On the Satisfiability of Modular Arithmetic Formulae

合集下载

【最新精选】了不起的盖茨比经典句子中英文

菲茨杰拉德的《了不起的盖茨比》，摘录书中比较经典的句子：Chapter 11. 每当你觉得想要批评什么人的时候，你切要记着，这个世界上的人并非都具备你禀有的条件。

Whenever you feel like criticizing any one, just remember that all the people in this world haven’t had the advantages that you’ve had.2.人们的善恶感一生下来就有差异。

A sense of the fundamental decencies is parceled out unequally at birth.3.人们的品行有的好像建筑在坚硬的岩石上，有的好像建筑在泥沼里，不过超过一定的限度，我就不在乎它建在什么之上了。

Conduct may be founded on the hard rock or the wet marshes, but after a certain point I don’t care what it’s founded on.Chapter 2这时，天色已经暗了下来，我们这排高高地俯瞰着城市的灯火通明的窗户，一定让街头偶尔抬头眺望的人感到了，人类的秘密也有其一份在这里吧，我也是这样的一个过路人，举头望着诧异着。

我既在事内又在事外，几杯永无枯竭的五彩纷呈的生活所吸引，同时又被其排斥着。

Yet high over the city our line of yellow windows must have contributed their share of human secrecy to the casual watcher in the darkening streets, and I was him too, looking up and wondering. I was within and without, simultaneously enchanted and repelled by the inexhaustible variety of life.Chapter 31. 他理解体谅地笑了——这笑比理解和体谅有更多的含义。

my model的英语作文

When writing an essay in English about My Model,its important to consider the context in which the term model is being used.Here are a few different approaches you might take,depending on the specific meaning of model in your essay:1.A Role Model:Begin by introducing who your role model is and why they are important to you. Discuss the qualities and achievements of your role model that you admire. Explain how their actions or life story has influenced your own life or goals.Example Paragraph:My role model is Malala Yousafzai,a Pakistani activist for female education and the youngest Nobel Prize laureate.Her courage and determination to fight for girls education rights in the face of adversity have deeply inspired me.Malalas story has taught me the importance of standing up for what I believe in,even when it is difficult.2.A Fashion Model:Describe the physical attributes and style of the model.Discuss the impact they have had on the fashion industry or their unique contributions to it.Explain why you find their work or presence in the industry notable.Example Paragraph:Kendall Jenner is a fashion model who has made a significant impact on the industry with her unique style and presence.Her tall and slender physique,combined with her ability to carry off diverse looks,has made her a favorite among designers and fashion enthusiasts alike.I admire her for her versatility and the way she uses her platform to promote body positivity.3.A Model in Science or Technology:Introduce the model as a theoretical framework or a practical tool used in a specific field.Explain the principles behind the model and how it is applied.Discuss the benefits or limitations of the model and its implications in the real world.Example Paragraph:The Standard Model in physics is a theoretical framework that describes three of the four known fundamental forces excluding gravity and classifies all known elementary particles.It has been instrumental in understanding the behavior of subatomic particles and predicting the existence of new particles,such as the Higgs boson.However,the models inability to incorporate gravity or dark matter has led to ongoing research for amore comprehensive theory.4.A Model in Business or Economics:Introduce the business or economic model and its purpose.Explain how the model works and the strategies it employs.Discuss the success or challenges associated with the model and its potential for future growth.Example Paragraph:The subscriptionbased business model has become increasingly popular in recent years, particularly in the software panies like Adobe have transitioned from selling packaged software to offering services on a subscription basis,allowing for continuous revenue streams and a more predictable income.This model has been successful in fostering customer loyalty and providing a steady income,although it requires ongoing innovation to maintain customer interest.5.A Model in Art or Design:Describe the aesthetic or functional qualities of the model.Discuss the creative process or design principles that inform the model.Explain the cultural or historical significance of the model and its influence on contemporary art or design.Example Paragraph:The Eames Lounge Chair,designed by Charles and Ray Eames,is a model of modern furniture that has become an icon of midcentury design.Its elegant form,made from molded plywood and leather,exemplifies the designers commitment to blending comfort with aesthetics.The chairs timeless appeal has made it a staple in both residential and commercial settings,influencing countless furniture designs that followed. Remember to structure your essay with a clear introduction,body paragraphs that develop your points,and a conclusion that summarizes your main e specific examples and evidence to support your claims,and ensure your writing is clear,concise, and engaging.。

门捷列夫英语作文

门捷列夫英语作文In the realm of scientific discovery, Dmitri Mendeleev's contribution stands as a monumental milestone. His innovative approach to organizing elements into a coherent system revolutionized the field of chemistry.Mendeleev's enduring legacy, the Periodic Table, was not just a compilation of facts but a testament to his analytical prowess and foresight. He meticulously arranged the elements, predicting their properties and even the existence of undiscovered ones.The Periodic Table, a simple yet profound tool, has guided generations of scientists. It is a universal language that transcends borders, connecting researchers in their quest for understanding the fundamental building blocks of our universe.Despite facing skepticism and resistance, Mendeleev's unwavering belief in his work paved the way for its eventual acceptance. His perseverance is a lesson in the importance of conviction in the face of doubt.Today, the Periodic Table is not just a staple in chemistry classrooms but a symbol of scientific progress. It serves as a reminder that the pursuit of knowledge is an ongoing journey, with each discovery a stepping stone to the next.Mendeleev's story is one of curiosity, ingenuity, and the relentless pursuit of truth. His work has left an indelible mark on the scientific community, inspiring countless minds to delve deeper into the mysteries of the natural world.In conclusion, Dmitri Mendeleev's life and work exemplify the spirit of scientific inquiry. His Periodic Table is more than just an educational tool; it is a beacon of human ingenuity, guiding us toward a deeper understanding of the elements that constitute our world.。

科技英语翻译1

驱动这些机器的动力装置是一台50马力的感应电动机。
► 2）通顺易懂 ► 译文的语言符合译语语法结构及表达习惯，容易为读者所理解和接受。
► A. When a person sees, smells, hears or touches something, then he is perceiving.
2. Cramped(狭窄的) conditions means that passengers’ legs cannot move around freely.
空间狭窄，旅客的两腿就不能自由活动。
3. All bodies are known to possess weight and occupy space.
忠实、通顺（普遍观点）
► 科技英语文章特点：(well-knit structure;tight logic;various styles)结构严谨，逻辑严密，文体多样
1. 科技翻译的标准：准确规范，通顺易懂，简洁明晰 1）准确规范
所谓准确，就是忠实地，不折不扣地传达原文的全部信息内容。所谓规范，就是译文要符合所涉及的科学技或某个专业领域的专业语言表
实验结果等，而不是介绍这是这些结果，理论或现象是由谁发明或发现的。
► In this section, a process description and a simplified process flowsheet are given for each DR process to illustrate the types of equipment used and to describe the flow of materials through the plant. The discussion does not mention all the variations of the flowsheet which may exist or the current status of particular plants. In the majority of the DR processes described in this section, natural gas is reformed in a catalyst bed with steam or gaseous reduction products from the reduction reactor. Partial oxidation processes which gasify liquid hydrocarbons, heavy residuals and coal are also discussed. The reformer and partial oxidation gasifier are interchangeable for several of the DR processes.

涌现优于权威英文原文

涌现优于权威英文原文"Emergence Trumps Authority"In today's rapidly changing and interconnected world, the concept of emergence is gaining increasing attention as a more effective way to tackle complex problems and drive innovation. Emergence refers to the phenomenon where new and unexpected patterns, properties, or behaviors emerge from the interactions of simpler elements within a system. This stands in stark contrast to the traditional top-down approach of authority, where decisions and solutions are handed down from a single source of power.The main advantage of emergence over authority is its ability to harness the collective intelligence and creativity of a group. Instead of relying on the expertise of a few individuals at the top, emergence draws on the diverse perspectives and experiences of many. This leads to more robust and innovative solutions, as well as greater buy-in and support from those involved in the process.Furthermore, emergence is better suited to navigate the complexities and uncertainties of modern challenges. With the pace of change accelerating and the interdependencies of various systems becoming more evident, no single authority figure can possibly possess all the knowledge and insight needed to address the diverse and evolving issues we face. In contrast, emergence allows for a more organic and adaptive approach, where solutions can emerge and evolve over time as new information and perspectives come to light.Additionally, emergence encourages participation and empowerment, as individuals feel a sense of ownership and responsibility for the outcomes of the collective efforts. This can lead to increased motivation, collaboration, and resilience within the group, as well as a greater sense of satisfaction and fulfillment for all involved.While authority certainly has its time and place, especially in situations requiring clear direction and decisive action, the benefits of emergence cannot be overlooked. By recognizing and harnessing the power of emergence, organizations, communities, and individuals can better adapt to the complexities and uncertainties of our modern world and drive more effective and sustainable solutions. Ultimately, "Emergence Trumps Authority."。

