The Spike-Type Solution of Second-Order Semilinear Differential Equation with Integral Boundary
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。
A new type of solution of the Schrodinger equation on a self-similar fractal potential
a r X i v :q u a n t -p h /0607097v 3 4 J u l 2008A new type of solution of the Schr¨o dinger equation on a self-similar fractal potentialN L Chuprikov and O V SpiridonovaTomsk State Pedagogical University,634041,Tomsk,RussiaAbstract.Scattering a quantum particle by a self-similar fractal potential on a Cantor set is investigated.We present a new type of solution of the functional equation for the transfer matrix of this potential,which was derived earlier from the Schr¨o dinger equation.PACS numbers:03.65.Ca,03.65.XpIn this Letter we address the model[1,2]of scattering a quantum particle by aself-similar fractal potential(SSFP)given on a Cantor set.This scattering problem is,perhaps,the most simple one to allow studying the influence of the scale invariance ofideal deterministic fractals on physical processes in continuous media to involve suchfractal structures.Note that the sharp attenuations,found in[1],in the spectrum of probability wavestransmitted through this ideal fractal potential have also been observed experimentally(see[3])in the transmission spectrum of electromagnetic waves propagating through areal fractal medium(a numerical modelling for the corresponding pre-fractals see in[4]).However,the problem is that the model[1,2]remans incomplete in some respects.Inthis Letter we present a new type of solution to the Schr¨o dinger equation on the SSFP,in addition to two types presented in[1,2].So,let V(x)be a SSFP on the generalized Cantor set in the interval[0,L];eachlevel of the SSFP consists of N(N≥2)SSFPs of the next level whose width isαtimes smaller than that of the former(see[2]).Let also W be a power of the SSFP,that is,its total area:W= ∞−∞V(x)dx.In line with[1,2],for a particle with a given energy E(E=¯h2k2/2m),the transfer matrix Z(φ)(φ=kL)of the SSFP must obeythe functional equationZ(φ)=Z(αφ)[D(γφ)Z(αφ)]N−1;(1)Z(φ)= q(φ)p(φ)p∗(φ)q∗(φ) ,D(φ)= e iφ00e−iφ ;q(φ)=1T(φ)exp[−iJ(φ)],p(φ)= T(φ)exp i πα(N−1);R=1−T;T(φ),J(φ)and F(φ)are,respectively,the transmissioncoefficient and phase characteristics of the SSFP;here F=0for the SSFP-barriers and F=πfor the SSFP-wells(see[2]).As it has turned out,Eq.(1)not uniquely determines the transfer matrix of the SSFP.Two different types of solutions of this equation have been presented in[1,2]. Remind that thefirst type was obtained for any values of W,αand N to characterize the SSFP.In this case,for small values ofφ, 2−J.The second type of solutions exists only for the SSFP-barriers,if W=3N¯h2T(φ)∼y(φ)∼φfor small values ofφ.In this Letter we present a new(third)type of solutions(found by Chuprikov),with a cardinally different behavior of the tunneling parameters in the asymptotic region. Namely,in this case,for small values ofφwe haveT(φ)= 1+cosh2[ω(ln(φ))]sinh2(cφ−s) −1,J(φ)=arctan{sinh[ω(ln(φ))]tanh(cφ−s)},(2)where c is a nonzero constant;ωis a nonzero real-valued function to obey the condition,ω[ln(φ)]=ω[ln(φ)+ln(α)].To extend this solution onto the whole ln(φ)-axis,one has to use the recurrence relations(18)and(19)presented in[2].As in[1,2],we display here ln(R/T)versus ln(φ).Figures(1)-(6)show this function for three values ofω-1,10and15-and three values of c-0.001,0.01and0.1.As is seen,there are three regions on the ln(φ)-axis, with a qualitatively different dependence of ln(R/T):in the left regionln ln(R/T)∼ln(2|c|)−s ln(φ);in the right oneln( R/T¨ˆ)∼−2s ln(φ),where R/T is the envelop of R(φ)/T(φ).As regards the middle region,two qualitatively different types of changing this function are possible here.Forω=1and all three values of c(seefigures(1)and(2)) ln(R/T)∼−2s ln(φ).Such behavior also occurs forω=10and c=0.001(seefigures(3)and(4)).At the same time,forω=15and all three values of c(seefigures(5)and(6)),as well as for ω=10and c=0.1(seefigures(3)and(4))we haveln(R/T)∼−2ln(φ).Note that the right region appears for all three types of solutions(see[2]).As regards the left one to follow from(2),such a behavior is a distinctive feature of the third type of solutions.It is also important to note here that for the solutions of thefirst and third types the phase path of the wave inside the out-of-barrier regions(i.e.,in the regions where the potentials are equal to zero)is infinitesimally small in comparison with the wave path in the barrier regions.This feature distinguishes these types of solutions from the second one.A simple analysis shows that the tunneling parameters are non-differentiable functions at the pointφ=0,whenωdepends onφ.Figure7shows the function R(φ)/T(φ)for this case.So,there are at least three types of the transfer matrices of the SSFP.As is seen, though all of them are nonzero only on the Cantor set,i.e.,the set of zero measure, we deal with different potentials.The Cantor set is a non-countable one,and,thus,it yet provides a much enough room for setting potentials with such different scattering properties.Of course,in this case,it is of great importance is tofind the sequences of pre-fractals to lead to the SSFPs,when the generation number of pre-fractals tends to infinity.Additionally,another open question regarding the model is that the parameters to enter the third type of solutions remain to be connected to the SSFP parameters.References[1]Chuprikov N L2000J.Phys.A:Math.Gen.334293[2]Chuprikov N L and D N Zhabin2000J.Phys.A:Math.Gen.334309[3]Takeda M W,Kirihara S,Miyamoto Y,Sakoda K and Honda K2004Phys.Rev.Lett.92093902[4]Honda K and Otobe Y2006J.Phys.A:Math.Gen.39L315Figure captions\Figure{\label{fig1}The$\ln(\phi)$-dependence of$\ln(R/T)$for$s=0.5$,$c=0.001$ and$\omega=1$;bold full curve-$N=2$,thin full curve-$N=4$;points show the asymptote$13-2s\ln(\phi)$.\Figure{\label{fig2}The$\ln(\phi)$-dependence of$\ln(R/T)$for$N=3$,$\alpha=13$ and$\omega=1$;broken curve-$c=0.1$;thin full curve-$c=0.01$;bold full curve-$c=0.001$;circles show the asymptote$10-2s\ln(\phi)$.\Figure{\label{fig3}The$\ln(\phi)$-dependence of$\ln(R/T)$for$s=0.5$,$c=0.001$ and$\omega=10$;bold full curve-$N=2$,thin full curve-$N=4$;points show the asymptote$3-2s\ln(\phi)$.\Figure{\label{fig4}The$\ln(\phi)$-dependence of$\ln(R/T)$for$N=3$,$\alpha=13$ and$\omega=10$;broken curve-$c=0.1$;thin full curve-$c=0.01$;bold full curve-$c=0.001$;points show the asymptote$8-2\ln(\phi)$;circles show the asymptote$4-2s\ln(\phi)$.\Figure{\label{fig5}The$\ln(\phi)$-dependence of$\ln(R/T)$for$s=0.5$,$c=0.001$ and$\omega=15$;bold full curve-$N=2$,thin full curve-$N=4$;points show the asymptote$5-2\ln(\phi)$.\Figure{\label{fig6}The$\ln(\phi)$-dependence of$\ln(R/T)$for$N=3$,$\alpha=13$ and$\omega=15$;broken curve-$c=0.1$;thin full curve-$c=0.01$;bold full curve-$c=0.001$;points show the asymptote$8-2\ln(\phi)$;circles show the asymptote$4-2s\ln(\phi)$.\Figure{\label{fig7}The$\ln(\phi)$-dependence of$\ln(R/T)$for$s=0.5$,$c=0.001$ and$\omega=15\left[\sin\left(2\pi\frac{\ln(\phi)}{\ln(\alpha)}\right)+1.001\right]$; bold full curve-$N=2$,thin full curve-$N=4$;points show the asymptotes$5-2\ln(\phi)$and$14-2s\ln(\phi)$.。
