the 3n+1 problem
创意数学题
创意数学题英文文档：Creative Mathematics ProblemsMathematics is not just about numbers and calculations.It can also be a creative field where students can apply their knowledge to solve interesting problems.Here are some creative mathematics problems that can challenge and engage students:1.The Pyramid Problem:Imagine a pyramid with a height of 10 units and each of its four triangular faces having an area of 20 square units.Calculate the total surface area of the pyramid.2.The Coin Problem:You have 10 coins, 5 of which are fake and weigh 1 gram each, while the remaining 5 are real and weigh 1.5 grams each.You have a balance scale and can measure the weight of up to 3 coins at a time.How can you determine which coins are fake using the scale only 3 times?3.The Magic Square Problem:Create a 3x3 square using the numbers 1 to 9, where the sum of the numbers in each row, column, and diagonal is the same.What is this common sum?4.The Time Problem:You have a 12-hour clock and you want to set it to 5:00 PM.However, you only have 5 minutes to do so.How can you adjust the clock to the desired time?5.The Area Problem:A farmer has a rectangular field with a length of 8 units and a width of 4 units.If the farmer wants to increase the area of the field by a factor of 4, what new dimensions should the field have?These creative mathematics problems encourage students to think critically, apply their knowledge, and come up with innovative solutions.By solving such problems, students can develop a deeper understanding of mathematical concepts and enhance their problem-solving skills.中文文档：创意数学题数学不仅仅是关于数字和计算。
牛津深圳版英语九年级上册Unit-1单元习题(无答案)
牛津版英语九年级上册Unit 1:Wise men in historyReading A单词：1 adj.金的，金色的2 n. 王冠，皇冠3 n.（pl.）奥运会4 n. 同意，应允5 n. 证实6 n. 罐7 v. 不能肯定，对……无把握8 adj. 真的，正宗的9 n. 真相，实情10 v. 好像，似乎短语：1 金王冠2 首先，起先3 对（某人或某事）满意4 发现，查明5 考虑，思考6 用……把……装满根据课文内容填空：One day in l Greece. King Hiero asked a crown maker to make him a 2 crown. At first, he was very 3 with it."It's a nice crown . isn’t .it?" he asked his men. 4 . however, he began to 5 that it was a real golden cro wn. “Is it made completely of gold?” he wondered. He sent it to Archimedes and asked him to find out the 6 ."This problem seems 7 to solve. What should I do?" thought Archimedes. Archimedes was still 8 about this problem as he filled his bath with water. When he got into the bath with water. When he got into the bath, some water ran 9 ."That’s it "shouted Archimedes. ”I know how to 10 the king 's problem ! "基础训练：I．单项选择l. The beautiful girl has _ hair.A gold B. golden C. iron D. metal2 I whether his statement is true.A. thinkB. believeC. doubtD. know3. We are agreement on this point.A. atB. underC. onD. in4. Please write about your____ experience.A. realB. reallyC. trueD. truly5. She seems the secret.A know B. knows C.. to know D. knowingⅡ单词拼写6. I want you to tell the t .7. Do you d that he will keep his word?8. I am in a with the decision.9. It is difficult to s the problem10. Please f this glass for me.Ⅲ完成句子11．她似乎从上周开始就一直生病。
ACM文档2
• The first student receives 10.00 from the third student • The second student doesn’t need to do anything. • The answer will be $10.00
数据类型大小的重要性
• 不同的数据类型占据的字节数是不一样的， 只有记住每一个的大小才能正确选择数据 类型。 • 例如，要存储一个数字为2548，可不可以 用char，可不可以用short，可不可以用 int？ • 如果某些情况下需要节约内存，那么能用 小的数据类型就不要选择过大的
字符类型
• char（character）：存储1字节，2^7~2^7-1 • unsigned char：存储1字节，0~2^8-1 • 以上可以做字符使用也可以作整数使用， 此数据类型建立了整数与字符之间的对应， 例如：97对应’a’，65对应’A’，32对应 空格，48对应’0’，还有一类成为转义字 符，例如10对应回车
ACM入门辅导
Tutorial Two
上2周小结
上2周在172.22.27.1完成题目 的10级同学排名：
ID：
10723110078 1006100071
Solved
30
习题
• http://172.22.27.1/ • Begginer的分类 • 题号：1000，1008，1013，1022， 1027，1031，1048，1056，1156， 1218，1229，1230，1231，1232， 1259 • 比较困难：1098，1137，1214
浮点型
• float：占4字节，7位有效数字 • double：占8字节，15位有效数字 • 一般来说用到浮点型的问题都是追求精度 的，而不是卡内存，所以在一般情况下我 们应当选择使用double，而很少用float
具体数学Concrete Mathematics(第2版)部分答案ch1p2
T0 = 0 T1 = 3*0 + 2 = 2 T2 = 3*2 + 2 = 8 T3 = 3*8 + 2 = 26 T4 = 3*26 + 2 = 80 Observation : Tn = 3n -1
Now we have to prove that Recurrence relation = Closed Formula Lets apply Mathematical induction : Recurrence : T0 = 0, Tn = 3 Tn-1 + 2 Closed Formula : Tn = 3n -1 , n >= 0
Basis case : T0 = 0 C.F : T0 = 30 – 1 = 1 – 1 = 0 Therefore, Recurrence = C.F for n=0 Let us assume that our closed formula is correct for values <= n – 1. So, now we need to prove : Tn = 3n - 1
GENERALIZE : Lets assume that the tower has “n” disks. Let Tn be the minimum number of moves that will transfer “n” disks from one peg (i.e. A to another (i.e. C). Clearly T0 = 0, because no moves at all are needed to transfer a tower of n = 0 disks.