Finiteness results for modular curves of genus at least 2

a r X i v :m a t h /0211394v 2 [m a t h .N T ] 22 D e c 2003FINITENESS RESULTS FOR MODULAR CURVES OF GENUS ATLEAST 2MATTHEW H.BAKER,ENRIQUE GONZ ´ALEZ-JIM ´ENEZ,JOSEP GONZ ´ALEZ,AND BJORN POONEN Abstract.A curve X over Q is modular if it is dominated by X 1(N )for some N ;if in addition the image of its jacobian in J 1(N )is contained in the new subvariety of J 1(N ),then X is called a new modular curve.We prove that for each g ≥2,the set of new modular curves over Q of genus g is ﬁnite and computable.For the computability result,we prove an algorithmic version of the de Franchis-Severi Theorem.Similar ﬁniteness results are proved for new modular curves of bounded gonality,for new modular curves whose jacobian is a quotient of J 0(N )new with N divisible by a prescribed prime,and for modular curves (new or not)with levels in a restricted set.We study new modular hyperelliptic curves in detail.In particular,we ﬁnd all new modular curves of genus 2explicitly,and construct what might be the complete list of all new modular hyperelliptic curves of all genera.Finally we prove that for each ﬁeld k of characteristic zero and g ≥2,the set of genus-g curves over k dominated by a Fermat curve is ﬁnite and computable.1.Introduction Let X 1(N )be the usual modular curve over Q .(See Section 3.1for a deﬁnition.)A curve 1X over Q will be called modular if there exists a nonconstant morphism π:X 1(N )→X over Q .If X is modular,then X (Q )is nonempty,since it contains the image of the cusp ∞∈X 1(N )(Q ).The converse,namely that if X (Q )is nonempty then X is modular,holds if the genus g of X satisﬁes g ≤1[9].In particular,there are inﬁnitely many modular curves over Q of genus 1.On the other hand,we propose the following:Conjecture 1.1.For each g ≥2,the set of modular curves over Q of genus g is ﬁnite.Remark 1.2.(i)When we speak of the ﬁniteness of the set of curves over Q satisfying some condition,we mean the ﬁniteness of the set of Q -isomorphism classes of such curves.(ii)For any ﬁxed N ,the de Franchis-Severi Theorem (see Theorem 5.5)implies theﬁniteness of the set of curves over Q dominated by X 1(N ).Conjecture 1.1can be2BAKER,GONZ´ALEZ-JIM´ENEZ,GONZ´ALEZ,AND POONENthought of as a version that is uniform as one ascends the tower of modular curves X1(N),provided that oneﬁxes the genus of the dominated curve.(iii)Conjecture1.1is true if one restricts the statement to quotients of X1(N)by sub-groups of its group of modular automorphisms.See Remark3.16for details.(iv)If X1(N)dominates a curve X,then the jacobian Jac X is a quotient2of J1(N):= Jac X1(N).The converse,namely that if X is a curve such that X(Q)is nonempty and Jac X is a quotient of J1(N)then X is dominated by X1(N),holds if the genusg of X is≤1,but can fail for g≥2.See Section8.2for other“pathologies.”(v)In contrast with Conjecture1.1,there exist inﬁnitely many genus-two curves over Q whose jacobians are quotients of J1(N)for some N.See Proposition8.2(5).(vi)In Section9,we use a result of Aoki[3]to prove an analogue of Conjecture1.1in which X1(N)is replaced by the Fermat curve x N+y N=z N in P2.In fact,such an analogue can be proved over arbitraryﬁelds of characteristic zero,not just Q.We prove many results towards Conjecture1.1in this paper.Given a variety X over aﬁeld k,letΩ=Ω1X/k denote the sheaf of regular1-forms.Call a modular curve X over Q newof level N if there exists a nonconstant morphismπ:X1(N)→X(deﬁned over Q)such that π∗H0(X,Ω)is contained in the new subspace H0(X1(N),Ω)new,or equivalently if the image of the homomorphismπ∗:Jac X→J1(N)induced by Picard functoriality is contained in the new subvariety J1(N)new of J1(N).(See Section3.1for the deﬁnitions of H0(X1(N),Ω)new, J1(N)new,J1(N)new,and so on.)For example,it is known that every elliptic curve E over Q is a new modular curve of level N,where N is the conductor of E.Here the conductor cond(A)of an abelian variety A over Q is a positive integer p p f p,where each exponent f p is deﬁned in terms of the action of an inertia subgroup of Gal(2Quotients or subvarieties of varieties,and morphisms between varieties,are implicitly assumed to be deﬁned over the sameﬁeld as the original varieties.If X is a curve over Q,and we wish to discuss automor-phisms over C,for example,we will write Aut(X C).Quotients of abelian varieties are assumed to be abelian variety quotients.MODULAR CURVES OF GENUS AT LEAST23 If we drop the assumption that our modular curves are new,we can still prove results,but (so far)only if we impose restrictions on the level.Given m>0,let Sparse m denote the set of positive integers N such that if1=d1<d2<···<d t=N are the positive divisors of N, then d i+1/d i>m for i=1,...,t−1.Deﬁne a function B(g)on integers g≥2by B(2)=13, B(3)=17,B(4)=21,and B(g)=6g−5for g≥5.(For the origin of this function,see the proofs of Propositions2.1and2.8.)A positive integer N is called m-smooth if all primes p dividing N satisfy p≤m.Let Smooth m denote the set of m-smooth integers.Theorem1.5.Fix g≥2,and let S be a subset of{1,2,...}.The set of modular curves over Q of genus g and of level contained in S isﬁnite if any of the following hold:(i)S=Sparse B(g).(ii)S=Smooth m for some m>0.(iii)S is the set of prime powers.Remark1.6.Since Sparse B(g)∪Smooth B(g)contains all prime powers,parts(i)and(ii)of Theorem1.5imply(iii).Remark1.7.In contrast with Theorem1.3,we do not know,even in theory,how to compute theﬁnite sets of curves in Theorem1.5.The reason for this will be explained in Remark5.10. If X is a curve over aﬁeld k,and L is aﬁeld extension of k,let X L denote X×k L.The gonality G of a curve X over Q is the smallest possible degree of a nonconstant morphism X C→P1C.(There is also the notion of Q-gonality,where one only allows morphisms over Q.By deﬁning gonality using morphisms over C instead of Q,we make the next theorem stronger.)In Section4.3,we combine Theorem1.3with a known lower bound on the gonality of X1(N)to prove the following:Theorem1.8.For each G≥2,the set of new modular curves over Q of genus at least2 and gonality at most G isﬁnite and computable.(We could similarly prove an analogue of Theorem1.5for curves of bounded gonality instead ofﬁxed genus.)Recall that a curve X of genus g over aﬁeld k is called hyperelliptic if g≥2and the canonical map X→P g−1is not a closed immersion:equivalently,g≥2and there exists a degree-2morphism X→Y where Y has genus zero.If moreover X(k)=∅then Y≃P1k, and if also k is not of characteristic2,then X is birational to a curve of the form y2=f(x) where f is a separable polynomial in k[x]of degree2g+1or2g+2.Recall that X(k)=∅is automatic if X is modular,because of the cusp∞.Taking G=2in Theorem1.8,weﬁnd that the set of new modular hyperelliptic curves over Q isﬁnite and computable.We can say more:Theorem1.9.Let X be a new modular hyperelliptic curve over Q of genus g≥3and level N.Then(i)g≤16.(ii)If Jac X is a quotient of J0(N),then g≤10.If moreover3|N,then X is the genus-3 curve X0(39).(iii)If Jac X is not a quotient of J0(N),then either g is even or g≤9.Further information is given in Sections6.3and6.5,and in the appendix.See Section3.1 for the deﬁnitions of X0(N)and J0(N).4BAKER,GONZ´ALEZ-JIM´ENEZ,GONZ´ALEZ,AND POONENAs we have already remarked,if we consider all genera g≥2together,there are inﬁnitely many new modular curves.To obtainﬁniteness results,so far we have needed to restrict either the genus or the gonality.The following theorem,proved in Section7,gives a diﬀerent type of restriction that again impliesﬁniteness.Theorem1.10.For each prime p,the set of new modular curves over Q of genus at least2 whose jacobian is a quotient of J0(N)new for some N divisible by p isﬁnite and computable. Question1.11.Does Theorem1.10remain true if J0(N)new is replaced by J1(N)new? Call a curve X over aﬁeld k of characteristic zero k-modular if there exists a nonconstant morphism X1(N)k→X(over k).Question1.12.Is it true that for everyﬁeld k of characteristic zero,and every g≥2,the set of k-modular curves over k of genus g up to k-isomorphism isﬁnite?Remark 1.13.If X is a k-modular curve over k,and we deﬁne k0=k∩k,then X=X0×k0k for some k0-modular curve X0.This follows from the de Franchis-Severi Theorem.Remark1.14.If k and k′areﬁelds of characteristic zero with[k′:k]ﬁnite,then a positive answer to Question1.12for k′implies a positive answer for k,since Galois cohomology and theﬁniteness of automorphism group of curves of genus at least2show that for each X′over k′,there are at mostﬁnitely many curves X over k with X×k k′≃X′.But it is not clear, for instance,that a positive answer forMODULAR CURVES OF GENUS AT LEAST25 Corollary 2.4.Let X be a curve of genus g≥2over aﬁeld k of characteristic zero. Then the image X′of the canonical map X→P g−1is the common zero locus of the set ofhomogeneous polynomials of degree4that vanish on X′.Proof.We may assume that k is algebraically closed.If X is hyperelliptic of genus g,saybirational to y2=f(x)where f has distinct roots,then we may choose{x i dx/y:0≤i≤g−1}as basis of H0(X,Ω),and then the image of the canonical map is the rational normalcurve cut out by{t i t j−t i′t j′:i+j=i′+j′}where t0,...,t g−1are the homogeneous coordinates on P g−1.If X is nonhyperelliptic of genus3,its canonical model is a planequartic.In all other cases,we use Petri’s Theorem.(The zero locus of a homogeneouspolynomial h of degree d<4equals the zero locus of the set of homogeneous polynomialsof degree4that are multiples of h.) Lemma2.5.Let X be a hyperelliptic curve of genus g over aﬁeld k of characteristic zero, and suppose P∈X(k).Let{ω1,...,ωg}be a basis of H0(X,Ω)such that ord P(ω1)<···< ord P(ωg).Then x:=ωg−1/ωg and y:=dx/ωg generate the functionﬁeld k(X),and there is a unique polynomial F(x)of degree at most2g+2such that y2=F(x).Moreover,F is squarefree.If P is a Weierstrass point,then deg F=2g+1and ord P(ωi)=2i−2for all i;otherwise deg F=2g+2and ord P(ωi)=i−1for all i.Finally,it is possible to replace eachωi by a linear combination ofωi,ωi+1,...,ωg to makeωi=x g−i dx/y for1≤i≤g. Proof.This follows easily from Lemma3.6.1,Corollary3.6.3,and Theorem3.6.4of[21]. Proof of Proposition2.1.Suppose that X,P,q,and the w i are as in the statement of the proposition.Letωi be the corresponding elements of H0(X,Ω).We will show that X is determined by the w i when B=max{8g−7,6g+1}.Since B>8g−8,Lemma2.2implies that the w i determine the set of homogeneouspolynomial relations of degree4satisﬁed by theωi,so by Corollary2.4the w i determine theimage X′of the canonical map.In particular,the w i determine whether X is hyperelliptic,and they determine X if X is nonhyperelliptic.Therefore it remains to consider the case where X is hyperelliptic.Applying Gaussianelimination to the w i,we may assume0=ord q(w1)<···<ord q(w g)≤2g−2and that theﬁrst nonzero coeﬃcient of each w i is1.We use Lemma2.5repeatedly in what follows.Thevalue of ord(w2)determines whether P is a Weierstrass point.Suppose that P is a Weierstrass point.Then w i=q2i−2(1+···+O(q B−2i+2)),where each“···”here and in the rest of this proof represents some known linear combination of positive powers of q up to but not including the power in the big-O term.(“Known”means “determined by the original w i.”)Deﬁne x=w g−1/w g=q−2(1+···+O(q B−2g+2)).Deﬁne y=dx/(w g dq)=−2q−(2g+1)(1+···+O(q B−2g+2)).Then y2=4q−(4g+2)(1+···+O(q B−2g+2)).Since B≥6g+1,we have −(4g+2)+(B−2g+2)>0,so there is a unique polynomial F(of degree2g+1)such that y2=F(x).A similar calculation shows that in the case where P is not a Weierstrass point,thenB≥3g+2is enough. Remark2.6.Let us show that if the hypotheses of Proposition2.1are satisﬁed except that thew i belong tok ,Ω),then the conclusion still holds.Let E be aﬁnite Galois extension of k containing all the coeﬃcients of the w i.The E-span of the w i must be stable under Gal(E/k)if they come6BAKER,GONZ´ALEZ-JIM´ENEZ,GONZ´ALEZ,AND POONENfrom a curve over Q,and in this case,we can replace the w i by a k-rational basis of this span.Then Proposition2.1applies.Remark2.7.We can generalize Proposition2.1to the case where q is not a uniformizing parameter on X:Fix an integer g≥2,and let k be aﬁeld of characteristic zero.Let B>0be the integer appearing in the statement of Proposition2.1,and let e bea positive integer.Then if w1,...,w g are elements of k[[q]]/(q eB),then upto k-isomorphism,there exists at most one curve X over k such that thereexist P∈X(k),an analytic uniformizing parameter q′∈ˆO X,P and a relationq′=c e q e+c e+1q e+1+...with c e=0,such that w1dq,...,w g dq are theexpansions modulo q eB of some basis of H0(X,Ω).The proof of this statement is similar to the proof of Proposition2.1,and is left to the reader. The rest of this section is concerned with quantitative improvements to Proposition2.1, and is not needed for the generalﬁniteness and computability results of Sections4and5. Proposition2.8.Proposition2.1holds with B=B(g),where B(2)=13,B(3)=17, B(4)=21,and B(g)=6g−5for g≥5.Moreover,if we are given that the curve X to be recovered is hyperelliptic,then we can use B(g)=4g+5or B(g)=2g+4,according as P is a Weierstrass point or not.Proof.For nonhyperelliptic curves of genus g≥4,we use Theorem2.3instead of Corol-lary2.4to see that B>6g−6can be used in place of B>8g−8.Now suppose that X is hyperelliptic.As before,assume ord q(w1)<···<ord q(w g)and that theﬁrst nonzero coeﬃcient of each w i is1.The value of ord(w2)determines whether P is a Weierstrass point.Suppose that P is a Weierstrass point.Then w i=q2i−2(1+···+O(q B−2i+2)).(As in the proof of Proposition2.1,···means a linear combination of positive powers of q,whose coeﬃcients are determined by the w i.)Deﬁne˜x=w g−1/w g=q−2(1+···+O(q B−2g+2)).For 1≤i≤g−2,the expression˜x g−i w g=q2i−2(1+···+O(q B−2g+2))is the initial expansion of w i+ g j=i+1c ij w j for some c ij∈k,and all the c ij are determined if2+(B−2g+2)>2g−2, that is,if B≥4g−5.Let w′i=w i+ g j=i+1c ij w j=q2i−2(1+···+O(q B−2i+2)).Deﬁne x=w′1/w′2=q−2(1+···+O(q B−2)).Deﬁne y=−2q−(2g+1)(1+···+O(q B−2))as the solution to w′1dq=x g−1dx/y.Then y2=4q−(4g+2)(1+···+O(q B−2)),and if−(4g+2)+B−2>0, we can recover the polynomial F of degree2g+1such that y2=F(x).Hence B≥4g+5 suﬃces.A similar proof shows that B≥2g+4suﬃces in the case that P is not a Weierstrass point.Hence max{6g−5,4g+5,2g+4}suﬃces for all types of curves,except that the6g−5 should be8g−7when g=3.This is the function B(g). Remark2.9.We show here that for each g≥2,the bound B=4g+5for the precision needed to recover a hyperelliptic curve is sharp.Let F(x)∈C[x]be a monic polynomial of degree2g+1such that X:y2=F(x)and X′:y2=F(x)+1are curves of genus g that are not birationally equivalent.Let q be the uniformizing parameter at the point at inﬁnity on X such that x=q−2and y=q−(2g+1)+O(q−2g).Deﬁne q′similarly for X′.A calculation shows that the q-expansions of the diﬀerentials x i dx/y for0≤i≤g−1are even power series in q times dq,and modulo q4g+4dq they agree with the corresponding q′-expansionsMODULAR CURVES OF GENUS AT LEAST27 for X′except for the coeﬃcient of q4g+2dq in x g−1dx/y.By a change of analytic parameter q=Q+αQ4g+3for someα∈C,on X only,we can make even that coeﬃcient agree.A similar proof shows that in the case that P is not a Weierstrass point,the bound2g+4 cannot be improved.Remark2.10.When studying new modular curves of genus g,we can also use the multi-plicativity of Fourier coeﬃcients of modular forms(see(3.7))to determine some coeﬃcients from earlier ones.Hence we can sometimes get away with less than B(g)coeﬃcients of each modular form.2.2.Descending morphisms.The next result will be used a number of times throughout this paper.In particular,it will be an important ingredient in the proof of Theorem1.9. Proposition2.11.Let X,Y,Z be curves over aﬁeld k of characteristic zero,and assume that the genus of Y is>1.Then:(i)Given nonconstant morphismsπ:X→Z andφ:X→Y such thatφ∗H0(Y,Ω)⊆π∗H0(Z,Ω),there exists a nonconstant morphism u′:Z→Y making the diagramXπuY u′Ycommute.Proof.The conclusion of(i)is equivalent to the inclusionφ∗k(Y)⊆π∗k(Z).It suﬃces to prove that every function inφ∗k(Y)is expressible as a ratio of pullbacks of meromorphic diﬀerentials on Z.If Y is nonhyperelliptic,then theﬁeld k(Y)is generated by ratios of pairs of diﬀerentials in H0(Y,Ω),so the inclusion follows from the hypothesisφ∗H0(Y,Ω)⊆π∗H0(Z,Ω).When Y is hyperelliptic,we must modify this argument slightly.We have k(Y)=k(x,y),where y2=F(x)for some polynomial F(U)in k[U]without double roots. Theﬁeld generated by ratios of diﬀerentials in H0(Y,Ω)is k(x),soφ∗k(x)⊆π∗k(Z).To show thatφ∗y∈π∗k(Z)too,write y=x dx/(x dx/y)and observe that x dx/y∈H0(Y,Ω). Now we prove(ii).The hypothesis on u∗lets us apply(i)withφ=π◦u to construct u′:Y→Y.Since Y has genus>1and k has characteristic zero,the Hurwitz formula implies that u′is an automorphism.Considering functionﬁelds proves uniqueness. Remark2.12.Both parts of Proposition2.11can fail if the genus of Y is1.On the other hand,(ii)remains true under the additional assumption that X→Y is optimal in the sense that it does not factor nontrivally through any other genus1curve.Remark2.13.Proposition2.11remains true if k hasﬁnite characteristic,provided that one assumes that the morphisms are separable.8BAKER,GONZ´ALEZ-JIM´ENEZ,GONZ´ALEZ,AND POONEN3.Some facts about modular curves3.1.Basic facts about X1(N).We record facts about X1(N)that we will need for the proof ofﬁniteness in Theorem1.3.See[55]for a detailed introduction.Let H={z∈C:Im z>0}.The group SL2(R)acts on H by a b c d z=az+bq for some g-dimensional C-subspaceS2(N)of q C[[q]].We will not deﬁne modular forms in general here,but S2(N)is known as the space of weight2cusp forms onΓ1(N).If M|N and d|NM .Similarly deﬁne the old subvariety J1(N)old of J1(N).The space S2(N)has a hermitian inner product called the Petersson inner product.Let S2(N)new denote the orthogonal complement to S2(N)old in S2(N).The identiﬁcations above also give us new and old subspaces of H0(X1(N)C,Ω)and H0(J1(N)C,Ω).Let J1(N)new=J1(N)/J1(N)old. There is also an abelian subvariety J1(N)new of J1(N)that can be characterized in two ways: either as the abelian subvariety such thatker(H0(J1(N)C,Ω)→H0((J1(N)new)C,Ω))=H0(J1(N)C,Ω)old,or as the abelian subvariety such that J1(N)=J1(N)old+J1(N)new with J1(N)old∩J1(N)new ﬁnite.(The latter description uniquely characterizes J1(N)new because of a theorem that no Q-simple quotient of J1(N)old is isogenous to a Q-simple quotient of J1(N)new;this theorem can be proved by comparing conductors,using[11].)The abelian varieties J1(N)new and J1(N)new are Q-isogenous.We deﬁne X0(N),J0(N),and J0(N)new similarly,starting withΓ0(N):= a b c d ∈SL2(Z) c≡0(mod N)instead ofΓ1(N).For n≥1,there exist well-known correspondences T n on X1(N),and they induce en-domorphisms of S2(N)and of J1(N)known as Hecke operators,also denoted T n.There is a unique basis New N of S2(N)new consisting of f=a1q+a2q2+a3q3+...such that a1=1and T n f=a n f for all n≥1.The elements of New N are called the newforms ofMODULAR CURVES OF GENUS AT LEAST29level N.(For us,newforms are always normalized:this means that a1=1.)Each a n is√an algebraic integer,bounded byσ0(n)k/k).The Galois group G Q acts on New N.For any quotient A of J1(N),let S2(A)denote the image of H0(A C,Ω)→H0(J1(N)C,Ω)≃S2(N)(the last isomorphism drops the dq/q);similarly for any nonconstant morphismπ:X1(N)→X of curves,deﬁne S2(X):=π∗H0(X C,Ω)q(3.1)Q-isogenyf→A f.Shimura proved that J1(N)is isogenous to a product of these A f,and K.Ribet[53]proved that A f is Q-simple.This explains the surjectivity of(3.1).The injectivity is well-known to experts,but we could notﬁnd a suitable reference,so we will prove it,as part of Proposi-tion3.2.The subﬁeld E f=Q(a2,a3,...)of C is a numberﬁeld,and dim A f=[E f:Q].Moreover, End(A f)⊗Q can be canonically identiﬁed with E f,and under this identiﬁcation the element λ∈End A f acts on f as multiplication byλ(considered as element of E f),and on each Galois conjugateσf by multiplication byσλ.(Shimura[57,Theorem1]constructed an injection End(A f)⊗Q֒→E f,and Ribet[53,Corollary4.2]proved that it was an isomorphism.)If A and B are abelian varieties over Q,let A Q∼B denote the statement that A and B are isogenous over Q.Proposition3.2.Suppose f∈New N and f′∈New N′.Then A f Q∼A f′if and only if N=N′and f=τf′for someτ∈G Q.Proof(K.Ribet).The“if”part is immediate from Shimura’s construction.Therefore it suﬃces to show that one can recover f,up to Galois conjugacy,from the isogeny class of A f. Letℓbe a prime.Let V be the Qℓ-Tate module Vℓ(A f)attached to A f.Let Qℓ. The proof of Proposition4.1of[53]shows thatQℓ[G Q]-module of dimension2over Qℓ.Moreover,for p∤ℓN,the trace of the p-Frobenius automorphism acting on Vσequalsσ(a p),where a p∈E f is the coeﬃcient of q p in the Fourier expansion of f.If f′∈New′N is another weight2newform,and A f Q∼A f′,then(using′in the obvious way to denote objects associated with f′),we have isomorphisms of G Q-modules V≃V′and V′.Fixσ:E f֒→V′,whereσ′is some embedding E f′֒→3Earlier,in Theorem7.14of[55],Shimura had attached to f an abelian subvariety of J1(N).10BAKER,GONZ´ALEZ-JIM´ENEZ,GONZ´ALEZ,AND POONENWe have parallel decompositionsS2(N)new= f∈G Q\New N τ:E f֒→C CτfJ1(N)new Q∼ f∈G Q\New N A fand parallel decompositionsS2(N)= M|N f∈G Q\New M d|Nc d f(q d)Mfor some M|N and f∈New M,where c d∈E f depends on f and d.Proof.By multiplying the quotient map J1(N)→A by a positive integer,we may assume that it factors through the isogenyJ1(N)→ M|N f∈G Q\New M A n f fof(3.4).We may also assume that A is Q-simple,and even that A is isomorphic to A fwith for some f,so that the quotient map J1(N)→A is the composition of J1(N)→A n ffa homomorphism A n ff→A.The latter is given by an n f-tuple c=(c d)of elements of End(A f),indexed by the divisors d of N/M.Underc→A,X1(N)֒→J1(N)→A n ffthe1-form on A C≃(A f)C corresponding to f pulls back to d|Nc d f(q d)Mfor some M|N and f∈New M,where c d∈E f depends on f and d.Proof.Apply Lemma3.5to the Albanese homomorphism J1(N)→Jac X.3.2.Automorphisms of X1(N).3.2.1.Diamonds.The action on H of a matrix a b c d ∈Γ0(N)induces an automorphism of X1(N)over Q depending only on(d mod N).This automorphism is called the diamond operator d .It induces an automorphism of S2(N).Letεbe a Dirichlet character modulo N,that is,a homomorphism(Z/N Z)∗→C∗.Let S2(N,ε)be the C-vector space{h∈S2(N):h| d =ε(d)h}.A form h∈S2(N,ε)is called a form of Nebentypusε.Every newform f∈New N is a form of some Nebentypus,and is therefore an eigenvector for all the diamond operators.Character theory gives a decompositionS2(N)= εS2(N,ε),whereεruns over all Dirichlet characters modulo N.Deﬁne S2(N,ε)new=S2(N,ε)∩S2(N)new.When we writeε=1,we mean thatεis the trivial Dirichlet character modulo N,that is,ε(n)= 1if(n,N)=10otherwise.A form of Nebentypusε=1is a form onΓ0(N).We recall some properties of a newform f= ∞n=1a n q n∈S2(N,ε).Let condεdenote the smallest integer M|N such thatεis a composition(Z/N Z)∗→(Z/M Z)∗→C∗. Throughout this paragraph,p denotes a prime,and(3.11)pv p(N)=1,ε=1=⇒f|W p=−a p f(3.12)(3.13)p∤N=⇒ε(p)a p.The equivalence(3.8)is trivial.For the remaining properties see[39]and[5].3.2.2.Involutions.For every integer M|N such that(M,N/M)=1,there is an automor-phism W M of X1(N)C inducing an isomorphism between S2(N,ε)new and S2(N,εM⊗f in S2(N,εM(p)a p if p∤M,a p if p|M.Then(3.14)f|W M=λM(f)(εN/M(−M)f,f|(W M′W M)=q.It is known thatS2(Γ(N,ε))=nk=1S2(N,εk),where n is the order of the Dirichlet characterε.The diamond operators and the Weil involution induce automorphisms of X(N,ε)C.If moreoverε=1,the curve X(N,1)is X0(N)and the automorphisms W M on X1(N)C induce involutions on X0(N)over Q that are usually called the Atkin-Lehner involutions.Remark 3.16.Deﬁne the modular automorphism group of X1(N)to be the subgroup of Aut(X1(N)Q ,which we continue to denote by d and W N respectively.Throughout the paper,D will denote the abelian subgroup of Aut(XQ )generated by Dand W N.If moreover X is hyperelliptic,and w is its hyperelliptic involution,then the group generated by D and W N.w will be denoted by D′N.Note that D is a subgroup of Aut(X),and the groups D N,D′N are G Q-stable by(3.15).If Jac X Q∼A f for some f with nontrivial Nebentypus,then D N is isomorphic to the dihedral group with2n elements,D2·n,where n is the order of the Nebentypus of f.For every nonconstant morphismπ:X1(N)→X of curves over Q such that S2(X)⊆⊕n i=1S2(N,εi)for some Nebentypusεof order n,there exists a nonconstant morphism π(ε):X(N,ε)→X over Q.This is clear when the genus of X is≤1,and follows from Proposition2.11(i)if the genus of X is>1.In particular,for a new modular curve X of genus>1,there exists a surjective morphism X0(N)→X if and only if D is the trivial group.More generally,we have the following result.Lemma3.17.Let X be a new modular curve of level N,and let G be a G Q-stable subgroup of Aut(XG:={φ∈Aut(X):φ∗ω=ωfor allω∈H0(X,Ω)G}.ThenG.Now supposeφ∈G⊆G as required.3.3.Supersingular points.We will use a lemma about curves with good reduction. Lemma3.19.Let R be a discrete valuation ring with fractionﬁeld K.Suppose f:X→Y is aﬁnite morphism of smooth,projective,geometrically integral curves over K,and X extends to a smooth projective model X over R(in this case we say that X has good reduction).If Y has genus≥1,then Y extends to a smooth projective model Y over R,and f extends to aﬁnite morphism X→Y over R.Proof.This result is Corollary4.10in[40].See the discussion there also for references to earlier weaker versions. The next two lemmas are well-known(to coding theorists,for example),but we could not ﬁnd explicit references,so we supply proofs.Lemma3.20.Let p be a prime.LetΓ⊆SL2(Z)denote a congruence subgroup of level N not divisible by p.Let XΓbe the corresponding integral smooth projective curve overF p-points on the reduction mapping to supersingular points on X(1)is at least(p−1)ψ/12.Proof.By[33],the curve X(N)admits a smooth model over Z[1/N],and has a rationalpoint(the cusp∞).Since p∤N and X(N)dominates XΓ,Lemma3.19implies that XΓhas good reduction at p,at least if XΓhas genus≥1.If XΓhas genus0,then the rational point on X(N)gives a rational point on XΓ,so XΓ≃P1,so XΓhas good reduction at p in any case.ReplacingΓby the group generated byΓand−id does not change XΓ,so withoutloss of generality,we may assume that−id∈Γ.Thenψ=(SL2(Z):Γ).If E is an elliptic curve,thenΓnaturally acts on theﬁnite set of ordered symplectic bases of E[N].The curve YΓ:=XΓ−{cusps}classiﬁes isomorphism classes of pairs(E,L),where E is an elliptic curve and L is aΓ-orbit of symplectic bases of E[N].Fix E.Since SL2(Z)acts transitively on the symplectic bases of E[N],the number of Γ-orbits of symplectic bases is(SL2(Z):Γ)=ψ.Two such orbits L and L′correspond to the same point of XΓif and only if L′=αL for someα∈Aut(E).Thenψis the sum of the sizes of the orbits of Aut(E)acting on theΓ-orbits,soψ= (E,L)∈XΓ#Aut(E)F p,we obtainsupersingular points(E,L)∈XΓ(#Aut(E,L)=ψ supersingular E/#Aut(E)=(p−1)ψF p).Thereforethe number of supersingular points on XΓmust be at least2(p−1)ψ/24=(p−1)ψ/12. Lemma3.21.Let p be a prime.Given a supersingular elliptic curve E overF pand the p2-power Frobenius endomorphism of E′equals−p.Proof.Honda-Tate theory supplies an elliptic curve E over F p such that the characteristicpolynomial of the p-power Frobenius endomorphism Frob p satisﬁes Frob2p=−p.All super-singular elliptic curves over Fp→E.The inseparable part of this isogeny is a power of Frob p,so without loss of generality,we mayassume thatφis separable.The kernel K ofφis preserved by−p=Frob2p,so K is deﬁned over F p2.Take E′=E Fp2/K. The following is a generalization of inequalities used in[48].Lemma3.22.Let X be a new modular hyperelliptic curve of level N and genus g over Q. If p is a prime not dividing N,then(p−1)(g−1)<2(p2+1).Proof.We may assume g≥2.Since p∤N,Lemma3.19implies that X1(N)and X have good reduction at p,and the morphismπ:X1(N)→X induces a corresponding morphism of curves over F p.By Proposition2.11(ii),the diamond automorphism −p of X1(N) induces an automorphism of X,which we also call −p .These automorphisms induces automorphisms of the corresponding curves over F p.For the rest of this proof,X1(N),X,π, −p represent these objects over F p.Also denote by −p the induced morphism P1→P1。