Analysis of the inexact Uzawa algorithm for saddle point problems
JAMES H. BRAMBLEy , JOSEPH E. PASCIAKz , AND APOSTOL T. VASSILEVx
Abstract. In this paper, we consider the so-called \inexact Uzawa" algorithm for iteratively solving block saddle point problems. Such saddle point problems arise, for example, in nite element and nite di erence discretizations of Stokes equations, the equations of elasticity and mixed nite element discretization of second order problems. We consider both the linear and nonlinear variants of the inexact Uzawa algorithm. We show that the linear method always converges as long as the preconditioners de ning the algorithm are properly scaled. Bounds for the rate of convergence are provided in terms of the rate of convergence for the preconditioned Uzawa algorithm and the reduction factor corresponding to the preconditioner for the upper left hand block. In the nonlinear case, the inexact Uzawa algorithm is shown to converge provided that the nonlinear process approximating the inverse of the upper left hand block is of su cient accuracy. Bounds for the nonlinear iteration are given in terms of this accuracy parameter and the rate of convergence of the preconditioned Uzawa algorithm. Applications to the Stokes equations and mixed nite element discretization of second order elliptic problems are discussed and nally, the results of numerical experiments involving the algorithms are presented. Key words. inde nite systems, iterative methods, preconditioners, saddle point problems, Stokes equations, Uzawa algorithm AMS(MOS) subject classi cations. 65N30, 65F10
安徽省合肥市第一中学2024-2025学年高二上学期期中考试英语试卷
安徽省合肥市第一中学2024-2025学年高二上学期期中考试英语试卷一、阅读理解Impressive exhibitions in the US worth traveling for in 2024 Here are several museum exhibitions across the USA that are worth traveling for in 2024.1. Yayoi Kusama: Infinite LoveSFMOMA, San FranciscoOn view: now through September 7For six decades now, Japanese polymath Yayoi Kusama has been exploring the concept of the “infinity room.” These meditations on perception, the universe and existence itself combine bold colors, three-dimensional forms and mirror-generated visual illusions to transport viewers to an inclusive aesthetic world. In the exhibition Yayoi Kusama: Infinite Love, they have landed in Northern California for the first time. Featured works including the brand-new Dreaming of Earth’s Sphericity, I Would Offer My Love (2023) and the famous LOVE IS CALLING (2013) will be on display at SFMOMA through next fall. Be sure to reserve advance tickets the minute they go on sale.2. Matisse and the SeaSt Louis Art Museum, St LouisOn view: February 17-May 12,2024Henri Matisse lived for decades near the Mediterranean, and a number of blues carry through his entire oeuvre (全部作品), largely inspired by the reflection of light of the water. With the artist’s Bathers with Turtle (1907–8) as a museum highlight, the exhibition travels across both Matisse’s works and the world itself, with works by this 20th-century master in various media, depicting the sea as a subject and as a theme.3. Georgia O’ Keeffe: “My New Yorks”Art Institute of Chicago, ChicagoOn view: June 2-September 24,2024This show at the Art Institute of Chicago will explore how Georgia O’ Keeffe - an artist soclosely associated with the Southwest and nature - spent her formative years in the USA’s biggest city. Before she turned her eye to flowers and desert sunsets, Georgia O’ Keeffe captured the distinctive perspectives of New York City, looking up at skyscrapers from street level and down from her 30th-floor apartment.4. Whitney Biennial 2024: Even Better Than the Real ThingWhitney Museum of American Art, New Y ork CityOn view: starting March 20,2024Some leave angry. Others emerge inspired. Yet however you react, it’s hard to forget any Whitney Biennial. Multimedia pieces and political themes are never hard to detect. Organized by Chrissi e Iles and Meg Onli, the lineup at this year’s -Biennial has yet to be announced. But whoever the participants are, their work is sure to make a statement.1．What can we learn from the artist Yayoi Kusama and his works?A．His work Infinite Love has been on display for decades.B．His works feature incorporating varied colors boldly into the works.C．Dreaming of Earth’s Sphericity was inspired by the light of water.D．Four-dimensional forms will transport viewers to the universe.2．Who is most likely to be the target audience for the last exhibition?A．people concerned with current political affairs.B．people having a passion for economy.C．people fond of pursuing old fashion.D．people enthusiastic about different reactions. 3．What do the exhibition 2 and 3 have in common?A．Both artists prefer using city landscape in the works.B．Both exhibitions need to be reserved in advance.C．Both artists’ works focus on themes concerning surroundings.D．Both artists’ works embody political themes.My husband and I fell in love when we would sit and talk in the living room of my old apartment in front of the windows drinking cups of black coffee, sometimes until sunrise. I was so extremely fortunate to have finally found that one special person.However, it was soon after our honeymoon that my husband climbed into the tomb called “the office” and buried himself in piles of paperwork and clients, and I just kept silent for fear ofturning into a complaining wife. It seemed as if overnight an invisible wall had been put up between us. He just lay beside snoring like a hibernating bear unaware of my winter.When our daughter was born, my life was centred on her and I no longer seemed to care that my husband was getting busier and spending less time at home. Somewhere between his work timetable and our home and young daughter, we were losing contact with each other. That invisible wall was now being hardened by the mortar (砂浆) of indifference.Then tragedy struck our lives, when my husband’s younger brother was killed in 2001, together with thousands of other innocent people. He was identified only by the engraving (雕刻) on the inside of his wedding ring. Attending our brother’s memorial service was an eye-opening experience for both of us. For the first time, we saw our own marriage was almost like my in-laws. At the tragic death of the youngest son they could not reach out to comfort one another. It seemed as if somewhere between the oldest son’s first tooth and the youngest son’s graduation they had lost each other.Later one night, my husband told of his fear of dying and I spoke of trying to find myself in the writings of my journal. It seemed as if each of us had been hiding our soul-searching from the other.We are slowly working toward building a bridge - not a wall, so that when we reach out to each other, we do not find a barrier we cannot pass through or retreat from the stranger on the other side.4．