The objective is to transfer the entire tower from A to C( in the diagram), if direct moves between A and C are disallowed. This pபைடு நூலகம்oblem is a variant of Tower of Hanoi problem.
电网方案及作业中N-1原则运用与设备的有关性疑问
电网方案及作业中N-1原则运用与设备的有关性疑问这些年，跟着电网的不断强壮，对电网安稳性及设备牢靠性的恳求越来越高，电网安稳导则和有关电网规划缔造的技能导则对电网安全作业的N-1原则均提出了纷歧样恳求。
可是，因为方案、方案、作业及运营等纷歧样有些、有关技能人员对这一原则的了解和知道纷歧样，供电牢靠性与电价未构成有用的利益互动联络，构成电网方案缔造、出产作业中不行避免地发作先天性安全风险，对电网的久远翻开也构成了晦气影响。
因而，有必要弄清对N-1原则的一些过错知道，行进电网出资效益和安全性、牢靠性，在确保电网可持续翻开的条件下，稳步行进供电牢靠性。
1N-1原则的实质依照电网安稳导则有关界说，N-1原则是指正常作业办法下电力体系中恣意一元件（如线路、发电机、变压器等）无缺点或因缺点断开后，电力体系应能坚持安稳作业和正常供电，别的元件不过负荷，电压和频率均在答应方案内。
N-1原则用于单一元件无缺点断开条件下电力体系静态安全剖析，或单一元件缺点断开后的电力体系安稳性剖析即动态安全剖析。
当发电厂仅有一回送出线路时，送出线路缺点或许致使失掉一台以上发电机组，此种状况也按N-1原则思考。
由此可见，N-1原则包含两层意义：一是确保电网的安稳；二是确保用户得到契合质量恳求的接连供电。
从如今状况看，确保电网的安稳因为触及悉数电网安全，不管在资金投入、作业办法仍是技能办法上均得到满意的注重，但在确保用户分外是边远区域用户接连供电方面仍存在知道上的短少，构成有些电网出产作业长时刻处于被逼局势。
2N-1与体系安稳性恳求电力体系安稳分为静态安稳、暂态安稳、动态安稳、电压安稳。
电力体系中单一元件无缺点断开后，直接影响其静态安稳和电压安稳，使正常输变电才干遭到束缚，其间以发电机组和输电线路停运较为显着，分外是单电源线路或单台主变压器供电的变电站，当线路或变压器停电修补时，影响最直接；电力体系中单一元件缺点断开后，直接影响其暂态安稳、动态安稳和电压安稳，其间以发电机组缺点、母线缺点和输电线路缺点较为出色，分外是纽带变电站母线和网间联络线路。
电网N-1问题[整理]
![电网N-1问题[整理]](https://img.taocdn.com/s3/m/2bbcfd2d590216fc700abb68a98271fe910eaf04.png)
正常运行方式下的电力系统中任一元件（如线路、发电机、变压器等）无故障或因故障断开，电力系统应能保持稳定运行和正常供电，其他元件不过负荷，电压和频率均在允许范围内。
这通常称为N－1原则。
N-1原则用于电力系统静态安全分析(单一元件无故障断开)，或动态安全分析(单一元件故障后断开的电力系统稳定性分析)。
当发电厂仅有一回送出线路时，送出线路故障可能导致失去一台以上发电机组，此种情况也按N－1原则考虑。
电网规划及运行中N-1准则应用与设备的相关性问题近年来，随着电网的不断壮大，对电网稳定性及设备可靠性的要求越来越高，电网稳定导则和有关电网规 划建设的技术导则对电网安全运行的N-1准则均提出了不同要求。
然而，由于规划、设计、运行及经营等不同部门、相关技术人员对这一准则的理解和认识不同，供电可靠性与电价未形成有效的利益互动关系，造成电网规划建设、生产运行中不可避免地产生先天性安全隐患，对电网的长远发展也造成了不利影响。
因此，有必要澄清对N-1准则的一些错误认识，提高电网投资效益和安全性、可靠性，在确保电网可持续发展的前提下，稳步提高供电可靠性。
1N-1准则的本质按照电网稳定导则有关定义，N-1准则是指正常运行方式下电力系统中任意一元件(如线路、发电机、变压器等)无故障或因故障断开后，电力系统应能保持稳定运行和正常供电，其他元件不过负荷，电压和频率均在允许范围内。
N-1准则用于单一元件无故障断开条件下电力系统静态安全分析，或单一元件故障断开后的电力系统稳定性分析即动态安全分析。
当发电厂仅有一回送出线路时，送出线路故障可能导致失去一台以上发电机组，此种情况也按N-1原则考虑。
由此可见，N-1准则包含两层含义：一是保证电网的稳定;二是保证用户得到符合质量要求的连续供电。
从目前情况看，保证电网的稳定由于涉及整个电网安全，无论在资金投入、运行方式还是技术措施上均得到足够的重视，但在保证用户特别是边远地区用户连续供电方面仍存在认识上的不足，造成局部电网生产运行长期处于被动局面。
2003-JCP-lowkineticenergy
Kinetic energy fix for low internal energy flowsX.Y.Hua,*,B.C.Khooa,baSingapore-MIT Alliance,National University of Singapore,4Engineering Drive 3,Singapore 117576,Singapore bDepartment of Mechanical Engineering,National University of Singapore,Singapore 119260,SingaporeReceived 11October 2002;received in revised form 26May 2003;accepted 11August 2003AbstractWhen the kinetic energy of a flow is dominant,numerical schemes employed can encounter difficulties due to negative internal energy.A case study with several commonly used conservative schemes (MUSCL,ENO,WENO and CE/SE)shows that high order schemes may have less ability to preserve positive internal energy (MUSCL and CE/SE),or present less accurate results (WENO and ENO)when the internal energy to kinetic energy ratio is low.By analyzing the positivity property for second-order conservative schemes with large fixed CFL number conditions for time step restriction,this paper proposes the energy consistency conditions for second-order Riemann-solver type schemes and CE/SE method.According to the said energy consistency conditions,a kinetic energy fix method which limits the magnitude of kinetic energy relative to the total energy is introduced.The numerical examples show that the kinetic energy fixed CE/SE method produces reasonable results and keeps positive internal energy for flows with very low internal energy even when a vacuum occurs.Ó2003Elsevier B.V.All rights reserved.Keywords:Low internal energy flow;Positivity property;Conservative scheme1.IntroductionThe gas dynamic problems are usually solved by conservative schemes,in which the internal energy is used to determine the pressure by subtracting the kinetic energy component from the total energy.For flows in which the ratio of internal energy to kinetic energy is low,the resulting internal energy obtained may be negative hence giving rise to negative pressure and thereby invalidating the ensuing computation.A useful test case to evaluate the ability of a computational method in handling such low internal energy flow is that of a one-dimensional tube containing a gas having diametrically opposite initial velocities,which is usually referenced as the ‘‘1–2–3problem’’[18].Einfeldt et al.