The solution conformation of triarylmethyl radicals

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希尔伯特的23个数学问题

BULLETIN(New Series)OF THEAMERICAN MATHEMATICAL SOCIETYVolume37,Number4,Pages407–436S0273-0979(00)00881-8Article electronically published on June26,2000MATHEMATICAL PROBLEMSDAVID HILBERTLecture delivered before the International Congress of Mathematicians at Paris in1900.Who of us would not be glad to lift the veil behind which the future lies hidden;to cast a glance at the next advances of our science and at the secrets of its development during future centuries?What particular goals will there be toward which the leading mathematical spirits of coming generations will strive?What new methods and new facts in the wide and richﬁeld of mathematical thought will the new centuries disclose?History teaches the continuity of the development of science.We know that every age has its own problems,which the following age either solves or casts aside as proﬁtless and replaces by new ones.If we would obtain an idea of the probable development of mathematical knowledge in the immediate future,we must let the unsettled questions pass before our minds and look over the problems which the science of to-day sets and whose solution we expect from the future.To such a review of problems the present day,lying at the meeting of the centuries,seems to me well adapted.For the close of a great epoch not only invites us to look back into the past but also directs our thoughts to the unknown future.The deep signiﬁcance of certain problems for the advance of mathematical science in general and the important rˆo le which they play in the work of the individual investigator are not to be denied.As long as a branch of science oﬀers an abundance of problems,so long is it alive;a lack of problems foreshadows extinction or the cessation of independent development.Just as every human undertaking pursues certain objects,so also mathematical research requires its problems.It is by the solution of problems that the investigator tests the temper of his steel;heﬁnds new methods and new outlooks,and gains a wider and freer horizon.It is diﬃcult and often impossible to judge the value of a problem correctly in advance;for theﬁnal award depends upon the grain which science obtains from the problem.Nevertheless we can ask whether there are general criteria which mark a good mathematical problem.An old French mathematician said:“A mathematical theory is not to be considered complete until you have made it so clear that you can explain it to theﬁrst man whom you meet on the street.”This clearness and ease of comprehension,here insisted on for a mathematical theory,I should still more demand for a mathematical problem if it is to be perfect;for what is clear and easily comprehended attracts,the complicated repels us.Moreover a mathematical problem should be diﬃcult in order to entice us,yet not completely inaccessible,lest it mock at our eﬀorts.It should be to us a guide408DA VID HILBERTpost on the mazy paths to hidden truths,and ultimately a reminder of our pleasure in the successful solution.The mathematicians of past centuries were accustomed to devote themselves to the solution of diﬃcult particular problems with passionate zeal.They knew the value of diﬃcult problems.I remind you only of the“problem of the line of quickest descent,”proposed by John Bernoulli.Experience teaches,explains Bernoulli in the public announcement of this problem,that lofty minds are led to strive for the advance of science by nothing more than by laying before them diﬃcult and at the same time useful problems,and he therefore hopes to earn the thanks of the mathematical world by following the example of men like Mersenne,Pascal, Fermat,Viviani and others and laying before the distinguished analysts of his time a problem by which,as a touchstone,they may test the value of their methods and measure their strength.The calculus of variations owes its origin to this problem of Bernoulli and to similar problems.Fermat had asserted,as is well known,that the diophantine equationx n+y n=z n(x,y and z integers)is unsolvable—except in certain self-evident cases.The attempt to prove this impossibility oﬀers a striking example of the inspiring eﬀect which such a very special and apparently unimportant problem may have upon science.For Kummer,incited by Fermat’s problem,was led to the introduction of ideal numbers and to the discovery of the law of the unique decomposition of the numbers of a circularﬁeld into ideal prime factors—a law which to-day in its generalization to any algebraicﬁeld by Dedekind and Kronecker,stands at the center of the modern theory of numbers and whose signiﬁcance extends far beyond the boundaries of number theory into the realm of algebra and the theory of functions.To speak of a very diﬀerent region of research,I remind you of the problem of three bodies.The fruitful methods and the far-reaching principles which Poincar´e has brought into celestial mechanics and which are to-day recognized and applied in practical astronomy are due to the circumstance that he undertook to treat anew that diﬃcult problem and to approach nearer a solution.The two last mentioned problems—that of Fermat and the problem of the three bodies—seem to us almost like opposite poles—the former a free invention of pure reason,belonging to the region of abstract number theory,the latter forced upon us by astronomy and necessary to an understanding of the simplest fundamental phenomena of nature.But it often happens also that the same special problemﬁnds application in the most unlike branches of mathematical knowledge.So,for example,the problem of the shortest line plays a chief and historically important part in the foundations of geometry,in the theory of curved lines and surfaces,in mechanics and in the calculus of variations.And how convincingly has F.Klein,in his work on the icosahedron,pictured the signiﬁcance which attaches to the problem of the regular polyhedra in elementary geometry,in group theory,in the theory of equations and in that of linear diﬀerential equations.In order to throw light on the importance of certain problems,I may also refer to Weierstrass,who spoke of it as his happy fortune that he found at the outset of his scientiﬁc career a problem so important as Jacobi’s problem of inversion on which to work.MATHEMATICAL PROBLEMS409 Having now recalled to mind the general importance of problems in mathematics, let us turn to the question from what sources this science derives its problems. Surely theﬁrst and oldest problems in every branch of mathematics spring from experience and are suggested by the world of external phenomena.Even the rules of calculation with integers must have been discovered in this fashion in a lower stage of human civilization,just as the child of to-day learns the application of these laws by empirical methods.The same is true of theﬁrst problems of geometry, the problems bequeathed us by antiquity,such as the duplication of the cube, the squaring of the circle;also the oldest problems in the theory of the solution of numerical equations,in the theory of curves and the diﬀerential and integral calculus,in the calculus of variations,the theory of Fourier series and the theory of potential—to say noting of the further abundance of problems properly belonging to mechanics,astronomy and physics.But,in the further development of a branch of mathematics,the human mind, encouraged by the success of its solutions,becomes conscious of its independence. It evolves from itself alone,often without appreciable inﬂuence from without,by means of logical combination,generalization,specialization,by separating and col-lecting ideas in fortunate ways,new and fruitful problems,and appears then itself as the real questioner.Thus arose the problem of prime numbers and the other problems of number theory,Galois’s theory of equations,the theory of algebraic invariants,the theory of abelian and automorphic functions;indeed almost all the nicer questions of modern arithmetic and function theory arise in this way.In the meantime,while the creative power of pure reason is at work,the outer world again comes into play,forces upon us new questions from actual experience, opens up new branches of mathematics,and while we seek to conquer these new ﬁelds of knowledge for the realm of pure thought,we oftenﬁnd the answers to old unsolved problems and thus at the same time advance most successfully the old theories.And it seems to me that the numerous and surprising analogies and that apparently prearranged harmony which the mathematician so often perceives in the questions,methods and ideas of the various branches of his science,have their origin in this ever-recurring interplay between thought and experience.It remains to discuss brieﬂy what general requirements may be justly laid down for the solution of a mathematical problem.I should sayﬁrst of all,this:that it shall be possible to establish the correctness of the solution by means of aﬁnite number of steps based upon aﬁnite number of hypotheses which are implied in the statement of the problem and which must always be exactly formulated.This requirement of logical deduction by means of aﬁnite number of processes is sim-ply the requirement of rigor in reasoning.Indeed the requirement of rigor,which has become proverbial in mathematics,corresponds to a universal philosophical necessity of our understanding;and,on the other hand,only by satisfying this requirement do the thought content and the suggestiveness of the problem attain their full eﬀect.A new problem,especially when it comes from the world of outer experience,is like a young twig,which thrives and bears fruit only when it is grafted carefully and in accordance with strict horticultural rules upon the old stem,the established achievements of our mathematical science.Besides it is an error to believe that rigor in the proof is the enemy of simplic-ity.On the contrary weﬁnd it conﬁrmed by numerous examples that the rigorous method is at the same time the simpler and the more easily comprehended.The410DA VID HILBERTvery eﬀort for rigor forces us toﬁnd out simpler methods of proof.It also fre-quently leads the way to methods which are more capable of development than the old methods of less rigor.Thus the theory of algebraic curves experienced a considerable simpliﬁcation and attained greater unity by means of the more rigor-ous function-theoretical methods and the consistent introduction of transcendental devices.Further,the proof that the power series permits the application of the four elementary arithmetical operations a well as the term by term diﬀerentiation and integration,and the recognition of the utility of the power series depending upon this proof contributed materially to the simpliﬁcation of all analysis,particularly of the theory of elimination and the theory of diﬀerential equations,and also of the existence proofs demanded in those theories.But the most striking example for my statement is the calculus of variations.The treatment of theﬁrst and second variations of deﬁnite integrals required in part extremely complicated calculations, and the processes applied by the old mathematicians had not the needful rigor. Weierstrass showed us the way to a new and sure foundation of the calculus of variations.By the examples of the simple and double integral I will show brieﬂy,at the close of my lecture,how this way leads at once to a surprising simpliﬁcation of the calculus of variations.For in the demonstration of the necessary and suﬃcient criteria for the occurrence of a maximum and minimum,the calculation of the sec-ond variation and in part,indeed,the wearisome reasoning connected with theﬁrst variation may be completely dispensed with—to say nothing of the advance which is involved in the removal of the restriction to variations for which the diﬀerential coeﬃcients of the function vary but slightly.While insisting on rigor in the proof as a requirement for a perfect solution of a problem,I should like,on the other hand,to oppose the opinion that only the concepts of analysis,or even those of arithmetic alone,are susceptible of a fully rigorous treatment.This opinion,occasionally advocated by eminent men,I con-sider entirely erroneous.Such a one-sided interpretation of the requirement of rigor would soon lead to the ignoring of all concepts arising from geometry,mechanics and physics,to a stoppage of theﬂow of new material from the outside world,and ﬁnally,indeed,as a last consequence,to the rejection of the ideas of the continuum and of the irrational number.But what an important nerve,vital to mathematical science,would be cut by the extirpation of geometry and mathematical physics! On the contrary I think that wherever,from the side of the theory of knowledge or in geometry,or from the theories of natural or physical science,mathematical ideas come up,the problem arises for mathematical science to investigate the principles underlying these ideas and so to establish them upon a simple and complete system of axioms,that the exactness of the new ideas and their applicability to deduction shall be in no respect inferior to those of the old arithmetical concepts.To new concepts correspond,necessarily,new signs.These we choose in such a way that they remind us of the phenomena which were the occasion for the formation of the new concepts.So the geometricalﬁgures are signs or mnemonic symbols of space intuition and are used as such by all mathematicians.Who does not always use along with the double inequality a>b>c the picture of three points following one another on a straight line as the geometrical picture of the idea “between”?Who does not make use of drawings of segments and rectangles enclosed in one another,when it is required to prove with perfect rigor a diﬃcult theorem on the continuity of functions or the existence of points of condensation?Who could dispense with theﬁgure of the triangle,the circle with its center,or with the crossMATHEMATICAL PROBLEMS411 of three perpendicular axes?Or who would give up the representation of the vector ﬁeld,or the picture of a family of curves or surfaces with its envelope which plays so important a part in diﬀerential geometry,in the theory of diﬀerential equations, in the foundation of the calculus of variations and in other purely mathematical sciences?The arithmetical symbols are written diagrams and the geometricalﬁgures are graphic formulas;and no mathematician could spare these graphic formulas,any more than in calculation the insertion and removal of parentheses or the use of other analytical signs.The use of geometrical signs as a means of strict proof presupposes the exact knowledge and complete mastery of the axioms which underlie thoseﬁgures;and in order that these geometricalﬁgures may be incorporated in the general treasure of mathematical signs,there is necessary a rigorous axiomatic investigation of their conceptual content.Just as in adding two numbers,one must place the digits under each other in the right order,so that only the rules of calculation,i.e.,the axioms of arithmetic,determine the correct use of the digits,so the use of geometrical signs is determined by the axioms of geometrical concepts and their combinations.The agreement between geometrical and arithmetical thought is shown also in that we do not habitually follow the chain of reasoning back to the axioms in arithmetical,any more than in geometrical discussions.On the contrary we ap-ply,especially inﬁrst attacking a problem,a rapid,unconscious,not absolutely sure combination,trusting to a certain arithmetical feeling for the behavior of the arithmetical symbols,which we could dispense with as little in arithmetic as with the geometrical imagination in geometry.As an example of an arithmetical theory operating rigorously with geometrical ideas and signs,I may mention Minkowski’s work,Die Geometrie der Zahlen.1Some remarks upon the diﬃculties which mathematical problems may oﬀer,and the means of surmounting them,may be in place here.If we do not succeed in solving a mathematical problem,the reason frequently consists in our failure to recognize the more general standpoint from which the problem before us appears only as a single link in a chain of related problems. Afterﬁnding this standpoint,not only is this problem frequently more accessible to our investigation,but at the same time we come into possession of a method which is applicable also to related problems.The introduction of complex paths of integration by Cauchy and of the notion of the ideals in number theory by Kummer may serve as examples.This way forﬁnding general methods is certainly the most practicable and the most certain;for he who seeks for methods without having a deﬁnite problem in mind seeks for the most part in vain.In dealing with mathematical problems,specialization plays,as I believe,a still more important part than generalization.Perhaps in most cases where we seek in vain the answer to a question,the cause of the failure lies in the fact that problems simpler and easier than the one in hand have been either not at all or incompletely solved.All depends,then,onﬁnding out these easier problems,and on solving them by means of devices as perfect as possible and of concepts capable of generalization. This rule is one of the most important leers for overcoming mathematical diﬃculties and it seems to me that it is used almost always,though perhaps unconsciously.412DA VID HILBERTOccasionally it happens that we seek the solution under insuﬃcient hypotheses or in an incorrect sense,and for this reason do not succeed.The problem then arises:to show the impossibility of the solution under the given hypotheses,or in the sense contemplated.Such proofs of impossibility were eﬀected by the ancients, for instance when they showed that the ratio of the hypotenuse to the side of an isosceles right triangle is irrational.In later mathematics,the question as to the impossibility of certain solutions plays a pre¨e minent part,and we perceive in this way that old and diﬃcult problems,such as the proof of the axiom of parallels,the squaring of the circle,or the solution of equations of theﬁfth degree by radicals haveﬁnally found fully satisfactory and rigorous solutions,although in another sense than that originally intended.It is probably this important fact along with other philosophical reasons that gives rise to the conviction(which every mathematician shares,but which no one has as yet supported by a proof) that every deﬁnite mathematical problem must necessarily be susceptible of an exact settlement,either in the form of an actual answer to the question asked, or by the proof of the impossibility of its solution and therewith the necessary failure of all attempts.Take any deﬁnite unsolved problem,such as the question as to the irrationality of the Euler-Mascheroni constant C,or the existence of an inﬁnite number of prime numbers of the form2n+1.However unapproachable these problems may seem to us and however helpless we stand before them,we have,nevertheless,theﬁrm conviction that their solution must follow by aﬁnite number of purely logical processes.Is this axiom of the solvability of every problem a peculiarity characteristic of mathematical thought alone,or is it possibly a general law inherent in the nature of the mind,that all questions which it asks must be answerable?For in other sciences also one meets old problems which have been settled in a manner most satisfactory and most useful to science by the proof of their impossibility.I instance the problem of perpetual motion.After seeking in vain for the construction of a perpetual motion machine,the relations were investigated which must subsist between the forces of nature if such a machine is to be impossible;2and this inverted question led to the discovery of the law of the conservation of energy,which,again,explained the impossibility of perpetual motion in the sense originally intended.This conviction of the solvability of every mathematical problem is a powerful incentive to the worker.We hear within us the perpetual call:There is the problem. Seek its solution.You canﬁnd it by pure reason,for in mathematics there is no ignorabimus.The supply of problems in mathematics is inexhaustible,and as soon as one problem is solved numerous others come forth in its place.Permit me in the fol-lowing,tentatively as it were,to mention particular deﬁnite problems,drawn from various branches of mathematics,from the discussion of which an advancement of science may be expected.Let us look at the principles of analysis and geometry.The most suggestive and notable achievements of the last century in thisﬁeld are,as it seems to me,the arithmetical formulation of the concept of the continuum in the works of Cauchy, Bolzano and Cantor,and the discovery of non-euclidean geometry by Gauss,Bolyai,MATHEMATICAL PROBLEMS413 and Lobachevsky.I thereforeﬁrst direct your attention to some problems belonging to theseﬁelds.1.Cantor’s problem of the cardinal number of the continuumTwo systems,i.e.,two assemblages of ordinary real numbers or points,are said to be(according to Cantor)equivalent or of equal cardinal number,if they can be brought into a relation to one another such that to every number of the one assemblage corresponds one and only one deﬁnite number of the other.The inves-tigations of Cantor on such assemblages of points suggest a very plausible theorem, which nevertheless,in spite of the most strenuous eﬀorts,no one has succeeded in proving.This is the theorem:Every system of inﬁnitely many real numbers,i.e.,every assemblage of numbers (or points),is either equivalent to the assemblage of natural integers,1,2,3,...or to the assemblage of all real numbers and therefore to the continuum,that is,to the points of a line;as regards equivalence there are,therefore,only two assemblages of numbers,the countable assemblage and the continuum.From this theorem it would follow at once that the continuum has the next cardinal number beyond that of the countable assemblage;the proof of this theorem would,therefore,form a new bridge between the countable assemblage and the continuum.Let me mention another very remarkable statement of Cantor’s which stands in the closest connection with the theorem mentioned and which,perhaps,oﬀers the key to its proof.Any system of real numbers is said to be ordered,if for every two numbers of the system it is determined which one is the earlier and which the later, and if at the same time this determination is of such a kind that,if a is before b and b is before c,then a always comes before c.The natural arrangement of numbers of a system is deﬁned to be that in which the smaller precedes the larger.But there are,as is easily seen,inﬁnitely many other ways in which the numbers of a system may be arranged.If we think of a deﬁnite arrangement of numbers and select from them a particular system of these numbers,a so-called partial system or assemblage,this partial system will also prove to be ordered.Now Cantor considers a particular kind of ordered assemblage which he designates as a well ordered assemblage and which is characterized in this way,that not only in the assemblage itself but also in every partial assemblage there exists aﬁrst number.The system of integers1,2,3,...in their natural order is evidently a well ordered assemblage.On the other hand the system of all real numbers,i.e.,the continuum in its natural order,is evidently not well ordered.For,if we think of the points of a segment of a straight line,with its initial point excluded,as our partial assemblage,it will have noﬁrst element.The question now arises whether the totality of all numbers may not be arranged in another manner so that every partial assemblage may have aﬁrst element,i.e., whether the continuum cannot be considered as a well ordered assemblage—a ques-tion which Cantor thinks must be answered in the aﬃrmative.It appears to me most desirable to obtain a direct proof of this remarkable statement of Cantor’s, perhaps by actually giving an arrangement of numbers such that in every partial system aﬁrst number can be pointed out.414DA VID HILBERT2.The compatibility of the arithmetical axiomsWhen we are engaged in investigating the foundations of a science,we must set up a system of axioms which contains an exact and complete description of the relations subsisting between the elementary ideas of that science.The axioms so set up are at the same time the deﬁnitions of those elementary ideas;and no statement within the realm of the science whose foundation we are testing is held to be correct unless it can be derived from those axioms by means of aﬁnite number of logical steps.Upon closer consideration the question arises:Whether,in any way,certain statements of single axioms depend upon one another,and whether the axioms may not therefore contain certain parts in common,which must be isolated if one wishes to arrive at a system of axioms that shall be altogether independent of one another.But above all I wish to designate the following as the most important among the numerous questions which can be asked with regard to the axioms:To prove that they are not contradictory,that is,that aﬁnite number of logical steps based upon them can never lead to contradictory results.In geometry,the proof of the compatibility of the axioms can be eﬀected by constructing a suitableﬁeld of numbers,such that analogous relations between the numbers of thisﬁeld correspond to the geometrical axioms.Any contradiction in the deductions from the geometrical axioms must thereupon be recognizable in the arithmetic of thisﬁeld of numbers.In this way the desired proof for the compatibility of the geometrical axioms is made to depend upon the theorem of the compatibility of the arithmetical axioms.On the other hand a direct method is needed for the proof of the compatibility of the arithmetical axioms.The axioms of arithmetic are essentially nothing else than the known rules of calculation,with the addition of the axiom of continuity.I recently collected them3and in so doing replaced the axiom of continuity by two simpler axioms,namely,the well-known axiom of Archimedes,and a new axiom essentially as follows:that numbers form a system of things which is capable of no further extension,as long as all the other axioms hold(axiom of completeness).I am convinced that it must be possible toﬁnd a direct proof for the compatibility of the arithmetical axioms,by means of a careful study and suitable modiﬁcation of the known methods of reasoning in the theory of irrational numbers.To show the signiﬁcance of the problem from another point of view,I add the following observation:If contradictory attributes be assigned to a concept,I say, that mathematically the concept does not exist.So,for example,a real number whose square is−1does not exist mathematically.But if it can be proved that the attributes assigned to the concept can never lead to a contradiction by the application of aﬁnite number of logical processes,I say that the mathematical existence of the concept(for example,of a number or a function which satisﬁes certain conditions)is thereby proved.In the case before us,where we are concerned with the axioms of real numbers in arithmetic,the proof of the compatibility of the axioms is at the same time the proof of the mathematical existence of the complete system of real numbers or of the continuum.Indeed,when the proof for the compatibility of the axioms shall be fully accomplished,the doubts which have been expressed occasionally as to the existence of the complete system of real numbers will become totally groundless.The totality of real numbers,i.e., the continuum according to the point of view just indicated,is not the totality of。

