Analysis of the binary Euclidean algorithm
Algebraic Bernouilli Equations
EngineeringParallel solution of Large-ScaleAlgebraic Bernouilli Equations via sign function iterationsS ERGIO B ARRACHINA M IR*-P ETER B ENNER**-M ARK B ULL*** M URRAY C OLE***-E NRIQUE Q UINTANA O RTIÂ**Dpt.of Computer Science and Engineering.Jaume I University.Spain.**Dpt.of Mathematics.Chemnitz University of Technology.Germany.***Edinburgh Parallel Computing Centre(EPCC).University of .***School of Informatics.University of .Abstract.±We investigate the solution of the algebraic Bernouilli equation via the Newton iteration for the matrix sign function.The algorithms proposed here are easily parallelizable and thus provide an efficient tool to solve large-scale problems.We report the efficiency and scalability of our parallel implementation on the HPCx system.1.IntroductionThe original name of the project we wanted to conduct during the HPC-Europa stay at EPCC was``Systematic Development of Parallel Algorithms for Dense Linear Algebra using FLAME''.We wanted to extend the existing tools for FLAME[4]with a new API for developing parallel dense linear algebra algorithms.In order to do so,we plan to follow the next stages:1.Become familiar with the hardware resources and the people in EPPC.2.Refine the interface of the API for coding numerical linear algebra algorithms.3.Develop the specific codes for the API using the numerical linear algebra libraries:LAPACK,BLAS,and PLAPACK.4.Benchmark parallel codes developed using the API for traditional operations in numerical linear algebra.Nevertheless,after concluding the analysis stage of the parallel extension of the FLAME methodology,we arrived to the conclusion that a far longer period of time was actually needed to satisfactorily achieve the stated objective.Therefore,we decided to make a better use of the available resources at EPCC by targeting a related,but more specific goal:developing a parallel solver of the Algebraic Bernoulli Equation(ABE). This equation has several applications in control theory.In particular,large-scale equations of this class need to be solved,e.g.,in model reduction of unstable dynamical linear systems corresponding to RLC circuits and VLSI devices[5].This way,our objectives turned to the development of a parallel ABE solver alongside with a C library that provided the algebraic operations required by the solver.In turn,this library should make use of the algebra libraries ScaLAPACK,LAPACK,BLAS,and PLAPACK.This report describes the proposed algorithm to parallel solve large-scale ABEs and the experimental results we have obtained on the HPCx system.2.Applied methodologyConsider the Algebraic Bernouilli Equation (ABE):A T X XA ÀXGX 0;1 where A ;G P R n Ân ,G G T ,and X P R n Ân is the sought-after solution.To solve the ABE we proceed by computing in first place the sign function of the following 2n Â2n matrix:H A G 0ÀA T!: 2 To compute sign (H )we use the Newton's root-finding iteration procedure,which leads to the following iteration:A 0X A ;A k 1X 121c k A k c k A À1k ;G 0X G ;G k 1X 121c k G k c k A À1k G k A ÀT k;k 0;1;2F F F 3 where,in order to accelerate the convergence,the determinantal scaling c k j det (A )j 1=n [3]is used.Finally,at convergence,after "k iterations,the solution of (1)is then obtained from the full-rank linear least squares problem:G "k I n ÀA T "k !X A "k I n 0n !: 4 The ABE solver described in this section is basically composed of traditional matrix computations.All these operations can be efficiently performed employing parallel linear algebra libraries for distributed memory computers.Here we have employed the parallel kernels in the ScaLAPACK library [2].ed resourcesThe experiments presented in the next section were performed on the HPCx system [1,6].The HPCx system comprises 50IBM POWER4+Regatta nodes,i.e.1600processors,delivering 10.8TeraFlop/s peak,or up to at least 6TeraFlops/s sustained.The system is equipped with 1.6TByte of memory and 36TByte of disk.256SERGIO BARRACHINA MIR -PETER BENNER -MARK BULL -ETC.Each Regatta system frame consists of 321.7GHz POWER4processors.The total main memory of 32GB per frame is shared between the 32processors of the frame.As we have used at most 20processors,we have employed a single Regatta frame for each one of our experiments.4.Results achievedOur first experiment reports the execution time of the parallel routines for an ABE of dimension n 3200.This is about the larger size we could solve on a single node of the platform considering the number of data matrices involved,the amount of workspace necessary for computations,and the size of the RAM per node 1.The left hand plot in Figure 1;reports the execution time of the parallel algorithm using different number of processors (n p ).The figure shows quite remarkable speed-ups when a small number of processors is employed.As expected,the efficiency decreases as n p gets larger.We next evaluate the scalability of the parallel algorithms when the problem size per node is constant.For that purpose,we fix the problem dimensions to n = n p 3200,and report the Mflops (millions of flops)per node.The right hand plot in Figure 1shows the Mflop rate per node of the parallel routine.These results demonstrate the scalability of our parallel kernels,as there is only a minor decrease in the performance of the algorithms when the number of processors is increased while the problem dimensions per node remains fixed.All the results shown here correspond to 10iterations of thealgorithm.F IGURE 1.