[3]analyzed the characteristics of low internal energy flows and proposed the HLLE scheme [6]to solve the ‘‘1–2–3problem’’.Toro [18]tested and found that several classic Riemann-solver type schemes can keep the density and internal energy positive and have/locate/jcpJournal of Computational Physics 193(2003)243–259*Corresponding author.Tel.:+65-68744797;fax:+65-67752920.E-mail addresses:smahx@.sg (X.Y.Hu),mpekbc@.sg (B.C.Khoo).0021-9991/$-see front matter Ó2003Elsevier B.V.All rights reserved.doi:10.1016/j.jcp.2003.08.007the so called positivity property.Gressier et al.[5]also discussed the positivity conditions of several classical flux vector splitting schemes.For high order conservative schemes,even for those constructed based on positivefirst-order schemes,as will be shown by a case study in Section4,may show less ability to maintain positivity or not being able to obtain results with sufficient accuracy for low internal energyflow problems(or those when the ratio of internal to kinetic energy is lower than3).On the positivity property,Linde and Roe[9]discussed about the positivity conditions for a second order multi-dimensional MUSCL-type scheme while Perthame[11]and Tang and Xu[17]studied the positivity conditions for second-order kinetic schemes.Perthame and Shu[12] provided a remarkable theorem which states that,given afirst-order positive conservation scheme such as Godunov and Lax–Friedrichs schemes,one can always build a higher order positive scheme under the conditions that(a)the cell wall values for numericalflux calculation satisfy positive density and pressure, and(b)sufficiently small CFL number be utilized to constrain the time step.As afixed CFL number is usually employed in practical computation,(c)additional constraints on the interpolation procedure are ually,thefixed CFL number permitted in practical computation is considerably small and therefore greatly decreased the efficiency.On the other hand,to increase the accuracy of computation for low internal energyflows,Cocchi et al.[2]suggested a second-order non-conservative formulation for the energy equation to decrease the non-physical temperature increase.However,the non-conservative scheme may result in an exponential error growth,and it is not known that if this non-conservative formula can handleflows with even lower internal to kinetic energy ratios.The CE/SE method[1],a non-Riemann-solver type conservative scheme,introduces less errors when computing for low internal energyflows with reasonably large CFL number for time step restriction.However,as shown in the case study below,it works well provided the internal energy of theflows is not very low.The motivation of this paper stems from the above mentioned difficulties of maintaining positivity and accuracy preserving for high order schemes when computing for low internal energyflows.Wefirst study the performances of several different high order conservative schemes(MUSCL,ENO,WENO and CE/SE) to determine the difficulties that each scheme may face with the decrease of internal energy in theflow.Then we analyze the positivity properties of second-order Riemann-solver type schemes and the CE/SE method for the cases with largefixed CFL number for high computational efficiency.Based on these analysis,we propose the energy consistency conditions,by which a kinetic energyfix method for a general second-order conservative scheme is introduced.As the CE/SE method is found from the case study to give more ac-curate results,it is modified with the said kinetic energyfix to compute for several numerical examples for flows with a large range of low internal energy.2.Euler equationsAssuming thefluid is inviscid and compressible,theflow is described by Euler equations