simultaneous equation method

Simultaneous Equation MethodIntroductionIn mathematics, simultaneous equations play a crucial role in solving real-world problems and modeling various phenomena. The simultaneous equation method is a powerful technique used to find solutions for a system of equations. This method involves solving multiple equations together to determine the values of unknown variables. In this article, we will explore the simultaneous equation method in detail and discuss its applications.Understanding Simultaneous EquationsDefinitionSimultaneous equations, also known as a system of equations, are a set of equations that share the same variables. The solutions of these equations simultaneously satisfy each equation in the system. The general form of simultaneous equations can be written as:a1x + b1y = c1a2x + b2y = c2Here, x and y are the variables, while a1, a2, b1, b2, c1, and c2 are constants.Types of Simultaneous EquationsSimultaneous equations can be classified into three types based on the number of solutions they have:1.Consistent Equations: These equations have a unique solution,meaning there is a specific set of values for the variables thatsatisfy all the equations in the system.2.Inconsistent Equations: This type of system has no solution. Theequations are contradictory and cannot be satisfied simultaneously.3.Dependent Equations: In this case, the system has infinitely manysolutions. The equations are dependent on each other and represent the same line or plane in geometric terms.To solve simultaneous equations, we employ various methods, with the simultaneous equation method being one of the most commonly used techniques.The Simultaneous Equation MethodThe simultaneous equation method involves manipulating and combining the given equations to eliminate one variable at a time. By eliminating one variable, we can reduce the system to a single equation with one variable, making it easier to find the solution.ProcedureThe general procedure for solving simultaneous equations using the simultaneous equation method is as follows:1.Identify the unknow n variables. Let’s assume we have n variables.2.Write down the given equations.3.Choose two equations and eliminate one variable by employingsuitable techniques such as substitution or elimination.4.Repeat step 3 until you have a single equation with one variable.5.Solve the single equation to determine the value of the variable.6.Substitute the found value back into the other equations to obtainthe values of the remaining variables.7.Verify the solution by substituting the found values into all theoriginal equations. The values should satisfy each equation.If the system is inconsistent or dependent, the simultaneous equation method will also lead to appropriate conclusions.Applications of Simultaneous Equation MethodThe simultaneous equation method finds applications in numerous fields, including:EngineeringSimultaneous equations are widely used in engineering to model and solve various problems. Engineers employ this method to determine unknown quantities in electrical circuits, structural analysis, fluid mechanics, and many other fields.EconomicsIn economics, simultaneous equations help analyze the relationship between different economic variables. These equations assist in studying market equilibrium, economic growth, and other economic phenomena.PhysicsSimultaneous equations are a fundamental tool in physics for solving complex problems involving multiple variables. They are used in areas such as classical mechanics, electromagnetism, and quantum mechanics.OptimizationThe simultaneous equation method is utilized in optimization techniques to find the optimal solution of a system subject to certain constraints. This is applicable in operations research, logistics, and resource allocation problems.ConclusionThe simultaneous equation method is an essential mathematical technique for solving systems of equations. By employing this method, we can find the values of unknown variables and understand the relationships between different equations. The applications of this method span across various fields, making it a valuable tool in problem-solving and modeling real-world situations. So, the simultaneous equation method continues to be akey topic in mathematics and its practical applications in diverse disciplines.。

化学化工英语试题及答案

化学化工英语试题及答案一、选择题（每题2分，共20分）1. Which of the following is a chemical element?A. WaterB. OxygenC. HydrogenD. Carbon答案：B, C, D2. The chemical formula for table salt is:A. NaOHB. NaClC. HClD. NaHCO3答案：B3. What is the process called when a substance changes from a solid to a liquid?A. SublimationB. VaporizationC. MeltingD. Condensation答案：C4. In the periodic table, which group contains alkali metals?A. Group 1B. Group 2C. Group 17D. Group 18答案：A5. What is the name of the process where a substance decomposes into two or more substances due to heat?A. CombustionB. OxidationC. ReductionD. Decomposition答案：D6. Which of the following is a physical property of a substance?A. ColorB. TasteC. SolubilityD. Reactivity答案：A7. What is the term for a compound that releases hydrogen ions (H+) when dissolved in water?A. BaseB. AcidC. SaltD. Neutral答案：B8. The law of conservation of mass states that in a chemical reaction:A. Mass is lostB. Mass is gainedC. Mass remains constantD. Mass can be converted into energy答案：C9. Which of the following is a type of chemical bond?A. Ionic bondB. Covalent bondC. Hydrogen bondD. All of the above答案：D10. What is the name of the process where a substance absorbs energy and changes from a liquid to a gas?A. MeltingB. VaporizationC. SublimationD. Condensation答案：B二、填空题（每题2分，共20分）1. The symbol for the element iron is ________.答案：Fe2. The pH scale ranges from ________ to ________.答案：0 to 143. A compound that produces a basic solution when dissolvedin water is called a ________.答案：base4. The smallest particle of an element that retains its chemical properties is called a ________.答案：atom5. The process of separating a mixture into its individual components is known as ________.答案：separation6. The study of the composition, structure, and properties of matter is called ________.答案：chemistry7. The process of a substance changing from a gas to a liquid is called ________.答案：condensation8. A(n) ________ reaction is a type of chemical reactionwhere two or more substances combine to form a single product. 答案：synthesis9. The volume of a gas at constant temperature and pressureis directly proportional to the number of ________.答案：moles10. The process of converting a solid directly into a gas without passing through the liquid phase is known as ________. 答案：sublimation三、简答题（每题10分，共30分）1. Explain what is meant by the term "stoichiometry" in chemistry.答案：Stoichiometry is the calculation of the relative quantities of reactants and products in a chemical reaction.It is based on the law of conservation of mass and involvesthe use of balanced chemical equations and the molar massesof substances to determine the amounts of reactants needed to produce a certain amount of product or the amounts ofproducts formed from a given amount of reactant.2. Describe the difference between a physical change and a chemical change.答案：A physical change is a change in the state or form of a substance without altering its chemical composition. Examples include melting, freezing, and boiling. A chemical change, on the other hand, involves a change in the chemical composition of a substance, resulting in the formation of new substances. Examples include combustion and rusting.3. What are the three main types of chemical bonds, and givean example of each.答案：The three main types of chemical bonds are ionic bonds, covalent bonds, and metallic bonds. An ionic bond is formed when electrons are transferred from one atom to another, resulting in the formation of oppositely charged ions. An example is the bond between sodium (Na) and chloride (Cl) in table salt (NaCl). A covalent bond is formed when two atoms share electrons, as seen in water (H2O) where hydrogen atoms share electrons with oxygen. Metallic bonds occur in metals, where a "sea" of delocalized electrons is shared among positively charged metal ions, as in sodium metal。

伊藤引理

The formal expression (1) is meant to be a simply way to express the integral relations. Integrate both sides over the time integral [T1, T2]. From the left side of (1) you get
where
u(XT , T ) − u(X0, 0) = ∆uk ,
tk <T
∆uk = u(Xk + ∆Xk, tk + ∆t) − u(Xk, tk) , ∆Xk = Xk+1 − Xk .
The Taylor expansion is
∆uk
=
∂x uk ∆Xk
+
1 2
∂x2uk
(∆Xk )2
+
3
≤ C∆t 2 .
2
up the terms (∆X)2 and adding up the terms v(Xt)∆t have the same limit as ∆t → 0. Both of these arguments use ideas from Lesson 3 and Assignment 3.
There also is an application of Borel Cantelli to show that the arguments are
∂t uk ∆t
+ O(|∆Xk|3) + O(∆t |∆Xk|) + O(∆t2) .
We sum over k. On the left side we get u(XT , t) − u(X0, 0). There are six sums on the right to consider.