好奇逐星波的英语作文

Curiosity has always been a driving force in human exploration and discovery.It is the very essence of our desire to understand the world around us and beyond.As we gaze up at the night sky,the twinkling stars beckon us with their silent call,igniting the spark of curiosity within us.This essay will delve into the concept of curiosity and its role in our pursuit of the stars.The human race has been captivated by the cosmos since time immemorial.Ancient civilizations studied the heavens,mapping constellations and predicting celestial events with remarkable accuracy.This innate curiosity led to the development of astronomy,a field that has expanded our understanding of the universe and our place within it.The invention of the telescope by Galileo Galilei in the early17th century marked a significant leap in our ability to observe and study celestial bodies,further fueling our curiosity about the stars.Curiosity is not merely a passive interest it is an active pursuit of knowledge.It propels us to ask questions,to challenge existing theories,and to seek answers that may lie beyond our current understanding.This inquisitive nature has led to numerous scientific breakthroughs and technological advancements.For instance,the curiosity about the nature of stars has led to the discovery of nuclear fusion,the process that powers our sun and other stars,and has opened up the possibility of harnessing this energy for human use.The quest for knowledge about the stars has also led to the development of space exploration.From the first human spaceflight by Yuri Gagarin in1961to the recent Mars missions,our curiosity has taken us beyond our own planet and into the vast expanse of space.This exploration has not only expanded our understanding of the universe but has also provided us with a new perspective on our own planet and its place in the cosmos. Moreover,curiosity about the stars has inspired countless individuals to pursue careers in science,technology,engineering,and mathematics STEM.It has motivated generations of students to learn about the cosmos,to dream of becoming astronauts,and to contribute to our collective understanding of the universe.This curiosity has also led to the creation of numerous educational programs and initiatives aimed at fostering interest in astronomy and space exploration.In conclusion,curiosity is the catalyst that drives our exploration of the stars.It is the spark that ignites our desire to learn,to discover,and to understand the universe.As we continue to gaze up at the night sky,let us remember that it is our curiosity that will lead us to new horizons and unlock the mysteries of the cosmos.Whether through the development of new technologies,the pursuit of scientific knowledge,or the inspirationof future generations,curiosity will always be the guiding force in our journey to explore the stars.。

(2024年高考真题含解析)2024年北京市普通高中学业水平等级性考试英语试卷(含解析)

2024年普通高等学校招生全国统一考试北京卷英语试卷养成良好的答题习惯，是决定成败的决定性因素之一。

做题前，要认真阅读题目要求、题干和选项，并对答案内容作出合理预测;答题时，切忌跟着感觉走，最好按照题目序号来做，不会的或存在疑问的，要做好标记，要善于发现，找到题目的题眼所在，规范答题，书写工整;答题完毕时，要认真检查，查漏补缺，纠正错误。

第一部分知识运用(共两节，30分)第一节(共10小题;每小题 1. 5分，共15分)阅读下面短文，掌握其大意，从每题所给的A、B、C、D 四个选项中，选出最佳选项。

I’d just arrived at school, ready for another school day. I was reading a book in the classroom when there was an 1 . “Today at 1: 10 there will be auditions (面试) for a musical.” My friends all jumped up in excitement and asked me, “Will you be going, Amy?” “Sure,” I said. I had no 2 in drama, but I’d try out because my friends were doing it.At 1:10, there was a 3 outside the drama room. Everyone looked energetic. I hadn’t expected I’d be standing there that morning. But now that I was doing it, I 4 felt nervous. What if I wasn’t any good?I entered the room and the teachers made me say some lines from the musical. They then 5 my singing skills and asked what role I wanted to play. The teachers were smiling and praising me. I felt like I had a 6 , so I said, “A big role.” They said they’d look into it. I started getting really nervous. What if I didn’t get a main role?Soon, the cast list was 7 . My friends checked and came back shouting, “Amy, you got the main role!” Sure enough, my name was at the top. I just stared at it and started to 8 . I was so happy.After two months we were all prepared and ready to go on stage. It was fun. And when people started 9 , that gave me a boost of confidence. It stayed with me and made me feel 10 . I realised that by trying something new, I can have fun — even if it means stepping out of my comfort zone.1．A．assignment B．initiative C．announcement D．interview2．A．hesitancy B．interest C．worry D．regret3．A．game B．show C．play D．line4．A．suddenly B．continuously C．originally D．generally5．A．advertised B．tested C．challenged D．polished6．A．demand B．credit C．dream D．chance7．A．traded B．posted C．questioned D．claimed8．A．well up B．roll in C．stand out D．go off9．A．whispering B．arguing C．clapping D．stretching10．A．funnier B．fairer C．cleverer D．braver第二节(共10小题;每小题1. 5分，共15分)A阅读下列短文，根据短文内容填空。

名人演讲国会大厦告别演讲

名人演讲：国会大厦告别演讲道格拉斯·麦克阿瑟，美国陆军五星上将。

出生于阿肯色州小石城的军人世家。

1899年中学毕业后考入西点军校，1903年以名列第一的优异成绩毕业，到工程兵部队任职，并赴菲律宾执勤。

麦克阿瑟有过50年的军事实践经验，被美国国民称之为“一代老兵”，而其自身的又曾是“美国最年轻的准将、西点军校最年轻的校长、美国陆军历史上最年轻的陆军参谋长”，凭借精妙的军事谋略和敢战敢胜的胆略，麦克阿瑟堪称美国战争史上的奇才。

提起这句话：“老兵永远不死，只会慢慢凋零”（Old soldiers never die，the just fade **），就不由得想起那个叼着玉米棒子烟斗的麦克阿瑟，和他在1951年4月19日被解职后在国会大厦发表的题为《老兵不死》著名演讲。

我即将结束五十二年的军旅生涯。

我从军是在本世纪开始之前，而这是我童年的希望与梦想的实现。

自从我在西点军校的教练场上宣誓以来，这个世界已经过多次变化，而我的希望与梦想早已消逝，但我仍记着当时最流行的一首军歌词，极为自豪地宣示“老兵永远不死，只会慢慢凋零”。

I am losing m 52 ears of militar servie。

When I joined the Arm， even before the turn of the entur， it as the fulfillment of all of m boish hopes and dreams。

The orld has turned over man times sine I took the oath on the plain at West Point， and the hopes and dreams have long sine vanished， but I still remember the refrain of one of the most popular barrak ballads of that da hih prolaimed most proudl that "old soldiers never die; thejust fade **。