what can we learn about the author’s husband From the second paragraph?A．He was fully involved in his work.B．He didn’t show any affection for her.C．He preferred his work to his family D．He got tired of his nagging wife5．What does the underlined word in Paragraph 4 mean?A．The author’s husband’s brothers.B．The author’s husband’s brothers-in-law.C．The author’s husband’s parents.D．The author’s husband’s sisters-in-law. 6．Which of the following best describe the author?A．Dependent and critical.B．Sensitive and sensible.C．Sympathetic and emotional.D．Ambitious and understanding.7．What can we infer from the passage about the couple?A．Attending the memorial service worsened their relationship.B．Their brother’s death set off their reflection on marriage.C．Communication was a most effective means to break the barrier.D．The fear of dying prevented the husband from reaching out.Nobel science prizes are awarded in three areas: physics, chemistry and physiology or medicine. But occasionally some noteworthy discovery comes along that does not really fit into any of them. Similar flexibility, though in an area with far more profound consequences than ethology (行为学), has been demonstrated with regard to this year’s physics prize.Showing a sense of timeliness not always apparent in its deliberations, Sweden’s Royal Academy of Science has stretched the definition of physics to include computer science, and given its recognition to two of the pioneers of the artificial-intelligence (AI) revolution.John Hopfield of Princeton University and Geoffrey Hinton of the University of Toronto both did their crucial work in the early 1980s, at a time when computer hardware was unable to take full advantage of it. Dr Hopfield was responsible for what has become known as the Hopfield network - a type of artificial neural network that behaves like a physical structure called a spin glass, which gave the academy a fa int reason to call the field "physics". Dr Hinton’s contribution was to use an algorithm (算法) known to train neural networks.Artificial neural networks are computer programs based loosely on the way in which real; biological networks of nerve cells are believed to work. In particular, the strengths of the connections between "nodes" (结点) in such networks are plastic. Hopfield networks, in which each node is connected to every other except itself, are particularly good at learning to extract patterns from sparse (稀疏的) or noisy data.Dr Hinton’s algorithm enhances neural networks’ learning ability by letting them work, in effect, in three dimensions. Hopfield networks and their types are, in essence, two-dimensional. Though they actually exist only as simulations in software, they can be thought of as a structure of physical layers of nodes. Dr Hinton adjusted Dr Hopfield’s networks using a branch of maths called statistical mechanics to create what are known as Boltzmann machines. Boltzmann machines can be used to create systems that learn in an unsupervised manner, spotting patterns in data without having to be explicitly taught.It is, then, the activities of these two researchers which have made machine learning reallysing. AI models can now not only learn, but create. Such tools have thus gone from being able to perform highly specific tasks, such as recognizing cancerous cells in pictures of tissue samples or streamlining particle-physics data, to anything from writing essays for lazy undergraduates to running robots.8．Why does the writer mention the three areas of Nobel science prizes?A．To inform readers of the specific information.B．To introduce the flexibility of this years’ Nobel physics prize.C．To share with readers the importance of the Nobel prizes.D．To highlight the critical role physics plays in the world.9．What can be the evidence that the two researcher’s activities can be called “physics”?A．The Hopfield Networks are two-dimensional.B．The nodes in the Hopfield Network connect each other.C．The Hopfield Network functions in a similar way to a spin glass.D．The Hopfield Network can extract patterns using a little data.10．How did Dr Hinton strengthen neural networks’ learning ability?A．He used special physical principles.B．He changed the function of the networks.C．He thought of a structure suitable for the networks.D．He made use of maths to transform their ways of working11．What can be the main idea of the passage?A．AI neural networks can be widely used.B．Two researchers will be awarded the Nobel Physics Prize.C．AI researchers have received the Nobel Prize for Physics.D．Physiology and medicine researchers are common in the Nobel Prize winners.The term parasocial interaction (虚拟社交) was introduced in the 1950s by the social scientists Donald Horton and R. Richard Wohl. It was the early days of home television, and they were seeing people form a close connection with actors who were appearing virtually in their home. Today, the definition is much broader. After all, actors, singers, comedians, athletes, and countless other celebrities are available to us in more ways than ever before. Forming parasocialbonds has never been easier.Psychologists document cases of parasocial relationships that can go much deeper, with severe consequences. Scholars note parasocial bonds range from casual talk about stars to intense emotions, to uncontrollable behavior and fantasies. At the deepest level, the parasocial relationship can be dangerous, such as when a fan loses touch with reality and secretly follows a star. It can also lead to confusion about one’s own identity, particularly in adolescents who are still forming their sense of self, as they may model themselves on the media figures with whom they have parasocial relationships.In 2021, two psychologists from York University, in Canada, found that forming parasocial bonds was strongly related to avoidant attachment. That is, people who tended to push others away in their day-to-day lives were more likely to relate to fictional characters. You can easily see how parasocial relationships could be a replacement when one finds real-life attachment difficult. This could start a feedback cycle, in which avoiding close relationships stimulates parasocial bonding, which in turn leads to reduced interactions with real-life family and friends as the fans spends their time and energy on someone who doesn’t know they exist.My purpose here is not to say that parasocial interactions are always bad for you, or even abnormal. Rather, it is to suggest that heavy parasocial bonding might be a signal that you are crowding out the real people who can give you the love you truly need. One way to address this is to get some more distance from your fictional friends, thus pausing the feedback cycle and giving yourself more space to pursue in-person connection.12．