常德“PEP”24年小学6年级上册J卷英语第6单元真题试卷

常德“PEP”24年小学6年级上册英语第6单元真题试卷考试时间：100分钟（总分:140）A卷考试人：_________题号一二三四五总分得分一、综合题(共计100题)1、听力题：The __________ can indicate areas of potential geological hazards.2、What do we call the part of the plant that grows below the ground?A. LeafB. StemC. RootD. Flower答案: C3、填空题：My mom encourages me to follow my __________ (热情).4、What is 2 + 2?A. 3B. 4C. 5D. 6答案：B5、听力题：The chemical formula for barium sulfate is _____.6、What do we call the process of converting a solid directly to gas?A. MeltingB. FreezingC. SublimationD. Evaporation答案:CSome _______ can be toxic to pets.8、填空题：The ________ is very friendly and enjoys company.9、What is a baby chicken called?A. DucklingB. GoslingC. ChickD. Lamb答案:C10、填空题：My pet dog is very __________. (可爱)11、听力题：The cat is ___ (chasing/hiding) from the dog.12、选择题：What is the name of the famous American author known for his short stories?A. Edgar Allan PoeB. Mark TwainC. Nathaniel HawthorneD. Ernest Hemingway13、听力题：The main component of enzymes is _____.14、What is the capital of Kenya?A. NairobiB. KampalaC. Addis AbabaD. Dar es Salaam答案:A. Nairobi15、选择题：What is the capital of Chile?A. SantiagoB. ValparaísoC. ConcepciónD. La Serena16、填空题：The sunflowers turn towards the ______.The ________ is a lovely flower.18、填空题：My favorite animal is a _______ (狐狸).19、What is the main purpose of a compass?A. Measure temperatureB. Tell timeC. Show directionD. Measure weight答案:C20、How many legs does a spider have?A. SixB. EightC. TenD. Twelve答案:B21、填空题：I planted some ______ (种子) in a pot. I hope they grow into beautiful ______ (花).22、What is the name of the famous playwright known for his tragedies?A. William ShakespeareB. Anton ChekhovC. Tennessee WilliamsD. Arthur Miller答案: A23、填空题：My dog always _______ (跟随) me on walks.24、填空题：The _______ (猴子) swings on vines.25、填空题：My favorite thing to do is ________ (玩游戏) with friends.26、听力题：The main component of essential oils is _____.27、听力题：The __________ is critical for sustaining life on earth.28、What do we call the annual celebration of a person's birth?A. AnniversaryB. HolidayC. BirthdayD. Festival答案:C29、听力题：The kitten is very ______ (curious).30、填空题：The ______ (小鱼儿) swims gracefully through the coral reefs.31、What is the name of the famous American musician known for "Counting Stars"?A. OneRepublicB. Maroon 5C. ColdplayD. Imagine Dragons答案：A32、What is the name of the famous bear from the jungle?A. BalooB. PaddingtonC. YogiD. Winnie答案:A33、听力题：In a reaction, the limiting reagent is the reactant that is completely _____ first.34、听力题：Chemical bonds can be ionic or ______.35、听力题：The dog is ________ in the grass.36、填空题：The _____ (clover) is a small plant with three leaves.37、What do you call the frozen form of water?A. SteamB. IceC. SnowD. Rain答案:BWhich animal has a pouch for carrying its babies?A. DogB. KangarooC. CatD. Elephant39、What is the name of the longest river in Africa?A. NileB. CongoC. ZambeziD. Niger答案:A40、Which of these is a vegetable?A. AppleB. CarrotC. BananaD. Grape41、What is the smallest continent?A. AfricaB. AsiaC. AustraliaD. Europe答案: C42、听力题：The sun is shining ________ today.43、Which instrument has keys and is played by pressing them?A. GuitarB. DrumsC. PianoD. Violin44、填空题：The ________ was a major turning point in the American Civil War.45、What is the capital of Afghanistan?A. KabulB. HeratC. KandaharD. Mazar-i-SharifI saw a ________ dancing in the yard.47、听力题：The ice cream is _____ melting. (slowly)48、What do we call the act of expressing opinions in writing?A. CommentaryB. EditorialC. ReviewD. All of the Above答案：D49、选择题：What do we call the place where we keep our money?A. BankB. StoreC. LibraryD. School50、听力题：The stars are ________ at night.51、填空题：In my opinion, being _______ (形容词) is a valuable trait in life. It helps us connect with others.52、填空题：The _____ (植物标本) helps preserve plant species for study.53、选择题：How many continents are there in the world?A. 5B. 6C. 7D. 854、What do we call the process of a seed developing into a new plant?A. GerminationB. PhotosynthesisC. PollinationD. Fertilization答案:A55、听力题：She is _____ (cooking) breakfast.What do you use to see in the dark?A. CandleB. FlashlightC. LampD. Mirror57、填空题：The dolphin is very _______ (聪明) and playful.58、听力题：The ________ (mountain) is difficult to climb.59、What do we call a plant that grows in water?A. TerrestrialB. AquaticC. MarineD. Hydroponic答案: B60、What do you call the largest land animal?A. RhinoB. ElephantC. HippoD. Giraffe答案：B61、Which shape has four equal sides?A. RectangleB. TriangleC. SquareD. Circle答案：C62、填空题：The zebra has black and white _______ (条纹).63、听力题：The _______ of a sound can be influenced by the environment.64、填空题：I hope to discover new passions as I continue to _______ (成长) and learn.65、填空题：The sun is ________ (明亮) today.I can ______ my homework. (finish)67、What do we call the small, sweet fruit that grows on a vine?A. RaspberryB. StrawberryC. GrapeD. Blueberry答案:C68、填空题：The garden has many _______ that blossom beautifully all year long.69、听力题：The process of electroplating deposits a layer of ______.70、填空题：The _____ (植物生命周期) includes several stages.71、填空题：I am learning to speak ________ (法语) because I want to visit ________ (法国).72、 gather ______ (坚果) for winter. 填空题：Squirrel73、What do you call a baby lion?A. CubB. PupC. CalfD. Kit答案:A74、填空题：A parakeet can have many different ______ (颜色).75、What is the name of the famous mountain range in South America?A. AndesB. RockiesC. HimalayasD. Alps答案：A76、听力题：The chemical formula for cyclopentane is ______.77、What do we call the study of the interaction between organisms and their environment?A. EcologyB. BiologyC. ZoologyD. Botany78、填空题：The ________ (城市发展) requires careful planning.79、Where do fish live?A. TreesB. MountainsC. WaterD. Grass答案:C80、What is the opposite of empty?A. FullB. HeavyC. LightD. Open答案:A81、听力题：The capital of Pakistan is _______.82、听力题：The flowers bloom in the _____ (spring).83、填空题：The __________ (历史学家) study past events and societies.84、n Conquest changed the course of ________ (英国历史). 填空题：The Nort85、填空题：The ancient Romans built _____ to celebrate their victories.86、听力题：The chemical symbol for tellurium is _______.87、What is the name of the famous artist known for his abstract paintings?A. PicassoB. Van GoghC. Da VinciD. Monet答案:A88、填空题：My _____ (邻居) is very friendly.89、听力题：She is _____ (playing/reading) a book right now.90、What is the opposite of tall?A. ShortB. LongC. HighD. Low答案:A91、填空题：The _______ (蝙蝠) hangs upside down.92、填空题：The tarantula is a type of ________________ (蜘蛛).93、What do we call a large body of freshwater?A. OceanB. SeaC. LakeD. River答案:C94、听力题：The Amazon __________ is the longest river in South America.95、填空题：The _____ (小鸟) made a nest in our backyard. It has three eggs.小鸟在我们后院筑了一个巢，它有三个蛋。

四川大学化工考研884复试面试英语题库翻译原文

1.When a fluid flows through a duct or over a surface, the velocity over a place at right angles tothe stream is not normally uniform. The variation of velocity can be shown by the use ofstreamlines which are lines so drawn that the velocity vector is always tangential to them. The flow rate between any two streamlines is always the same. Constant velocity over a cross-section is shown by equidistant streamlines and increase in velocity by closer spacing ofstreamlines.2.It is found the heat transfer rate per unit area q is dependent on those physical propertieswhich affect flow pattern. The thermal properties of the fluid and the velocity of flow the fluid over the surface, the temperature difference t and a factor deternming the natural circulation effect cause by expansion of the fluid on heating.3.The oil distilled in a very large steel tower, the technical names of which is the fractionatingtower. The tower is thirty to fifty meters tall and its diameter is one to three meters. It isdivided into chambers, each of which contains a layer of trays. There are holes in the trays. The chanbers are at different heights, and temperature at each height is different.4. A group of operation for separating the components of mixture is based on the transfer ofmaterial from one homogeneous phase to anther. Unlike purely mechanical separation, these methods utilize differences in vapor pressure or solubility, not density or particle size. The driving force for transfer is a conoantration differce to or a concentration gradient, much as a temperature difference or a temperature gradient provides the driving force heat transfer.5.The three most important characteristics of an individual particle are its composition, its sizeand its shape. Composition determine such properties as density and conductivity， provided that the particle is completely uniform. A particle shape may be regular such as spherical or cubic, or it may be irregular as, for example, with a piece of broken glass.6.homogeneous catalysis is the industrical oxo process for manufacturing normal iosobuty(异丁醛)。

1 Introduction to the Galois Theory of Linear Differential Equations

∂ ∂ , . . . , ∂x }) = inﬁnitely diﬀerExamples 1.2.2 1) (C ∞ (Rm ), ∆ = { ∂x 1 m m entiable functions on R . ∂ ∂ 2) (C(x1 , . . . , xm ), ∆ = { ∂x , . . . , ∂x }) = ﬁeld of rational functions 1 m ∂ 3) (C[[x]], ∂x ) = ring of formal power series 1 ] C((x)) = quotient ﬁeld of C[[x]] = C[[x]][ x ∂ 4) (C{{x}}, ∂x ) = ring of germs of convergent series 1 C({x}) = quotient ﬁeld of C{{x}} = C{{x}}[ x ] ∂ ∂ 5) (MO , ∆ = { ∂x , . . . , ∂x }) = ﬁeld of functions meromorphic on 1 m open,connected m O ⊂C
1.2 What is a Linear Diﬀerential Equation? I will develop an algebraic setting for the study of linear diﬀerential equations. Although there are many interesting questions concerning diﬀerential equations in characteristic p [MvdP03a, MvdP03b, dP95, dP96], we will restrict ourselves throughout this paper, without further mention, to ﬁelds of characteristic 0. I begin with some basic deﬁnitions.