历年考研英语阅读真题及答案解析

历年考研英语阅读真题及答案解析历年考研英语阅读真题及答案解析多做做历年来的考研英语阅读理解，让自己发现阅读的规律。

下面是店铺给大家整理的历年考研英语阅读真题及答案解析，供大家参阅!1985年考研英语阅读真题及答案解析Section III Reading ComprehensionEach sentence or passage below is followed by four statements. One of the statements is a suggestion which can be made from the information given in the original sentence or passage. Read them carefully and make your choice. Put your choice in the brackets on the left. (10 points)EXAMPLE:[A] You should get up when he comes in.[B] You should support him.[C] You shouldn't be afraid to argue with him.[D] You must be of the same height as he is.ANSWER: [B]26. Watch your step when your turn comes to have an interview with the general manager.[A] When you are asked to see the general manager, be sure not to step into his office without his permission.[B] Watch the steps when you go upstairs to see the general manager at his office.[C] Be sure to be careful when it is your turn to go to the general manager's office for an interview with him.[D] Watch out and don't step into the general manager's office until it is your turn to have an interview with him.27. Since no additional fund is available, the extension of thebuilding is out of the question.[A] The extension of the building is impossible because we are unable to get extra fund for the purpose.[B] There is some problem about the extension of the building owing to lack of fund.[C] Since no additional fund is available, we have to solve the problem regarding the extension of the building with our own resources.[D] We can undertake the extension of the building even without additional fund. It is no problem at all.28. All along he has been striving not to fall short of his parents' expectations.[A] He has been trying hard all the time to live up to what his parents expect of him.[B] His parents have been expecting him to work hard.[C] All the time he has been trying hard to balance himself so as not to fall down as his parents thought he would.[D] All the time, as his parents expect him to do, he has been trying hard to save and not to be short of money.29. The various canals which drain away the excessive water have turned this piece of land into a highly productive agricultural area.[A] The canals have been used to water the land.[B] The canals have been used to raise agricultural production.[C] Excessive water has been helpful to agricultural production.[D] The production has been mainly agricultural.30. The replacement of man by machines has not led to unemployment. On the contrary, the total numbers engaged inthe textile industry have continued to rise. The fact should not be ignored by those who maintain that unemployment and machinery are inseparable companions.[A] The belief that the use of machinery causes unemployment is unfounded.[B] The use of machinery results in a rise in production.[C] Many people lose their jobs when machines are introduced.[D] Contrary to general belief, machinery and unemployment are inseparable companions.答案解析Section III: Reading Comprehension (10 points)26.[C]27.[A]28.[A]29.[B]30.[A]1986年考研英语阅读真题及答案解析Section III Reading ComprehensionEach of the two passages below is followed by five questions. For each question there are four answers. Read the passages carefully and choose the best answer to each of the questions. Put your choice in the brackets on the left. (10 points) Text 1There are a great many careers in which the increasing emphasis is on specialization. You find these careers in engineering, in production, in statistical work, and in teaching. But there is an increasing demand for people who are able to take in great area at a glance, people who perhaps do not know too much about any one field. There is, in other words, a demand for people who are capable of seeing the forest rather than the trees, of making general judgments. We can call these people “generalists.” And these “generalists” are particularly needed for positions in administration, where it is their job to see thatother people do the work, where they have to plan for other people, to organize other people’s work, to begin it and judge it.The specialist understands one field; his concern is with technique and tools. He is a “trained” man; and his educationa l background is properly technical or professional. The generalist -- and especially the administrator -- deals with people; his concern is with leadership, with planning, and with direction giving. He is an “educated” man; and the humanities are his strongest foundation. Very rarely is a specialist capable of being an administrator. And very rarely is a good generalist also a good specialist in particular field. Any organization needs both kinds of people, though different organizations need them in different proportions. It is your task to find out, during your training period, into which of the two kinds of jobs you fit, and to plan your career accordingly.Your first job may turn out to be the right job for you -- but this is pure accident. Certainly you should not change jobs constantly or people will become suspicious of your ability to hold any job. At the same time you must not look upon the first job as the final job; it is primarily a training job, an opportunity to understand yourself and your fitness for being an employee.26. There is an increasing demand for ________.[A] all round people in their own fields[B] people whose job is to organize other people’s work[C] generalists whose educational background is either technical or professional[D] specialists whose chief concern is to provide administrative guidance to others27. The specialist is ________.[A] a man whose job is to train other people[B] a man who has been trained in more than one fields[C] a man who can see the forest rather than the trees[D] a man whose concern is mainly with technical or professional matters28. The administrator is ________.[A] a “trained” man who is more a specialist than a generalist[B] a man who sees the trees as well as the forest[C] a man who is very strong in the humanities[D] a man who is an “educated” specialist29. During your training period, it is important ________.[A] to try to be a generalist[B] to choose a profitable job[C] to find an organization which fits you[D] to decide whether you are fit to be a specialist or a generalist30. A man’s first job ________.[A] is never the right job for him[B] should not be regarded as his final job[C] should not be changed or people will become suspicious of his ability to hold any job[D] is primarily an opportunity to fit himself for his final jobText 2At the bottom of the world lies a mighty continent still wrapped in the Ice Age and, until recent times, unknown to man. It is a great land mass with mountain ranges whose extent and elevation are still uncertain. Much of the continent is a complete blank on our maps. Man has explored, on foot, less than one per cent of its area. Antarctica differs fundamentally from the Arcticregions. The Arctic is an ocean, covered with drifting packed ice and hemmed in by the land masses of Europe, Asia, and North America. The Antarctic is a continent almost as large as Europe and Australia combined, centered roughly on the South Pole and surrounded by the most unobstructed water areas of the world -- the Atlantic, Pacific, and Indian Oceans.The continental ice sheet is more than two miles high in its centre, thus, the air over the Antarctic is far more refrigerated than it is over the Arctic regions. This cold air current from the land is so forceful that it makes the nearby seas the stormiest in the world and renders unlivable those regions whose counterparts at the opposite end of the globe are inhabited. Thus, more than a million persons live within 2,000 miles of the North Pole in an area that includes most of Alaska, Siberia, and Scandinavia -- a region rich in forest and mining industries. Apart from a handful of weather stations, within the same distance of the South Pole there is not a single tree, industry, or settlement.31. The best title for this selection would be ________.[A] Iceland[B] Land of Opportunity[C] The Unknown Continent[D] Utopia at Last32. At the time this article was written, our knowledge of Antarctica was ________.[A] very limited[B] vast[C] fairly rich[D] nonexistent33. Antarctica is bordered by the ________.[A] Pacific Ocean[B] Indian Ocean[C] Atlantic Ocean[D] All three34. The Antarctic is made uninhabitable primarily by ________.[A] cold air[B] calm seas[C] ice[D] lack of knowledge about the continent35. According to this article ________.[A] 2,000 people live on the Antarctic Continent[B] a million people live within 2,000 miles of the South Pole[C] weather conditions within a 2,000 mile radius of the South Pole make settlements impractical[D] only a handful of natives inhabit Antarctica答案解析Section III: Reading Comprehension (10 points)26.[B]27.[D]28.[C]29.[D]30.[B]31.[C]32.[A]33.[D]34.[A]35.[C]1987年考研英语阅读真题及答案解析Section II Reading ComprehensionEach of three passages below is followed by five questions. For each question there are four answers, read the passages carefully and choose the best answer to each of the questions. Put your choice in the ANSWER SHEET. (15 points)Text 1For centuries men dreamed of achieving vertical flight. In 400 A.D. Chinese children played with a fan-like toy that spun upwards and fell back to earth as rotation ceased. Leonardo da Vinci conceive the first mechanical apparatus, called a “Helix,” which could carry man straight up, but was only a design and wasnever tested.The ancient-dream was finally realized in 1940 when a Russian engineer piloted a strange looking craft of steel tubing with a rotating fan on top. It rose awkwardly and vertically into the air from a standing start, hovered a few feet above the ground, went sideways and backwards, and then settled back to earth. The vehicle was called a helicopter.Imaginations were fired. Men dreamed of going to work in their own personal helicopters. People anticipate that vertical flight transports would carry millions of passengers as do the airliners of today. Such fantastic expectations were not fulfilled.The helicopter has now become an extremely useful machine. It excels in military missions, carrying troops, guns and strategic instruments where other aircraft cannot go. Corporations use them as airborne offices, many metropolitan areas use them in police work, construction and logging companies employ them in various advantageous ways, engineers use them for site selection and surveying, and oil companies use them as the best way to make offshore and remote work stations accessible to crews and supplies. Any urgent mission to a hard-to-get-to place is a likely task for a helicopter. Among their other multitude of used: deliver people across town, fly to and from airports, assist in rescue work, and aid in the search for missing or wanted persons.11. People expect that ________.[A] the airliners of today would eventually be replaced by helicopters[B] helicopters would someday be able to transport large number of people from place to place as airliners are now doing[C] the imaginations fired by the Russian engineer’sinvention would become a reality in the future[D] their fantastic expectations about helicopters could be fulfilled by airliners of today12. Helicopters work with the aid of ________.[A] a combination of rotating devices in front and on top[B] a rotating device topside[C] one rotating fan in the center of the aircraft and others at each end[D] a rotating fan underneath for lifting13. What is said about the development of the helicopter?[A] Helicopters have only been worked on by man since 1940.[B] Chinese children were the first to achieve flight in helicopters.[C] Helicopters were considered more dangerous than the early airplanes.[D] Some people thought they would become widely used by average individuals.14. How has the use of helicopters developed?[A] They have been widely used for various purposes.[B] They are taking the place of high-flying jets.[C] They are used for rescue work.[D] They are now used exclusively for commercial projects.15. Under what conditions are helicopters found to be absolutely essential?[A] For overseas passenger transportation.[B] For extremely high altitude flights.[C] For high-speed transportation.[D] For urgent mission to places inaccessible to other kinds of craft.Text 2In ancient Greece athletic festivals were very important and had strong religious associations. The Olympian athletic festival held every four years in honor of Zeus, king of the Olympian Gods, eventually lost its local character, became first a national event and then, after the rules against foreign competitors had been abolished, international. No one knows exactly how far back the Olympic Games go, but some official records date from 776 B.C. The games took place in August on the plain by Mount Olympus. Many thousands of spectators gathered from all parts of Greece, but no married woman was admitted even as a spectator. Slaves, women and dishonored persons were not allowed to compete. The exact sequence of events uncertain, but events included boy’s gymnastics, boxing, wrestling, horse racing and field events, though there were fewer sports involved than in the modern Olympic Games.On the last day of the Games, all the winners were honored by having a ring of holy olive leaves placed on their heads. So great was the honor that the winner of the foot race gave his name to the year of his victory. Although Olympic winners received no prize money, they were, in fact, richly rewarded by their state authorities. How their results compared with modern standards, we unfortunately have no means of telling.After an uninterrupted history of almost 1,200 years, the Games were suspended by the Romans in 394 A.D. They continued for such a long time because people believed in the philosophy behind the Olympics: the idea that a healthy body produced a healthy mind, and that the spirit of competition in sports and games was preferable to the competition that caused wars. It was over 1,500 years before another such international athletic gathering took place in Athens in 1896.Nowadays, the Games are held in different countries in turn. The host country provides vast facilities, including a stadium, swimming pools and living accommodation, but competing courtiers pay their own athletes’ expenses.The Olympics start with the arrival in the stadium of a torch, lighted on Mount Olympus by the sun’s rays. It is carried by a succession of runners to the stadium. The torch symbolized the continuation of the ancient Greek athletic ideals, and it burns throughout the Games until the closing ceremony. The well-known Olympic flag, however, is a modern conception: the five interlocking rings symbolize the uniting of all five continents participating in the Games.16. In ancient Greece, the Olympic Games ________.[A] were merely national athletic festivals[B] were in the nature of a national event with a strong religious colour[C] had rules which put foreign participants in a disadvantageous position[D] were primarily national events with few foreign participants17. In the early days of ancient Olympic Games ________.[A] only male Greek athletes were allowed to participate in the games[B] all Greeks, irrespective of sex, religion or social status, were allowed to take part[C] all Greeks, with the exception of women, were allowed to compete in Games[D] all male Greeks were qualified to compete in the Games18. The order of athletic events at the ancient Olympics ________.[A] has not definitely been established[B] varied according to the number of foreign competitors[C] was decided by Zeus, in whose honor the Games were held[D] was considered unimportant19. Modern athletes’ results cannot be compared with those of ancient runners because ________.[A] the Greeks had no means of recording the results[B] they are much better[C] details such as the time were not recorded in the past[D] they are much worse20. Nowadays, the athletes’ expenses are paid for ________.[A] out of the prize money of the winners[B] out of the funds raised by the competing nations[C] by the athletes themselves[D] by contributions焦点导航考研英语完型 | 考研英语真题 | 考研英语阅读 | 考研英语翻译 | 考研英语经验交流考研英语作文 | 考研常见问题 | 专家解读Text 3In science the meaning of the word “explain” suffers with civilization’s every step in search of reality. Science cannot really explain electricity, magnetism, and gravitation; their effects can be measured and predicted, but of their nature no more is known to the modern scientist than to Thales who first looked into the nature of the electrification of amber, a hard yellowish-brown gum. Most contemporary physicists reject the notion that man can ever discover what these mysterious forces “really” are. “Electricity,” Bertrand Russell says, “is not a thing, like St.Paul’s Cathe dral; it is a way in which things behave. When we have told how things behave when they are electrified, and under what circumstances they are electrified, we have told all there is to tell.” Until recently scientists would have disapproved of such an idea. Aristotle, for example, whose natural science dominated Western thought for two thousand years, believed that man could arrive at an understanding of reality by reasoning from self-evident principles. He felt, for example, that it is a self-evident principle that everything in the universe has its proper place, hence one can deduce that objects fall to the ground because that’s where they belong, and smoke goes up because that’s where it belongs. The goal of Aristotelian science was to explain why things happen. Modern science was born when Galileo began trying to explain how things happen and thus originated the method of controlled experiment which now forms the basis of scientific investigation.21. The aim of controlled scientific experiments is ________.[A] to explain why things happen[B] to explain how things happen[C] to describe self-evident principles[D] to support Aristotelian science22. What principles most influenced scientific thought for two thousand years?[A] the speculations of Thales[B] the forces of electricity, magnetism, and gravity[C] Aristotle’s natural science[D] Galileo’s discoveries23. Bertrand Russell’s notion about electricity is ________.[A] disapproved of by most modern scientists[B] in agreement with Aristotle’s theo ry of self-evidentprinciples[C] in agreement with scientific investigation directed toward “how” things happen[D] in agreement with scientific investigation directed toward “why” things happen24. The passage says that until recently scientists disagreed with the idea ________.[A] that there are mysterious forces in the universe[B] that man cannot discover what forces “really” are[C] that there are self-evident principles[D] that we can discover why things behave as they do25. Modern science came into being ________.[A] when the method of controlled experiment was first introduced[B] when Galileo succeeded in explaining how things happen[C] when Aristotelian scientist tried to explain why things happen[D] when scientists were able to acquire an understanding of reality of reasoning[C] grants[D] credits答案解析Section II: Reading Comprehension (15 points)11.[B]12.[B]13.[D]14.[A]15.[D]16.[B]17.[A]18.[A]19.[C]20.[B]21.[B]22.[C]23.[C]24.[B]25.[A]1988年考研英语阅读真题及答案解析Section II Reading ComprehensionEach of the three passages below is followed by some questions. For each question there are four answers. Read thepassages carefully and chose the best answer to each of the questions. Put your choice in the ANSWER SHEET. (20 points) Text 1It doesn’t com e as a surprise to you to realize that it makes no difference what you read or study if you can’t remember it. You just waste your valuable time. Maybe you have already discovered some clever ways to keep yourself from forgetting.One dependable aid that does help you remember what you study is to have a specific purpose or reason for reading. You remember better what you read when you know why you’re reading.Why does a clerk in a store go away and leave you when your reply to her offer to help is, “No, thank you. I’m just looking”? Both you and she know that if you aren’t sure what you want, you are not likely to find it. But suppose you say instead, “Yes, thank you. I want a pair of sun glasses.” She says, “Right this way, please.” And you and she are off -- both eager to look for exactly what you want.It’s quite the same with your studying. If you chose a book at random, “just looking” for nothing in particular, you are likely to get just that -- nothing. But if you do know what you want, and if you have the right book, you are almost sure to get it. Your reasons will vary; they will include reading or studying “to find out more about”, “to understand the reasons for”, “to find out how”. A good student has a clear purpose or reason for what he is doing.This is the way it works. Before you start to study, you say to yourself something like this, “I want to know why Stephen Vincent Benet happened to write about America. I’m reading this article to find out.” Or, “I’m going to skim this story tosee what lif e was like in medieval England.” Because you know why you are reading or studying, you relate the information to your purpose and remember it better.Reading is not one single activity. At least two important processes go on at the same time. As you read, you take in ideas rapidly and accurately. But at the same time you express your own ideas to yourself as you react to what you read. You have a kind of mental conversation with the author. If you expressed your ideas orally, they might sound like this: “Ye s, I agree. That’s my opinion too.” or “Ummmm, I thought that record was broken much earlier. I’d better check those dates,” or “But there are some other facts to be considered!” You don’t just sit there taking in ideas -- you do something else, and that something else is very important.This additional process of thinking about what you read includes evaluating it, relating it to what you already know, and using it for your own purposes. In other words, a good reader is a critical reader. One part of critical reading, as you have discovered, is distinguishing between facts and opinions. Facts can be checked by evidence. Opinions are one’s own personal reactions.Another part of critical reading is judging sources. Still another part is drawing accurate inferences.16. If you cannot remember what you read or study, ________.[A] it is no surprise[B] it means you have not really learned anything[C] it means you have not chosen the right book[D] you realize it is of no importance17. Before you start reading, it is important ________.[A] to make sure why you are reading[B] to relate the information to your purpose[C] to remember what you read[D] to choose an interesting book18. Reading activity involves ________.[A] only two simultaneous processes[B] primarily learning about ideas and evaluating them critically[C] merely distinguishing between facts and opinions[D] mainly drawing accurate inferences19. A good reader is one who ________.[A] relates what he reads to his own knowledge about the subject matter[B] does lots of thinking in his reading[C] takes a critical attitude in his reading[D] is able to check the facts presented against what he has already knownText 2If you live in a large city, you are quite familiar with some of the problems of noise, but because of some of its harmful effects, you may not be aware of the extent of its influence on human behavior. Although everyone more or less knows what noise is, i.e., it is sounds that one would rather not hear, it is perhaps best to define it more precisely for scientific purposes. One such definition is that noise is sounds that are unrelated to the task at hand. Thus stimuli that at one time might be considered relevant will at another time be considered noise, depending on what one is doing at the moment. In recent years there has been a great deal of interest in the effects of noise on human behavior, and concepts such as “noise pollution” have arisen, together with movements to reduce noise.Exposure to loud noises can definitely produce a partial or complete loss of hearing, depending on the intensity, duration, and frequency composition of the noise. Many jobs present noise hazards, such as working in factories and around jet aircraft, driving farm tractors, and working (or sitting) in music halls where rock bands are playing. In general, continuous exposure to sounds of over 80 decibels (a measure of the loudness of sound) can be considered dangerous. Decibel values correspond to various sounds. Sounds above about 85 decibels may, if exposure is for a sufficient period of time, produce significant hearing loss. Actual loss will depend upon the particular frequencies to which one is exposed, and whether the sound is continuous or intermittent.Noise can have unexpected harmful effects on performance of certain kinds of tasks, for instance, if one is performing a watch keeping task that requires vigilance, in which he is responsible for detecting weak signals of some kind (e.g., watching a radar screen for the appearance of aircraft).Communicating with other people is unfavorably affected by noise. If you have ridden in the rear of a jet transport, you may have noticed that it was difficult to carry on a conversation at first, and that, eventually, you adjusted the loudness of your speech to compensate for the effect. The problem is noise.20. Noise differs from sound in that ________.[A] it is sounds that interfere with the task being done[B] it is a special type of loud sound[C] it is usually unavoidable in big cities[D] it can be defined more precisely than the latter21. One of the harmful effects of noise on human performance is that ________.[A] it reduces one’s sensitivity[B] it renders the victim helpless[C] it deprives one of the enjoyment of music[D] it drowns out conversations at worksites22. The purpose of this passage is ________.[A] to define the effects of noise on human behavior[B] to warn people of the danger of noise pollution[C] to give advice as to how to prevent hearing loss[D] to tell the difference between noise and soundText 3The traditional belief that a woman’s place is in the home and that a woman ought not to go out to work can hardly be reasonably maintained in present conditions. It is said that it is a woman’s task to care for the children, but families today tend to be small and with a year or two between children. Thus a woman’s whole period of childbearing may occur within five years. Furthermore, with compulsory education from the age of five or six her role as chief educator of her children soon ceases. Thus, even if we agree that a woman should stay at home to look after her children before they are of school age, for many women, this period would extend only for about ten years.It might be argued that the house-proud woman would still find plenty to do about the home. That may be so, but it is certainly no longer necessary for a woman to spend her whole life cooking, cleaning, mending and sewing. Washing machines take the drudgery out of laundry, the latest models being entirely automatic and able to wash and dry a large quantity of clothes in a few minutes. Refrigerators have made it possible to store food for long periods and many pre-cooked foods are obtainable in tins. Shopping, instead of being a daily task, can be completed。

