TWO-DIMENSIONAL RIEMANN PROBLEMS： FROM SCALAR CONSERVATION LAWS TO COMPRESSIBLE EULER EQUATIONS

In this paper we survey the authors＇ and related work on two-dimensional Riemann problems for hyperbolic conservation laws, mainly those related to the compressible Euler equations in gas dynamics. It contains four sections： 1. Historical review. 2. Scalar conservation laws. 3. Euler equations. 4. Simplified models.     

CONVERGENCE RATE OFDISTRIBUTIONS OF TRIMMED SUMS

1.IntroductionLet{Xn,n2l}beasequenceofi.i.d.randomvariableswithnondegeneratedistributionF.Fornonnegativeintegersk.andrn,k. rn相似文献   

PHANTOM MAPS AND SPACES OF THE SAME n-TYPE FOR ALL n

1.IntroductionLetXbeaconnectedCWcomplexandX.denoteitsn-skeleton.Amapf:X-YiscalledaphantommapifitsrestrictiontoeachskeletonXuisnullhomotopic.ItiseasytoknowthateveryphantommapfromafiniteCWcomplexortoaspacewithonlyfinitenootrivialhomotopygroupsisnecessarilytrivialuptohomotopy.HenceessentialphantommapcanoccuronlywhenthedomainXisaninfinitedimensionalspaceorthetargetisaspacewithinfinitenontrivialhomotopygroups.Suchmapsappeartobenullhomotopicfromanumberofdifferentpoilltsofview;e.g.,theyinducethe…     

POLES OF ZETA FUNCTIONS OF COMPLETE INTERSECTIONS

51.IntroductionLetF.beafinitefieldofqele~swithcharederisticp.LetXbeanThdilnensionalalgebraicsetdeadedoverF..ThezetafunctionofX/F.isdennedbywhereXOdenotesthesetofclosedpointsOfX/F.and#X(F.d)denotesthenUInberof.F.d-rationalpointsonX.ItiseasytoseethatZ(X,T)isapowerserieswithintegercoefficients.Dwork'srationalltytheoremIg]showsthatthezetafunctionZ(X,T)isrationalinT.ThuS,therearealgebraicintegerspll'.3Pr,pl,'3basuchthatThereisagoodreasonthatwemoantheabovezetafunctionbythepower(--1)"-…     

仿酉对称矩阵的构造及对称正交多小波滤波带的参数化

仿酉矩阵在小波、多小波、框架的构造中发挥了重要的作用.本文给出仿酉对称矩阵(简记为p.s.m.)的显式构造算法,其中仿酉对称矩阵是元素为对称或反对称多项式的仿酉矩阵.基于已构造的p.s.m.和已知的正交对称多小波(简记为o.s.m.),给出o.s.m.的参数化.恰当地选择一些参数,可得到具有一些优良性质的o.s.m.,例如Armlet.最后作这一个算例,构造出一类对称的Chui-Lian Armlet滤波带.     

Improvement of convergence to steady state solutions of Euler equations with weighted compact nonlinear schemes

The convergence to steady state solutions of the Euler equations for weighted compact nonlinear schemes (WCNS) [Deng X. and Zhang H. (2000), J. Comput. Phys. 165, 22-44 and Zhang S., Jiang S. and Shu C.-W. (2008), J. Comput. Phys. 227, 7294-7321] is studied through numerical tests. Like most other shock capturing schemes, WCNS also suffers from the problem that the residue can not settle down to machine zero for the computation of the steady state solution which contains shock waves but hangs at the truncation error level. In this paper, the techniques studied in [Zhang S. and Shu. C.-W. (2007), J. Sci. Comput. 31, 273-305 and Zhang S., Jiang S and Shu. C.-W. (2011), J. Sci. Comput. 47, 216-238], to improve the convergence to steady state solutions for WENO schemes, are generalized to the WCNS. Detailed numerical studies in one and two dimensional cases are performed. Numerical tests demonstrate the effectiveness of these techniques when applied to WCNS. The residue of various order WCNS can settle down to machine zero for typical cases while the small post-shock oscillations can be removed.     

CONTINUOUS L-DOMAINS

61.IntroductionConiinuousDCPOisoneofthemostimportantstructuresin-domaintheory.Unfor-tunatelythecategoryofcolltinuousDCPOandtheScottcontinuousfunctionsfailstobeCartesianclosed,soitisaverysignificantproblemtofinditsmakimalCartesianclosedfullsubcategories.ThecategoryofL-domain'swasdiscoveredbyA.JungwhilesolvingtheproblemoffindingmtalmalCartesianclosedcategoriesofalgebraicDCPO's[1l,andrecentworksbyA.JungandG.Gund.r[2]showsthatthecategoryofL-domain'sisaninterestingcategoryOntheotherhand,i…     

名人名著谈统计学

W.E.Denting(1940),Discussion of Professor Hottelling's Paper "The Teaching of Statistics"(Ann.Math.Stat.Vol.11,457-470): Above all,a statistician must be a scientist. 最重要的是,统计学家应该是科学家.     