宁波2024年05版小学六年级下册T卷英语上册试卷(含答案)

宁波2024年05版小学六年级下册英语上册试卷(含答案)考试时间：80分钟（总分:120）A卷考试人：_________题号一二三四五总分得分一、综合题(共计100题)1、填空题：I can ______ (定期) reflect on my progress.2、填空题：The turtle swims slowly in the _______ (水).3、填空题：A __________ (催化循环) enhances reaction efficiency in chemical processing.4、What do you call a person who plays a musical instrument?A. MusicianB. SingerC. ComposerD. Conductor答案:A5、听力题：The chemical formula for calcium hydroxide is _______.6、How do you say "hand" in Spanish?A. ManoB. MainC. HandD. Maño7、听力题：The main gas in the air we breathe is _____.8、填空题：The ________ was a significant period in the evolution of human rights.The ________ (jacket) keeps me warm.10、填空题：Certain plants can ______ (抵抗) pests naturally.11、听力题：A _______ can be a beautiful centerpiece for a table.12、听力题：We eat ______ (snacks) during recess.13、听力题：A liquid that can dissolve a solute is called a _______.14、What is 3 + 5?A. 6B. 7C. 8D. 9答案:C15、填空题：Julius Caesar was a famous Roman _______. (统治者)16、听力题：A solution that contains the maximum amount of solute is _____ (saturated).17、What is the name of the famous ancient city in Iraq?A. BabylonB. NinevehC. UrD. All of the above18、听力题：The Magna Carta was signed in _______.19、What do we call the process of a caterpillar becoming a butterfly?A. MetamorphosisB. TransformationC. EvolutionD. Development答案:A20、填空题：I can ______ (表达) my thoughts clearly.The chemical symbol for rubidium is __________.22、听力题：The study of chemicals and their reactions is known as _______.23、What is the name of the famous explorer who sailed across the ocean in 1492?A. Vasco da GamaB. Ferdinand MagellanC. Christopher ColumbusD. Marco Polo答案:C24、What is the name of the famous ancient city in Jordan?A. PetraB. BabylonC. AthensD. Rome答案:A25、填空题：I want to grow _____ (蔬菜) this year.26、填空题：We can play with a ________ outside.27、选择题：What do you call the part of the plant that absorbs water?A. LeafB. StemC. RootD. Flower28、听力题：We enjoy going to the ___. (beach) every summer.29、听力题：A mixture that contains two or more phases is called a ______.30、What is the capital of Canada?A. TorontoB. OttawaC. VancouverD. MontrealI like to _______ (与朋友一起)去健身房。

【最新】PDE工具箱简介毕业论文外文翻译整理

附录A 英文原文PDE toolbox IntroductionBasics of the Finite Element Method：The solutions of simple PDEs on complicated geometries can rarely be expressedin terms of elementary functions. You are confronted with two problems: First youneed to describe a complicated geometry and generate a mesh on it. Then you needto discretize your PDE on the mesh and build an equation for the discreteapproximation of the solution. The pdetool graphical user interface providesyou with easy-to-use graphical tools to describe complicated domains and generatetriangular meshes. It also discretizes PDEs, finds discrete solutions and plots results. You can access t he mesh structures and the discretization functions directlyfrom the command line (or from a file) and incorporate them into specialized applications.Here is an overview of the Finite Element Method (FEM). The purpose of this presentation is to get you acquainted with the elementary FEM notions. Here you findthe precise equations that are solved and the nature of the discrete solution.Different extensions of the basic equation implemented in Partial DifferentialEquation Toolbox software are presented. A more detailed description can be foundin Chapter 3, “Finite Element Method”．You start by approximating the computational domain Ωwith a union of simple geometric objects, in this case triangles. The triangles form a mesh and each vertex iscalled a node. You are in the situation of an architect designing a dome. He has tostrike a balance between the ideal rounded forms of the original sketch and the limitations of his simple building-blocks, triangles or quadrilaterals. If the resultdoes not look close enough to a perfect dome, the architect can always improve hiswork using smaller blocks.Next you say that your solution should be simple on each triangle. Polynomialsare a good choice: they are easy to evaluate and have good approximation propertieson small domains. You can ask that the solutions in neighboring triangles connect toeach other continuously across the edges. You can still decide how complicated the polynomials can be. Just like an architect, you want them as simple as possible. Constants are the simplestchoice but you cannot match values on neighboring triangles. Linear functions come next. This is like using flat tiles to build a waterproof dome, which is perfectly possible.A Triangular Mesh (left) and a Continuous Piecewise Linear Function on That Meshpdetool GUIIn this section.-27“Introduction” on page 1“The Menus” on page 1-29-30“The Toolbar” on page 1“The GUI Modes” on page 1-31-32“The CSG Model and the Set Formula” on page 1-33“Creating Rounded Corners” on page 1-35“Suggested Modeling Method” on page 1-39“Object Selection Methods” on page 1-40“Display Additional Information” on page 1page 1-40“Entering Parameter Values as MATLAB Expressions” on“Using Earlier Version Partial Differential Equation Toolbox Model Files”on page 1-41IntroductionPartial Differential Equation Toolbox software includes a complete graphicaluser interface (GUI), which covers all aspects of the PDE solution process.You start it by typingpdetoolat the MATLAB command line. It may take a while the first time you launchpdetoolduring a MATLAB session. The following figure shows the pdetoolGUIas it looks when you start it.At the top, the GUI has a pull-down menu bar that you use to control the modeling.It conforms to common pull-down menu standards. M enu item followed by a right arrow lead to a submenu. Menu items followed by an ellipsis lead to a dialog box.Stand-alone menu items lead to direct action. Below the menu bar, a toolbar with icon buttons provide quick and easy access to some of the most important functions.To the right of the toolbar is a pop-up menu that indicates the currentapplication mode. You can also use it to change the application mode. The upper right part of the GUI also provides the x- and y-coordinates of the current cursor position. It is updated when you move the cursor inside the main axes area in the middle of the GUI. The edit box for the set formula contains the active set formula. In the main axesyou draw the 2-D geometry, display the mesh, plot the solution, etc. At the bottomof the GUI, an information line provides information about the current activity. Itcan also display help information about the toolbar buttons.The MenusThere are 11 different pull-down menus in the GUI. See Chapter 2, “Graphical User Interface” for a more detailed description of the menus and the dialog boxes: ?File menu. From the File menu you can Open and Save model files thatcontain a command sequence t hat reproduces your modeling session. Youcan alsoprint the current graphics and exit the GUI.?Edit menu. From the Edit menu you can cut, clear, copy, and paste the solid objects. There is also a Select All option.?Options menu. The Options menu contains options such as toggling theaxis grid, a “snap-to-grid” feature, and zoom. You can also adjust the axis limits andthe grid spacing, select the application mode, and refresh the GUI.?Draw menu. From the Draw menu you can select the basic solid objectssuch as circles and polygons. You can then draw objects of the selected type usingthe mouse. From the Draw menu you can also rotate the solid objects and export the geometry to the MATLAB main workspace.?Boundary menu. From the Boundary menu you access a dialog box whereyou define the boundary conditions. Additionally, you can labeedges and subdomains, remove borders between subdomains, and export the decomposed geometry and the boundary conditions to the workspace.?PDE menu. The PDE menu provides a dialog box for specifying the PDE,and there are menu options for labeling subdomains and exporting PDE coefficients tothe workspace.?Mesh menu. From the Mesh menu you create and modify the triangular mesh. You can initialize, refine, and jiggle the mesh, undo previous mesh changes,label nodes and triangles, display the mesh quality, and export the mesh to the workspace.?Solve menu. From the Solve menu you solve the PDE. You can also open a dialog box where you can adjust the solve parameters, and you can export the solutionto the workspace.?Plot menu. From the Plot menu you can plot a solution property. A dialogbox lets you select which property to plot, which plot style to use andseveral otherplot parameters. If you have recorded a movie (animation) of the solution, you canexport it to the workspace.? Window menu. The Window menu lets you select any currently open MATLAB figure window. The selected window is brought to the front? Help menu. The Help menu provides a brief help window.The ToolbarThe toolbar underneath the main menu at the top of the GUI contains icon buttonsthat provide quick and easy access to some of the most important functions.The five leftmost buttons are draw mode buttons and they represent, fromleft toright:The draw mode buttons can only be activated one at the time and they all work thesame way: single-clicking a button allows you to draw one solid object of theselected type. Double-clicking a button makes it “stick,” and you can then continue to draw solid objects of the selected type until yousingle-click the button to “release”　it. Using the right mouse button or Ctrl+click, the drawing is constrained to a squareor a circle.The second group of six buttons includes the following analysis buttons.The button toggles the zoom function on/offThe GUI Modes1 Define the geometry (2-D domain).2 Define the boundary conditions.3 Define the PDE.4 Create the triangular mesh.5 Solve the PDE.6 Plot the solution and other physical properties calculated from the solution (post processing).The pdetool GUI is designed in a similar way. You work in six differentmodes, each corresponding to one of the steps in the PDE solving process:?In draw mode, you can create the 2-D geometry using the constructive solidgeometry (CSG) model paradigm. A set of solid objects (rectangle, circleellipse, and polygon) is provided. These objects can be combined using set formulas in a flexible way.?In boundary mode, you can specify the boundary conditions. You can have different types of boundary conditions on different boundaries. In this mode, the original shapes of the solid objects constitute borders between subdomains of the model. Such borders can be eliminated in this mode.? In PDE mode, you can interactively specify the type of PDE problem, and the PDE coefficients. You can specify the coefficients for each subdomain independently. This makes it easy to specify, e.g., various materialproperties in a PDE model.? In mesh mode, you can control the automated mesh generation and plot the mesh.? In solve mode, you can invoke and control the nonlinear and adaptive solver for elliptic problems. For parabolic and hyperbolic PDE problems, you can specify the initial values, and the times for which the output should be generated. For the eigenvalue solver, you can specify the interval in which to search for eigenvalues.? In plot mode, there is a wide range of visualization possibilities. You can visualize both in the pdetool GUI and in a separate figure window. You can visualize three different solution properties at the same time, usingcolor, height, and vector field plots. There are surface, mesh, contour, and arrow (quiver) plots available. For parabolic and hyperbolic equations, youcan animate the solution as it changes with time.The CSG Model and the Set FormulaPartial Differential Equation Toolbox functions use the Constructive Solid Geometry (CSG) model paradigm for modeling. You can draw solid objects that can overlap. There are four types of solid objects:?Circle object — Represents the set of points inside and on a circle.?Polygon object — Represents the set of points inside and on a polygon given by a set of line segments.?Rectangle object — Represents the set of points inside and on a rectangle.?Ellipse object —Represents t he set of points inside and on an ellipse. The ellipse can be rotated.Each solid object is automatically given a unique name by the GUIdefault names are C1, C2, C3, etc., for circles; P1, P2, P3, etc. for polygons; R1,R2, R3, etc., for rectangles; E1, E2, E3, etc., for ellipses. Squares, although a special case of rectangles, are named SQ1, SQ2, SQ3, etc. The name is displayed on the solid object itself. You can use any unique name, as long as it contains no blanks. In draw mode, you can alter the names and the geometries of the objects by double-clicking them, which opens a dialog box. The following figure shows an object dialog box for a circle.You can use the name of the object to refer to the corresponding set of points in a set formula. The operators +, *, and - are used to form the set of points Ωin the plane over which the differential equation is solved. The operators +, the set union operator, and *, the set intersection operator, have the same precedence. The operator -, the set difference operator, has higher precedence. The precedence can be controlled by using parentheses. The resulting geometrical model, Ω, is the set of points for which the set formula evaluates to true. By default, it is the union of all solid objects. We often refer to the area Ωas the decomposed geometry.Creating Rounded CornersAs an example of how to use the set formula, let us model a plate with roundedfeature corners (fillets). Start the GUI and turn on the grid and the “snap-to-grid”　using the Options menu. Also, change the grid spacing to -1.5:0.1:1.5 for the x-axis and -1:0.1:1 for the y- axis.Select Rectangle/square from the Draw menu or click the button with the rectangle icon. Then draw a rectangle with a width of 2 and a height of 1 using the mouse, starting at (-1,0.5). To get the round corners, add circles, one in each corner. The circles should have a radius of 0.2 and centers ata distance that is 0.2 units from the left/right and lower/upper rectangle boundaries ((-0.8,-0.3), (-0.8,0.3), (0.8,-0.3), and (0.8,0.3)). To draw several circles, double-click the button for drawing ellipses/circles (centered). Then draw the circles using the right mouse button or Ctrl+click starting at the circle centers. Finally, at each of the rectangle corners, draw four small squares with a side of 0.1.附录B英文翻译PDE工具箱简介有限元方法的基础：简单的偏微分方程的复杂的几何形状的解决方案很少能被表示为初等函数。

Notes for the Physics-Based Calculus workshop

Proof. Plugging x = 0 into both sides gives A = A . Let f (x) = A cos(ωx) + B sin(ωx). Case 1. If (A, B ) = (0, 0), then f is constant, and since ω = 0, A = B = 0 by Corollary 1.13. Case 2. If (A, B ) = (0, 0), then f is not constant (using ω = 0 and Corollary 1.13). By Theorem 1.11, f (x) = M cos(ωx + φ), with M > 0, and since ω = 0, M is the maximum value of f . Applying Theorem 1.11 to f (x) = A cos(ω x) + B sin(ω x) gives f (x) = M cos(ω x + φ ), with maximum value M > 0. Since f has a unique maximum value, M = M > 0. We can conclude A2 + B 2 = M 2 = A2 + (B )2 , so B = ±B . We also get cos(ωx + φ) = cos(ω x + φ ) for all x. Taking the derivative of both sides gives −ω sin(ωx + φ) = −ω sin(ω x + φ ). Since ω = 0 and ω = 0, these are non-constant functions with maximum values |ω | and |ω |, respectively, which must be equal, so ω = ±ω . If ω > 0 and ω > 0, ω = ω follows immediately, and the hypothesis becomes: A cos(ωx) + B sin(ωx) = A cos(ωx) + B sin(ωx), and the cosine terms cancel, so (B − B ) sin(ωx) is the constant function 0 and by Corollary 1.13, B − B = 0. Deﬁnition 1.15. Given a function of the form f (x) = A cos(ωx) + B sin(ωx), which is not identically zero, the constant |ω | is uniquely deﬁned, and we deﬁne ω| the number |2 π to be the “frequency” of f (x).