in one di-mension aso U o t þo FðUÞo x¼0;ð1Þwhere U¼ðq;q u;EÞT and FðUÞ¼ðq u;q u2þp;ðEþpÞuÞT.The equation of state is defined as p¼ðcÀ1Þq e;ð2Þwhere c¼1:4is the heat ratio for an ideal gas.This set of equations describes the conservation of the conservative variables:density q,momentum q u and total energy density E¼q eþ1q u2,where e is the internal energy per unit mass.The ratio of internal energy to kinetic energy can be defined as244X.Y.Hu,B.C.Khoo/Journal of Computational Physics193(2003)243–259K¼2q EÀðq uÞ2ðq uÞ:ð3ÞIn a low internal energyflow,the internal energy is close or even smaller than the kinetic energy,i.e.K<1. However,the internal energy and density must always be kept positive,i.e.K>0,to maintain a positive pressure p from Eq.(2)for a real physical state.As the state is derived from the conservative variables when solving Eq.(1),we say that the conservative variables are energy consistent when the system possesses a positive internal energy hence giving rise to a positive pressure.If K<0,the energy consistency fails,and negative internal energy and pressure occur.3.Conservative schemes3.1.Riemann-solver typeAn explicit Riemann-solver type conservation scheme of Eq.(1)on cell j can be written asU nþ1 j ¼ U njÀkð^F jþ1=2À^F jÀ1=2Þ;ð4Þwhere U nj and U nþ1jare the cell average values at n th and nþ1th time steps,^F jÆ1=2are numericalfluxes onthe respective cell walls.The scheme is stable under a Courant–Friedrich–Lewy(CFL)time step re-strictionk¼D tD x<N CFLmaxðj u i jþc iÞ;i¼1;N;ð5Þwhere N CFL<1is called CFL number.For afirst-order conservative scheme,^F jÆ1=2is defined directly based on the cell average values,i.e.^Fjþ1=2¼^Fð U j; U jþ1Þ;^F jÀ1=2¼^Fð U jÀ1; U jÞ:ð6ÞSeveral classic conservative schemes,such as Godunov,Lax–Friedrichs,HLLE,have positivity property under a CFL number conditionN CFL<a0;ð7Þwhere a061.The positivity property ensures positive density and pressure from U nþ1when U n satisfies positive density and pressure.A higher order scheme is usually constructed based on a positive preservingfirst-order scheme,such as MUSCL is based on Godunov scheme while ENO or WENO are based on Lax–Friedrichs scheme.The difference is that the^F jÆ1=2in a high order scheme are defined with the values of the forward and backward conservative variables approximated by a piecewise function U nðxÞon the walls of cell,say x jÀ1=2and x jþ1=2. Specifically,the second-order profiles can be written asq¼ q jþk q x;q u¼ð q uÞjþk q u x;E¼ E jþk E x;ð8Þin which the cell average values also represent the values on the node.The numericalfluxes of a high order scheme are^Fjþ1=2¼^FðUÀjþ1=2;Uþjþ1=2Þ;^F jÀ1=2¼^FðUÀjÀ1=2;UþjÀ1=2Þ:ð9ÞX.Y.Hu,B.C.Khoo/Journal of Computational Physics193(2003)243–259245On the conservative property,these high order approximations are always conserved pertaining to the local mean on the cell,i.e. U n j ¼ð1=D x ÞR x j þ1=2x j À1=2U n ðx Þd x .For second-order schemes,the linear interpolation approximations shown in Eq.(8)automatically ensure cell average values are conserved.3.2.CE/SE methodThe CE/SE method [1]is a non-Riemann-solver type conservative scheme and has second-order accu-racy in both time and spatial directions.In the CE/SE method,the time integration is performed on a staggered grid and each full time step (satisfying Eq.(5))is divided into two half time steps.From the initial conditions,smooth regions are defined in the cells near the nodes,such as ðx j À1=2;x j þ1=2Þ,with the linear profiles of Eq.(8).When the first half time step is calculated,the cell is shifted by D x =2,which is equivalentto defining for the region between nodes,such as ðx j ;x j þ1Þ;hence new cell average values U n j þ1=2are defined.As the values at the nodes are smooth with first-order derivatives,the fluxes on the new cell walls are physical.Hence,a conservative scheme similar to Eq.