±Performance of the ABE solver.1Although a single processor could use 32GB if the other processors in the same frame are not using any memory,we have considered 1GB as the maximum RAM per processor.PARALLEL SOLUTION OF LARGE-SCALE ALGEBRAIC BERNOUILLI ETC.257258SERGIO BARRACHINA MIR-PETER BENNER-MARK BULL-ETC.Acknowledgment.This work has been performed under the Project HPC-EUROPA (RII3-CT-2003-506079),with the support of the European Community-Research Infrastructure Action under the FP6``Structuring the European Research Area'' Programme.References[1]HPCx Home Page.http://www.hpcx-ac-uk/.[2]L.B LACKFORD,J.C HOI,A.C LEARY,E.D'A ZEVEDO,J.D EMME,I.D HILLON,J.D ONGARRA,S.H AMMARLIMG,G.H ENRY,A.P ETITET,K.S TANLEY,D.W ALKER and R.W HALEY.ScaLAPACKUsers'Guide.SIAM,Philadelphia,PA,1997.[3]R.B YERS.Solving the algebraic Riccati equation eith the matrix sign function.LinearAlgebra Appl.,85:267-279,1987.[4]J.G UNNELS,F.G USTAVSON,G.H ENRY and R.VAN DE G EIJN.FLAME:Formal linear algebramethods enviroment.TOMS,27(4):422-455,December2001.[5]M.K AMON,F.W ANG and J.W HITE.Recent improvements for fast inductance extraction andsimulation[packaging].In Proc.of the IEEE7th Topical Meeting on Electrical Performance of Electronic Packaging,pages281.284,1998.[6]M.A SHWORTH,I.J.B USH,M.F.G UEST,M.P LUMMER,A.G.S UNDERLAND,S.B OOTH,D.S.H ENTY,L.S MITH and K.S TRATFORD.HPCx;A new resource for UK computational science.In Proceedings of HPCS'2003,Canada,May2003.Numerical investigation oftrabecular bone behaviourJ IRI B ROZOVSKYDepartment of Structural Mechanics,USB-Technical University of Ostrava,Czech Republic There is considerable interest in the medical community in the mechanical properties of bone,particularly trabecular bone.The interest arises from the need to treat and repair damage to the skeleton due to trauma,disease,and aging.However,a better understanding of the relationships between structure and function of bone are needed in order to develop such diagnostic tools and treatments.The results of experimental investigation have to be supplemented by results of numerical investigations.There are several possibilities of numerical investigation. The most common is the high resolution finite element modelling.The high resolution finite element models are based on high resolution images of pieces of real trabecular bone.But these data are usually unavailable from living patients.It is also near impossible to use this approach to make models of larger pieces of bone.F IGURE1.±Example of one of smallest models.Because of these limitations we have introduced a more simple numerical model.This model respects the anisotropic behaviour of the bone material but it requires much lower number of finite elements.It is based on the combination of finite elements with different (but isotropic)material properties.It means that we are able to create suitable computational models of bone (which still respect the behaviour of material)with much lower numbers of finite elements.It should allow the numerical analysis of a very large pieces of bone.The main advantage is that our model can be used without needs for previous destructive experimental investigation (which is often unaccep-table for living patients).We also have confirmed that these models can be used for both linear and nonlinear analysis.But there are also disadvantages:the number of finite elements is still very high (putation requires usage of the high-performance computing to be usefull)and it is necessary to select reliable material indices to be able to select a suitable numerical model (e.g.distribution of material properties and values of these properties).We have verified that the traditional bone volume to total volume ratio and bone density data can not give all the necessary informations for a selection of an optimal numerical model.Because of this we have started to use a tensor scale parameters (which has been used as an parameter of bone anisotropy in recent papers from number of authors)for this task.References[1]B ROZOVSKY ,J .,P ANKAJ ,P.:Modelling of Trabecular Bone,poster presentation,TAM 2004,17.9.2004,Edinburgh,UK.[2]B ROZOVSKY ,J .,P ANKAJ ,P .,M ATERNA A.:Contribution to Numerical Investigation ofTrabecular Bone Behaviour,3rd International Conference on New Trends in Statics and Dynamics of Buildings,STU Bratislava,21.10.2004,Bratislava,Slovak Republic.260JIRI BROZOVSKYUsing ACO Metaheuristics on LoadBalancing AlgorithmsC ATALIN B ULANCEA*-B EN P AECHTER**-A DAM C ARTER****Institute for Computer Science,Romanian Academy,Iasi Branch**Centre for Emergent Computing,Napier University,Edinburgh***Edinburgh Parralel Computing Centre,The University of EdinburghAbstract.