the majority of the sculpture must be inflated

the majority of the sculpture must be inflated The Art of Inflatable SculpturesIn recent years, inflatable sculptures have gained popularity in the art world. These sculptures are created using materials such as plastic or rubber and are inflated using an air pump. The majority of the sculpture must be inflated in order to achieve the desired shape and size.One of the benefits of inflatable sculptures is their portability. They can be easily deflated and transported to different locations, making them ideal for temporary installations or exhibitions. Additionally, inflatable sculptures are often cheaper to produce than traditional sculptures made from materials such as bronze or marble.Inflatable sculptures can take on a variety of shapes and sizes, ranging from small, intricate designs to large, towering structures. Some artists use inflatable sculptures to create immersive installations that visitors can walk through or interact with. Others use them as a way to explore themes such as consumer culture, identity, or the environment.One of the most well-known inflatable sculptures is "Cloud Gate," located in Chicago's Millennium Park. This massive sculpture, which measures 66 feet long and 33 feet high, is made from stainless steel and is covered in a mirror finish. When visitors stand beneath it, they can see their reflection distorted in the curved surface of the sculpture.While inflatable sculptures may seem like a whimsical addition to the art world, they also serve as a commentary on our society's obsession with consumerism and disposable culture. Many inflatable sculptures are designed to be temporary, lasting only a few days or weeks before being deflated and discarded. This serves as a reminder that even the most seemingly permanent structures are ultimately temporary.In conclusion, inflatable sculptures are a unique and innovative addition to the art world. They offer artists a new way to express their creativity and engage with audiences, while also serving as a commentary on our society's values and priorities. Whether you're a fan of contemporary art or simply appreciate the whimsy of inflatable sculptures, there is no denying their impact on the art world.。

毅力创佳绩的英语作文

Perseverance is a vital trait that can lead to remarkable achievements.It is the unwavering determination to continue in the face of adversity,challenges,and setbacks. Here are some key points that can be discussed in an essay about the importance of perseverance in creating outstanding results:1.Definition of Perseverance:Begin by defining what perseverance means and how it differs from other traits like resilience or determination.2.Historical Examples:Provide examples of historical figures who demonstrated perseverance and achieved great things,such as Thomas Edison with his numerous attempts to invent the light bulb,or Abraham Lincoln,who faced many defeats before becoming the President of the United States.3.Overcoming Failure:Discuss how perseverance allows individuals to learn from their failures and use those lessons to improve and progress.4.Growth Mindset:Explain the concept of a growth mindset,which is the belief that abilities can be developed through dedication and hard work.Perseverance is a key component of this mindset.5.Longterm Goals:Perseverance is essential for achieving longterm goals.It helps individuals to stay focused and motivated over extended periods.6.Coping with Challenges:Perseverance helps people to cope with challenges and to find solutions to problems that may seem insurmountable at first.7.Personal Development:Perseverance contributes to personal development by building character,selfdiscipline,and selfconfidence.8.Innovation and Creativity:In the field of innovation,perseverance is crucial for pushing through the creative process and bringing new ideas to fruition.9.Teamwork:Perseverance is not only an individual trait but also a collective one. Teams that persevere together can achieve more than the sum of their individual efforts.10.Cultural Significance:Discuss how different cultures value perseverance and how it is often celebrated in literature,art,and folklore.11.ModernDay Applications:Provide examples of how perseverance is applied in modern contexts,such as in business,sports,and education.12.Encouraging Perseverance:Offer suggestions on how individuals and society can foster a culture of perseverance,including setting realistic goals,celebrating small victories,and providing support systems.13.Conclusion:Summarize the importance of perseverance and its impact on achieving excellence.Reiterate that while perseverance is challenging,it is a key ingredient in the recipe for success.Remember to use a variety of sentence structures and vocabulary to make your essay engaging and to support your points with evidence and examples.。