(4)can be written asU n þ1=2j þ1=2¼ U n j þ1=2Àk 2F ðU n þ1=4j þ1Þh ÀF ðU n þ1=4j Þi ;ð10Þwhere U n þ1=2j þ1=2are the cell average values after the first half time step ,F ðU n þ1=4j þ1Þand F ðU n þ1=4j Þare the physical fluxes on node points at time ðn þ1=4ÞD t .After the first half time step,the linear profile near j þ1=2is then constructed by a weighted average interpolation approach with the new cell average valuesU n þ1=2j þ1=2and the new node point values U n þ1=2j þ1and U n þ1=2j at time ðn þ1=2ÞD t .The values U n þ1=4j and U n þ1=2j at each node point are defined by the first-order Taylor expansionU n þn j ¼ U n jþn D t o U o t n j ;ð11Þwhere n ¼1=2or 1=4.With the relation of Eq.(1)and linear approximation,Eq.(11)can be written asU n þn j ¼ U n j Ànk F ðU Àj þ1=2Þh ÀF ðU þj À1=2Þi;ð12Þwhere U Àj þ1=2and U þj À1=2are the initial cell wall values which are given by a linear profile such as Eq.(8).When the second half time step is calculated,the difference form can be written in a similar way as for Eq.(10),but all the terms are shifted by D x =2spatially.After a full time step,the cell locations are shifted back to the original locations again.4.Case study of low internal energy flows:‘‘1–2–3problem’’The ‘‘1–2–3problem’’[18]gives a series of Riemann problems for an ideal gas U ðx ;0Þ¼U l if x <x 0;U rif x >x 0;ð13Þwhere the two constant initial conservative variables are U l ð1;À2;E 0Þand U r ð1;2;E 0Þand total energy E 0>2.E 0can be changed according to the different initial internal energy ratio.According to Smoller [15],if the initial internal to kinetic energy ratio K 0P 1(E 0P 16),the theoretical solution consists of two rar-efaction waves propagating in opposite directions and a static low density region exists between the rar-efaction regions.As K 0decreases,the density of the static region decreases.If the initial internal energy ratio K 0617,a vacuum occurs in the solution.However,there arises difficulties for numerical methods when 246X.Y.Hu,B.C.Khoo /Journal of Computational Physics 193(2003)243–259X.Y.Hu,B.C.Khoo/Journal of Computational Physics193(2003)243–259247 the initial condition is far less critical than a vacuum solution.Einfeldt et al.[3]found that,if K064 (E0610),the solution is not linearizable and may lead to numerical difficulties.In this paper,four com-monly used high order conservative schemes are tested:•Second-order MUSCL Hancock scheme(MUSCL)[13].•Third-order ENO-LF scheme(ENO)[14].•Fifth-order WENO-LF scheme(WENO)[7].•Second-order CE/SE method(CE/SE)[1].First,a case which has also been considered by Cocchi et al.[2]is tested.The initial total energy and in-ternal energy ratio are E0¼6and K0¼2.Here,the initial condition is much less critical than that asso-ciated with a vacuum solution.While the ENO scheme and the WENO scheme have no difficulty for various CFL numbers used,both the MUSCL scheme and the CE/SE methodÔexplodeÕor grow uncon-trollably in the presence of negative internal energy when the CFL number of0.9is applied.Because of this, lower CFL numbers of0.5and0.8are used,respectively.Fig.1shows the calculated pressure,density,248X.Y.Hu,B.C.Khoo/Journal of Computational Physics193(2003)243–259Table1Requirement of CFL number for the four numerical schemes with different E0aE0=K0MUSCL Hancock-2ENO-LF-3b WENO-LF-5b CE/SE-28.0/3.00.90.90.90.96.0/2.00.50.90.90.83.0/0.5–0.90.90.62.5/0.25–0.90.90.42.25/0.125c–0.70.70.3a It does not necessarily imply that the solution is accurate compared to analysis.b Usually,a CFL number bigger than0.8is not recommended.c A vacuum occurs in exact solution.velocity and temperature profiles at time t¼0:1.For all theflow variables considered,it is found that all the four schemes give reasonable results for the rarefaction regions.By comparing the low temperature and low density static region at the center of the domain with the exact solution,all the four schemes show much larger deviations than for the rarefaction regions.One can still discern the CE/SE method gives the smallest errors,especially for the temperature and density distributions.The ENO and WENO schemes give the largest deviations;further numerical tests on the ENO and WENO show results with smaller CFL number,even at0.1,the errors cannot be reduced much(not shown here).For the MUSCL scheme,the predicted temperature is comparatively much lesser,however the deviation is still much larger than of the CE/SE method.From the above,one can suggest that the CE/SE method gives the best solution,which predicts the density fairly accurately and considerably reduces the increase of the temperature in the static region.It may be noted that these results by the CE/SE method are already better than that shown by Cocchi et al.