±Dynamic computing environments require adaptive algorithms for load balancing.Ant Colony Optimisation is a metaheuristic which uses system experience in order to reach a stable state.To increase system flexibility,mobile agents are used for encapsulating tasks.During successive migration phases,those agents have the possibility to explore,to learn and to share information about load.Finally, this information is used in maintaining an equitable task load for each processing unit.MPI implementations were used to make simulations and comparisons with different deterministic nearest neighbour load balancing techniques.Results have shown some improvements in terms of convergence stability and task distribution compared with classical algorithms.IntroductionLoad balancing algorithms distribute tasks and communications activity evenly across multi-processing units systems so that no single device is overwhelmed. Equitable task allocation on physical processors is hard to achieve in optimum time. This is the reason that we should choose efficient algorithms for different system topologies and configurations.This problem can be compared with problem arising in naturalwork distributed system in terms of schedul ing,efficient use of workers and constraints between tasks.The aim of a Load Balancing(LB)algorithm can be stated as follows:optimising workload per processor in order to minimize the total execution time and maintain the communication costs to acceptable levels.Ants are simple organisms,also known to have a social behaviour.Ant colonies[1] have inspired the ant algorithms that were used to solve some difficult combinatorial optimisation problems such as the Travelling Salesman Problem(TSP)and the Quadratic Assignment Problem(QAP)[2].Ant algorithms allow adaptation to a environment,one of the features required to balance dynamic environments.In this paper a mobile agent based algorithm for LB in distributed memory message-passing computers will be presented and comparisons will be made between balancing an arbitrary graph topology computing environment using deterministic(DASUD)[3] and adaptive algorithms.The agent algorithm uses Ant Colony Optimisation(ACO)and a pheromone trailfor routing the tasks[4].This strategy can be stated as a nearest262CATALIN BULANCEA-BEN PAECHTER-ADAM CARTERneighbour LB method which tries to make a global balance using local neighbour-to-neighbour[5]transfers.To cover medium and large grain parallelism the workload will be represented by an integer.The Load Balancing ProblemThe generalized problem of LB is known to be NP complete[6].The LB problem has two important aspects,quality of load distribution and scheduling length.A good LB strategy has to produce an equitable task distribution over the system's processes in an acceptable time.Considering T execution1(the time of execution without balance and T execution2 (time of execution with balance)we will presume that:T execution1>T execution2(1)Assuming T execution1=T max_load_bb and T execution2=T max_load_ab+T balance,then (2)T max load bb>T max load ab T balancewhere T max_load_bb±time of execution for maximum loaded processor before balance, T max_load_ab is the time of execution for maximum loaded processor after balance and T balance is the time to re-balance the system.An efficient balancing scheme should follow:(3)T balance T max load bbÀT max load abIf T balance does not satisfy(3),then the time to transfer load units is too high or the LB scheme has a great time complexity and it is better to not reschedule tasks.In some cases,the LB algorithm is followed prior to the execution and,it is performed only once;this is called static load balancing or static mapping.This is effective for computation which has predictive run time behaviour and the dependen-cies graph is well known.If the computation run-time is not predictable,performing LB only once at the beginning is not sufficient.For these cases it might be better re-balancing the system more than once or periodically during run-time to ensure that the dynamic behaviour more closely matches the available resources.This way of scheduling is called dynamic LB.The dynamic LB procedures can be either optimal,when we can be sure that we have the best configuration,or sub-optimal,when heuristics can be used for com-puting sub-optimal solutions in an acceptable time.Description of agent based LB algorithmWe state that each agent is encapsulating[7]a single task.An execution queue is considered for holding the agents until they are executed so each node has a logical architecture as drawn in figure1:The queue's management thread should be kept within reasonable limits of complexity to avoid system overhead.Agents use a probabilistic system [8]to decide migration to another physical processor.During migrations,they will accumulate information about the load in the distributed system and will share this load with other rmation will be used while trying to estimate the global average load and system error.So,each agent will compute his own average load(a_l )according to:a l 21Á(a l cr n l )(4)and a l 212Áa l 1m