How has parasocial interaction changed according to Paragraph 1?A．It has become more accessible.B．It has affected more celebrities.C．It has lost much of its significance.D．It has turned into a two-way process. 13．What is Paragraph 2 mainly about?A．Reasons behind celebrity following.B．Origins of dangerous relationships.C．Different types of parasocial relationships.D．Potential harm of parasocialrelationships.14．Which of the following can lead to parasocial relationships?A．Socializing with strangers.B．Having strong family support.C．Participating in group activities.D．Struggling with relationships in reality.15．What might the author suggest for those with heavy parasocial relationships?A．Meeting fictional friends in real life.B．Seeking guidance from professionals.C．Hanging out more with real friends.D．Creating more space for being alone.We are overwhelmed by an unprecedented volume of information. 16 if we don’t actively engage with it.In order to stay focused and retain more information, it’s important to be highly engaged with the content. 17 It mostly relies on critical thinking. Active reading transforms passive absorption into an interactive, analytical process. There are many active reading strategies, but here are some of the most immediately useful.Understand the author’s purpose. 18 Take a few minutes to read the introduction or any other material available to become aware of the reason and intent of writing.Adjust your reading rate. Instead of using a constant rate, adapt yourself to the content you’re reading. 19 , and speeding up when it’s information you are already familiar with.Annotate the content. Taking notes is a great way to stay engaged with the content. Use the margins to write ideas that pop into your mind when reading something.Paraphrase. Whenever a new concept seems a bit more complex to grasp, stop reading and try to paraphrase it using your own words. This will force you to assess your level of understanding.Organize the information visually. Map the content into a graphic to better visualize it and make it your own. You can craft a simple mind map, or be creative with collages and other forms of visual thinking.Evaluate the content. Every so often, take a step back and think critically about what you’re reading. 20Consult a reference. Whenever you’ re in doubt, use a dictionary or another external reference to make sure you understand a new concept or an unfamiliar word’s meaning and have all the necessary background information.Summarize the ideas. Once you’ re done reading a book, sit down and write your own summary. Get bonus points if you publish it online to learn in public and get feedback and additional perspectives from other readers.Active reading will help you make the most of the time you spend reading books and blog posts by ensuring you retain more of the relevant content and can apply it in your day-to-day life and work.A．This means slowing down to comprehend better new or more complex information. B．Yet, research suggests that we forget up to 70% of new information within 24 hours.C．It matters for you to assess what you read.D．Active reading basically means reading something with the determination to understand, evaluate, and remember relevant aspects of what you read.E．Is it well structured, are there gaps in the argument, does the author sound biased?F．Is the goal of the author to inform, entertain, or advertise their product or services?G．Our life is packed with varied information.二、完形填空Michael Surrell and his wife had just parked the car when they got a call from their daughter, “The house next door is on fire!” He immediately went to 21 and saw an old woman cried. “The baby is inside!” “The baby” was 8-year-old Tiara Roberts, the woman’s 22 .Though the fire department had been called, Surrell 23 rushed into the burning house. The thick 24 caused him to stumble blindly around and made it impossible to 25 . After a few minutes in the smoke-filled house, he moved outside to 26 his breath.“Where is Tiara?” he asked 27 .“The second floor,” her grandma shouted back.Taking a deep breath, Surrell went in a second time. Because the house had a 28 layout to his, he found the stairs 29 and made it to the second floor.But the darkness was overwhelming. All he could feel was the crackling and popping of burning wood. Then a soft but 30 moan emerged. He crawled toward the sound, feeling around for any 31 of the little girl. Finally, he 32 something. He scooped Tiara into his arms, 33 through the smoke.Fortunately, Surrell managed to help Tiara out; she was 34 from the hospitalafter a few days. However, the fire worsened Surrell’s pulmonary (肺的) condition, which he suffered before, and he feels the effects even two years later. “It’s a small 35 to pay,” he says. “I would do it again without a second thought.”21．A．stimulate B．witness C．investigate D．innovate 22．A．niece B．granddaughter C．cousin D．daughter 23．A．consciously B．passionately C．instantly D．occasionally 24．A．mist B．smoke C．dust D．smog 25．A．escape B．distinguish C．see D．breathe 26．A．hold B．save C．waste D．catch 27．A．randomly B．cautiously C．nervously D．desperately 28．A．opposite B．similar C．different D．striking 29．A．mysteriously B．thrillingly C．threateningly D．effortlessly 30．A．distinct B．loud C．massive D．sharp 31．A．sense B．symbol C．sound D．sign 32．A．touched B．found C．explored D．got 33．A．running B．breaking C．struggling D．going 34．A．rescued B．composed C．suspended D．released 35．A．fee B．bill C．check D．price三、语法填空阅读下面短文，在空白处填入一个适当的单词或括号内单词的正确形式。
THE EXISTENCE OF POSITIVE SOLUTIONS TO BOUNDARY VALUE PROBLEM OF SECOND ORDER DIFFERENTIAL EQUAT
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No.1
J. Dai, et c., P OSITIVE SO LUTIONS TO BVP
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2
Main Results
Th e fi xe d p oint in de x h as the followin g p rop erties.