单纯形法中三个系数名称的英文名

单纯形法中三个系数名称的英文名In the simplex method of linear programming, there are three key coefficients that play a crucial role: the objective function coefficient, the right-hand side coefficient, and the slack variable coefficient. These coefficients are essential for determining the optimal solution to a linear programming problem.1. Objective Function Coefficient:The objective function coefficient, often denoted as "c" or "Cb," represents the coefficients of the decision variables in the objective function. In the simplex method, the objective function is typically written as a linear combination of the decision variables, with thecoefficients indicating the relative importance or value of each variable. The goal of the simplex method is to maximize (or minimize) the objective function by adjusting the decision variables.2. Right-Hand Side Coefficient:The right-hand side coefficient, often denoted as "b," represents the coefficients of the right-hand side constants in the constraints of the linear programming problem. These coefficients define the upper or lower bounds on the decision variables, ensuring that the solution satisfies all the constraints. In the simplex method, the right-hand side coefficients are used to calculate the values of the slack variables, which represent the amount of slack or wiggle room available in the constraints.3. Slack Variable Coefficient:The slack variable coefficient, often denoted as "s," represents the coefficients of the slack variables introduced in the simplex method. Slack variables are artificial variables added to the linear programming problem to convert the constraints into equations. They allow the simplex method to explore the feasible region of the problem by moving along the edges of the polyhedrondefined by the constraints. The slack variable coefficients are typically set to zero initially and are updated during the simplex iterations to reflect the current state of the solution.The simplex method utilizes these three coefficients to navigate the feasible region of the linear programming problem and find the optimal solution. By adjusting the decision variables and slack variables based on the values of these coefficients, the simplex method cansystematically move towards a solution that maximizes or minimizes the objective function while satisfying all the constraints.。

费马大定理英文作文

费马大定理英文作文英文：The Fermat's Last Theorem has been a fascinating topic for mathematicians and scholars for centuries. It states that there are no three positive integers a, b, and c that satisfy the equation an + bn = cn for any integer value of n greater than 2. This theorem was first conjectured by Pierre de Fermat in 1637 and it remained unproven for over 350 years until it was finally proven by Andrew Wiles in 1994.The journey to proving Fermat's Last Theorem was long and arduous. Wiles spent seven years working in secrecy on the proof, and when he finally presented it to the world, it was a groundbreaking moment in the history of mathematics. The proof involved complex mathematical concepts and required a deep understanding of number theory and algebraic geometry.As an example, let's consider the case when n = 3. In this case, the equation becomes a³ + b³ = c³. Fermat's Last Theorem states that there are no three positive integers a, b, and c that satisfy this equation. This means that no matter how large the numbers a, b, and c are, it is impossible to find a solution to this equation.The significance of Wiles' proof goes beyond just solving a centuries-old mathematical problem. It demonstrated the power of perseverance and dedication in the pursuit of knowledge. It also opened up new avenues for research in mathematics and inspired a new generation of mathematicians to push the boundaries of what is possible.中文：费马大定理一直是数学家和学者们几个世纪以来的一个迷人话题。

成都2024年05版小学3年级下册B卷英语第一单元自测题[含答案]

成都2024年05版小学3年级下册英语第一单元自测题[含答案]考试时间：100分钟（总分:140）A卷考试人：_________题号一二三四五总分得分一、综合题(共计100题)1、听力题：A ____ is a friendly pet that often wags its tail.2、填空题：The country famous for its glaciers is ________ (冰岛).3、What is the main source of energy for the Earth?A. MoonB. SunC. StarsD. Wind答案:B. Sun4、填空题：The ice cream is ________ (冷).5、听力题：The chemical formula for sodium sulfite is ______.6、填空题：The flowers in the garden are _______ and cheerful, making me smile.7、How many zeros are in one thousand?A. OneB. TwoC. ThreeD. Four答案:C8、听力题：A saturated solution cannot dissolve any more ______.9、What do we call an animal that only eats meat?A. HerbivoreB. CarnivoreC. OmnivoreD. Insectivore答案：B10、填空题：My cat loves to chase _______ (影子).11、填空题：I love to watch _______ at the zoo (我喜欢在动物园看_______).12、What is the name of the famous detective created by Arthur Conan Doyle?A. Hercule PoirotB. Sherlock HolmesC. Sam SpadeD. Philip Marlowe答案：B13、填空题：The squirrel is an expert at climbing _______ (树木).14、填空题：I admire my grandparents for their ____.15、听力题：They are _____ (celebrating) a birthday.16、听力题：In chemistry, a _______ is a substance made of two or more elements.17、What do you call a place where you can borrow books?A. SchoolB. LibraryC. StoreD. Park18、听力题：We are going to ___ dinner. (eat)19、Which animal is known as man's best friend?A. CatB. HamsterC. DogD. Goldfish答案：C20、选择题：What do you call the process of changing from a liquid to a gas?A. CondensationB. EvaporationC. FreezingD. Melting21、听力题：A __________ is a part of the periodic table that contains metals.22、What is the main purpose of a flag?A. DecorationB. Symbol of identityC. Indicator of weatherD. None of the above答案: B23、听力题：A ______ is a type of fish that can be found in rivers and streams.24、填空题：The country famous for its architecture is ________ (西班牙).25、What do we call the study of weather?A. BiologyB. MeteorologyC. GeographyD. Astronomy答案:B26、Which instrument is played by blowing air into it?A. GuitarB. DrumsC. FluteD. Piano答案:C27、What is the main ingredient in ice cream?A. MilkB. WaterC. JuiceD. SodaThe teacher gives us ______ for being good. (stickers)29、What is the capital of Russia?A. KyivB. MoscowC. MinskD. Tbilisi答案:B30、填空题：The duck swims ______ (在) the pond.31、填空题：I enjoy cooking ______ with my mom.32、填空题：The __________ (工厂制度) started during the Industrial Revolution.33、听力题：A ____ is known for its ability to find food in the dark.34、听力题：_______ can grow in both sunny and shady places.35、填空题：I like to help my dad ________ (清理院子).36、What is the name of the famous American landmark in South Dakota?A. Statue of LibertyB. Mount RushmoreC. Golden Gate BridgeD. Grand Canyon答案:B37、填空题：My __________ (玩具名) is a great __________ (名词) for learning.38、What is the capital of Iceland?A. ReykjavikB. OsloC. HelsinkiD. Copenhagen答案: A. ReykjavikWhat do you call a group of lions?A. PackB. PrideC. FlockD. Herd40、What is the capital of France?A. BerlinB. MadridC. ParisD. Rome41、填空题：My _______ (猫) scratches the furniture.42、听力题：The tree is _______ (full) of leaves.43、听力题：A _______ is a chemical process that produces nutrients.44、听力题：A __________ has a shell and can live in water or on land.45、Which of these is a popular fruit?A. TomatoB. CarrotC. AppleD. Lettuce答案: C46、选择题：What is the name of the famous rock formation in Australia?A. Ayers RockB. UluruC. Great Barrier ReefD. Blue Mountains47、填空题：We will have ________ (晚餐) together tonight.48、填空题:I love learning about different ________ around the world.49、What is the name of the ocean that is between Africa and Australia?A. AtlanticB. IndianC. ArcticD. Pacific50、填空题：I enjoy sharing ______ with my neighbors.51、填空题：The __________ (晨曦) over the mountains is stunning.52、What is the capital of the Czech Republic?A. PragueB. BratislavaC. BudapestD. Ljubljana答案:A53、填空题：The __________ (历史的反击) challenge established norms.54、填空题：A rabbit has long _______ to listen carefully.55、填空题：A ______ (蜜蜂) is important for pollinating plants.56、填空题：The ________ loves to explore and discover new things.57、填空题：My family has a ______ pet. (我的家里有一只______宠物。

兰州2024年04版小学四年级上册L卷英语第一单元综合卷[有答案]

兰州2024年04版小学四年级上册英语第一单元综合卷[有答案]考试时间：90分钟（总分:100）A卷考试人：_________题号一二三四五总分得分一、综合题(共计100题共100分)1. 选择题：What is the opposite of ‘cold’?A. WarmB. HotC. CoolD. Chilly2. 听力题：I enjoy ___ (reading) before bed.3. 选择题：What is the capital of France?A. BerlinB. MadridC. ParisD. Rome4. 选择题：What is the term for a scientist who studies rocks?A. BiologistB. GeologistC. ChemistD. Physicist答案:B5. 听力题：The chemical formula for sodium nitrate is _____.6. 填空题：The lynx is known for its tufted _________ (耳朵).7. 填空题：My dog enjoys going for _______ (散步) with me.8. 填空题：My favorite thing to do at night is ______.9. 听力题：I paint with _____ (油漆).10. 选择题：What do we call the art of folding paper into shapes?A. OrigamiB. PaintingC. SculptingD. Drawing答案:A11. 填空题：The _______ (青蛙) croaks loudly at night.12. 听力题：I want to ________ (dance) at the party.13. 填空题：The country known for its historical significance is ________ (以历史重要性闻名的国家是________).14. 选择题：What is the term for the study of stars and planets?A. BiologyB. ChemistryC. AstronomyD. Geology15. 选择题：What is the primary ingredient in sushi?A. RiceB. NoodlesC. BreadD. Potatoes16. 选择题：What do we call a story that is not true?a. Factb. Fictionc. Legendd. History答案:b17. 听力题：A ______ is a representation of an experiment's outcome.18. 选择题：Which animal is famous for its long migrations?A. ElephantB. SalmonC. LionD. Tiger答案: B19. 填空题：Her dress is _______ (漂亮的).根据图片提示，选出正确的答案。