À1m À1j 1oth ag l j23(5)where m is the number of agents in current node and oth_ag_l j is the average load of agent j from current node.The estimated system error will be:est error 12Áa l À1n n i 1neigh n l i a l a l À1m À1 m À1j 1oth ag l j a lH f f f d I g g g e (6)where n is the number of adjacent nodes,neigh_n_li is the load of adjacent node i,m -number of agents in current node and oth_ag_l j is the average load of agent j from the current node.Each agent holds a position (where position P f 0;1;2;F F F ;m g )in the execution queue in accordance with its priority.Therefore,the agent which holds the second position has a greater probability to be executed in a shorter time that an agent is in the fifth position.This is expressed in the probabilistic decision function which is stated as:p leave position Àk(7)F IGURE 1.±Node's computational architecture.USING ACO METAHEURISTICS ON LOAD BALANCING ALGORITHMS 263wherek12Áa l;position a l(1Àtf)À12Áa l;position>a l(1Àtf) Vb b b`b b bX(8)The tf variable is used for making agents more active if the system's error is considerable.It is expressed below:t f exp(est error)À1(9)where est_error was defined in(6).In a synthetic way,during successive migrations through various nodes,agents try to learn the global average load of the system and try to estimate the error.Then,they will try to reach the global balance state,while making probabilistic decisions using those values as thresholds.To increase the efficiency of their migrations,a modified ACO pheromone trail metaheuristic will be used.Trails will have directional attributes,so each node will have his own pheromone table;a local process of normalisation was also consider-ed(each trail<1and S trails=1).Assuming that the node degree is d,each pheromone table will have d+1entries, where d of them are for adjacent nodes and one is provided for current node and can be stated as node_capacity(n_c).If the agent decides to migrate and chooses the entry corresponding to current node,then it will remain in current node.Considering node i,with three adjacent nodes and a task which is coming in,as shown in next figure2:The following equations for the pheromone table will be considered:t a2t a 1 D t1(10)F IGURE2.±Task coming in node i.264CATALIN BULANCEA-BEN PAECHTER-ADAM CARTERt x 2t xD t 1d 1(11)n c 2n c D t 1d 1 D t 1(12)where D t 1is the trailamount used to modify the pheromone tabl e when an agent is coming in current node,t a ,t x are the corresponding trails for routes a and x ,x P f b ;c g ,d is degree of node i .If a task is leaving node i (as in Figure 3)then another set of equations for updating the pheromone trails should be considered.t a 2t a D t 2d 1 D t (13)t x 2t x 1 D t 2(14)n c 2n c1 D t 2(15)where D t 2is the quantity used to modify the pheromone table when an agent is leaving current node.The main reason for equations (10-15)can be stated as follows:if tasks are coming into the current node from a certain direction,then the probability of sending other tasks in that direction should be minimized in order to avoid overloading for those computationalresources;al so,for increased efficiency in routing agents through under loaded nodes,we should consider D t 1>D t 2(experimentally was noticed that the optimum ratio was D t 1D t 21:2)F IGURE 3.±Task leaving node i.USING ACO METAHEURISTICS ON LOAD BALANCING ALGORITHMS 265266CATALIN BULANCEA-BEN PAECHTER-ADAM CARTER The probabilistic migration mechanism and the directional pheromone trails were combined in a LB strategy which offers sub-optimalheuristic sol ution for arbitrary graph topology computing systems.For simulations,the message-passing paradigm was considered,implementing agent-carrying tasks as messages exchanged between processes.However,a single thread was considered for comparing the efficiency of the agent based algorithm with other methods.for each node parallel do:for each agent in current node do://this is a lifecycleagent.gath_inf_adj_node_lds()agent.gath_inf_oth_ag_avglds()if(agent.prec_position<=position)thenagent.choose_node(pheromone_table);agent.reconsider_prec_position()//stabilization fnelseagent.decide(position);if(agent.decision==leave)thenagent.choose_node(pheromone_table);agent.prec_position=position;end_for;population.update();go_update_pheromon_tables(pheromon_tables);exchange_agents();arrive_update_pheromon_tables(pheromon_tables);if(agents arrived in current node)thenpopulation.update()end_parallel_for;Acknowledgment.The work has been performed under the Project HPC-EUROPA (RII3-CT-2003-506079),with the support of the European Community-Research Infrastructure Action under the FP6``Structuring the European Research Area'' Programme.Experiments included tests of convergence and execution simulations on16,32, 64nodes architectures.There were made comparative tests with DASUD(Diffuse Algorithm Searching Unbalanced Domains)algorithms,considering tasks about100-1000cycles.The simulation considered only the logical efficiency of named algorithms, time for execution being measured on cycles.Main tests were made using an inconstant environment topology with32 