Lemm a 1 Let K be a closed con vex set in a Banach space E , D be a boun ded open set such t hat D k := D ∩ K �= ∅ an d T : D k → K be a completely con tin uou s map . S uppose that x �= T x for all x ∈ ∂ D k . Then we obtai n the follow ing conclu sion s: (D1 )( Existen ce) if i ( T, D k , K ) �= 0 , th en T has a fixed poin t in D k ; (D2 )( Normalizat ion) if u 0 ∈ D k , t hen i( b u , D k , K ) = 1 , where b u( x ) = u 0 , for x ∈ D k ; (D3 )( Homot opy ) let h : [0 , 1] × D k → K be a complet ely cont inu ous map such that x �= h (t, x ) for x ∈ ∂ D k an d t ∈ [0 , 1] , then i( h (0 , · ) , D k , K ) = i (h (1 , · ) , D k , K ) . For fu rthe r in form ation on the classical fi xe d p oint in de x for comp letely c ontinuou s map s, see [7]. Now we state an d p rove a ne w fixe d p oint the orem in C 1 [0, 1] sp ace usin g th e p rope rtie s of th e fi xe d p oint in de x. Theorem 1 Let E = C 1 [0 , 1] be en dow ed w ith t he no rm � x� = max{� x � 0 , � x� � 0 } , wh ere � x � 0 = max | x ( t) | , an d P = { x ∈ E | x (t ) ≥ 0 , t ∈ [0 , 1] } be a co ne in E . S uppose
MATH1003_Integral Calculus and Modelling_Summer School 2013_tut13
The University of SydneyMath1003Integral Calculus and ModellingAssumed KnowledgeObjectives(12a)To be able to rewrite two coupledfirst-order differential equations as a single second-order differential equation.(12b)To be able to sketch the solutions of second-order differential equations with constant coefficients.Preparatory Questions1.Two species,struggling to compete against each other in the same environment,havepopulations at time t of x(t)and y(t),satisfying the equationsx′(t)=3x(t)−4y(t),y′(t)=−2x(t)+y(t).Find the second-order differential equation satisfied by x(t).Practice Questions2.Find x(t)and y(t)in Preparatory Question1.Solution For x′′−4x′−5x=0the auxiliary equation is m2−4m−5=0with roots m=5or m=−1.So x=Ae5t+Be−t.Hence x′=5Ae5t−Be−t,and from x′=3x−4y we obtainy=1[3(Ae5t+Be−t)−(5Ae5t−Be−t)]4=−1(i)Show that x′′(t)−2x′(t)+5x(t)=0.(ii)Find x(t)if x(0)=100and x′(0)=100.(Take t=0to be the time at which monitoring of the population sizes begins.)(iii)Hencefind y(t).(iv)Sketch x(t)and y(t)and then X(t)and Y(t)as a function of t.What does the model predict will happen to the original populations?Solution(i)x′=3x−2yx′′=3x′−2y′=3x′−2(4x−y)=3x′−8x+2y=3x′−8x+(3x−x′)=2x′−5x.That is,x′′−2x′+5x=0.(ii)The auxiliary equation is m2−2m+5=0,with roots m=1+2i and m=1−2i.Therefore x=e t(A cos2t+B sin2t).If x=100when t=0,then A=100and x=e t(100cos2t+B sin2t).Hence,x′=e t(−200sin2t+2B cos2t)+e t(100cos2t+B sin2t).When t=0, x′=2B+100=100,and so B=0.Therefore x=100e t cos2t.(iii)y(t)=1(300e t cos2t−100e t cos2t+200e t sin2t)2=100e t(cos2t+sin2t).(iv)x,X,YThe populations will oscillate with increasingly large swings.Eventually theswings will get so big that X(t)=3000+x(t)will be zero.Once this hap-pens the model,and hence the solutions,will no longer be valid.The prey willthen become extinct.The predator will then presumably also become extinctunless it has an alternative source of food.More Questions4.Find the general solution of the following system of equations:dxdt=3x−y.Solution We can rearrange this system to obtain one second-order differential equation.Differentiate thefirst equation with respect to t to obtain d2xdt+dydt from the second equation then givesd2xdt+3x−y.Finally,we can use thefirst equation of the system to replace y bydxdt2=dxdt−x .Collecting terms on the left-hand-side then gives the second-order linear differential equationd2xdt −x=2Ae2t−2Be−2t−Ae2t−Be−2t=Ae2t−3Be−2t.So the general solution of the system isx=Ae2t+Be−2t,y=Ae2t−3Be−2twhere A and B are arbitrary constants.5.Find the general solution of the pair of differential equationsdxdt=2y,byfirst solving the second equation and then substituting into thefirst.(There are two equations,so you should have two arbitrary constants of integration at the end.) Find the particular solution satisfying the initial conditions x=1,y=2when t=0.Solution The second equation is separable and has the general solution ln|y|=2t+C, or y=Ae2t.Substitute this result into thefirst equation,and rearrange to obtain the equation dx/dt−5x=−3Ae2t.This is afirst-order linear equation,and has the general solution x=Ae2t+Be5t.If y=2when t=0,we have A=2.If x=1when t=0,we have1=A+B,and so B=−1.The required particular solution is therefore x=2e2t−e5t,y=2e2t.6.(i )For each of the following systems of differential equations for x (t )and y (t ),findan equivalent second-order differential equation.(You do not need to find any solutions.)(a)x ′=y +4x,y ′=6x .(b)x ′=y,y ′=x +5y .(ii )Eliminate y (t )from the following system to obtain a nonlinear second-order equa-tion for x (t ).(You do not need to find any solutions.)x ′=y,y ′=x −xy +3tSolution(i )(a)Differentiating the first equation gives x ′′=y ′+4x ′.We can then replace y ′by 6x to obtain x ′′=6x +4x ′.Thus we obtain the equivalent second-order equation x ′′−4x ′−6x =0.Note that elimination of x (t )gives the same equation for y (t ).(b)Differentiating the second equation gives y ′′=x ′+5y ′.We can then replacex ′by y to obtain y ′′=y +5y ′.Thus we obtain the equivalent second-order equation y ′′−5y ′−y =0.Note that elimination of y (t )gives the same equation for x (t ).(ii )Differentiating the first equation gives x ′′=y ′.The second equation tells usy ′=x −xy +3t ,so we obtain x ′′=x −xy +3t .We can then replace y by x ′to get x ′′=x −xx ′+3t .This then gives the nonlinear second-order differential equation x ′′+xx ′−x =3t .7.Find whether each of the following first-order equations is separable,linear or neither.If the equation is separable or linear find its general solution.(i )dyx(ii )dyx (1+x )(iii )dyx −cos y(iv )dy2(1+x )ySolution(i )The equation is not separable,however we can rearrange the equation asdyx=x,which we now recognise as first-order linear.Multiplying by the integrating factore(1/x )dx=e ln x =x.we obtainx dydx(xy )=x 2.Integrating both sides with respect to x ,we getxy =x 33+C(ii)The equation is linear(for y a function of x)dyx(1+x)=−1dx =y−1x(1+x) .Separating and integrating,dy x(1+x)= 11+x dx(by partial fractions). Soln|y−1|=ln|x|−ln|1+x|+C=ln x1+x,where A=±e C.So the general solution is y=1+Axdy −2yxy2−1,and separabledy2(1+x)y= y2−11+x .Separating and integrating:2y1+x·This gives ln|y2−1|=ln|1+x|+C,and so y2−1=A(1+x),where A=±e C.Thus we obtain the general solution y2=1+A(1+x).Answers to Preparatory Questions1.Differentiating x′=3x−4y gives x′′=3x′−4y′.But y′=−2x+y,sox′′=3x′−4(−2x+y)=3x′+8x−4y=3x′+8x+(x′−3x)(since−4y=x′−3x)=4x′+5x.That is,x′′−4x′−5x=0.。
2024-2025学年四川省成都市第七中学高二上学期期中考试英语试卷