[2]with non-conservative modifications.However,the CE/SE method must be computed with smaller CFL number to keep the internal energy positive,just as for the MUSCL scheme.Table1shows the nominally initial conditions of imposed largest CFL number which can be used with the different internal energy ratio for the four schemes such that subsequent computations can continue and still maintain a strictly non-negative internal energy.From Table1,it is clear that the MUSCL scheme shows the poorest ability to keep the internal energy positive.It ÔexplodesÕwhen K0¼1or smaller,even when the CFL number is decreased to a very small value such as 0.1.The CE/SE method behaves better,but the CFL number also shows a need to be decreased as the initial internal energy ratio decreases.The ENO and WENO schemes are much more robust than both the MUSCL and CE/SE schemes;numerical results show that the ENO and WENO schemes can maintain positivity with much larger CFL number than that in Perthame and Shu[12].On the other hand,the results of the second-order MUSCL scheme and CE/SE scheme in particular seem to exhibit higher accuracy for such low internal energyflows.The above results show,for these conservative schemes,positivity and accuracy are difficult to be maintained at the same time.In the following section,the positivity of second-order Riemann-solver type schemes and the CE/SE method are analyzed to show that the difficulties/problems associated with low internal energy is mainly caused by energy consistency failure.A kinetic energyfix method is then proposed to ensure the positivity property while still maintaining reasonably largefixed CFL number and high accuracy at the same time.5.Analysis of low internal energyflowsAs the MUSCL scheme and CE/SE method have no difficulty in maintaining positive internal energy for the‘‘1–2–3problem’’if they degenerate to thefirst-order forms,and the main difference between the highorder conservative scheme andfirst-order scheme is the interpolations used to predict the profiles inside the computational cell j concerned,it is quite logical to study how the interpolations affect the positivity property.5.1.Positivity property of Riemann-solver type schemesTheorem1.Assume thefirst-order form of a Riemann-solver type conservative scheme satisfies the positivity property under a CFL condition of Eq.(7).If the numericalflux is defined by Eq.(9)and the cell wall values take on positive density and pressure,the full second-order scheme is positivity preserving under the CFL number conditionN CFL<12a0:ð14ÞRemark1.This theorem is similar to Theorem1in[12].However,it further ensures the positivity of a second-order scheme under afixed CFL number without any other constraint than the cell wall values.Remark2.For third or higher order schemes,if the numericalflux is defined with cell wall values,similar results to Theorem1in Perthame and Shu[12]can also be obtained.However,the positivity cannot be ensured under the conditions in Theorem1.Proof.For a second-order scheme,the conservative variables in the cell j can be approximated by Eq.(8). The cell average values and cell wall values always have the relationU j ¼12ðUÀjþ1=2þUþjÀ1=2Þ:ð15ÞHence,Eq.(4)can be written asU nþ1 j ¼12UÀjþ1=2nÀ2k^FðUþjþ1=2;UÀjþ1=2ÞhÀ^FðUÀjþ1=2;UþjÀ1=2Þioþ12UþjÀ1=2nÀ2k^FðUÀjþ1=2;UþjÀ1=2ÞhÀ^FðUþjÀ1=2;UÀjÀ1=2Þioð16Þto give a linear combination of twofirst-order schemes.As these twofirst-order schemes have positiveproperty while UÀjþ1=2and UþjÀ1=2assume positive density and pressure under the condition of Eq.