nodes.In the figure below is shown a relation between tasks characteristic and the T dasud/T agent_algorithm ratio.This measure give a comparison of the performance of the adaptive algorithm against the DASUD deterministic algorithm.Each task was considered for about200cycles and initially the load was disposed in four nodes.In first chart (Figure 5)T dasud /T agent_algorithm ratio is shown correlated to task number.Each value is an average result from three different experiments.In the second chart (Figure 6),the relation between the T dasud /T agent_algorithm and task size is expressed.The initial load was about 88tasks,initially disposed in four nodes as shown in Figure 4.It was used the same experimentalscheme that the one used for the chart in Figure 6.In the last chart (Figure 7)the same ratio T dasud /T agent_algorithm is shown,related to the number of processors.The load was about 120tasks,each lasting about 200cycles.F IGURE 4.±32nodes graphenvironment.F IGURE 5.±Comparative tests while increasing number of tasks.USING ACO METAHEURISTICS ON LOAD BALANCING ALGORITHMS 267Conclusions and future workThere is need for LB in dynamic environments (here,the ``platform effect''[4]is amplified in most cases)and those strategies require adaptation.Because of the probabilistic decision mechanism,the agent algorithm has a slower speed of conver-gence than DASUD as it can be seen in the chart in Figure 7.However,when the task's time requirement is increased,their allocation on physical processor is generally better than that given by the DASUD algorithm (Figure6).F IGURE 6.±Comparative tests while increasing tasksize.F IGURE 7.±Comparative test while increasing the number of nodes.268CATALIN BULANCEA -BEN PAECHTER -ADAM CARTERUSING ACO METAHEURISTICS ON LOAD BALANCING ALGORITHMS269 The next steps in researching this LB technology will include a combined strategy random/deterministic for speeding up the converging process and a mixed simulator which will use MPI and OpenMP libraries and allow real-time comparative tests.Publications[1]D ORIGO M.and G.D I C ARO(1999).The Ant Colony Optimization Meta-Heuristic.In D.Corne,M.Dorigo and F.Glover,editors,New Ideas in Optimization,McGraw-Hill,11-32.[2]M.D ORIGO,G.D I C ARO and L.M.G AMBARDELLA,Ant Algorithms for discrete optimization,ArtificialLife,Vol.5,No.3,(1999)pp.137-172.[3] A.C ORTEÂS,A.R IPOLL,M.A.S ENAR,F.C EDOÂand E.L UQUE,On the convergence of SID andDASUD load-balancing algorithms,TechnicalReport,Universitat Autonoma de Barcelona(1998)Spain.[4]R UUD S CHOONDERWOERD,O WEN H OLLAND,J ANET B RUTEN,Ant-like agents for load balancingin telecommunications network s,InternationalConference on Autonomous Agents, Proceedings of the first internationalconference on Autonomous agents,Marina delRey, California,United States,1997,ISBN0-89791-877-0,pp.209±216[5] A.C ORTEÂS,A.R IPOLL,M.A.S ENAR,F.C EDOÂand E.L UQUE,Performance Comparison ofDynamic Load-Balancing Strategies for Distributed Computing,32nd Hawaii Interna-tionalConference of System Sciences,1999.[6]D AN G RIGORAS,Parallel Computing±Systems and Applications,Computer Libris Agora,2000,ISBN973-97534-6-9,pp.338-360.[7]Y.W ANG,J.L IU and X.J IN,Modeling Agent-Based Load Balancing with Time Delays,InProceeding of WI/IAT2003-IEEE/WIC International Conference on Intelligent Agent Technology,October13-17(2003)Halifax,Canada.[8]M.C RAUS,C.B ULANCEA,An agent based model for real-time load balancing in non-uniformconnected computing environments,Third European Conference on Intelligent Systems and Technologies ECIT'200421-23July2004,Iasi,Romania.Soft-Tissue Deformation Predictionfor Maxillo-Facial Surgical PlanningH UGUES F ONTENELLE*-A LESSANDRO S ARTI**R OBERTO G ORI***-C LAUDIO L AMBERTI****Dept.of Medical Physics,School of Medicine,University of Patras,Greece**Dept.of Electronics,Computer Science and Systems,University of Bologna,Italy***CINECA,Interuniversity Consortium,Bologna,Italy***Dept.of Electronics,Computer Science and Systems,University of Bologna,ItalyAbstract.±In aesthetical maxillo-facial surgery,both the surgeon and the patient would like to foresee the results of a particular surgery.We present a software which helps the surgeon planning the operation,tracing osteotomy lines and deciding of bone displacements.We model the behavior of the skin and solve its new shape according to planning inputs.We extend the theory for a novel approach for solving Elliptic Partial Differential Equations(PDE's)on irregular domains,in3D.We implement it and show how it could be applied to a specific area of medicine.IntroductionThe purpose of the V I S U(Virtual Surgery)system is to plan virtually(i.e.by means of a software)the essential steps of a maxillo-facial surgical procedure.It allows the surgeon to evaluate different kind of procedures,and find the best according to an aesthetical point of view.It gives the patient the possibility to appreciate the hypothetical outcome of the real surgical intervention.The organ geometry is obtained from CT medical images.The simulation is performed directly on the grid of the3D CT, by Finite-Differences.This