不可压缩均匀各向同性湍流直接数值 1024 POF Gotoh_Fukayama_Nakano

Velocityﬁeld statistics in homogeneous steady turbulence obtained using a high-resolution direct numerical simulationToshiyuki Gotoh a)Department of Systems Engineering,Nagoya Institute of Technology,Showa-ku,Nagoya466-8555,JapanDaigen FukayamaInformation and Mathematical Science Laboratory,Inc.,2-43-1,Ikebukuro,Toshima-ku,Tokyo171-0014,JapanTohru NakanoDepartment of Physics,Chuo University,Kasuga,Bunkyo-ku,Tokyo112-8551,JapanVelocityﬁeld statistics in the inertial to dissipation range of three-dimensional homogeneous steadyturbulentﬂow are studied using a high-resolution DNS with up to Nϭ10243grid points.The rangeof the Taylor microscale Reynolds number is between38and460.Isotropy at the small scales ofmotion is well satisﬁed from half the integral scale͑L͒down to the Kolmogorov scale͑␩͒.TheKolmogorov constant is1.64Ϯ0.04,which is close to experimentally determined values.The thirdorder moment of the longitudinal velocity difference scales as the separation distance r,and itscoefﬁcient is close to4/5.A clear inertial range is observed for moments of the velocity differenceup to the tenth order,between2␭Ϸ100␩and L/2Ϸ300␩,where␭is the Taylor microscale.Thescaling exponents are measured directly from the structure functions;the transverse scalingexponents are smaller than the longitudinal exponents when the order is greater than four.Thecrossover length of the longitudinal velocity structure function increases with the order andapproaches2␭,while that of the transverse function remains approximately constant at␭.Thecrossover length and importance of the Taylor microscale are discussed.©2002AmericanInstitute of Physics.͓DOI:10.1063/1.1448296͔I.INTRODUCTIONKolmogorov studied the statistical laws of a velocity ﬁeld for small scales of turbulent motion at high Reynolds numbers.1,2Two hypotheses were introduced in his theory ͑hereafter K41for short͒:local isotropy and homogeneity exists;and there is an inertial range in the energy spectrum of theﬂow that is independent of viscosity and large-scale properties at sufﬁciently high Reynolds numbers.The most prominent conclusion of his theory is the presence of the Kolmogorov spectrum E(k)ϭK⑀¯2/3kϪ5/3in the inertial range,where⑀¯is the average rate of energy dissipation per unit mass and K is a universal constant.Since K41,there has been a considerable amount of ef-fort made to study the turbulent velocityﬁeld statistics in the inertial range,and the energy spectrum has been a central quantity of interest.The Kolmogorov spectrum and constant have been measured inﬁeld and laboratory experiments.3–7 The exponent for the inertial range spectrum is now widely accepted asϪ5/3,with a small correction to account forﬂow intermittency.The Kolmogorov constant K is between1.5 and 2.After studying the results of many experiments, Sreenivasan stated that K is1.62Ϯ0.17.7The spectral theory of turbulence has also been used to predict the Kolmogorov constant.The value of K is1.77when the Lagrangian history direct interaction approximation͑LHDIA͒is used,8,9and 1.72when the Lagrangian renormalized approximation ͑LRA͒is used.10,11These are fully systematic theories that do not contain any ad hoc parameters.Direct numerical simulations͑DNSs͒of turbulentﬂows are now performed at higher Reynolds numbers,due to the recent dramatic increase in computational power.In the early 90’s,the resolution of DNS reached Nϭ5123grid points with a Taylor microscale Reynolds number R␭of 210ϳ240.12–20Most high-resolution DNSs have been per-formed for steady turbulence conditions to achieve high Rey-nolds numbers and obtain reliable statistics.Although results were reported with R␭greater than200,an inertial range spectrum was observed only for the lowest narrow wave number band at which forcing was applied.The Kolmogorov constant was inferred to be about1.5ϳ2in Ref.14and1.62 in Ref.16,but these results are not convincing,due to the insufﬁcient width of the scaling range,anisotropy of theﬂow ﬁeld,limited ensemble size,forcing techniques used,and numerical limitations of the simulations.Intermittency has also attracted the interest of research-ers.Since Kolmogorov’s intermittency theory͑hereafter K62͒,21many theoretical and statistical models of intermit-tency have been developed.4,22,23The scaling exponents of higher order structure functions for velocity differences in the inertial range were studied intensively.Intermittency in-creases with a decrease in the size of the scales of motion. The small-scale statistics gradually deviate from a Gaussian distribution,and the scaling exponents differ from those pre-dicted by K41.Experiments at very high Reynolds numbers have beena͒Electronic mail:gotoh@system.nitech.ac.jpPHYSICS OF FLUIDS VOLUME14,NUMBER3MARCH200210651070-6631/2002/14(3)/1065/17/$19.00©2002American Institute of Physicsperformed in the atmospheric boundary layer and in huge wind tunnels,and the measured scaling exponents were found to deviate from K41scaling.5,6,24,25,84However,there have been arguments made about the lack of small-scaleﬂow isotropy and homogeneity in these experiments,which might be affected by the large-scale shear.25,26For experiments at moderate Reynolds numbers under relatively well-controlled laboratory conditions,the width of the scaling range is usually not large enough to determine the scaling exponents precisely.Extended self-similarity͑ESS͒has been exploited to overcome this difﬁculty and applied to various turbulentﬂows in both experiments and DNSs.28–31 The idea is to measure the scaling exponents of the structure functions when they are plotted against the third order lon-gitudinal structure function,rather than to use the separation distance.The width of the scaling range is longer than that obtained with the usual method at low to moderate Reynolds numbers.The scaling exponents are anomalous,but do agree with those obtained from high Reynolds number experiments up to a certain order.24,28–31However,there is no consensus as to why the structure functions give a longer inertial range, or what is missing from theﬂow statistics as a result.Also, there is no unique way to determine the scaling exponents for the transverse and mixed velocity structure functions,be-cause those higher order structure functions can be plotted against other types of third order structure functions as well as the third order longitudinal structure function.There also have been arguments about whether the scal-ing exponents for the longitudinal and transverse structure functions at small scales are equal.25–27,32–38Many experi-ments and DNSs have reported that higher order longitudinal scaling exponents are larger than transverse ones.However, some researchers have argued that the difference is due to deviation from the assumed conditions,such as local homo-geneity,isotropy,and the independence of small scales from macroscale parameters.They have suggested that when the Reynolds number becomes large enough,the difference will vanish.36,37,39,40In many aspects of turbulence research,there have been questions posed about the extent to which the local homoge-neity and isotropy of the turbulent velocityﬁeld are attained. This will affect the small-scale statistics signiﬁcantly.Recent experimental studies have shown that local isotropy is par-tially satisﬁed for lower order moments.25,26,37However,it is not sufﬁcient to examine only the conditions assumed in the above studies,and only a limited knowledge of the trueﬂow conditions is available so far.26,37,38A DNS with a sufﬁciently large grid size provides abetter opportunity to examine the points raised above.It hasthe advantage that any physical quantity can be measureddirectly without deforming theﬂowﬁeld.In the presentstudy,a series of large scale DNSs have been performed at ahigh resolution of up to Nϭ10243and R␭ϭ460.41–45The inertial range of the turbulenceﬁeld has a considerablelength,and useful velocity statistics can be extracted such asthe Kolmogorov constant,the energy spectrum,velocitystructure functions up to the tenth order,their scaling expo-nents,and probability density functions for velocity differ-ences.To the authors’best knowledge,these are theﬁrstDNS data in the inertial range;the data provide new insightinto the inertial and dissipation ranges.The main purposes of the present paper are to describethe statistics of the velocityﬁeld in an incompressible steadyturbulentﬂow obtained from the DNS,and to reexaminecurrent knowledge of turbulence,developed since K41.Thepaper is organized as follows.The numerical aspects of thepresent DNS are described in Sec.II,and the energy spec-trum is examined in Sec.III.The variation of single pointquantities and probability density functions͑PDFs͒with theReynolds number is discussed in Sec.IV.The isotropy of thesecond and third order moments of the velocity difference isexamined in Sec.V,and the energy budget is examined interms of the Ka´rma´n–Howarth–Kolmogorov equation inSec.VI.The structure functions and scaling exponents arediscussed in Sec.VII.Section VIII presents an analysis ofthe crossover lengths of the structure functions.Finally,asummary and conclusions are provided in Sec.IX.II.NUMERICAL SIMULATIONThe Navier–Stokes equations are integrated in Fourierspace for unit density:ͩץץtϩ␯k2ͪuϭP͑k͒•F͓uÃ␻͔kϩf,͑1͒͗f͑k,t͒f͑Ϫk,s͒͘ϭP͑k͒F͑k͒4␲k2␦͑tϪs͒,͑2͒where␻is the vorticity vector,P͑k͒is the projection opera-tor,F denotes a Fourier transform,and f is a solenoidal Gaussian random force that is white in time.The spectrum of the random force F(k)is constant over the low wave number band and zero otherwise;the force is normalized asTABLE I.DNS parameters and statistical quantities of the runs.T eddya v is the period used for the time average.R␭N k max␯c f Forcing range T eddya v E⑀¯L␭␩(ϫ10Ϫ2)K 38128360 1.50ϫ10Ϫ2 1.30ͱ3рkрͱ1222.6 1.99 1.190.8910.501 4.10¯5425631217.00ϫ10Ϫ30.70ͱ3рkрͱ1214.9 1.390.6270.8290.393 2.72¯702563121 4.00ϫ10Ϫ30.50ͱ3рkрͱ1249.7 1.160.4570.7850.318 1.93¯1255123241 1.35ϫ10Ϫ30.50ͱ3рkрͱ12 5.52 1.250.4920.7440.1850.841¯2845123241 6.00ϫ10Ϫ40.501рkрͱ6 3.03 1.960.530 1.2460.1490.449 1.64 38110243483 2.80ϫ10Ϫ40.511рkрͱ6 4.21 1.740.499 1.1390.09890.258 1.63 46010243483 2.00ϫ10Ϫ40.511рkрͱ6 2.14 1.790.506 1.1500.08410.199 1.64 1066Phys.Fluids,Vol.14,No.3,March2002Gotoh,Fukayama,and Nakano͵ϱF ͑k ͒dk ϭ⑀¯in ,͑3͒where ⑀¯in is the average rate of the energy input per unit mass.A pseudo-spectral code was used to compute the con-volution sums,and the aliasing error was effectively re-moved.The time integration was performed using the fourth order Runge–Kutta–Gill method.Physical quantities of turbulent ﬂow include the total energyE ͑t ͒ϭ12͗u 2͘ϭ32u ¯2ϭ͵ϱE ͑k ͒dk ,͑4͒the average energy dissipation per unit mass⑀¯ϭ2␯͵ϱk 2E ͑k ͒dk ,͑5͒the integral scaleL ϭͩ3␲4͵ϱk Ϫ1E ͑k ͒dkͪͲE ,͑6͒the Taylor microscale␭ϭͩ5EͲ͵ϱk 2E ͑k ͒dkͪ1/2,͑7͒the Taylor microscale Reynolds numberR ␭ϭu ¯␭␯,͑8͒and the Kolmogorov scale␩ϭͩ␯3⑀¯ͪ1/4.͑9͒The range of the Taylor microscale Reynolds number was 38to 460.The characteristic parameters of the DNS are listed in Table I.43Most of these are identical to Gotoh and Fukayama,43but the averaging time for R ␭ϭ381was ex-tended to 4.21large eddy turnover times.A statistically steady state was conﬁrmed by observing the time evolutionof the total energy,the total enstrophy,and the skewness of the longitudinal velocity derivative.The statistical averages were computed as time averages over tens of large eddy turnover times for the lower Reynolds number ﬂows,and over a few large eddy turnover times for the higher Reynolds number ﬂows.The resolution condition k max ␩Ͼ1was satis-ﬁed for most runs,except for R ␭ϭ460in which k max ␩was slightly less than unity (k max ␩ϭ0.96).This does not ad-versely affect the results in the inertial range.The computational time required for runs at a N ϭ10243resolution varied,depending on the statistical data that was gathered.Typically,60h was required for one large eddy turnover time.The total time of the computations was more than 500h for the longest run (R ␭ϭ381).Data col-lected during the transition period to steady state ͑about six large eddy turnover times ͒were discarded.The relatively long time required to attain steady state was due to the low wave number band forcing.This imposes a severe computa-tional putations with R ␭р284were per-formed on a Fujitsu VPP700E parallel vector machine with 16processors at RIKEN.Simulations of higher R ␭were per-formed on a Fujitsu VPP5000/56with 32processors at the Nagoya University Computation Center.III.ENERGY SPECTRUMFigure 1shows the three-dimensional energy spectrum calculated for each run.All of the curves are scaled to the Kolmogorov units and multiplied by k 5/3.As the Reynolds number increases,the curves extend toward lower wave numbers.The curves of ﬂows with Reynolds numbers larger than R ␭ϭ284contain a ﬁnite plateau,which indicates that E (k )ϰk Ϫ5/3.There is a bump when 0.04рk ␩р0.3at the high end of the inertial range,which is consistent with pre-vious experimental and numerical observations.6,16The nor-malized energy transfer ﬂux,deﬁned by1⑀¯⌸͑k ͒ϭ1⑀¯͵kϱT ͑k Ј͒dk Ј͑10͒is shown in Fig.2,where T (k )is a nonlinear energy transfer function in the energy spectrum equation.4,22Between0.007рk ␩р0.04,⌸(k )/⑀¯is approximately constantand FIG.1.Scaled energy spectra,⑀¯Ϫ1/4␯Ϫ5/4(k ␩)5/3E (k ).The inertial range is 0.007рk ␩р0.04and K ϭ1.64Ϯ0.04.A horizontal line indicates K ϭ1.64.FIG.2.Normalized energy transfer ﬂux,⌸(k )/⑀¯for R ␭ϭ381and 460.1067Phys.Fluids,Vol.14,No.3,March 2002Velocity ﬁeld statistics in homogeneousclose to unity;thus the ﬂow is in an equilibrium state over the inertial range of the energy spectrum,corresponding to the plateaus in Fig.1.The Kolmogorov constant given in Table I is determined using a least square ﬁt between 0.007рk ␩р0.04on the R ␭Ͼ284curves.In Ref.43,the Kolmog-orov constant was reported as K ϭ1.65Ϯ0.05.However,the averaging time has since been extended for the R ␭ϭ381run.The R ␭ϭ478run differs slightly from statisticalequilibrium,since ⌸(k )/⑀¯is not exactly one;for this reason,the R ␭ϭ478data were not used for this analysis.The Kol-mogorov constant,computed using the data only from the R ␭ϭ381and 460runs,isK ϭ1.64Ϯ0.04,͑11͒which is in good agreement with experimental values and recent DNS data.7,16There are many DNSs reporting the Kolmogorov constant higher than the value 1.64.However,the length of the inertial range in those DNSs is not long enough to clearly observe the k Ϫ5/3range,and the top of the bump of the compensated