D.C.隶属函数模糊集及其应用(I)--D.C.隶属函数模糊集基本概念与性质

本文是D.C.隶属函数模糊集及其应用系列研究的第一部分.建立了D.C.隶属函数模糊集的基本概念.探讨了D.C.隶属函数模糊集的基本性质和D.C.隶属函数模糊集对一些常见的重要t模、余模和伪补的封闭性.并以此建立了丰富的模糊数学应用模型.     

FINITENESS CONDITIONS FOR GENERALIZED EXPONENTS OF DIGRAPHS

1.IntroductionR.A.BrualdiandBolianLiuintroducedin199011]theconceptofgeneralizedexponentsforprimitivedigraphs.Thisconceptisageneralizationofthetraditionalconceptoftheexponentsforprimitivedigraphsandhasbackgroundsinmemorylesscommunicationsystemsassociatedwithdigraphs.Inthispaperweshowthatgeneralizedexponentscanalsobedefinedfordigraphswhicharenotnecessarilyprimitive.WewillgivenecessaryandsufficientconditionsforthefinitenessofthegeneralizedexponentsexpD(k),f(D,k)andF(D,k).Theseideasandresultss…     

A Revision of the Proof of the Kepler Conjecture

The Kepler conjecture asserts that no packing of congruent balls in three-dimensional Euclidean space has density greater than that of the face-centered cubic packing. The original proof, announced in 1998 and published in 2006, is long and complex. The process of revision and review did not end with the publication of the proof. This article summarizes the current status of a long-term initiative to reorganize the original proof into a more transparent form and to provide a greater level of certification of the correctness of the computer code and other details of the proof. A final part of this article lists errata in the original proof of the Kepler conjecture.     

关于“费尔马最后定理的证明”一文的评注

本文对“费尔马最后定理的证明”一文作出几点评注，主要结论是该证明仅仅是对费尔马最后定理的部分情形的证明，即并没有完全证明费尔马最后定理     

Impacts of inquiry pedagogy on undergraduate students conceptions of the function of proof

Mathematicians and mathematics educators agree that proof is an important tool in mathematics, yet too often undergraduate students see proof as a superficial part of the discipline. While proof is often used by mathematicians to justify that a theorem is true, many times proof is used for another purpose entirely such as to explain why a particular statement is true or to show mathematics students a particular proof technique. This paper reports on a study that used a form of inquiry-based learning (IBL) in an introduction to proof course and measured the beliefs of students in this course about the different functions of proof in mathematics as compared to students in a non-IBL course. It was found that undergraduate students in an introduction to proof course had a more robust understanding of the functions of proof than previous studies would suggest. Additionally, students in the course taught using inquiry pedagogy were more likely to appreciate the communication, intellectual challenge, and providing autonomy functions of proof. It is hypothesized that these results are a response to the pedagogy of the course and the types of student activity that were emphasized.     

Euler’s Pentagonal Number Theorem and the Rogers-Fine Identity

Euler discovered the pentagonal number theorem in 1740 but was not able to prove it until 1750. He sent the proof to Goldbach and published it in a paper that finally appeared in 1760. Moreover, Euler formulated another proof of the pentagonal number theorem in his notebooks around 1750. Euler did not publish this proof or communicate it to his correspondents, probably because of the difficulty of clearly presenting it with the notation at the time. In this paper we show that the method of Euler??s unpublished proof can be used to give a new proof of the celebrated Rogers-Fine identity.     

On the hyperbolicity of the period-doubling fixed point

We give a new proof of the hyperbolicity of the fixed point for the period-doubling renormalization operator using the local dynamics near a semi-attractive fixed point (in a Banach space) and the theory of holomorphic motions. We also give a new proof of the exponential contraction of the Feigenbaum renormalization operator in the hybrid class of the period-doubling fixed point: our proof uses the non-existence of invariant line fields in the period-doubling tower (C. McMullen), the topological convergence (D. Sullivan), and a new infinitesimal argument.

     

An elementary geometric nonstandard proof of the Jordan curve theorem

We give an elementary proof, using nonstandard analysis, of the Jordan curve theorem. We also give a nonstandard generalization of the theorem. The proof is purely geometrical in character, without any use of topological concepts and is based on a discrete finite form of the Jordan theorem, whose proof is purely combinatorial.Some familiarity with nonstandard analysis is assumed. The rest of the paper is self-contained except for the proof a discrete standard form of the Jordan theorem. The proof is based on hyperfinite approximations to regions on the plane.Research of the first author partially supported by FONDECYT Grant # 91-1208 and of the second author, by FONDECYT Grant # 90-0647.     

A sample-path approach to the optimality of echelon order-up-to policies in serial inventory systems

We present a new proof of the optimality of echelon order-up-to policies in serial inventory systems, first proved by Clark and Scarf. Our proof is based on a sample-path analysis as opposed to the original proof based on dynamic programming induction.     

On the proof of Sullivan's eventual periodicity theorem

In this paper,we give a modified proof of Sullivan's eventual theorem for rational dynamics. Our proof is based on some idea of the Sullivan's proof,but does not make use of the Teichmuller theory.     

An easy proof of the Rogers-Ramanujan identities

A proof of the Rogers-Ramanujan identities is presented which is brief, elementary, and well motivated; the “easy” proof of whose existence Hardy and Wright had despaired. A multisum generalization of the Rogers-Ramanujan identities is shown to be a simple consequence of this proof.