approach saves the usual time required for building the tetrahedral mesh,as in the Finite-Elements approach.The inputs for the model,i.e.the bone displacements,are planned interactively by the surgeon who traces osteotomy lines,defining anatomical regions using the mouse or a haptic device.The software is a collection of tools for data acquisition,3D reconstruction,input visualization,surgical planning,numerical simulation and output visualization as described earlier in[1]and [2].Here we address the numerical simulation part,and seek to improve the current solver.Earlier works,such as[4]of mechanical engineering,demonstrate that a Finite Difference approach can be as good as a finite element one,but run significantly faster. This is especially the case for applications of medical imaging because3D datasets are already discretized in a regular fashion.The V I S U software and its previous numerical solver already gave satisfying results,but we hoped that,after validation,the presented novel approach based on EJIIM will further reduce the error margin.MethodsWe choose the Lame Âequation of elasticity to describe the behavior of the soft-tissue.In [3],Wiegmann introduces the Explicit Jump Immersed Interface Method (EJIIM),which solves PDEs on irregular domain.With to this new approach,our Elasticity equation is discretized on the whole grid.The irregular interface is introduced by jumps,along with the traction-free Boundary Conditions,then em-bedded with the normal equation to form a linear system:A D T ÀC IU J F 1F 2 where A is the discretized Lame Âequation matrix,C distributes jump coefficients of the interface,D T perform grid-to-interface extrapolation for unknown interface values (by tri-variate quadratic polynomials),U is the solution,J holds the jump coefficients,F 1is zeros,F 2is zero for unknown jumps or u 0 x for known displacements.More can be found in [6].ResultsThe EJIIM theory has been extended to the third dimension.A C++library has been developed for solving this class of problems.It makes use of the Portable Extensible Toolkit for Scientific Computation (PETSc)for handling the sparse matrices and solving the linear systems.We computed the error between the simulation prediction and the post-operative CT scans of patients three months after surgery,as the validation of the previous model.Unfortunately,our solver does not perform better.We identified this bad behavior as being due to D T .This grid-to-interface extrapolation needs enough points to perform well,but in our particular case,the domain of computation is too ``thin''.ConclusionsAlthough we did not improve the error of the previous solver,we extended a theory and implemented a class for solving Partial Differential Equations with Finite Differences on Irregular Domain in three-dimensions.We know that this approach must find other applications in which the domain of computation is ``thick''enough.A work-around however is possible,using adaptive meshing,and further research in this direction might lead to better results.Acknowledgments.The Inter-University Post-Graduate Course on Bio-Medical Engineering of the University of Patras,Greece.The Bioimaging Group,Biomedical Engineering Laboratory,DEIS,University of Bologna,Italy.The work has been performed under the Project HPC-EUROPA (RII3-CT-2003-506079),with the support SOFT-TISSUE DEFORMATION PREDICTION FOR MAXILLO-FACIAL SURGICAL PLANNING 271。
流形及分形上分析及偏微分方程国际会议
会议报道流形及分形上分祈及偏微分方暇国际尝议孙玉华(南开大学数学学院，天津300071)流形及分形上分析及偏微分方程国际会议于2019年9月22〜26 H在南开大学陈省身研究所举办。
会议邀请了国内外知名教授参会并作报告。
此外，还有国内外的众多青年学者在本次会议上受邀作了报告。
参加此次会议的人数逾90位。
流形及分形上分析及偏微分方程国际会议主要围绕“流形分析”、“分形分析”及“偏微分方程”3个主题展开。
会议邀请报告涉及到的课题方向均为以上三个主题交叉的研究方向，如热核估计、度量空间上的偏微分方程、流形的随机几何等等。
美国华盛顿大学陈振庆教授带来了最新的研究成果，关于对称狄氏型热核估计的稳定型研究及Harnark不等式的研究，该问题主要针对在一般度量空间上既有扩散又有跳跃的马尔科夫过程。
在一般的体积条件下及一些比较弱的假设下如跳跃核、容度条件及庞加莱不等式下建立了热核的双边估计；同时对相关的抛物型Harnack不等式的稳定性进行了刻画。
纽约城市大学Dodziuk教授带来了关于具有正曲率连通和流形的报告。
他回顾了基于Gromov和Lawson想法如何在和流形上基于度量下构造正曲率的例子。
这种改进可以在任何大于等于三维的流形上操作，并且在检验正曲率流形的极限的性质上非常有用。
香港中文大学丰德军教授带来了关于带有重叠的自相似测度维数的估计的报告。
他介绍了如何在带重叠自相似测度空间上计算维数的上下界估计的办法。
利用这种办法，他介绍了如何在伯努利卷积上去估算维数的做法。
比勒菲尔德大学Grig〇r‘y a n教授作了题为“分孙玉华：副教授。
，收稿日期：2019-10-30 48Tel:185****7104形及流形上分析及偏微分方程”的报告，介绍了体积估计与型问题、随机完备性问题、热核估计及薛定谔方程、半线性椭圆方程以及布朗运动逃逸速率的联系。
他的报告完美契合了我们此次会议的题目。
美国西北大学徐佩教授带来了关于流形上倒向随机微分方程的几何的报告。
欧几里得几何推理 英文
欧几里得几何推理英文Euclidean Geometry ReasoningGeometry has been a fundamental branch of mathematics since ancient times, and Euclidean geometry, named after the renowned Greek mathematician Euclid, has played a pivotal role in its development. Euclid's work, compiled in his seminal text "Elements," laid the foundation for our understanding of the properties and relationships of geometric shapes and their behavior in a two-dimensional plane.At the heart of Euclidean geometry lies the concept of axioms, which are self-evident truths that serve as the starting point for logical deductions. These axioms, combined with a set of postulates, form the basis for the construction of a coherent and consistent system of geometric principles. From these fundamental building blocks, Euclid and his successors were able to