2019届虹口区及黄浦区高三英语一模试题及答案

Ⅱ. Grammar and VocabularySection ADirections：After reading the passage below, fill in the blanks to make the passage coherent and grammatically correct. For the blanks with a given word, fill in each blank with the proper form of the given word; for the other blanks, use one word that best fits each blank.Just How Buggy is Your Phone?What item in your home crawls with the most germs? If you say ___21___ toilet seat, you’re wrong. Kitchen sponges top the list. But cell phones are pretty dirty too. They contain around 10 times as many germs as toilet seats. People touch their phones, laptops, and other digital devices all day long, yet rarely clean them.In one incident, a thief paid a terrible price for stealing a germy cell phone. He stole it from a hospital in Uganda during a widespread of the deadly disease Ebola. The phone’s owner reported the theft before ___22___（die）from the disease. Soon, the thief began showing symptoms and finally ___23___（confess）to the crime.___24___ in that unusual case a cell phone carried dangerous bacteria, not all germs are bad. Most cause no harm. In fact, they could provide helpful information. Look at the surface of your phone carefully. Do you see some dirty mars?“That's all you,”says microbial ecologist Jarrad Hampton-Marcell.“That’s biological information.”It turns out that the types of germs that you apply all over your phone or tablet are different from ___25___ of your friends and family. They’re like a fingerprint that could identify you. Some day in the future, investigators may use these microbial fingerprints to solve crimes. Phones and digital devices may be one of the best places to look for buggy clues.In a 2017 study, researchers sampled a range of surfaces in 22 participants’ homes, ___26___ countertops and floors to computer keyboards and mice. Then they tried to match the microbial fingerprints on each object to its owner. The office equipment was easiest to match to its owner. In an ___27___（early）study, a different group of researchers found that they could use microbial fingerprints to identify the person who ___28___（use）a computer keyboard even after the keyboard sat untouched for two weeks at room temperature.One day, microbial signatures might show ___29___ people have gone and what they have touched. They could prove ___30___ an unmarked device is yours. So, sure, your phone is pretty germy. Does that inspire you, or does it just bother you?Section BDirections：Complete the following passage by using the words in the box. Each word can only be used once. Note that there is one word more than you need.A. measurementB. similarC. remarkablyD. monetaryE. astronomyF. alteredG. civilization H. defined I. independence J. invariably K. dominatedThe NileThe ancient Greek writer Herodotus once described Egypt-with some envy-as‘the gift of the Nile’. The Egyptians depend on the river for food, for water and for life. The Ancient Egyptians were able to control and use the Nile, creating the earliest irrigation systems and developing a prosperous ___31___.Snaking through the deserts, the Nile would flood almost ___32___ each year in June. Once the water subsided, a rich deposit of sand was left behind, making an excellent topaoil. Seeds were sown, yielding wheat, barley, beans, lentils and leeks.Drought could spell disaster for the Egyptians, so during the dry seasons, they dug basins and channels to deliver water to their land. They also devised simple channels to transfer water at the peak of the flood.An early system of ___33___ a Nilometer, was used to de determine the size of the floods. Later, during the New Kingdom, a lifting system called a shaduf was used to raise water from the river--___34___ to the way in which a well is used today. The Egyptians took up some of the earliest trading missions. Without a(n) ___35___ system they exchanged goods, bringing back timber, precious stones, pottery, spices and animals. Their efforts in medicine were also ___36___ advanced: surgeons performed operations to remove cysts（囊肿）. Mummification gave them great understanding of the human body-yet they also relied heavily on various medicines to prevent disease, and discoveries were often confused with superstition （迷信）. And while a great deal of time was dedicated to ___37___ the Egyptians thought the stars were gods.By the 16th century Egypt was under the Ottoman Empire until Britain seized control in 1882. What is now mostly Arabic Egypt only won ___38___ from Britain after World War Ⅱ. The Suez Canal, opened in 1869, __________the country as a center for world transportation. But it, and the completion of the Aswan High Dam in 1971 ___40___ the ecology of the Nile, which now struggles to satisfy the country’s rapidly growing population, currently more than 76 million-the largest in the Arab world.Ⅲ. Reading ComprehensionSection ADirections：For each blank in the following passages there are four words or phrases marked A, B, C, and D. Fill in each blank with the word or phrase that best fits the context.Keeping The Taps Running in Thirsty CitiesWater covers 71% of Earth’s surface yet only 2% of it is accessible as a source of fresh water. ___41___ on this limited resources is rising, a trend likely to continue.It is important to recognize that it is not just city residents who ___42___ water. Agriculture, industry and tourism often require more water than the municipal water supply. Globally, 70% of fresh water is ___43___ for agriculture, but locally in heavily irrigated（灌溉）areas this can increate to 90%. A healthy environment also requires fresh water, and the quality of available water is as important as its ___44___.Water stress is not always caused by physical shortages in dry areas. ___45___ for water resources between different users within river catchments or basins can also be a cause.Every thirsty city operates within its own context, ___46___ to the challenge of providing adequate water supplies. Cape Town, ___47___, has faced three years of drought during which winter rains failed to materialize. At the end of the 2017 rainy season the city faced the ___48___ of its dams running dry during 2018. The dams were only 37% full—in the same week four years before they were full to the top. In January 2018, it was ___49___ that Cape Town would reach Day Zero, when it would be forced to turn off the taps, in April. This was despite the city reducing its water use by more than half, from 1.2 billion litres a day in 2015 to fewer than 600 million litres, and working ___50___ with industry and agriculture to reduce demand.On February 1, the authorities put in place a strict limit of 50 litres of water per person per day. ___51___, in Britain this is considered enough for a five-minute shower of half a washing machine cycle on full load.In addition, a ban was placed on using ___52___ water for gardens, water management devices were installed at householdwith a high water use and the water pressure was reduced to cut demand and leaks. At the same, the city launched a media ___53___ to change habits and introduced higher duties. This is not without its costs; agriculture and tourism, both significant areas of employment, have ___54___. It is a classic example of the problem of water economics-the cost of water is low but the cost of a lack of water is very high.Crises such as the Cape Town drought are in danger of becoming the new norm. The ___55___ of Day Zero must serve as a wake-up call for cities across the world to develop cost-effective water management strategies to cope with an uncertain future.41.A. Impact B.Pressure C.Impression D.Observation42.A. recycle B.waste C.consume D.apply43.A. restored B.abstracted C.separated D.preserved44.A. change B.source C.origin D.volume45.A. Competition B.Protection C.Construction D.Regulation46.A. contributing B.regarding C.responding D.referring47.A. in addition B.for example C.on the contrary D.as a result48.A. prospect B.illustration C.symptom D.security49.A. reported B.presented C.predicted D.explained50.A. respectively B.increasingly C.restrictively D.extensively51.A. By comparison B. In other words C.To our surprise D.What’s more52.A. feasible B.drinkable C.inevitable D.influential53.A. campaign B.statement C.presentation D.advertisement54.A. invaded B.liberated C.suffered D.proceeded55.A. change B.theory C.record D.threatSection B(A)Despite an advertisement campaign suggesting wall-to-wall special effects, “Bridge of Terabithia” is grounded in reality far more than in fantasy. Adapting Katherine Paterson’s award-winning novel, the screenwriters David Paterson and Jeff Stockwell have produced a thoughtful and extremely affecting story of a transformative friendship between two unusually gifted children. The result is a movie whose emotional depth could appeal more to adults than to their children.Jess Aarons (Josh Hutcherson) is a sixth grader with four sisters, financially tensed parents and a talent for drawing. An introverted(内向的) kid who is regularly picked on by the school buses, Jess forms a bond with a new student named Leslie (Anna Sophia Robb), a free spirit whose parents, both writers, are fondly neglectful. An attraction between outsiders, their friendship feeds on her words and his pictures; together they create an imaginary kingdom in the woods behind their homes, a world they can control and where their minds can wander free.Beautifully capturing a time when a bully in school can occur as large as a monster in a nightmare and the encouragement of a teacher can alter the course of a life, “Bridge to Terabithia” keeps the fantasy in the background to find magic in the everyday. Gabor Csupo directs this, his first feature, like someone close to the pain of being different, fascinated in tiny, perfect details.With strong performances from all the leads, “Bridge to Terabithia” is able to handle adult topics with sensitivity. As theemotional landscape darkens, those who haven’t read the book may be surprised at the sorrow the filmmakers cause without ever resorting to horror or terror. In other words, your children may cry, but they won’t be traumatized so badly.Consistently smart and delicate as a spider web, “Bridge to Terabithia” is the kind of children’s movie rarely seen nowadays. At a time when many public schools are being forced to cut music and art from the curriculum, the story’s insistence on the healing power of a cultivated imagination is both welcome and essential.56.The second paragraph indicates that Jess and Leslie ________.A.lost their control over the imaginary kingdomB.looked down on their individual realitiesC.formed a good friendship despite their different talentsD.wrote a book about a magical land called Terabithia57.Which of the following words is most likely to replace “traumatized” (paragraph 4)?A.criticizedB.ignoredC.delightedD.shocked58.The two children most likely ________.A.skipped school to play in the woods behind their campusB.created an imaginary world as an escape from realityC.disappointed their parents with their over-active imaginationsD.won against the bullies at school with strong performances59.Which of the following statements will the author most probably agree with?A.The fantasy components of the movie were too over-done.B.The movie is motional but not much too dramatic.C.“Bridge to Terabithia” has a negative impact on public school education.D.Children shouldn’t watch the film as they are too young to understand the topics.(B)Hot Air BalloonsA hot air balloon is madeup of 3 main parts:The EnvelopeThe actual fabric balloonwhich holds the airThe BurnerThe unit which pushes theheat up into the envelopeThe BasketWhere the passengers andpilot standThe basis of how the balloon works is that warmer air rises in cooler air. This is because hot air is lighter than cool air as it has less mass per unit of volume. Mass can be defined by the measure of how much matter something contains. The actual balloon has to be large as it takes a large amount of heated air to lift it off theground.The burner uses propane gas to heat up the air in the envelope to move the balloon off the ground and into the air. The pilot must keep firing the burner at regular intervals throughout the flight to ensure that the balloon continues to the stable. Naturally, the hot air will not escape from the hot at the very bottom of the envelop as firstly, hot air rises and secondly, the floating power keeps it moving up.To move the balloon upwards, the pilot opens up the propane value which lets the propane flow to the burner which in turn frees the flame up into the envelope. It works in much the same way as a gas grill: the more you open the valve, the bigger the flame to beat the air and the faster the balloon rises.The “Parachute Valve” at the very top of the balloon is what is used to bring the balloon down towards the ground. It is a circle of fabric cut out of the top of the envelop which is controlled by a rope which runs down through the middle of the envelope to the basket. If the pilot wants to bring the balloon down, he or she simply pulls on the rope which will open the valve, letting hot air escape, decreasing the inner air temperature. This cooling of air causes the balloon to slow its rise.The pilot can operate horizontally by changing the vertical position of the balloon because the wind blows in different directions at different altitudes. If the pilot wants to move in a particular direction, he or she simply arises and falls to the appropriate level and rides with the wind.60.The purpose of this article is to __________.A.explain how hot air balloons workB.illustrate why hot air balloons are usefulC.describe hot air balloons’ structurerm readers about how hot air balloons are made61.What would happen if the “Parachute Valve” could not be released after it was opened?A.The inside of the balloon would continue to heat up.B.The balloon would climb up more rapidlyC.The self-sealing valve would need to take over the role of the Parachute Valve.D.The balloon would begin to move down more rapidly.62.Which of the following skills or knowledge would be the most useful to a balloon pilot?A.The ability to sew the panels of fabric together to make a balloon.B. An understanding of how propane gas is manufactured.C.A knowledge of the background of passengers who are travelling in the balloon.D. A knowledge of air currents and wind directions in the area where he is piloting the balloon.(C)The surface of Venus has never seemed very hospitable. Temperatures change around 470°C(900°F), the result of a runway greenhouse effect, and the pressure of its atmosphere, thick with carbon dioxide and sulfuric acid(硫酸), is some 90 times that of Earth’s. Lead(铅) would flow like water on Venus, and water cannot have existed in liquid form for perhaps a billion years.Now NASA’S Magellan spacecraft seems to have found one more horror in the nasty landscape: active volcanoes. Last week the space agency released the first detailed map of Venus and the most dramatic images ever made of its surface. The picture offer the best evidence to date that a planet once assumed dead is actually a lively pot of geological change.The most amazing image is of Venus’s second tallest mountain, Maat Mons, which rises 8km(5 miles) . Most of the planet’s many peaks, including 9.5-km-(6-mile-) high Maxwell Montes, look bright in the radar pictures Magellan takes from its orbit above the permanent could cover. That means they are strong reflectors of radar waves. But Maat Mons is dark; like the Stealth bomber, it absorbs much of the radar falling on it.This interesting fact, say project scientists, is a strong hint that the mountains has recently been covered with lava(熔岩). Rock that sits on the surface of mountaintops appears to weather quickly in the hot , chemically reactive atmosphere, creating a soil that is rich in iron sulfide(硫化铁). It is this mineral, the scientists believe, that can easily be seen on radar. If Maat Mons doesn’t have any, it has probably been resurfaced, perhaps within the past few years.Such resurfacing has undoubtedly taken place in Venus lowlands: earlier images of the planet showed vast areas that are remarkably free of craters(火山坑). That would be easy to explain on a Planet like Earth, where cratering from meteor strikes is erased by steady erosion. But while there is some evidence of wind erosion on Venus, the best explanation for the lack of cratering is periodic lava flow. Magellan has found direct evidence of such flows, including domelike upwellings and hardened streamed of rock trailing down the sides of Venusian peaks. There are also signs of other geologic activities, including dramatic faulting and several distinct incidents of mountain building. But the evidence can’t indicate whether they really occurred millions of years ago. The case for active Venusian volcanoes is not yet proved, but Magellan, which is now well into its second complete survey of the planet’s surface, may eventually settle the issue.63.Which of the following has NO possibility to be found on Venus now?A. Carbon dioxideB.Sulfuric acidC.Liquid waterD.Active volcanoes64.The scientists believe that _________ shows up easily on radar.A.geological changeB.iron sulfideC.mountain mineralD. lava flow65.Which of the following is TRUE according to the passage?A.The resurfacing has changed the images of the vast areas in Venus lowlands.B.The wind erosion on Venus is caused by periodic lava flowsC.Streams of rock trailing down the side of Venusian peaks can be seen on EarthD.Other geologic activities have caused dramatic and unbelievable climate phenomenon.66.What can be inferred from the passage?A.NASA’S Magallan spacecraft fails to stand the environment of Venus.B.There is clear and confirmed evidence for the active Venusian volcanoes on Venus.C.Some evidence of periodic lava flows has been found by NASA astronauts.D.Magellan will conduct a follow-up complete survey of the Venus’ surface.Section CDirections: Read the following passage. Fill in each blank with a proper sentences given in the box. Each sentence can be used only once. Note that there are two more sentences than you need.A.However, facial recognition seems merely to encode them.B.Research show that artificial intelligence can reconstruct the facial structures of people.C. Anyone with a phone can take a picture for facial-recognition programs to use.D.Technology is rapidly catching up with the human ability to read faces.E.Continuous facial recording that paints computerized data onto the real world might changethe texture of social interactions.F.The astonishing variety of facial features helps people recognize each other and is crucial tothe formation of complex societies.Nowhere To Hide:What Machines Can Tell From Your FaceThe human face is a remarkable piece of work. 67 So is the face’s ability to send emotional signals, whether through the unconscious shame or the trick of a false smile. People spend much of their waking lives, in the office and the courtroom as well as the bar and the bedroom, reading faces, for signs of attraction, hostility, trust and deceit. They also spend plenty of time trying to hide their feelings, intentions or nature.68 In America facial recognition is used by churches to track worshippers’ attendance; in Britain, by retailers to spot past shoplifters. This year Welsh police used it to arrest a suspect outside a football game. In China it confirms the identities of ride-hailing drivers, permits tourists to enter attractions and lets people pay for things with a smile. Apple’s new iPhone is expected to use it to unlock the homescreen.Set against human skills, such applications might seem enhancive. Some breakthroughs, such as flight or the internet, obviously transform human abilities. 69 Although faces are peculiar to individuals, they are also public, so technology does not, at first sight, intrude on something that is private. And yet the ability to record, store and analyse images of facescheaply, quickly and on a vast scale promises one day to bring about fundamental changes to notions of privacy, fairness and trust.70 Masking true feelings helps fix the wheels of daily life. If your partner can spot every prohibited yawn, and your boss every hint of annoyance, marriages and working relationships will be more truthful, but less harmonious. The basis of social interactions might change, too, from a set of commitments founded on trust to calculations of risk and reward derived from the information a computer attaches to someone’s face. Relationships might become more reasonable, but also transactional.IV.Summary Writing71.Directions: Read the following passage. Summarize the main idea and the main point(s) of the passage in no more than 60 words. Use your own words as far as possible.Sport TourismTourism is the world’s largest industry and is predicted to grow well into the years to come. Increasingly, the economic importance of tourism has been recognized by governments around the world. At the same time, the tourism industry has become more complicated in its development and marketing new forms of tourism. One of the fastest growing parts of the tourism industry is travel related to sport and physical activity. A recent survey found that while the traditional beach and sight-seeing vacations continue to predominate, 22% of those surveyed reported that opportunities to participate in sports were important when selecting a vacation.The term sport tourism has been adopted in recent years to describe sport-related leisure travel. It is generally recognized that three are three broad categories of sport tourism. The first category. Watching sporting events or Sports Event Tourism includes hallmark events such as FIFA World Cup Football Championships, and the Olympic games. Tournament sponsored by the Professional Golf Association or the World Tennis Association are also part of the spectator-centered sector of sport tourism.The second type of sport tourism, celebrity and nostalgia sport tourism involves visiting famous sports-related attractions. Visits of the sports halls of fame fall into this category. Another form of celebrity and nostalgia sport tourism that has emerged in recent years is meeting famous sports personalities. The cruise industry has been experienced in this area. Sports theme cruise such as “the NBA basketball cruise” arrange for passengers to meet personalities from sports while on board.Active participation is the third category of sports tourism. This is composed of individuals who travel to participate in golf, skiing, and tennis in particular, although other sports such as fishing, and scuba diving are popular in the US.第II卷（共40分）V.TranslationDirections: Translate the following sentences into English, using the words given in the brackets.72.很多人对他们的潜能一无所知。

the-damned-human-race

ቤተ መጻሕፍቲ ባይዱ
2. For it obliges me to renounce my allegiance to the Darwinian theory of the Ascent of Man from the Lower Animals and to name it the Descent of Man from the Higher Animals.
to cheat sb out of his money or job or land,etc
Compare: to talk sb out of sth; to trick sb out of sth
the ignorant and helpless: the uneducated and powerless people; the poor laboring people in general
Detailed Discussion of the Text
Paragraph 1
1. I have been studying the traits and dispositions of the “lower animals”, and contrasting them with the traits and dispositions of man. I find the result humiliating to me.
brought the cat’s looseness with him but had left the
unconsciousness behind—the saving grace which
excuses the cat.
Cats are immoral, but they do not know it. They just can’t help it. Man has inherited cats’ looseness, but not their innocence, which is what excuses the cat for its low morals.

高二英语阅读理解哲学思考题单选题20题

高二英语阅读理解哲学思考题单选题20题1. In the philosophical text, "All that is solid melts into air," the meaning implies that:A. Change is constantB. Stability is the keyC. Nothing ever changesD. Air is more important than solid答案：A。

本题考查对哲学中变化观念的理解。

选项A 体现了变化是持续不断的这一观点，与文本含义相符。

选项B 强调稳定性是关键，与原文强调变化的主旨相悖。

选项C 表示没有任何东西会改变，与原文表达的意思相反。

选项D 是对原文的错误解读，原文并非在比较空气和固体的重要性。

2. The philosopher said, "The unexamined life is not worth living." This statement suggests that:A. Self-reflection is essentialB. Living without thinking is fineC. Actions speak louder than thoughtsD. Thought has no value答案：A。

此题目旨在考察对哲学中自我反思重要性的理解。

选项 A 指出自我反思是必不可少的，符合哲学家言论的含义。

选项B 认为不思考地生活没问题，与原文意思相反。

选项C 说行动比思想更有力量，与题干所强调的思考的重要性无关。

选项D 表示思想没有价值，与原文强调的思想的重要性相悖。

3. "The only true wisdom is in knowing you know nothing." This philosophical idea indicates that:A. Knowledge is limitlessB. We should stop learningC. We know everythingD. Ignorance is bliss答案：A。