energy spectrum k 5/3E (k )is un-derstood as the inertial range,so that the Kolmogorov con-stant is read as about 2as seen in Fig.1.16The Kolmogorov constant 1.64is also close to the value obtained using the LHDIA ͑1.77͒,8,9the LRA ͑1.72͒.10,11These spectral theories of turbulence are consistent with Lagrangian dynamics,arederived systematically,and contain no ad hoc parameters.Figure 3shows the one-dimensional energy spectrum ob-tained from the present DNS with R ␭ϭ460,from experi-ments,and from the LRA.The agreement between the curves is satisfactory.Therefore we conclude that the present DNS has successfully calculated a homogeneous turbulent ﬂow ﬁeld in the inertial range of the energy spectrum.IV.ONE-POINT STATISTICS A.MomentsSome one-point moments of the velocity ﬁeld areS 3͑u ͒ϵ͗u 3͗͘u 2͘3/2,S 3͑u x ͒ϵ͗u x 3͗͘x 2͘3/2,͑12͒K 4͑u ͒ϵ͗u 4͗͘u 2͘2,K 4͑u x ͒ϵ͗u x 4͗͘u x 2͘2,K 4͑u y ͒ϵ͗u y 4͗͘u y 2͘2,͑13͒where u is the velocity component in the x direction.The variation of these moments with the Reynolds number is shown in Fig.4and listed in Table II.The general behavior of the curves is consistent with previous DNS and experi-mental data.13,14,18,19,26,46,47There are small effects of rela-tively low resolution on S 3and K 4for the velocity deriva-tives for R ␭ϭ381and 460data.The skewness factor of the velocity u is very small for runs with the R ␭р125,and isofparison of one-dimensional energy spectra.Symbols:experi-ments,solid line:present DNS (R ␭ϭ460),dashed line:statistical theory ͑LRA and MLRA ͒.FIG.4.Variation of the moments of the velocity and velocity gradient withthe Reynolds number.Line:present DNS,circle:K 4(u y )͑Jime´nez et al.,Ref.13͒,solid square:K 4(u )͑Jime ´nez et al.,Ref.13͒,square:K 4(u x )͑Wang et al.,Ref.14͒,plus:K 4(u x )͑Vedula and Yeung,Ref.18͒,star:ϪS 3(u x )͑Wang et al.,Ref.14͒.TABLE II.Moments of the velocity and velocity derivatives.R ␭S 3(u )K 4(u )S 3(ץu /ץx )K 4(ץu /ץx )K 4(ץu /ץy )380.0227 2.89Ϫ0.520 4.14 5.16540.00563 2.86Ϫ0.517 4.47 6.00700.00473 2.93Ϫ0.519 4.81 6.621250.0820 2.94Ϫ0.529 5.658.192840.0231 2.77Ϫ0.531 6.6310.1381Ϫ0.246 2.98Ϫ0.5747.9012.2460Ϫ0.1682.89Ϫ0.5457.9111.71068Phys.Fluids,Vol.14,No.3,March 2002Gotoh,Fukayama,and Nakanothe order of 0.2for runs with the R ␭у284.The relatively large values of the velocity skewness are caused by the shorter averaging time used compared to the low Reynolds number runs.Since most of the energy resides in the lowest wave number band,there are persistent large ﬂuctuations of the large scales of motion over longer time period.The longer time average or the forcing at larger wave numbers would yield smaller velocity skewness.The ﬂatness factor of the velocity ﬁeld is close to three,which is the Gaussian value.The skewness factor of the longitudinal velocity deriva-tives is very insensitive to the Reynolds number,S 3͑u x ͒ϰR ␭0.0370,͑14͒where the exponent is determined by a least square ﬁt.Theaverage value is Ϫ0.53,which is consistent with experimen-tal observations over the range of Reynolds numbers studied in the present work.However,the exponent is smaller than indicated by the experimental data.26,46The ﬂatness factors for the longitudinal and transverse velocity derivatives in-crease with the Reynolds number asK 4͑u x ͒ϰR ␭0.266,K 4͑u y ͒ϰR ␭0.335.͑15͒The exponent of K 4(u y )is larger than that of K 4(u x );thus,the PDF for the transverse velocity derivative has longer tails than those of the longitudinal velocity derivative.From ex-perimental observations,Shen and Warhaft reported thatK 4(u x )ϰR ␭0.37and K 4(u y )ϰR ␭0.25.26Since there is scatter in the experimetal data,the exponents in Eq.͑15͒by the present DNS are not inconsistent with the experimental data.Van Atta and Antonia studied the Reynolds number dependence of S 3(u x )and K 4(u x ),46and found thatS 3͑u x ͒ϰR ␭0.12,K 4͑u x ͒ϰR ␭0.32for ␮ϭ0.2,͑16͒S 3͑u x ͒ϰR ␭0.15,K 4͑u x ͒ϰR ␭0.41for ␮ϭ0.25,͑17͒where ␮is the exponent deﬁned by ͗⑀r 2͘ϰr Ϫ␮for the locally averaged energy dissipation rate.4,21Generally,the Reynolds number dependency of S 3and K 4in our DNSs is weaker than observed in the experiments,irrespective of the type of forcing used.We believe this is because the range of Rey-nolds numbers in DNS is smaller than experimental ﬂows,and there remain small-scale anisotropy effects in the experi-ments.B.Probability density functionsThe probability density function conveys information about single-point velocity statistics.It has been one of the central issues of turbulence research in the last decade.Single-point PDFs for the velocity and its derivatives are shown in Figs.5–7.A longer time period was necessary for the time average to obtain well-converged PDF for the ve-locity Q (u ).The distribution Q (u )is close to Gaussian,and its tail extends to very low values of the order of 10Ϫ10.Such values have not been reported in the literature.The Q (u )curve for R ␭ϭ381is skewed negatively,but this is attributed to the insufﬁcient time-averaging period ͑four large eddy turnover times ͒that was used.The overall trend is that Q (u )decays faster than a Gaussian distribution at large ampli-tudes.This behavior was also observed in one-dimensional decaying and forced Burgers turbulence.48,49Jime´nez has shown that the PDF Q (u )is slightly sub-Gaussian as the energy spectrum decays faster than k Ϫ1.50FIG.5.Variation of velocity PDF with the Reynoldsnumber.FIG.6.Variation of the longitudinal velocity derivative PDF with the Rey-noldsnumber.FIG.7.Variation of the transverse velocity derivative PDF with the Rey-nolds number.1069Phys.Fluids,Vol.14,No.3,March 2002Velocity ﬁeld statistics in homogeneousThis is consistent with the present DNS results.Studies of the Q (u )tail predict that Q (w )ϰexp(Ϫc ͉w ͉3)when the forc-ing has a short correlation time.51,52Here,w ϭu /͗u 2͘1/2is the normalized velocity amplitude and c is a nondimensional constant.The asymptotic form of Q (u )was examined by plotting ln ͓Ϫln(Q (w )͔against ln ͉w ͉;however,the Q (w )tails were too short to determine the true asymptotic form.The PDF for the longitudinal velocity derivative is slightly skewed,as expected from the ﬁnite negative value of the skewness factor.The tail becomes longer as the Reynolds number increases.Figure 7shows that the PDF of the trans-verse derivative is symmetric and has a longer tail than the longitudinal derivative.There are many theories for the PDF of the velocity derivative.The asymptotic tail of Q (ץu /ץy )is presented in Fig.8,in which both the positive and negative sides are plotted by assuming that the PDF is symmetric.The tails gradually become longer as the Reynolds number increases;therefore,Q (s )is Reynolds-number dependent,and cannot be represented in a single stretched exponential form as Q (s )ϰexp(Ϫb ͉s ͉h ),where s is the normalized amplitude of ץu /ץy and b is a nondimensional constant that is a function of the Reynolds number.53V.ISOTROPYThe hypothesis of isotropy of the ﬂow ﬁeld is one of the key components of K41.There are various methods to ex-amine the degree of isotropy.One measure of isotropy can be obtained from the relations between the second and third order longitudinal and transverse velocity structure func-tions.These areD LL ϵ͗͑␦u r ͒2͘,D TT ϵ͗͑␦v r ͒2͘,͑18͒D LLL ϵ͗͑␦u r ͒3͘,D LTT ϵ͗␦u r ͑␦v r ͒2͘,͑19͒where␦u r ϵ͑u ͑x ϩr ͒Ϫu ͑x ͒͒•r /r ,͑20͒␦v r ϵ͑u ͑x ϩr ͒Ϫu ͑x ͒͒•͑I Ϫrr /r 2͒•e Ќ,͑21͒and e Ќis the unit vector perpendiculer to r ,and I is the unit tensor.Then the isotropy and incompressibility relations areD TT ͑r ͒ϭD LL ͑r ͒ϩr 2dD LL ͑r ͒dr ,͑22͒D LTT ͑r ͒ϭ16ddrrD LLL ͑r ͒.͑23͒In DNS,the solenoidal property of the Fourier amplitude velocity vector u ͑k ͒is always satisﬁed to the level of nu-merical error,which is smaller than 10Ϫ15.Thus,the accu-racy of the above relations depends solely on the deviation from isotropy.The two sides of Eqs.͑22͒and ͑23͒are com-pared for R ␭ϭ125,381,and 460in Figs.9and 10.The curves in the ﬁgures are divided by r 2/3and r ,respectively,and the vertical axes of the plots are linear.The thick lines represent the left hand sides of Eqs.͑22͒and ͑23͒,and the thin lines correspond to the right-hand sides.The isotropy of the second and third order moments is excellent for scales less than L /2.The difference at larger separations is caused by the anisotropy due to the small number of energy-containing Fourier modes.The curves for R ␭ϭ381and460FIG.8.Variation of the asymptotic tail of the transverse velocity derivative PDF with the Reynolds number.Both positive and negative sides are plot-ted.The rightmost curve corresponds to R ␭ϭ460.FIG.9.Isotropy relation at the second order.Thin line:D TT (r )r Ϫ2/3,thick line:(D LL (r )ϩ(r /2)(dD LL (r )/dr ))r Ϫ2/3.L /␩and ␭/␩are shown for R ␭ϭ460.FIG.10.Isotropy relation at the third order.Thin line:D LTT (r )r Ϫ1,thick line:((1/6)(d /dr )rD LLL (r ))r Ϫ1.L /␩and ␭/␩are shown for R ␭ϭ460.1070Phys.Fluids,Vol.14,No.3,March 2002Gotoh,Fukayama,and Nakanoin Fig.9are not horizontal,suggesting that the second order structure function does not scale as r 2/3.The scaling expo-nents will be examined later in this paper.The isotropic re-lations,such as D 1122ϭD 1133and D 2222ϭ3D 2233ϭD 3333,and Hill’s higher order relations were not computed.54VI.KA´RMA ´N–HOWARTH–KOLMOGOROV EQUATION The energy budget for various scales is described by the Ka´rma ´n–Howarth–Kolmogorov ͑KHK ͒equation,45⑀¯r ϭϪD LLL ϩ6␯ץD LL ץrϩZ ͑24͒for steady turbulence,4,55,56where Z (r )denotes contributions due to the external force given by Z ͑r ,t ͒ϭ͵Ϫϱt͗␦f ͑r ,t ͒•␦f ͑r ,s ͒͘dsϭ12r͵0ϱͩ115ϩsin kr ͑kr ͒3ϩ3cos kr ͑kr ͒4Ϫ3sin kr ͑kr ͒5ͪF ͑k ͒dk .͑25͒Since the external force spectrum F (k )is localized in arange of low wave numbers,the asymptotic form of Z (r )for small separations is given asZ ͑r ͒ϭ235⑀¯in k f 2r 3,k f 2ϵ͐0ϱk 2F ͑k ͒dk͐0ϱF ͑k ͒dk.͑26͒A generalized Ka´rma ´n–Howarth–Kolmogorov equation has also been derived:57–6343⑀¯r ϭϪ͑D LLL ϩ2D LTT ͒ϩ2␯ץץr͑D LL ϩ2D TT ͒ϩW ,͑27͒where W ͑r ͒ϭ4r͵0ϱͩ13ϩcos kr ͑kr ͒2Ϫsin kr͑kr ͒3ͪF ͑k ͒dk ,Ϸ215⑀¯in r 3k f 2for ͉k f r ͉Ӷ1.͑28͒Equation ͑24͒is recovered by substituting Eqs.͑22͒and ͑23͒into Eq.͑27͒.Figure 11shows the results obtained when each term of Eq.͑24͒is divided by ⑀¯r for R ␭ϭ460.Curves in which r /␩is larger than r /␩ϭ1200are not shown,because the sign of D LLL changes.A thin horizontal line indicates the Kolmog-orov value 4/5.When the separation distance decreases,the effect of the large scale forcing used in the present DNS decreases quickly,while the viscous term grows gradually.The third order longitudinal structure function D LLL quickly rises to the Kolmogorov value,remains there over the iner-tial range ͑between r /␩Ϸ50and 300͒,and then decreases.In the inertial range,the force term decreases as r 3according to Eq.͑26͒,while the viscous term increases as r ␨2Ϫ1(␨2Ͻ1)when r decreases.͓Since each term in the ﬁgure is divided by (⑀¯r ),the slope of each curve is 2and ␨2Ϫ2,respectively.͔The sum of the three terms in the right hand side of Eq.͑24͒divided by ⑀¯r is close to 4/5,the Kolmogorov value.The deviation of the sum from the 4/5law at the smallest scales is due to the slightly lower resolution of the data at these scales ͑k max ␩is close to one ͒.At larger scales greater than r /␩ϭ700,the deviation is caused by the ﬁniteness oftheFIG.11.Terms in the Ka´rma ´n–Howarth–Kolmogorov equation when R ␭ϭ460.Thin solid line:4/5.FIG.12.Kolmogorov’s 4/5law.L /␩and ␭/␩are shown for R ␭ϭ460.The maximum values of the curves are 0.665,0.771,0.781,and 0.757for R ␭ϭ125,284,381,and 460,respectively.FIG.13.Terms in the generalized Ka´rma ´n–Howarth–Kolmogorov equation for R ␭ϭ460.Thin solid line:4/3.1071Phys.Fluids,Vol.14,No.3,March 2002Velocity ﬁeld statistics in homogeneousensemble,which indicates the persistent anisotropy of the larger scales.The above ﬁndings are consistent with the cur-rent knowledge of turbulence developed since Kolmogorov,although conﬁrmation of some aspects of turbulence using actual data is new from both a numerical and experimental point of view.56,59–65It is interesting and important to observe when the Kol-mogorov 4/5law is satisﬁed as the Reynolds numberincreases.6,66–69Figure 12shows curves of ϪD LLL (r )/(⑀¯r )for various Reynolds numbers.In this ﬁgure,the 4/5law applies when the curves are horizontal.The portion of the curves in which r /␩Ͼ1200is not shown.Although there is a small but ﬁnite horizontal range when R ␭Ͼ284,the level of the plateau is still less than the Kolmogorov value.The maximum values of the curves are 0.665,0.771,0.781,and 0.757for R ␭ϭ125,284,381,and 460,respectively.The value 0.781for R ␭ϭ381is 2.5%less than 0.8.An asymptotic state is approached slowly,which is consistent with recent studies.However,the asymptote is approached faster than predicted by the theoretical estimate.66,69Theslow approach is due to the fact that D LLL (r )is the third order structure function and most positive contributions are canceled by negative ones.Thus only the slight asymmetry of the ␦u r PDF contributes to D LLL .The level of the plateau of the R ␭ϭ460curve is slightly less than the others.A higher value would be expected if the time average period used for the R ␭ϭ460run were longer.The generalized Ka´rma ´n–Howarth–Kolmogorov equa-tion Eq.͑27͒is also examined in a similar fashion.Figure 13shows each term of the equation divided by ⑀¯r ;a horizontal line indicates the 4/3law.The agreement between the present data and theory is satisfactory.The third order moment slowly approaches the Kolmogorov value 4/3,as shown in Fig.14.The maximum values of the curves of the 4/3law are 0.564,1.313,1.297,and 1.259for R ␭ϭ125,284,381,and 460,respectively.VII.STRUCTURE FUNCTIONS AND SCALING EXPONENTSThe velocity structure functions are deﬁned asS p L ͑r ͒ϭ͉͗␦u r ͉p͘,S p T ͑r ͒ϭ͉͗␦v r ͉p͘,FIG.14.Kolmogorov’s 4/3law.L /␩and ␭/␩are shown for R ␭ϭ460.The maximum values of the curves for the 4/3law are 0.564,1.313,1.297,and 1.259for R ␭ϭ125,284,381,and 460,respectively.FIG.15.Variation of the ␦u r PDF with r for R ␭ϭ381.From the outermostcurve,r n /␩ϭ2n Ϫ1dx /␩ϭ2.38ϫ2n Ϫ1,n ϭ1,...,10,where dx ϭ2␲/1024.The inertial range corresponds to n ϭ6,7,8.Dotted line:Gaussian.FIG.16.Variation of PDF for ␦v r with r at R ␭ϭ381.The classiﬁcation of curves is the same as in Fig.17.FIG.17.Convergence of the tenth order accumulated moments C 10(␦u r )at R ␭ϭ381for various separations r n /␩ϭ2.38ϫ2n Ϫ1,n ϭ1,...,10.Curves are for n ϭ1,...10from the uppermost,and the inertial range corresponds to n ϭ6,7,8.1072Phys.Fluids,Vol.14,No.3,March 2002Gotoh,Fukayama,and Nakano。