2024-2025学年四川省成都市第七中学高二上学期期中考试英语试卷Are you dreaming of overcoming Mount Kilimanjaro, but not sure which route to take? Here’s an overview of four Kilimanjaro routes for you.Marangu RouteThe Marangu Route, also known as the “Coca-Cola Route”, is distinguished by its cottage accommodations and is a popular choice for climbers aiming to reach the top of Kilimanjaro.Providing a more comfortable option compared to camping, these simple cottages offer basic facilities and can be a welcoming sight after a day of challenging hiking.Machame RouteThe Machame Route, often referred to as the “Whiskey Route”, is famous for its splendid landscapes and challenging summit (顶峰) night that tests climbers’determination and tolerance.The night before the summit push, is a laborious test, characterized by sharp, rocky terrain (地形) and freezing temperatures, where climbers rely on their mental strength and physical preparedness to overcome the final barriers to Uhuru Peak.Lemosho RouteThe Lemosho Route offers breathtaking scenic views and owns one of the highest summit success rates among the Kilimanjaro routes, making it a favorite among climbers.What sets the Lemosho Route apart is its gradual ascent (上升) profile, allowing climbers to accustom effectively and increase their chances of reaching the summit successfully.Rongai RouteThe Rongai Rout e provides a quiet hiking experience, allowing climbers to adapt gradually while impressing themse lves with the untouched wilderness of Kilimanjaro’s northern side.Adaptation becomes more manageable due to the route’s gentle ascent, allowing climbers to adjust to the increasing height comfortably. The unique advantage of this path is its relatively lower traffic, providing a peaceful experience in harmony with nature’s patterns.1. What makes Marangu Route special?A．The accommodations. B．The free Coca-Cola.C．The camping sight. D．The challenging hiking.2. What is most required when you choose Machame Route?A．Climbing equipment. B．Help of Whiskey.C．Teamwork of climbers. D．Strong willpower.3. What do the last two routes have in common?A．They have lower traffic. B．They offer vast wilderness.C．They are easier to adapt to. D．They are planned for the old.I was sitting in a chemistry lab class during my first year of university, nervous about the experiment we were to perform. I grabbed a pipette and, as I feared, my hand started to shake. The experience was disheartening. I was hoping to pursue a career in science, but I started to wonder whether that would be possible. I thought my dreams had crashed to the ground.I was a boy born with brain damage. My family managed to find good doctors where we lived, in Leningrad (now St. Petersburg), Russia, and I took part in clinical trials testing new treatments. Shortly after my first birthday, I started walking and it became clear that my intelligence function was unaffected. So, in some sense, I was lucky. Still, I couldn’t do some things growing up. Both hands shook, especially when I was nervous or embarrassed. My left hand was much worse than my right, so I learned to write and do simple tasks with my right hand, but it wasn’t easy to do anything precisely.As a teenager, I faced a lot of bullyi ng at school. Feeling alone, I joined a study group called “The natural world”. I thought that getting into the world of animals would keep me away from people. That’s how I came into the field of biology. At university, I enjoyed the lectures in my scienc e classes. Many lab tasks proved impossible, however. As I struggled with my mood, I read a book about depression. From then on, the physiology of mental disorders became my scientific passion. I looked into what was being done locally and was excited to discover a lab that did behavioral experiments on rats to study depression.At the end of my second year, I approached the professor of the lab to see whether I could work with her. I was afraid to admit I couldn’t do some lab tasks. To my relief, she was c ompletely supportive. She set me to work performing behavioral experiments for others in the lab with the help of colleagues. I loved the supportive atmosphere and stayed there to complete my master’s and Ph. D.I’ve come to realize that my hands aren’t th e barrier I thought they were. By making use of my abilities and working as part of a team, I’ve been able to follow my passions. I’ve also realized that there’s much more to being a scientist than performing the physical labor. I may not collect all the d ata in my papers, but I’m fully capable of designing experiments and interpreting results, which, to me, is the most exciting part of science.4. What was the author’s dream?A．To live a normal life. B．To become a scientist.C．To get a master’s degree.D．To recover from depression.5. Why did the author say he was lucky in Paragraph 2?A．Because he didn’t lose the function of both hands.B．Because he learned how to walk at the age of one.C．Because his family could afford to see good doctors.D．Becau se his brain damage didn’t affect his intelligence.6. What can you learn from the passage?A．The team in the lab urged him to further his study.B．The author finally finished the lab tasks on his own.C．The author’s experience inspired him to help others.D．The author’s loneliness led him to the world of biology.7. What message does the author want to express?A．Loving yourself makes a difference.B．Opportunity follows prepared people.C．A bright future begins with a small dream.D．The sun somehow shines through the storm.Faced with an attempt by a new chatbot to imitate (模仿) his own songs, the musician Nick Cave delivered a strong response: it was “an absolutely horrible attempt”. He understood that AI was in its babyhood, but could only co nclude that the true horror might be that “it will forever be in its babyhood”. While a robot might one day be able to create a song, he wrote, it would never grow beyond “a kind of burlesque (滑稽的模仿)”, because robots-being composed of data-are unable to suffer, while songs arise out of suffering.Fans of Cave and his band will agree that his music is inimitable, but that doesn’t mean they would necessarily be able to tell the difference. A few days before Cave’s remarks, experts were asked to distinguish between four genuine artworks and their AI imitations. Their conclusions were wrong five times out of 12, and they were only unitedly right in one of the four picture comparisons. These are party games, but they point to an unfolding challenge that must be managed as a matter of urgency because, like it or not, AI art is upon us. The arrival of the human-impersonating ChatGPT might have increased general awareness, but artists across a wide range of disciplines are already exploring its potential, with the da ncer Wayne McGregor and London’s Young Vic Theatre among those who have created AI-based works.A strongly-worded report from Communications and Digital Committee (CDC) issued a wake-up call to the government, urging it to raise its game in educating future generations of tech-savvy professionals, and tackling key regulatory challenges. These included reviewing reforms to intellectual property law, strengthening the rights of performers and artists, and taking action to support the creative sector in adapting to the disturbances caused by swift and stormy technological change.While developing AI is important, it should not be pursued at all costs, the CDC stressed. It deplored the failure of the Department for Digital, Culture, and Media to offer a defence against proposed changes to intellectual property law that would give copyright exemption (版权豁免) to any work, anywhere in the world, involving AI text and data mining.The challenges of AI are both philosophical, as Cave suggested, and practical. They will unfold over the short and long term. State-of-the-art creative industries have a key role to play in shaping andexploring the philosophical ones, but they must have the practical help they require to survive and be successful. They need it now.8. Why does the author mention the four picture comparisons in Paragraph 2?A．To stress the similarities between AI art and human art.B．To argue that human art will be replaced by AI art.C．To prove AI is stretching the boundaries of art.D．To imply AI art cannot be underestimated.9. What does the underlined word “deplored” in Paragraph 5 probably mean?A．Clearly analyzed. B．Bravely suffered.C．Strongly criticized. D．Accurately perceived.10. What can be inferred from the passage?A．Some artists see AI as a tool even though it is a threat.B．Creative industries are responsible for causing the AI problem.C．