(14),Eq.(16)implies that U nþ1j has the same positivity property for pressure,which is indeed a concave functionof U.Ã5.2.Positivity property of the CE/SE methodTheorem2.If the cell wall values of the initial condition satisfy positive density and pressure,the CE/SE method preserves positivity without further time step restriction than Eq.(5).Remark1.As the CE/SE method usually prefers and allows for large CFL number for computation,this theorem implies that the same CFL number as applied in higher internal energy region can be used in the low internal energy region of the same computational domain.Remark2.The positivity property for the second order in time discretization is also enforced directly by this theorem.X.Y.Hu,B.C.Khoo/Journal of Computational Physics193(2003)243–259249Proof.As the two half steps of the CE/SE method have the same difference form,we only need to prove thepositivity condition for thefirst half step.One canfind the condition is equivalent to ensuring that U nþ1=2jþ1=2,U nþ1=4j and U nþ1=2jhave positive density and pressure from Eqs.(10)and(12).For Eq.(10),it can be written as the linear combination of two exact solversU nþ1=2 jþ1=2¼12Ujþ1=2Àk2FðU nþ1=4jþ1ÞÀF12Ujþ1=2þ12Ujþ1=2Àk2F12Ujþ1=2ÀFðU nþ1=4jÞ:ð17ÞSince the two exact solvers preserve positivity,Eq.(17)implies that,when U jþ1=2,and U nþ1=4j for every nodesatisfy positive density and pressure, U nþ1=2jþ1=2has the same property.As Eq.(17)is calculated only for a halftime step,the interaction of waves for the exact solvers can be avoided;this indicates that the CE/SE method has positivity property for thefirst half time step without further CFL number restriction.For U jþ1=2,with relations as specified in Eq.(8),we always haveUjþ1=2¼14ð U jþUÀjþ1=2þ U jþ1þUþjþ1=2Þ;ð18Þwhere UÇjÆ1=2are the two cell wall values from the initial condition.As this equation is a linear combi-nation of conservative variables,we need only to ensure the cell wall values satisfy positive density andpressure.On the other hand,we need U nþ1=4j for every node to satisfy positive density and pressure,and wealso require the values U nþ1=2j for every node to have the same property for the linear constructions as usedin the second half time step.Similarly,Eq.(12)can also be transformed into the combination of two exact solversU nþnj ¼12UjÀnk FðUÀjþ1=2ÞÀF12Ujþ12UjÀnk F12UjÀFðUþjÀ1=2Þ:ð19ÞEq.(19)has similar form as Eq.(17)and thereby ensures positivity for U nþ1=2j when the cell wall valuesUÇjÆ1=2satisfy positive density and pressure.As U nþ1=4jis equivalent to the solution of Eq.(19)att¼ðnþ1=4ÞD t,the condition that preserves positivity for U nþ1=2j also satisfies for U nþ1=4jfor the smallertime step of D t=4.From the above analysis,we can conclude that(a)the CE/SE has positivity property without further CFL number restriction,(b)the positivity condition is that the cell wall values satisfy positive density and pressure.Ã5.3.Energy consistency conditionsFor a positive preserving scheme,as the cell average values at the n th time step ensure positive density and pressure,from Eq.(3)these conservative variables are energy consistent;that is K>0.Hence,we have1 2ð q uÞ2jqjEj<1:ð20ÞFrom the analysis of positivity properties,one canfind that besides the CFL number constraint,the cell wall values are also required to take on positive density and pressure for preserving positivity on the new cell average values.For second-order schemes,as the density is interpolated from the near cell average values with minmod,weighted average or other limiters,the cell wall density is bounded by these near cell average values and always be positive.Therefore,one only needs to maintain the energy consistency for these cell wall conservative variables,i.e.250X.Y.Hu,B.C.Khoo/Journal of Computational Physics193(2003)243–2591 2ð q u jÆk q u dÞ2ð q jÆk q dÞð E jÆk E dÞ<1;ð21Þwhere d¼D x=2.If the cell wall values are energy consistent with positive density and pressure,the value at locationÆe,d>e>0,in the cell can be written asUðÆeÞ¼dÀedUjþedUÇjÆ1=2:ð22ÞOne can thenfind that the conservative variables at every location inside the cell have the same property. Hence,the energy consistency condition for the whole cell is satisfied by1 2 E j Z x jþ1=2x jÀ1=2ðq uÞ2q D xd x<1:ð23ÞThe integral on the left-hand side is the cell average kinetic energy,which can be calculated/expressed numerically by different approximations.One of the possible(approximated)energy consistency condition can be simply written as1 2ð q uÞ2jþk2q ud2qjEj<1:ð24ÞBy comparing Eq.