济南2024年02版小学4年级上册第13次英语第六单元真题

济南2024年02版小学4年级上册英语第六单元真题考试时间：90分钟（总分:100）A卷考试人：_________题号一二三四五总分得分一、综合题(共计100题)1、填空题：A _____ (植物工作坊) can teach valuable skills.2、选择题：What do we call a collection of stars?A. GalaxyB. UniverseC. PlanetD. Solar System3、选择题：What is the opposite of tight?A. LooseB. FirmC. StiffD. Rigid4、听力题：We play _____ (soccer) on Sundays.5、ts can grow in _____ (干燥的) areas. 填空题：Some pla6、听力题：The chemical formula for hydrochloric acid is _______.7、听力题：In a neutralization reaction, an acid and a base react to form _____ and water.8、填空题：The _____ (蜜蜂) buzzes around flowers collecting nectar.9、填空题：I want to plant a ________ to brighten my room.10、What is the name of the first artificial satellite launched into orbit?A. Sputnik 1B. Explorer 1C. Vanguard 1D. Luna 111、选择题：What do we call the first meal of the day?A. BreakfastB. LunchC. DinnerD. Snack12、填空题：The teacher plans fun _____ (活动) for the class.13、填空题：My birthday is in ______ (七月). I want to have a big ______ (派对) with cake and ______ (气球).14、听力题：The _______ of light can create various effects in art and design.15、填空题：The bison is a symbol of the _______ (美国) wilderness.16、听力题：The chemical symbol for chromium is ______.17、填空题：My sister is a talented __________ (编程员).18、What is the currency used in the United States?A. EuroB. DollarC. YenD. Pound答案:B19、Which sport involves kicking a ball?A. BasketballB. SoccerC. BaseballD. Tennis答案: B20、听力题：I like to ________ (evaluate) options carefully.21、选择题：Where do birds build their nests?A. In the waterB. In the groundC. In treesD. In caves22、What is the main purpose of a map?A. To show directionsB. To show storiesC. To show historyD. To show pictures答案：A23、填空题：中国的________ (legends) 经常包含神话与历史交织的故事。

昆明2024年10版小学4年级下册第十二次英语第4单元寒假试卷[含答案]

昆明2024年10版小学4年级下册英语第4单元寒假试卷[含答案]考试时间：80分钟（总分:120）B卷考试人：_________题号一二三四五总分得分一、综合题(共计100题)1、填空题：The butterfly has beautiful _________. (翅膀)2、What do we call the act of creating a work of art?A. ArtistryB. InspirationC. CreationD. Craftsmanship答案：C3、What is the name of the first spacecraft to fly by Saturn?A. Pioneer 10B. Voyager 1C. Voyager 2D. Cassini4、听力题：The chemical formula for iron(III) sulfate is _____.5、What do you call the art of folding paper into shapes?A. OrigamiB. CalligraphyC. PaintingD. Sculpture答案:A6、填空题：The turtle is hiding in its ______.7、填空题：A _____ (植物管理) plan can optimize resources and efforts.8、填空题：I enjoy watching ______ with my family. (我喜欢和家人一起看______。

)9、填空题：My dad loves to ________ (园艺).10、填空题：We visit the ______ (历史遗址) to understand our culture.11、听力题：The teacher is _____ the students to listen. (asking)12、听力题：The butterfly is ___ (landing) on the flower.13、填空题：A _____ (狗) can be a great friend. They love to play and go for walks.狗可以成为很好的朋友。

重庆2024年11版小学四年级下册第15次英语第4单元寒假试卷

重庆2024年11版小学四年级下册英语第4单元寒假试卷考试时间：90分钟（总分:120）B卷考试人：_________题号一二三四五总分得分一、综合题(共计100题)1、What is the name of the fruit that is yellow and curved?A. BananaB. MangoC. PearD. Peach答案: A2、听力题：We go to school _____ (everyday/never) in the morning.3、填空题：The ______ (植物的营养需求) varies greatly between species.4、Road was important for __________ (贸易). 填空题：The Silk5、听力题：The chemical formula for lithium hydroxide is ______.6、听力题：The chemical structure of DNA contains ______ bases.7、听力题：A __________ is a measurement of the angle of a slope.8、听力题：The chemical formula for gadolinium oxide is _____.9、填空题：The leaves fall from the _______ in autumn.10、What is the name of the famous waterfall in South America?A. Iguazu FallsB. Niagara FallsC. Victoria FallsD. Angel Falls答案:A. Iguazu Falls11、填空题：My brother is my best _______ who plays games with me.12、填空题：A ______ (海星) lives in the ocean and is colorful.13、填空题：A healthy garden requires good __________ (照顾).14、听力题：The bird sings ___. (sweetly)15、填空题：My best friend's name is . (我最好的朋友叫)16、听力题：My cousin is a ______. She loves to teach children.17、填空题：We need to _______ (提高) our skills.18、What do we call a large body of saltwater?a. Seab. Lakec. Riverd. Ocean答案:d19、选择题：Which animal is known for its long ears?A. CatB. DogC. RabbitD. Mouse20、听力题：Chlorine is commonly used as a _____ to disinfect water.The ________ is a special friend that I cherish.22、What is the name of the first spacecraft to fly by Jupiter?A. Pioneer 10B. Voyager 1C. Voyager 2D. Galileo23、填空题：The ant has a strong _______ (合作) ability.24、听力题：The __________ is a large area of land that is not flat.25、听力题：A polymer is a large molecule made up of many ______.26、填空题：I love to eat ________ (水果) every day.27、What is the name of the place where we can see wild animals in their natural habitat?A. ZooB. SafariC. AquariumD. Farm答案:B28、填空题：My best friend is __________ (聪明的).29、填空题：My _____ (舅舅) is visiting next month.30、What is the sound a cow makes?A. MeowB. MooC. BarkD. Roar31、What do you call the president of the United States?A. Prime MinisterB. ChancellorC. PresidentD. KingA _______ is a chemical compound formed from two different elements.33、填空题：I enjoy ______ (学习) about space.34、听力题：The city of Dublin is the capital of _______.35、填空题：My friend loves to write __________ (诗歌).36、填空题：A rabbit can hop over tall ______ (障碍).37、听力题：A reaction that occurs spontaneously is labeled as a ______ reaction.38、听力题：The sun is ______ in the sky. (shining)39、填空题：The peacock's feathers are stunning during ______ (求偶).40、填空题：The _____ (蟑螂) is not a very popular insect.41、听力题：Organic chemistry is the study of carbon-containing _____.42、听力题：I have a ______ of crayons in my art box. (set)43、填空题：The musician plays in a _____ (乐队).44、听力题：When you mix vinegar and baking soda, you get __________.45、听力题：I want to ________ a new toy.46、What is the name of the fairy tale about a girl who lost her shoe?A. RapunzelB. CinderellaC. Snow WhiteD. Little Red Riding Hood47、填空题：I love reading __________ with my family. (书籍)48、填空题：My aunt loves __________ (运动).49、What is the name of the famous river that flows through Egypt?A. NileB. AmazonC. MississippiD. Yangtze答案:A50、听力题：The ______ can move by swimming or crawling.51、填空题：在中国，农历新年是一个重要的________ (holiday)。

广州2024年10版小学6年级第十五次英语第三单元暑期作业

广州2024年10版小学6年级英语第三单元暑期作业考试时间：100分钟（总分:110）B卷考试人：_________题号一二三四五总分得分一、综合题(共计100题)1、听力题：A ______ can be very protective of its territory.2、填空题：The iguana is often seen basking in the ______ (阳光).3、填空题：Kittens are baby _______ (猫).4、What is the capital city of Mexico?A. CancunB. GuadalajaraC. Mexico CityD. Tijuana答案:C5、填空题：My friend loves __________ (探索新的领域).6、听力题：The chemical formula for -decanoic acid is ______.7、听力题：The ancient Romans used ________ to build their famous aqueducts.8、What is the name of the main character in "Harry Potter"?A. Ron WeasleyB. Hermione GrangerC. Harry PotterD. Draco Malfoy答案: CThe _______ provides a habitat for insects.10、听力题：She has a big _____ (狗).11、填空题：A vulture plays an important role in ______ (生态系统).12、听力题：The __________ is important for balancing ecosystems.13、填空题：My favorite thing to do with my friends is ______.14、听力题：My brother has a ______ (new) bike.15、s were known for their advancements in ________. 填空题：The Mayf16、听力题：The unit for measuring mass is ______.17、What do we call a large mass of ice that moves slowly?A. GlacierB. IcebergC. SnowbankD. Frost答案:A18、填空题：The ________ (农业与生态合作) benefits both fields.19、填空题：The ________ was an ancient civilization that lived along the Nile River.20、听力题：The ______ teaches us about community health.21、听力题：She is ___ her bike. (riding)22、听力题：Mars is known as the ______ Planet.My teacher is very ________ (友善). She helps us with our ________ (作业) and makes learning ________ (有趣).24、填空题：I can ______ (制定) plans for my future.25、What is the freezing point of water?A. 0°CB. 100°CC. 50°CD. 25°C答案:A26、填空题：A ________ (植物) can provide oxygen.27、听力题：I need to _____ (buy/sell) groceries.28、填空题：I like to ______ (参与) in cultural celebrations.29、听力题：Chemical bonds store ______ potential energy.30、选择题：What do you call the process of changing from a liquid to a gas?A. CondensationB. EvaporationC. FreezingD. Melting31、听力题：The _____ (ball/box) is round.32、What is the capital of the Netherlands?A. AmsterdamB. RotterdamC. The HagueD. Utrecht答案: A. Amsterdam33、听力题：She is _______ (learning) to play the guitar.34、Which fruit is known for its high vitamin C content?A. BananaB. OrangeC. AppleD. Grape答案: B35、听力题：Organic compounds always contain _____.36、听力题：A saturated solution is at _____ with respect to solute.37、听力题：The bison is a symbol of strength in the ____.38、What do you call a person who writes music?A. ComposerB. LyricistC. ArrangerD. All of the above答案: D39、听力题：The __________ is the largest ocean on Earth.40、What do we call a baby cat?A. PuppyB. KittenC. CalfD. Chick答案:B41、What is the main purpose of a map?A. To tell timeB. To show directionsC. To measure distanceD. To show weather答案:B42、Which fruit is known for keeping the doctor away?A. BananaB. OrangeC. AppleD. Grape答案:CA ______ is a symbol of freedom.44、y of Versailles ended ________ (第一次世界大战). 填空题：The Trea45、填空题：I love to _______ (写) letters to my friends.46、选择题：What do we use to cut paper?A. GlueB. ScissorsC. TapeD. Ruler47、听力题：I make _____ (晚餐) for my family.48、填空题：The cat loves to chase a _________. (激光点)49、听力题：He is reading a _____ (book/magazine) in the library.50、填空题：My puppy loves to chew on ______ (玩具).51、What do we call the art of folding paper into shapes?A. OrigamiB. CalligraphyC. PaintingD. Sculpture答案:A52、What do we call a scientist who studies the structure and function of biological molecules?A. BiochemistB. BiologistC. ChemistD. Geneticist答案: A53、填空题：A __________ (科学家社区) supports networking and mentorship for young researchers.A sloth is known for being very ______.55、填空题：I want to grow a ________ to share with my friends.56、听力题：A __________ is a type of reaction that occurs when heat is absorbed.57、填空题：The squirrel collects _________ (坚果) in the fall.58、听力题：The bird is ___ in the sky. (soaring)59、选择题：What is the main language spoken in Spain?A. EnglishB. SpanishC. FrenchD. Italian60、听力题：The chemical formula for potassium permanganate is _______.61、听力题：She is learning to ___. (sing)62、听力题：The _____ (sand/gravel) is warm.63、填空题：I enjoy making crafts with my ________ (姐妹) using recycled materials.64、填空题：My ________ (玩具) is a gift from my grandmother.65、听力题：The dog is ______ with its favorite toy. (playing)66、听力题：A _______ is a good choice for beginners.67、填空题：The rabbit is hiding in the _______ (兔子藏在_______里).My mom is a great __________ (影响者).69、community resilience framework) supports recovery from challenges. 填空题：The ____70、填空题：The bus driver, ______ (公交车司机), is very friendly.71、听力题：I want to _____ (go) to the circus.72、填空题：The rabbit is ________ (跳) in the garden.73、听力题：The ______ helps us learn about environmental issues.74、What is the capital city of Italy?A. VeniceB. RomeC. FlorenceD. Naples75、填空题：A flower has _____ (花瓣) that are colorful.76、听力题：There are five ______ in a hand. (fingers)77、选择题：What do we call a story that is passed down through generations?A. FableB. LegendC. MythD. Folktale78、What do we call the layer of the Earth that is made up of molten rock?A. CrustB. MantleC. CoreD. Lithosphere79、填空题：I like to read about _______ (我喜欢读关于_______的书).The __________ is the science of studying Earth’s features. (地球科学)81、听力题：In a decomposition reaction, a compound breaks down into two or more _____.82、听力题：A solution that contains more solute than it can theoretically hold is called a _______ solution.83、听力题：The ________ (activity) enhances learning.84、What is the most common pet?A. FishB. CatC. BirdD. Hamster答案:B85、What is the opposite of "big"?A. LargeB. HugeC. SmallD. Tall86、What is the name of the small, winged insect that produces honey?A. AntB. FlyC. BeeD. Wasp答案:C87、听力题：A mixture can be separated by ______.88、填空题：Understanding the basics of plant care can lead to a successful ______.(了解植物护理的基本知识可以导致成功的园艺。