Tech professionals need more training to better understand AI art.D．The quality of AI art dismisses concerns about intellectual property.11. Which would be the best title for the passage?A．The Creative Thief: AI Makes Perfect ArtB．AI in Art: A Battle That Must Be FoughtC．Threat or Opportunity: The Impact of AI on ArtD．The Rise of AI Art: What It Means to Human ArtistsThat dinosaurs ate the mammals that ran beneath their feet is not in doubt. Now an extraordinary fossil newly described in Scientific Reports, unearthed by a team led by Gang Han at Hainan Vocational University of Science and Technology in China, shows that sometimes the tables were turned.The fossil - dated to about 125 million years ago, during the Cretaceous period-was formed when a flow of boiling volcanic mud swallowed two animals seemingly locked in a life-and-death fight. The one on top is a mammal. The other animal is a herbivorous species closely related to the Triceratops (三角恐龙). Animal interactions such as this are exceptionally rare in the fossil record.One possibility is that the mammal was eating something already dead, rather than hunting live prey. These days it is uncommon for small mammals to attack much larger animals. But it is not unheard of. And Dr. Han and his colleagues point out that those mammals which eat dead bodies typically leave tooth marks all over the bones of the animals. The dinosaur’s rem ains show no such marks.There is also a chance the fossil could be a fake. More and more convincing fakes have emerged, as this one did — though Dr. Han and his colleagues argue that the complex and tangled nature of the skeletons (骨骼) makes that unlikely, too.Assuming it is genuine, the discovery serves as a reminder that not all dinosaurs were enormous during the Cretaceous and not all mammals were tiny. From nose to tail, the dinosaur is just 1.2 meters long. The mammal is a bit under half a meter in length. Despite being half the size, the mammal has one pa w firmly wrapped around one of its prey’s limbs, and another pulling on its jaw. It is biting down on the dinosaur’s chest, and has ripped off two of its ribs. Before they were interrupted, it seems that the mammal was winning.12. What does the author imply in Paragraph 1?A．The fittest survives. B．The hunters become hunted.C．All dinosaurs ate mammals. D．The truth always comes to light.13. Why does the author mention the “tooth mark” in Paragraph 3?A．To prove the fossil was fake.B．To show the forming of the fossil.C．To illustrate the process of hunting.D．To suggest the dinosaur was hunted alive.14. What makes Dr. Han think the fossil is genuine?A．The size of the fossil.B．The absence of fake fossils.C．The agreement of the opinions.D．The complexity of the skeletons.15. What is the function of the last paragraph?A．It offers a likely cause.B．It highlights a solution.C．It justifies the discovery.D．It provides a new discovery.If anyone had told me three years ago that I would be spending most of my weekends camping, I would have laughed heartily. Campers, in my eyes, were people who enjoyed insect bites, ill-cooked meals, and uncomfortable sleeping bags. They had nothing in common with me. 16The friends who introduced me to camping thought that it meant to be a pioneer. 17 We slept in a tent, cooked over an open fire, and walked a long distance to take the shower and use the bathroom.This brief visit with Mother Nature cost me two days off from work, recovering from a bad case of sunburn and the doctor’s bill for my son’s food poisoning.I was, nevertheless, talked into going on another fun-filled holiday in the wilderness. 18 Instead, we had a pop-up camper with comfortable beds and an air conditioner. My nature-loving friends had remembered to bring all the necessities of life.19 We have done a lot of it since. Recently, we bought a twenty-eight-foot travel trailer complete with a bathroom and a built-in TV set. There is a separate bedroom, a modern kitchen with a refrigerator. The trailer even has matching carpet and curtains.20 It must be true that sooner or later, everyone finds his or her way back to nature. I recommend that you find your way in style.I had always been warmly praised for my basketball shooting ability when I was in high school. But when I went to Ohio State, I discovered that everyone on the team was _________ in his hometown.To win a starting job on the team, I had figured I would have to _________ the coach with my shooting ability. But it turned out that the team was _________ full of attacking players and what it needed was someone to _________ on defense. Unwillingly, I decided to take that role, but didn’t expect the _________ was to make all the difference to me later.One day, when we were _________ a game against the Bucks, I was called out by the coach, Milwaukee, who gave me the _________ to guard our court. Though feeling a bit _________, I accepted. Throughout the game, I continued playing the defense role.Then near the end of the game, in a(n) _________ to widen the score gap, Milwaukee gave me a precious __________. He asked me to organize an attack. Running to the center of the court, I__________ the ball. It was a __________ three-point play and the championship was ours.Standing there in that circle of cheering audience, I came to __________ the importance of teamwork. Just as Milwaukee said, “__________ teams maybe have one or two players who stand out; good teams have five who work together. It is amazing what can be achieved when no one cares who gets his own __________.”21.A．confident B．excellent C．famous D．inexpert22.A．show B．provide C．impress D．depress 23.A．naturally B．already C．powerfully D．officially 24.A．focus B．call C．wait D．rely25.A．decision B．approach C．comment D．solution 26.A．checking in B．packing up C．applying for D．preparing for 27.A．partner B．credit C．responsibility D．movement 28.A．disappointed B．satisfied C．awkward D．cheerful 29.A．goal B．order C．effort D．exchange 30.A．lecture B．opportunity C．request D．strategy 31.A．kicked B．held C．caught D．shot32.A．key B．suitable C．typical D．formal33.A．explore B．realize C．organize D．view34.A．Formal B．Professional C．Poor D．Best35.A．present B．improvement C．power D．honor阅读下面短文，在空白处填入1个适当的单词或括号内单词的正确形式。
小学下册第13次英语第六单元真题(含答案)
小学下册英语第六单元真题(含答案)英语试题一、综合题(本题有50小题，每小题1分，共100分.每小题不选、错误，均不给分)1 The __________ is a famous river in India. (恒河)2 The chemical symbol for nitrogen is _______.3 What do we call the movement of the Earth around the sun?A. RotationB. RevolutionC. CirculationD. Orbit4 The _____ (兔子) is known for its long ears and speed.5 A ______ is a small creature that can be found in gardens.6 The ancient Greeks contributed to the fields of _____ and math.7 I enjoy taking ______ (课外活动) to explore my interests beyond school subjects.8 Which vegetable is orange and long?A. PotatoB. CarrotC. BroccoliD. Tomato答案:B9 The ability of a substance to react with oxygen is called ______.10 Carbon dioxide is produced during ______ respiration.11 We have a ______ (丰富的) curriculum that includes arts.12 What is the term for a person who studies animals?A. ZoologistB. BotanistC. BiologistD. Geologist答案: A13 She is a good ________.14 __________ (化学反应速率) can change based on conditions like temperature.15 How many wheels does a car have?A. TwoB. ThreeC. FourD. Five16 Which animal can fly?A. CatB. DogC. BirdD. Elephant答案：C17 A ______ is a geological feature that can provide insights into history.18 What is the main ingredient in bread?A. WaterB. FlourD. Salt19 When you push an object, you apply ______.20 What do you call a place where animals are kept for public display?A. ParkB. ZooC. FarmD. Aquarium答案: B21 What is the capital of South Africa?A. JohannesburgB. Cape TownC. PretoriaD. Durban答案: C22 I saw a _______ (小恐龙) at the museum.23 The continent where Egypt is located is ________ (非洲).24 I like to __________ (动词) my __________ (玩具名) at night.25 _____ (aloe) is great for soothing skin.26 My grandma loves to _______ (动词) with her friends. 