep moving forward.5.Developing Perseverance:Offer suggestions on how one can develop the habit of perseverance.This could include setting realistic goals,maintaining a positive attitude, and seeking support from others.6.Personal Stories:If you have personal experiences with perseverance,share them.This adds a personal touch and makes the essay more relatable.7.Quotes and Sayings:Incorporate quotes from famous individuals who have spoken about perseverance.For instance,The only limit to our realization of tomorrow will be our doubts of today.Franklin D.Roosevelt.8.Conclusion:Summarize the main points of your essay and reiterate the significance of perseverance.Encourage readers to embrace this quality in their own lives.9.Call to Action:End your essay with a call to action,urging readers to apply the lessons of perseverance in their own lives and to inspire others to do the same.Remember to structure your essay with a clear introduction,body paragraphs that explore each point in depth,and a strong conclusion that leaves a lasting impression on your e evidence and examples to support your arguments and make your essay more persuasive.。

Unit 3 Born to WinBorn to WinYou cannot teach a man anything. You can only help him discover it within himself.— Galileo[1] Each human being is born as something unique , something that never existed before. Each person is born with what he needs to win at life. A normal person can see, hear, touch, taste, and think for himself. Each has his own unique potentials —his capabilities and limitations. Each can be an important, thinking, aware, and creatively productive person in his own right — a winner.[2] The words “winner” and “loser” have many meanings. When we refer to a person as a winner, we do not mean one who defeats the other person by dominating and making him lose. Instead a winner is one who responds genuinely by being trustworthy and responsive , both as an individual and as a member of a society. A loser is one who fails to respond genuinely.[3] Few people are winners or losers all the time. It's a matter of degree . However, once a person has the capacity to be a winner, his chances are greater for becoming even more so…[4] Achievement is not the most important thing for winners; genuineness is. The genuine person realizes his own uniqueness and appreciates the uniqueness of others.[5] A winner is not afraid to do his own thinking and to use his own knowledge. He can separate facts from opinion and doesn't pretend to have all the answers. He listens to others, evaluates what they say, but comes to his own conclusions.[6] A winner is flexible . He does not have to respond in known, rigid ways. He can change his plans when the situation calls for it. A winner has a love for life. He enjoys work, play, food, other people, and the world of nature. Without guilt he enjoys his own accomplishments. Without envy he enjoys the accomplishments of others.[7] A winner cares about the world and its people. He is not separated from the general problems of society. He tries to improve the quality of life. Even in the face of national and international difficulty, he does not see himself as helpless . He does what he can to make the world a better place.[8] Although people are born to win, they are also born totally dependent on their environment. Winners successfully make the change from dependence to independence . Losers do not. Somewhere along the line losers begin to avoid becoming independent . This usually begins in childhood. Poor nutrition , cruelty , unhappy relationships, disease, continuing disappointments, and inadequate physical care are among the manyexperiences that contribute to making people losers.[9] A loser is held back by his low capacity to appropriately express himself through a full range of possible behavior. He may be unaware of other choices for his life if the path he chooses goes nowhere. He is afraid to try new things. He repeats not only his own mistakes and often repeats those of his family and culture.[10] A loser has difficulty giving and receiving love. He does not enter into close, honest, direct relationships with others. Instead, he tries to manipulate them into living up to his expectations and channels his energies into living up to their expectations.生而成功任何事都不可能由别人来教你，只能在别人的帮助下靠自己去发现。

考博英语（词汇）历年真题试卷汇编46(总分50, 做题时间90分钟)1. Structure and Vocabulary1.It is not too late, but ______ action is needed.(2002年春季上海交通大学考博试题)SSS_SINGLE_SELA fightB urgentC hurryD prompt分值: 2答案:D解析：本题空格处是说需要立即行动。

D项“prompt迅速的、及时的”，如：This mechanic is always prompt in his duties．(这个机修工人做工作一向是迅速爽快的。

)其他三项“right对的、恰当的；urgent紧急的：hurry匆忙、仓促”都不正确。

2.In November 1987 the government ______ a public debate on the future direction of the official sports policy.(2008年四川大学考博试题) SSS_SINGLE_SELA initiatedB designedC inducedD promoted分值: 2答案:D解析：initiate开始，创始，design设计，induce 引诱，招致，promote促进，发起。

根据后面的a public debate可知，正确答案为D，即“发起公众讨论”。

3.The______action of the policemen saved the people in the house from being burnt. (厦门大学2011年试题)SSS_SINGLE_SELA supremeB significantC promptD vital分值: 2答案:C解析：句子的大意为：警察迅速的行动救了房子中的人，使他们免受烧伤。

A项supreme“最高的，至高的，最重要的”；B项significant“重大的，有效的，有意义的”；C项prompt“及时的，迅速的，敏捷的”：D项vital“至关重要的，生死攸关的，有活力的”。

英语作文写日期The art of writing the date in English is a fundamental skill that is often overlooked but holds great importance in our daily lives. Whether it's filling out forms, organizing schedules, or corresponding with others, the proper formatting and expression of the date is crucial for clear and effective communication. In this essay, we will explore the various ways to write the date in English, the significance of this practice, and the nuances that one should be mindful of.One of the most common methods of expressing the date in English is the numerical format, which typically follows the pattern of month/day/year. For example, July 4, 2023 would be written as7/4/2023. This format is widely used in the United States and is a familiar sight on documents, calendars, and digital interfaces. However, it's important to note that in some parts of the world, the day/month/year format is more common, such as 4/7/2023. Understanding the local conventions is essential when communicating with individuals from different regions.Another popular way to write the date is by spelling out the month,day, and year. This format provides a more formal and eloquent expression of the date, such as July 4, 2023. This style is often preferred in professional settings, academic writing, and formal correspondence. It adds a touch of elegance and clarity to the date, making it easier to read and understand at a glance.In addition to the numerical and spelled-out formats, there are also variations that incorporate the use of ordinal numbers for the day. For instance, July 4th, 2023 or the 4th of July, 2023. This method is particularly common when referring to specific dates, such as holidays or significant events. The inclusion of the ordinal number helps to emphasize the uniqueness of the day and can add a sense of importance or formality to the date.The significance of accurately writing the date in English cannot be overstated. Proper date formatting ensures clear and unambiguous communication, preventing potential misunderstandings or confusion. In the business world, accurate date recording is crucial for maintaining accurate records, tracking deadlines, and coordinating schedules. In the academic realm, proper date formatting is essential for citing sources, submitting assignments, and ensuring the chronological integrity of research and documentation.Moreover, the way we write the date can also convey cultural andregional differences. As mentioned earlier, the month/day/year format is predominant in the United States, while theday/month/year format is more common in other parts of the world. Understanding and respecting these cultural nuances is important when communicating with individuals from diverse backgrounds.Furthermore, the choice of language and terminology used to express the date can also carry symbolic and emotional significance. For example, the use of the term "the 4th of July" to refer to Independence Day in the United States evokes a sense of national pride and historical significance that goes beyond the mere numerical expression of the date.In conclusion, the art of writing the date in English is a multifaceted and essential skill that encompasses both practical and cultural considerations. From the numerical formats to the more formal and eloquent expressions, the way we write the date reflects our attention to detail, our respect for conventions, and our ability to communicate effectively. By mastering the nuances of date writing, we can enhance our professional and personal interactions, maintain accurate records, and contribute to a more cohesive and interconnected global community.。