derive a vast array of theorems and proofs that have stood the test of time and remain integral to our understanding of the physical world.One of the most fundamental principles in Euclidean geometry is the concept of congruence. Two geometric figures are said to becongruent if they have the same size and shape, and can be superimposed on one another without any distortion. This idea of congruence is essential in many areas of geometry, from the construction of triangles and quadrilaterals to the study of transformations and symmetry.Another key concept in Euclidean geometry is the notion of parallel lines. Parallel lines are those that never intersect, no matter how far they are extended. Euclid's fifth postulate, often referred to as the parallel postulate, states that given a line and a point not on that line, there exists a unique line passing through the point that is parallel to the original line. This seemingly simple yet profound statement has been the subject of much debate and exploration throughout the history of mathematics.The study of triangles is a fundamental aspect of Euclidean geometry, and many of the theorems and proofs within this field revolve around the properties and relationships of these three-sided figures. From the Pythagorean theorem, which relates the lengths of the sides of a right triangle, to the various congruence and similarity criteria, the study of triangles has provided a rich tapestry of geometric understanding.Quadrilaterals, too, play a crucial role in Euclidean geometry. The classification and properties of these four-sided figures, such asrectangles, squares, rhombi, and trapezoids, have led to a deeper understanding of the underlying symmetries and relationships within the geometric plane.Beyond the study of individual shapes, Euclidean geometry also explores the concept of transformations, which are the various ways in which a figure can be moved, rotated, or scaled without altering its essential properties. These transformations, such as translations, reflections, and rotations, have important applications in fields ranging from art and design to computer graphics and engineering.The elegance and logical rigor of Euclidean geometry have made it a cornerstone of mathematical education and a powerful tool for problem-solving and reasoning. Its principles have been applied in a wide range of disciplines, from architecture and engineering to physics and astronomy. Moreover, the study of Euclidean geometry has paved the way for the development of more advanced geometric systems, such as non-Euclidean geometries, which have expanded our understanding of the nature of space and the universe.In conclusion, Euclidean geometry, with its foundational axioms, theorems, and proofs, remains a vital and dynamic field of study. Its ability to provide a coherent and logical framework for understanding the properties and relationships of geometric shapes has made it an enduring and essential component of mathematicaleducation and research. As we continue to explore the depths of this ancient discipline, we uncover new insights and applications that deepen our appreciation for the beauty and power of geometric reasoning.。
Fourier Transform
Fourier transformIn mathematics, the Fourier transform is the operation that decomposes a signal into its constituent frequencies. Thus the Fourier transform of a musical chord is a mathematical representation of the amplitudes of the individual notes that make it up. The original signal depends on time, and therefore is called the time domain representation of the signal, whereas the Fourier transform depends on frequency and is called the frequency domain representation of the signal. The term Fourier transform refers both to the frequency domain representation of the signal and the process that transforms the signal to its frequency domain representation.More precisely, the Fourier transform transforms one complex-valued function of a real variable into another. In effect, the Fourier transform decomposes a function into oscillatory functions. The Fourier transform and its generalizations are the subject of Fourier analysis. In this specific case, both the time and frequency domains are unbounded linear continua. It is possible to define the Fourier transform of a function of several variables, which is important for instance in the physical study of wave motion and optics. It is also possible to generalize the Fourier transform on discrete structures such as finite groups. The efficient computation of such structures, by fast Fourier transform, is essential for high-speed computing.DefinitionThere are several common conventions for defining the Fourier transform of an integrable function ƒ: R→ C (Kaiser 1994). This article will use the definition:for every real number ξ.When the independent variable x represents time (with SI unit of seconds), the transform variable ξ representsfrequency (in hertz). Under suitable conditions, ƒ can be reconstructed from by the inverse transform:for every real number x.For other common conventions and notations, including using the