她觉得这个活动很 _______ (形容词).27 My uncle loves to __________. (钓鱼)28 What do we call the seasonal change when trees lose their leaves?A. SpringB. SummerD. Winter答案: C29 What do we call the study of weather?a. Biologyb. Meteorologyc. Geographyd. Astronomy答案:B30 I can ______ (修理) my bike.31 The boiling point of water is __________ degrees Celsius.32 _______ can be used for cooking and seasoning.33 An emulsion is a mixture of two ________ that do not usually mix.34 The turtle is a symbol of _______ (长寿).35 What do we call a journey by train?A. TripB. RideC. Train rideD. Travel36 A solution with a pH of is very ______.37 The __________ is a famous city known for its historical sites. (耶路撒冷)38 I like ________ (exploring) new ideas.39 The bird is _____ (chirping/singing).40 Every evening, I read a story to my toy ____. (玩具名称)41 We have a _____ (晚会) planned for the end of the year.42 s can live for __________ (几百年). Some tre43 What is the capital of Japan?A. BeijingB. SeoulC. TokyoD. Bangkok答案：C44 The garden is alive with _______ and buzzing bees.45 What do we call a story that is told through dialogue and action?A. NovelB. PlayC. PoemD. Short Story46 What do we use to measure distance?A. ScissorsB. RulerC. ClockD. Brush47 What is the main ingredient in bread?A. WaterB. FlourC. YeastD. All of the above48 In my family, we believe it’s important to call each other ______. (在我的家庭中，我们相信称呼彼此为____是重要的。
好参考Second-Order Sliding-Mode Observer for Mechanical Systems
model-based observers are usually restricted to the cases when the model is exactly known; • high-gain differentiators [2] are not exact with any fixed finite gain and feature the peaking effect with high gains: The maximal output value during the transient grows infinitely as the gains tend to infinity (see, for example, [3], [5], [12], [15], and [16]). The sliding mode observers are widely used due to the finite-time convergence, robustness with respect to uncertainties and the possibility of uncertainty estimation (see, for example, the bibliography in the recent tutorials [3], [5], and [12]). A new generation of observers based on the second-order sliding-mode algorithms has been recently developed. In particular, asymptotic observers [13] and the asymptotic observer for systems with Coulomb friction [1], [11] were designed based on the second-order sliding-mode. These observers require the proof of a separation principle theorem due to the asymptotic convergence of the estimated values to the real ones. A robust exact differentiator [9] featuring finite-time convergence was designed as an application of the super-twisting algorithm [8]. Its implementation does not need the separation principle to be proved. These differentiators were, for example, successfully applied in [14], [4], and [10]. A new differentiator [7] was developed, based on it. Straightforward application of such a differentiator does not benefit from the knowledge of a mathematical model of the process. If such a model is known, or the system parameters and uncertainties can be estimated (which is common for the case of mechanical systems with Coulomb friction), it is reasonable to design a system-specific observer. An observer is proposed in this paper, which reconstructs the velocity from the position measurements, using the modification of the second-order sliding-mode super-twisting algorithm [8] with finitetime convergence. The separation principle theorem is trivial in this case, and the observer can be designed separately from the controller. Only partial knowledge of the system model is required. The discrete version of the of the proposed observer is considered, and the corresponding accuracy of the proposed observer is estimated. II. PROBLEM STATEMENT The general model of second-order mechanical systems has the form
从新的角度阐释物有本末英语作文
从新的角度阐释物有本末英语作文Rethinking the Dichotomy of Roots and Tips: A Holistic Perspective on the Interdependence of Beginnings and Endings.In the tapestry of life, every object, entity, and concept exists within an intricate web of relationships, where beginnings and endings are inextricably intertwined. The ancient Chinese adage "物有本末" encapsulates this profound interconnectedness, suggesting that all things possess both a root (本) and a tip (末). While this maxim has traditionally been interpreted as a hierarchical relationship, where the root represents the foundation and the tip the outgrowth, a more holistic understanding reveals a dynamic and fluid interplay between these two poles.From a scientific standpoint, the concept of beginnings and endings is often approached through the lens of thermodynamics and entropy. The first law of thermodynamicsdictates that energy cannot be created or destroyed, only transformed. Thus, every process or event is a manifestation of energy flowing through a system. The second law of thermodynamics introduces the concept of entropy, which measures the degree of disorder or randomness in a system. As a system undergoes change, its entropy tends to increase.In the context of object permanence, the root can be seen as the initial state of an object's existence, while the tip represents its current state. As time progresses, the object's original form may undergo various transformations, but its essence remains unaltered. The root, in this sense, provides a stable foundation for the object's identity, while the tip embodies its evolutionary journey.The root-tip dichotomy also finds resonance in the realm of human experience. Our lives are marked by countless beginnings and endings: the birth of a child, the graduation from school, the commencement of a new career, the end of a relationship, and ultimately, the conclusionof our own mortal journey. Each of these milestones represents a transition, a shifting of perspectives, and a redefinition of our selfhood.As we navigate these transitions, it is essential to acknowledge the interconnectedness of beginnings and endings. The root of our past experiences shapes our present and future aspirations, while the tips of our current endeavors hold the seeds of our potential growth. By embracing this holistic view, we cultivate a sense of continuity and purpose amidst the flux of life.Furthermore, the root-tip relationship can be extended to the broader concept of interconnectedness within the natural world. Every organism, from the smallest microbe to the largest whale, is part of a vast ecological tapestry. The roots of one species provide sustenance for others, while the tips of their branches create habitats and foster biodiversity. In this intricate web of life, each element plays a vital role in maintaining the delicate balance of the ecosystem.In conclusion, the adage "物有本末" offers a profound insight into the interconnectedness of beginnings and endings. By transcending the traditional hierarchical view, we recognize the dynamic and fluid interplay between these two poles. Whether in the realm of physics, biology, or human experience, the root-tip dichotomy reminds us thatall things are interconnected, and that the seeds of our future are sown in the soil of our past. By embracing this holistic perspective, we cultivate a deeper appreciationfor the journey itself and the profound unity that underlies all existence.。