GDEL AND SET THEORY

The Bulletin of Symbolic LogicVolume00,Number0,XXX0000G¨ODEL AND SET THEOR YAKIHIRO KANAMORI∗Kurt G¨odel(1906–1978)with his work on the constructible universe L established the relative consistency of the Axiom of Choice(AC)and the Continuum Hypothesis(CH).More broadly,he ensured the ascendancy of ﬁrst-order logic as the framework and a matter of method for set theory and secured the cumulative hierarchy view of the universe of sets.G¨odel thereby transformed set theory and launched it with structured subject matter and speciﬁc methods of proof.In later years G¨odel worked on a variety of set-theoretic constructions and speculated about how problems might be settled with new axioms.We here chronicle this development from the point of view of the evolution of set theory as aﬁeld of mathematics.Much has been written,of course,about G¨odel’s work in set theory,from textbook exposi-tions to the introductory notes to his collected papers.The present account presents an integrated view of the historical and mathematical development as supported by his recently published lectures and correspondence.Beyond the surface of things we delve deeper into the mathematics.What emerges are the roots and anticipations in work of Russell and Hilbert,and most prominently the sustained motif of truth as formalizable in the“next higher system”.We especially work at bringing out how transforming G¨odel’s work was for set theory.It is diﬃcult now to see what conceptual and technical dis-tance G¨odel had to cover and how dramatic his re-orientation of set theory was.What he brought into set theory may nowadays seem easily explicated, but only because we have assimilated his work as integral to the subject. Much has also been written about G¨odel’s philosophical views about sets and his wider philosophical outlook,and while these may have larger sig-niﬁcance,we keep the focus on the motivations and development of G¨odel’s actual mathematical constructions and contributions to set theory.Leaving2AKIHIRO KANAMORIhis“concept of set”alone,we draw out how in fact he had strong mathe-matical instincts and initiatives,especially as seen in his last,1970attempt at the continuum problem.§1.From truth to set theory.G¨odel’s advances in set theory can be seen as part of a steady intellectual development from his fundamental work on completeness and incompleteness.Two remarkably prescient passages in his early publications serve as our point of departure.His incompleteness paper [1931],submitted for publication17November1930,had a footnote48a: As will be shown in Part II of this paper,the true reason for theincompleteness inherent in all formal systems of mathematics isthat the formation of ever higher types can be continued into thetransﬁnite(cf.D.Hilbert,“¨Uber das Unendliche”,Math.Ann.95,p.184),while in any formal system at most denumerably many ofthem are available.For it can be shown that the undecidable propo-sitions constructed here become decidable whenever appropriatehigher types are added(for example,the type to the system P).An analogous situation prevails for the axiom system of set theory. This passage has been made much of,1whereas the following has not.It appeared in a summary[1932],dated22January1931,of a talk on the incompleteness results given in Karl Menger’s colloquium.Notably,matters in a footnote,perhaps an afterthought then,have now been expanded to take up fully one-third of an abstract on incompleteness:If we imagine that the system Z is successively enlarged by the in-troduction of variables for classes of numbers,classes of classes ofnumbers,and so forth,together with the corresponding compre-hension axioms,we obtain a sequence(continuable into the trans-ﬁnite)of formal systems that satisfy the assumptions mentionedabove,and it turns out that the consistency( -consistency)of anyof those systems is provable in all subsequent systems.Also,theundecidable propositions constructed for the proof of Theorem1[the G¨odelian sentences]become decidable by the adjunction ofhigher types and the corresponding axioms;however,in the highersystems we can construct other undecidable propositions by thesame procedure,and so forth.To be sure,all the propositionsthus constructed are expressible in Z(hence are number-theoreticpropositions);they are,however,not decidable in Z,but only inhigher systems,for example,in that of analysis.In case we adopta type-free construction of mathematics,as is done in the axiomG¨ODEL AND SET THEOR Y3 system of set theory,axioms of cardinality(that is,axiom postu-lating the existence of sets of ever higher cardinality)take the placeof type extensions,and it follows that certain arithmetic propo-sitions that are undecidable in Z become decidable by axioms ofcardinality,for example,by the axiom that there exist sets whosecardinality is greater than everyαn,whereα0=ℵ0,αn+1=2αn.The salient points of these passages is that the addition of the next“higher type”to a formal system leads to newly provable propositions of the system; the iterative addition of higher types can be continued into the transﬁnite; and in set theory,new propositions become analogously provable from“ax-ioms of cardinality”.The transﬁnite heritage from Hilbert[1926],cited in footnote48a,will be discussed in§5.Here we discuss the connections with the frameworks of types and of truth,which can be associated respectively with Bertrand Russell and Alfred Tarski.Mathematical logic was emerging from the Russellian world of orders and types,and G¨odel’s work would reﬂect and transform Russell’s initiatives. Russell’s ramiﬁed theory of types is a scheme of logical deﬁnitions based on orders and types indexed by the natural numbers.Russell proceeded “intensionally”;he conceived this scheme as a classiﬁcation of propositions based on the notion of propositional function,a notion not reducible to membership(extensionality).Proceeding however in modern fashion,we may say that the universe is to consist of objects stratiﬁed into disjoint types T n,where T0consists of the individuals,T n+1⊆{X|X⊆T n},and the types T n for n>0are further ramiﬁed into orders O i n with T n= i O i n. An object in O i n is to be deﬁned either in terms of individuals or of objectsin someﬁxed O j m for some j<i and m≤n,the deﬁnitions allowing for quantiﬁcation only over O j m.This precludes Russell’s Paradox and other “vicious circles”,as objects can only consist of previous objects and are built up through deﬁnitions only referring to previous stages.However,in this system it is impossible to quantify over all objects in a type T n,and this makes the formulation of numerous mathematical propositions at best cumbersome and at worst impossible.So Russell was led to introduce his axiom of Reducibility,which asserts that for each object there is a predicative object having exactly the same constituents,where an object is predicative if its order is the least greater than that of its constituents.This axiom in eﬀect reduced consideration to individuals,predicative objects consisting of individuals,predicative objects consisting of predicative objects consisting of individuals,and so on—the simple theory of types.2The above quoted G¨odel passages can be considered a point of transition from type theory to set theory.The system P of footnote48a is G¨odel’s4AKIHIRO KANAMORIstreamlined version of Russell’s theory of types built on the natural numbers as individuals,the system used in[1931].The last sentence of the footnote calls to mind the other reference to set theory in that paper;Kurt G¨odel[1931, p.178]wrote of his comprehension axiom IV,foreshadowing his approach to set theory,“This axiom plays the role of[Russell’s]axiom of reducibility(the comprehension axiom of set theory).”The system Z of the quoted[1932] passage is already the more modernﬁrst-order Peano arithmetic,the system in which G¨odel in his abstract described his incompleteness results.The passage envisages the introduction of higher-type variables,which would have the eﬀect of re-establishing the system P,but as one proceeds to higher and higher types,that“all the[unprovable]propositions constructed are expressible in Z(hence are number-theoretic propositions)”is an important point about incompleteness.The last sentence of the[1932]passage is G¨odel’sﬁrst remark on set theory of substance,and signiﬁcantly,his example of an“axiom of cardinality”to take the place of type extensions is essentially the one that both Abraham Fraenkel[1922]and Thoralf Skolem[1923]had pointed out as unprovable in Ernst Zermelo’s[1908]axiomatization of set theory and used by them to motivate the axiom of Replacement.We next face head on the most signiﬁcant underlying theme broached in our two quoted passages.G¨odel’s engagement with truth at this time, whether with conviction or caution,3could be viewed as his entr´e e into full-blown set theory.In later,speciﬁc terms,ﬁrst-order satisfaction involves canvassing arbitrary variable assignments,and higher-order satisfaction re-quires,in eﬀect,scanning all arbitrary subsets of a domain.In the introduction to his dissertation on completeness G¨odel[1929]had already made informal remarks about satisfaction,discussing the meaning of“‘A system of relations satisﬁes[erf¨u llt]a logical expression’(that is,the sentence obtained through substitution is true[wahr]).”In a letter to Paul Bernays of2April1931G¨odel4described how to deﬁne the unary predicate that picks out the G¨odel numbers of the“correct”[“richtig”]sentences of ﬁrst-order arithmetic.G¨odel then remarked,as he would in similar vein several times in his career,“Simultaneously and independently of me(as I gathered from a conversation),Mr.Tarski developed the idea of deﬁning the concept‘true proposition’in this way(for other purposes,to be sure).”Finally,G¨odel emphasized the“decidability of the undecidable propositions in higher systems”speciﬁcally through the use of the truth predicate.The semantic,recursive deﬁnition of the satisfaction relation,bothﬁrst-order and higher-order,wasﬁrst systematically formulated in set-theoretic terms by Tarski[1933][1935],to whom is usually attributed the undeﬁnabilityG¨ODEL AND SET THEOR Y5 of truth for a formal language within the language.5However,evident in G¨odel’s thinking was the necessity of a higher system to capture truth,and in fact G¨odel maintained to Hao Wang[1996,p.82]that he had come to the undeﬁnability of arithmetical truth in arithmetic already in the summer of 1930.6In a letter to Zermelo of12October1931G¨odel7pointed out that the undeﬁnability of truth leads to a quick proof of incompleteness:The class of provable formulas is deﬁnable and the class of true formulas is not, and so there must be a true but unprovable formula.G¨odel also cited his [1931]footnote48a,and this suggests that he himself invested it with much signiﬁcance.8Higher-order satisfaction is particularly relevant both for footnote48a and the[1932]abstract.Rudolf Carnap at this time was working on his Logical Syntax of Language,and in a manuscript attempted a deﬁnition of“analyt-icity”for a language that subsumed the theory of types.Working upward, he provided an adequate deﬁnition of truth forﬁrst-order arithmetic.In a letter to Carnap of11September1932G¨odel9pointed out however that Car-nap’s attempted recursive deﬁnition for second-order formulas contained a circularity.G¨odel wrote:...this error may only be avoided by regarding the domain ofthe function variables not as the predicates of a deﬁnite language,but rather as all sets and relations whatever.1On the basis of thisidea,in the second part of my work[1931]I will give a deﬁnitionfor“truth”,and I am of the opinion that the matter may not bedone otherwise....1This doesn’t necessarily involve a Platonistic standpoint,for I as-sert only that this deﬁnition(for“analytic”)be carried out within6AKIHIRO KANAMORIa deﬁnite language in which one already has the concepts“set”and“relation”.The semantic deﬁnition of second-order truth requires“all sets and relations whatever”and must be carried out where one“already has the concepts‘set’and‘relation’”.10A succeeding letter of28November1932from G¨odel to Carnap elaborated on G¨odel’s footnote48a.11G¨odel never actually wrote a Part II to his[1931] and laconically admitted in the letter that such a sequel“exists only in the realm of ideas”.G¨odel then clariﬁed how the addition of an inﬁnite type to the[1931]system P would render provable the unprovable propositions he had constructed—speciﬁcally since a truth deﬁnition can now be provided. Signiﬁcantly,G¨odel wrote however:...the interest of this deﬁnition does not lie in a clariﬁcation of theconcept‘analytic’since one employs in it the concepts‘arbitrarysets’,etc.,which are just as problematic.Rather I formulate itonly for the following reason:with its help one can show thatundecidable sentences become decidable in systems which ascendfarther in the sequence of types.The deﬁnition of truth is not itself clariﬁcatory,but it does serve a mathe-matical end.Tarski,of course,did put much store in his systematic deﬁnition of truth for formal languages,and Carnap would be much inﬂuenced by Tarski’s work on truth.Despite their contrasting attitudes toward truth,G¨odel’s and Tarski’s approaches had similarities.Tarski’s[1933][1935]undeﬁnability of truth result is couched in terms of languages having“inﬁnite order”, analogous to G¨odel’s[1931]system P having inﬁnite types,and G¨odel’s inﬁnite type is analogous to Tarski’s“metalanguage”.In a postscript in his[1935,p.194,n.108],Tarski acknowledged G¨odel’s footnote48a.In a lecture[1933]G¨odel expanded on the themes of our quoted passages. He propounded the axiomatic set theory“as presented by Zermelo,Fraenkel, and von Neumann”as“a natural generalization of the[simple]theory of types,or rather,what becomes of the theory of types if certain superﬂuous restrictions are removed.”First,instead of having separate types with sets of type n+1consisting purely of sets of type n,sets can be cumulative in the sense that sets of type n can consist of sets of all lower types.If S n is the collection of sets of type n,then:S0is the type of the individuals,and recursively, S n+1=S n∪{X|X⊆S n}.Second,the process can be continued into the transﬁnite,starting with the cumulation S = n S n,proceeding through successor stages as before,and taking unions at limit stages.G¨odel[1933,G¨ODEL AND SET THEOR Y7 p.46]again credited Hilbert for opening the door to the formation of types beyond theﬁnite types.As for how far this cumulative hierarchy of sets is to continue,the“ﬁrst two or three types already suﬃce to deﬁne very large ordinals”([1933,p.47])which can then serve to index the process,and so on,in an“autonomous progression”in later terminology.In a prophetic remark for set theory and new axioms,G¨odel observed:“We set out toﬁnd a formal system for mathematics and instead of that found an inﬁnity of systems,and whichever system you choose out of this inﬁnity,there is one more comprehensive,i.e.,one whose axioms are stronger.”Further echoing the quoted[1932]passage G¨odel[1933,p.48]noted that for any formal system S there is in fact an arithmetical proposition that cannot be proved in S,unless S is inconsistent.Moreover,if S is based on the theory of types, this arithmetical proposition becomes provable if to S is adjoined“the next higher type and the axioms concerning it.”G¨odel’s approach to set theory,with its emphasis on hierarchical truth, should be set into the context of the axiomatic development of the subject.12 Zermelo[1908]had provided the initial axiomatization of“the set theory of Cantor and Dedekind”,with characteristic axioms Separation,Inﬁnity, Power Set,and of course,Choice.Work most substantially of John von Neumann[1923][1928]on ordinals led to the incorporation of Cantor’s transﬁnite numbers as now the ordinals and the axiom schema of Replace-ment for the formalization of transﬁnite recursion.Von Neumann[1929] also formulated the axiom of Foundation,that every set is well-founded, and deﬁned the cumulative hierarchy in his system via transﬁnite recursion: The axiom entails that the universe V of sets is globally structured through a stratiﬁcation into cumulative“ranks”Vα,where with P(X)={Y|Y⊆X} denoting the power set of X,V0=∅;Vα+1=P(Vα);V = α< Vαfor limit ordinals ;andV= αVα.Zermelo in his remarkable[1930]subsequently provided hisﬁnal axiomati-zation of set theory,proceeding in a second-order context and incorporating both Replacement(which subsumes Separation)and Foundation.These axioms rounded out but also focused the notion of set,with theﬁrst pro-viding the means for transﬁnite recursion and induction and the second making possible the application of those methods to get results about all sets.G¨odel’s coming work would itself amount to a full embrace of Replace-ment and Foundation but alsoﬁrst-order deﬁnability,which would vitalize the earlier initiative of Skolem[1923]to establish set theory on the basis of8AKIHIRO KANAMORIﬁrst-order logic.13The now standard axiomatization ZFC is essentially the ﬁrst-order version of the Zermelo[1930]axiomatization,and ZF is ZFC without AC.§2.The constructible universe L.Set theory was launched on an indepen-dent course as a distinctiveﬁeld of mathematics by G¨odel’s formulation of the class L of constructible sets through which he established the relative consis-tency of AC in mid-1935and CH in mid-1937.14In hisﬁrst announcement, communicated9November1938,G¨odel[1938]wrote:“[The]‘constructible’sets are deﬁned to be those sets which canbe obtained by Russell’s ramiﬁed hierarchy of types,if extended toinclude transﬁnite orders.The extension to transﬁnite orders hasthe consequence that the model satisﬁes the impredicative axiomsof set theory,because an axiom of reducibility can be proved forsuﬃciently high orders.”This points to two major features of the construction of L:(i)G¨odel had reﬁned the cumulative hierarchy of sets described in his 1933lecture to a hierarchy of deﬁnable sets which is analogous to the orders of Russell’s ramiﬁed theory.Despite the broad trend in mathematical logic away from Russell’s intensional intricacies and toward versions of the simple theory of types,G¨odel had assimilated the ramiﬁed theory and its motiva-tions as of consequence and now put the theory to a new use,infusing its intensional character into an extensional context.(ii)G¨odel continued the indexing of the hierarchy through all the ordi-nals as given beforehand to get a class model of set theory and thereby to achieve relative consistency results.His earlier[1933,p.47]idea of using large ordinals deﬁned in low types for further indexing in a bootstrapping process would not suﬃce.That“an axiom of reducibility can be proved for suﬃciently high orders”is an opaque allusion to how Russell’s problematic axiom would be rectiﬁed in the consistency proof of CH(see§3)and more broadly to how the axiom of Replacement provided for new sets and enough ordinals.15Von Neumann’s ordinals would be the spine for a thin hierarchy of sets,and this would be the key to both the AC and CH results.G¨ODEL AND SET THEOR Y9 In a brief account[1939b]G¨odel informally presented L much as is done today:For any set x let def(x)denote the collection of subsets of x deﬁnable over x,∈ via aﬁrst-order formula allowing parameters from x.Then deﬁneL0=∅;Lα+1=def(Lα),L = α< Lαfor limit ordinals ;and the constructible universeL= αLα.Toward the end G¨odel[1939b,p.31]pointed out that L“can be deﬁned and its theory developed in the formal systems of set theory themselves.”This is a remarkable understatement of arguably the central feature of the construction of L:(iii)L is a class deﬁnable in set theory via a transﬁnite recursion that could be based on the formalizability of def(x),the deﬁnability of deﬁnabil-ity.G¨odel had not embraced the deﬁnition of truth as itself clariﬁcatory,16 but through his work he in eﬀect drew it into mathematics to a new mathe-matical end.Though understated in G¨odel’s writing,his great achievement here as in arithmetic is the submergence of metamathematical notions into mathematics.In the proof of the incompleteness theorem,G¨odel had encoded provabil-ity—syntax—and played on the interplay between truth and deﬁnability. G¨odel now encoded satisfaction—semantics—with the room oﬀered by the transﬁnite indexing,making truth,now deﬁnable for levels,part of the formalism and part of the subject matter.In modern parlance,an inner model of ZFC is a transitive(deﬁnable)class containing all the ordinals such that,with membership and quantiﬁcation restricted to it,the class satisﬁes each axiom of ZFC.G¨odel in eﬀect argued in ZF to show that L is an inner model of ZFC,and moreover that L satisﬁes CH.He thus established the relative consistency Con(ZF)implies Con(ZFC+CH).In what follows, we describe his proofs that L is an inner model of ZFC and in§3that L satisﬁes CH.In his sketch[1939b]G¨odel simply argued for the ZFC axioms holding in L as evident from the construction,with the extent of the ordinals and the sets provided by def(x)suﬃcient to establish Replacement in L.Only at the end when he was attending to formalization did he allude to the central issue of relativization.For here and later,recall that for a formulaϕand classes C and M,ϕM and C M denote the relativizations to M ofϕand C respectively,i.e.,ϕM denotesϕbut with the quantiﬁers restricted to the elements of M,and C M denotes the class deﬁned by the relativization to M of a deﬁning formula for C.G¨odel’s[1939b]arguments for relative consistency amount to establishingϕL as theorems of set theory for various10AKIHIRO KANAMORIϕstarting with the axioms of set theory themselves,and could only work if def L(x)=def(x)for x∈L.This absoluteness ofﬁrst-order deﬁnability is central to the proof if L is to be formally deﬁned via the def(x)operation, but notably G¨odel himself would never establish this absoluteness explicitly, preferring in his one rigorous published exposition of L to take an approach that avoids def(x)altogether.In his monograph[1940a],based on1938lectures,G¨odel provided a spe-ciﬁc,formal presentation of L in a class-set theory developed by Paul Bernays [1937],a theory based in turn on a theory of von Neumann[1925].First, G¨odel carried out a paradigmatic development of“abstract”set theory through the ordinals and cardinals with features that have now become com-mon fare,like his particular well-ordering of pairs of ordinals.17G¨odel then used eight binary operations,producing new classes from old,to generate L set by set via transﬁnite recursion.This veritable“G¨odel numbering”with ordinals bypassed the def(x)operation and made evident certain aspects of L.Since there is a direct,deﬁnable well-ordering of L,choice functions abound in L,and AC holds there.Much of the analysis of L would have to be devoted to verifying Replace-ment or at least Separation there,this requiring an analysis of theﬁrst-order formalization of set properties.It has sometimes been casually asserted that G¨odel[1940a]through his eight operations provided aﬁnite axiomatization of Separation,but this cannot be done.Through closure under the opera-tions one does get Separation for bounded formulas,i.e.,those formulas all of whose quantiﬁers can be rendered as∀x∈y and∃x∈y.G¨odel estab-lished using Replacement(in V)that for any set x⊆L,there is a y∈L such that x⊆y(9.63of[1940a]).He then established that a wide range of classes C⊆L satisfy the condition that for any x∈L,x∩C∈L,that C is“amenable”in later terminology.With this,he established L for every axiom of ZFC,the relativized instances of Replacement being the most crucial to conﬁrm.18Having bypassed def(x),this argumentation makes no appeal to absoluteness.§3.Consistency of the Continuum Hypothesis.G¨odel’s proof that L sat-isﬁes CH consisted of two separate parts.He established the implication V=L→CH,and,in order to apply this implication within L,the ab-soluteness L L=L to establish the desired(CH)L.That V=L→CH es-tablished a connection between two quite non-absolute concepts,the power set and successor cardinality of an inﬁnite set,and the absoluteness L L=L eﬀected the requisite relativization.That L L=L had been asserted in hisﬁrst announcement[1938],and follows directly from def L(x)=def(x)for x∈L,which was broached in the sketch[1939b].In[1940a],his approach to L L=L was rather through the evident absoluteness of the eight generating operations which in particular entailed that being a(von Neumann)ordinal is absolute and ensured the internal integrity of the generation of L.There is a nice resonance here with G¨odel[1931],in that there he had catalogued a series of functions to be primitive recursive whereas now he catalogued a series of set-theoretic operations to be absolute—the submergence of prov-ability(syntax)for arithmetic evolved to the submergence of deﬁnability (semantics)for set theory.The argument in fact shows that for any inner model M of ZFC,L M=L.Decades later many inner models based on ﬁrst-order deﬁnability would be investigated for which absoluteness consid-erations would be pivotal,and G¨odel had formulated the canonical inner model.G¨odel’s argument for V=L→CH rests,as he himself wrote in a brief summary[1939a],on“a generalization of Skolem’s method for constructing enumerable models.”This was theﬁrst signiﬁcant use of Skolem functions since Skolem’s own[1920]to establish the L¨owenheim–Skolem theorem. G¨odel[1939b]speciﬁcally established:(∗)For inﬁniteα,every constructible subset of Lαbelongs to some L for a of the same cardinality asα. It is straightforward to show that for inﬁniteα,Lαhas the same cardinality as that ofα.It follows from(∗)that in L,the power set of L is included ,and so CH follows.(G¨odel emphasized the Generalized Continuum in L1Hypothesis(GCH),that2ℵα=ℵα+1for allα,and V=L→GCH follows by analogous reasoning.)G¨odel[1939b]proved(∗)for an X⊆Lαsuch that X∈L by getting a set M⊆L containing X and suﬃciently many ordinals and deﬁnable sets so that M will be isomorphic to some L ,the construction of M ensuring that has the same cardinality asα.G¨odel’s approach to M,diﬀerent from the usual approach taken nowadays,can be seen as proceeding through layers deﬁned recursively,a new layer being deﬁned via closure according to new Skolem functions and ordinals based on the preceding layer.This was indeed a“generalization of Skolem’s method”, being an iterative application of Skolem closures.M having been suﬃciently bolstered,G¨odel then conﬁrmed that M is isomorphic with respect to∈to some L ,making theﬁrst use of the now familiar Mostowski transitive collapse.G¨odel in his monograph[1940a],having proceeded without def(x),for-mally carried out his[1939b]argument in terms of his eight operations,and this had the eﬀect of obscuring the Skolem deﬁnability and closure.There is,however,an economy of means that can be seen from G¨odel[1940a]:The arguments there demonstrated that absoluteness is not necessary to establisheither that L is an inner model of ZFC or that V=L→CH;absoluteness is only necessary where it is intrinsic,to establish L L=L.Until the1960s accounts of L dutifully followed G¨odel’s[1940a],presen-tation,and papers generally in axiomatic set theory often used and referred to G¨odel’s speciﬁc listing and grouping of his class-set axioms.However, modern expositions of L proceed in ZFC with the direct formalization of def(x),ﬁrst formulating satisfaction-in-a-structure and coding this in set theory.They then establish Replacement or Separation in L by appealing to an L analogue of the ZF Reﬂection Principle,drawn from Richard Mon-tague[1961,p.99]and Azriel Levy[1960,p.234].19Moreover,they establish V=L→CH via some version of the Condensation Lemma:If is a limit ordinal and X is an elementary substructure of L ,then there is a such that X is isomorphic to L .Instead of G¨odel’s hand-over-hand algebraic approach to get(∗),one incorporates the satisfaction-in-a-structure rela-tion and takes at least aΣ1-elementary substructure of an ambient L in a uniform fashion using its Skolem functions.This higher-level approach is indicative of how the satisfaction relation has been assimilated into modern set theory but also of what G¨odel’s approach had to encompass.One is left to speculate why,and perhaps to rue that,G¨odel did not himself articulate a reﬂection principle for use in L or some version of the Con-densation Lemma based on the model-theoretic satisfaction-in-a-structure relation.The requisite Skolem closure argument would have served as a mo-tivating entr´e e into his[1939b]proof of CH in L.Moreover,this approach would have provided a thematic link to G¨odel’s later advocacy of the heuris-tic of reﬂection,described in§7.Finally,with satisfaction-in-a-structure becoming the basis of model theory after Tarski–Vaught[1957]and the ZF Reﬂection Principle emerging only through the infusion of model-theoretic methods into set theory around1960,a fuller embrace by G¨odel of the satisfaction relation might have accelerated the process.That infusion was stimulated by Tarski through his students,and this sets in new counterpoint G¨odel’s indirect engagement with truth and satisfaction.20。