(24)to(20),one canfind that,for a second-order scheme,the whole cell has an addi-tional kinetic energy term,k2q uD x2,compared to that of the cell average value.6.Kinetic energyfix methodFrom the discussions in the last section,one can observe that the kinetic energy is non-physical when the energy consistency condition of Eq.(21)or(23)is not satisfied.Hence,the j k q u j as utilized in Eq.(8)is larger than that of a real physical profile existing at the cell.Therefore,a kinetic energyfix can be proposed to control the kinetic energy level for a physically reasonable profile.By comparing Eq.(20)to(21)or(23), one can suggest that the magnitude of kinetic energy,density and total energy befixed/limited to enforce or ensure energy consistency condition required for a positive pressure.In practice,thefixes can be proposed based on energy consistency condition for the cell wall values or the whole cell values.If the kinetic energy fix is based on the the cell wall values,we can limit all the k q u;k q;k E at the same time by introducing a parameter1>a>0to Eq.(21).That is,we solve for the equation1 2ð q u jÆa k q u dÞ2ð q jÆa k q dÞð E jÆa k E dÞ<1ð25Þto obtain a and thenfix the slopes viak0 q <a k q;k0q u<a k q u;k0E<a k E:ð26ÞIf the kinetic energyfix is based on the whole cell values,similarly a parameter1>b>0can be introduced to Eq.(24)so that1 2ð q uÞ2jþb2k2q ud2qjEj<1:ð27ÞX.Y.Hu,B.C.Khoo/Journal of Computational Physics193(2003)243–259251We solve for b and the slopes arefixed viak0 q <b k q;k0q u<b k q u;k0E<b k E:ð28ÞEven as Eqs.(26)and(28)suggest the bounds for the slopes,usually a slightly larger a or b,i.e.1:1a$1:3a or1:1b$1:3b,can be used to achieve even better results.It is noted that the abovefixes,Eqs.(26)and(28), are only applied to those cells which did not satisfy the energy consistency conditions caused by interpo-lations for non-physical profile.The accuracy in the smooth region is still preserved.It is also noted that there may be other kineticfix methods derived from Eq.(21)and(23);for example one of the alternativefix method is further outlined in Appendix A.However,our numerical results show the present simplefix method is sufficient for a large range of low internal energyflows even when a vacuum occurs.As the kinetic energyfix methods from Eqs.(25)–(28)are not scheme specific and many second-order conservative schemes have a linear slope prediction step,the said methods may be applicable to different schemes easily.The detailed procedure can be given as follows:252X.Y.Hu,B.C.Khoo/Journal of Computational Physics193(2003)243–259。
角谷猜想的证明
角谷猜想一简介考拉兹猜想，又称为3n+1猜想、角谷猜想、哈塞猜想、乌拉姆猜想或叙拉古猜想，是由日本数学家角谷静夫发现，是指对於每一个正整数，如果它是奇数，则对它乘3再加1，如果它是偶数，则对它除以2，如此循环，最终都能够得到1。
取一个数字如n = 6，根据上述公式，得出6→3→10→5→16→8→4→2→1。
（步骤中最大的数是16，共有7个步骤）如n = 11，根据上述公式，得出11→34→17→52→26→13→40→20→10→5→16→8→4→2→1。
（步骤中最大的数是52，共有13个步骤）如n = 27，根据上述公式，得出:27→82→41→124→62→31→94→47→142→71→214→107→322→161→484→242→12 1→364→182→91→274→137→412→206→103→310→155→466→233→700→350→175→526→263→790→395→1186→593→1780→890→445→1336→668→334→167→502→251→754→377→1132→566→283→850→425→1276→638→319→958→479→1438→719→2158→1079→3238→1619→4858→2429→7288→3644→1822→911→2734→1367→4102→2051→6154→3077→9232→4616→2308→1154→577→1732→866→433→1300→650→325→976→488→244→1 22→61→184→92→46→23→70→35→106→53→160→80→40→20→10→5→16→8→4→2→1。
（步骤中最大的数是9232，共有111个步骤）考拉兹猜想称，任何正整数，经过上述计算步骤後，最终都会得到1。
注意：与角谷猜想相反的是蝴蝶效应，初始值极小误差，会造成巨大的不同；而3x+1恰恰相反，无论多么大的误差，都是会自行的恢复。