亚里士多德的形而上学英文

亚里士多德的形而上学英文Aristotle's MetaphysicsAristotle, one of the most influential philosophers in history, is widely recognized for his contributions to various fields of knowledge. Among his numerous works, one of the most renowned is his treatise on metaphysics. In this article, we will delve into Aristotle's Metaphysics, exploring its key concepts and examining its influence on Western philosophy.Metaphysics, as defined by Aristotle, is the branch of philosophy that deals with the study of existence, reality, and the fundamental nature of things. It seeks to understand the ultimate principles that govern the universe and the essence of being. Aristotle's Metaphysics, also known as "The First Philosophy," is a profound exploration of these topics.In his treatise, Aristotle begins by addressing the concept of being. He argues that being is the basic substance that encompasses everything existing in the world. According to him, being has different levels of reality, ranging from potentiality to actuality. Potentiality refers to the inherent capacity of an object to become something else, while actuality signifies the fulfillment of that potentiality. Aristotle emphasizes the significance of actuality, asserting that it is the ultimate goal and purpose of every existing entity.A central concept in Aristotle's Metaphysics is his theory of causality. Aristotle posits that everything in the world can be explained by four types of causes: the material cause, the formal cause, the efficient cause, and the final cause. The material cause pertains to the substance or matter from which something is made. The formal cause refers to the structure, pattern,or form that determines an object's essence. The efficient cause denotes the agent or force that brings about the change or movement. Lastly, the final cause represents the ultimate purpose or goal for which an object exists. According to Aristotle, these causes operate in a hierarchical manner, ultimately leading to the realization of the final cause.Another notable aspect of Aristotle's Metaphysics is his exploration of potentiality and actuality in relation to change and motion. Aristotle argues that change is the actualization of potentiality, and it is through the process of change that an object or entity realizes its full potential. He discusses various types of change, including qualitative change, quantitative change, and substantial change. Aristotle's insights into the nature of change and motion have influenced subsequent philosophical and scientific discourses.In addition to these fundamental concepts, Aristotle's Metaphysics also addresses the notions of substance, essence, and form. Substance, according to Aristotle, is the underlying reality that endures through various changes. It is the substratum of existence, and everything else is predicated upon it. Essence, on the other hand, refers to the defining characteristics or qualities that make an object what it is. Form, closely related to essence, determines the essential nature and identity of an entity.The influence of Aristotle's Metaphysics on Western philosophy cannot be overstated. His framework of metaphysics has shaped philosophical inquiries for centuries, providing a basis for understanding and contemplating the most profound questions regarding existence and reality. Aristotle's emphasis on the role of causality, the nature of change, and theconcept of substance has been influential not only in philosophy but also in theology, science, and many other disciplines.In conclusion, Aristotle's Metaphysics stands as a cornerstone of philosophical thought. With its profound exploration of being, causality, change, and substance, it has significantly contributed to our understanding of the nature of reality. The influence of Aristotle's insights continues to resonate in contemporary philosophy, inspiring further reflections on the metaphysical foundations of our world.。

一个伟大的人英语作文

一个伟大的人英语作文When we speak of a great person we often think of someone who has made significant contributions to society culture or science. A great person is not only characterized by their achievements but also by their character values and the positive impact they have on others.Title A Great PersonIntroductionIn the vast tapestry of human history there are figures who stand out as beacons of inspiration guiding us with their wisdom and actions. These individuals through their dedication and perseverance have left an indelible mark on the world.Body Paragraph 1A great person is often defined by their unwavering commitment to a cause greater than themselves. For instance Mahatma Gandhi a leader who fought for Indias independence from British rule demonstrated extraordinary courage and conviction. His philosophy of nonviolence and civil disobedience not only influenced the course of Indian history but also inspired civil rights movements around the globe.Body Paragraph 2In addition to their dedication great individuals often possess a unique vision that propels them to challenge the status quo. Take for example Albert Einstein whose groundbreaking theories in physics revolutionized our understanding of the universe. His intellectual curiosity and relentless pursuit of knowledge have made him a symbol of scientific genius.Body Paragraph 3Moreover a great persons character is often marked by humility and a genuine concern for the welfare of others. Mother Teresa known for her tireless work with the poor and the sick exemplified selflessness and compassion. Her life was a testament to the power of love and service to humanity.Body Paragraph 4Furthermore the influence of a great person extends beyond their lifetime. Their legacies inspire future generations to strive for excellence and to make a difference. Nelson Mandela who fought against apartheid and became South Africas first black presidentcontinues to inspire people to fight for justice and equality.ConclusionIn conclusion a great person is a complex amalgamation of vision dedication character and impact. They are the architects of change the bearers of hope and the catalysts for progress. Their stories remind us of the potential within each of us to make a significant contribution to the world leaving a legacy that transcends time and space.。

Alexander Pope亚历山大·蒲柏英文介绍

蒲柏出生于一个罗马天主教家庭，他幼年时期患有结核性脊椎炎，造成驼背，身高没有超过1.37米，寿命不长（56岁去世）。

1727年的蒲柏.蒲柏从12岁即开始发表诗作，他的第一部重要作品是1711年他23岁时出版的诗体《批评论》，其中许多名句已经成为英语成语。
Alexander Pope was born in London as the son of Alexander Pope, a Roman Catholic linen-merchant, and Edith (Turner) Pope, who was forty-four when Alexander, her only child, was born. Edith Pope belonged to a large Yorkshire family, which divided along Catholic and Protestant lines. His early years Pope spent at Binfield on the edge of Windsor Forest, and recalled this period as a golden age: "Thy forests, Windsor, and thy green retreats, / At once the monarch's and the Muse's seats, / Invite my lays. Be present, sylvan maids! Unlock your springs, and open all your shades." Anecdotes from Pope's life were deemed worthy of collecting during his lifetime. Joseph Spence, a critic, minor poet, and Pope's biographer, tells that Pope was "a child of a particularly sweet temper and had a great deal of sweetness in his look when he was a boy".

1、下载文档前请自行甄别文档内容的完整性，平台不提供额外的编辑、内容补充、找答案等附加服务。
2、"仅部分预览"的文档,不可在线预览部分如存在完整性等问题,可反馈申请退款(可完整预览的文档不适用该条件!)。
3、如文档侵犯您的权益，请联系客服反馈,我们会尽快为您处理(人工客服工作时间：9:00-18:30)。

O(n4 )
1ቤተ መጻሕፍቲ ባይዱ
Introduction
Modular arithmetic is widely used in the design of cryptosystems and pseudo random number generators [22, 12]. Since integers use a ﬁnite binary representation in conventional programming languages such as C, modular arithmetic is often required in software veriﬁcation as well. Indeed, many algorithms are designed to avoid overﬂow in modular arithmetic explicitly. Veriﬁcation tools therefore need to support modular arithmetic to check these algorithms. In this paper, we discuss the satisﬁability problem of propositional formulae in modular arithmetic. All arithmetic computation in the formulae is over the ﬁnite ring Z2ω for some ﬁxed ω . In addition to linear terms, non-linear terms such as multiplications and modulo operations of arbitrary terms are allowed. We show that the satisﬁability problem is NP-complete for formulae of linear modular arithmetic. The problem is still in NP for full modular arithmetic. We give an eﬃcient reduction to integer programming to have a practical decision procedure for modular arithmetic. Several issues have to be addressed in our construction. Firstly, modular computation must be simulated by linear constraints, as well as all logical operations. Furthermore, non-linear multiplications and modulo operations need to be expressed in the form of linear constraints. Most importantly, we would not like our reduction to increase the size of the
On the Satisﬁability of Modular Arithmetic Formulae
Bow-Yaw Wang
⋆
Institute of Information Science Academia Sinica Taiwan
Abstract. Modular arithmetic is the underlying integral computation model in conventional programming languages. In this paper, we discuss the satisﬁability problem of propositional formulae in modular arithmetic can be obover the ﬁnite ring Z2ω . Although an upper bound of 22 tained by solving alternation-free Presburger arithmetic, it is easy to see that the problem is in fact NP-complete. Further, we give an eﬃcient reduction to integer programming with the number of constraints and variables linear in the length of the given linear modular arithmetic formula. For non-linear modular arithmetic formulae, an additional factor of ω is needed. With the advent of eﬃcient integer programming packages, our algorithm could be useful to software veriﬁcation in practice.
⋆
This work was partly supported by NSC under grants NSC 94-2213-E-001-003- and NSC 95-2221-E-001-024-MY3.
problem signiﬁcantly. Our construction should not use more than a linear number of constraints and variables in the length of the modular arithmetic formula. It is well-known that the ﬁrst-order non-linear arithmetic theory is undecidable [8]. Presburger arithmetic is a decidable ﬁrst-order linear arithmetic theory [6, 15, 19]. In [15], Oppen shows an upper bound of 22 for determining the truth of Presburger arithmetic formula of length n. If the number O(nm+4 ) of quantiﬁer alternation is m, the problem can be solved in time 22 and m+4 ) space 2O(n [19]. Although Presburger arithmetic can express ﬁrst-order linear arithmetic properties, it does not allow modular arithmetic nor non-linear operations. Integer programming optimizes a given linear objective function subject to a set of linear constraints [16]. The problem is known to be NP-complete. Unlike Presburger arithmetic, it does not allow arbitrary logical combinations of constraints but their conjunction. It does not allow modular arithmetic either. Other decision procedures for linear arithmetic are available. In [4], a survey of the automata-theoretic approach is given. For a special class of quantiﬁer-free Presburger arithmetic, [21] gives an eﬃcient reduction to Boolean satisﬁability. The tool CVC Lite [2] contains a decision procedure to check validity of linear arithmetic formula. Similar to [6, 15, 19], none of them considers modular arithmetic nor non-linear operations. In [1], a decision procedure for systems of modular arithmetic inequalities is proposed. Although the authors use an algebraic approach to check the satisﬁability of (in)equalities in a system. It is unclear whether the logical and modulo operations can be added within their framework. We note that our reduction may serve as a reduction to Presburger arithmetic. Since Presburger arithmetic does not allow modular arithmetic, encoding it in linear constraints allows us to solve the problem by various decision procedures for Presburger arithmetic. However, solving the corresponding Presburger O(n4 ) arithmetic formula requires 22 in the length of the modular arithmetic formula. Our reduction is more eﬃcient asymptotically. The remaining of paper is organized as follows. Section 2 contains the background. It is followed by the syntax and semantics of linear modular arithmetic in Section 3. The algorithm for the satisﬁability of linear modular arithmetic is presented in Section 4. The syntax and semantics of modular arithmetic formulae are deﬁned in Section 5. Section 6 discusses the satisﬁability problem for non-linear modular arithmetic. Applications of our algorithm are discussed in Section 7. We report our preliminary experimental results in Section 8. Finally, Section 9 concludes the paper.