angular frequency ω instead of the frequency ξ, see Other conventions and Other notations below. The Fourier transform on Euclidean space is treated separately, in which the variable x often represents position and ξ momentum.IntroductionThe motivation for the Fourier transform comes from the study of Fourier series. In the study of Fourier series, complicated functions are written as the sum of simple waves mathematically represented by sines and cosines. Due to the properties of sine and cosine it is possible to recover the amount of each wave in the sum by an integral. In many cases it is desirable to use Euler's formula, which states that e2πiθ = cos 2πθ + i sin 2πθ, to write Fourier series in terms of the basic waves e2πiθ. This has the advantage of simplifying many of the formulas involved and providing a formulation for Fourier series that more closely resembles the definition followed in this article. This passage from sines and cosines to complex exponentials makes it necessary for the Fourier coefficients to becomplex valued. The usual interpretation of this complex number is that it gives both the amplitude (or size) of the wave present in the function and the phase (or the initial angle) of the wave. This passage also introduces the need for negative "frequencies". If θ were measured in seconds then the waves e2πiθ and e−2πiθ would both complete one cycle per second, but they represent different frequencies in the Fourier transform. Hence, frequency no longer measures the number of cycles per unit time, but is closely related.There is a close connection between the definition of Fourier series and the Fourier transform for functions ƒ which are zero outside of an interval. For such a function we can calculate its Fourier series on any interval that includes the interval where ƒ is not identically zero. The Fourier transform is also defined for such a function. As we increase the length of the interval on which we calculate the Fourier series, then the Fourier series coefficients begin to look like the Fourier transform and the sum of the Fourier series of ƒ begins to look like the inverse Fourier transform. To explain this more precisely, suppose that T is large enough so that the interval [−T/2,T/2] contains the interval onis given by:which ƒ is not identically zero. Then the n-th series coefficient cnComparing this to the definition of the Fourier transform it follows that since ƒ(x) is zero outside [−T/2,T/2]. Thus the Fourier coefficients are just the values of the Fourier transform sampled on a grid of width 1/T. As T increases the Fourier coefficients more closely represent the Fourier transform of the function.Under appropriate conditions the sum of the Fourier series of ƒ will equal the function ƒ. In other words ƒ can be written:= n/T, and Δξ = (n + 1)/T − n/T = 1/T. where the last sum is simply the first sum rewritten using the definitions ξnThis second sum is a Riemann sum, and so by letting T → ∞ it will converge to the integral for the inverse Fourier transform given in the definition section. Under suitable conditions this argument may be made precise (Stein & Shakarchi 2003).could be thought of as the "amount" of the wave in the Fourier series of In the study of Fourier series the numbers cnƒ. Similarly, as seen above, the Fourier transform can be thought of as a function that measures how much of each individual frequency is present in our function ƒ, and we can recombine these waves by using an integral (or "continuous sum") to reproduce the original function.The following images provide a visual illustration of how the Fourier transform measures whether a frequency is present in a particular function. The function depicted oscillates at 3 hertz (if t measures seconds) and tends quickly to 0. This function was specially chosen to have a real Fourier transform which can easily be plotted. The first image contains its graph. In order to calculate we must integrate e−2πi(3t)ƒ(t). The second image shows the plot of the real and imaginary parts of this function. The real part of the integrand is almost always positive, this is because when ƒ(t) is negative, then the real part of e−2πi(3t) is negative as well. Because they oscillate at the same rate, when ƒ(t) is positive, so is the real part of e−2πi(3t). The result is that when you integrate the real part of the integrand you get a relatively large number (in this case 0.5). On the other hand, when you try to measure a frequency that is not present, as in the case when we look at , the integrand oscillates enough so that the integral is very small. The general situation may be a bit more complicated than this, but this in spirit is how the Fourier transform measures how much of an individual frequency is present in a function ƒ(t).Original function showingoscillation 3 hertz.Real and imaginary parts of integrand for Fourier transformat 3 hertzReal and imaginary parts of integrand for Fourier transformat 5 hertz Fourier transform with 3 and 5hertz labeled.Properties of the Fourier transformAn integrable function is a function ƒon the real line that is Lebesgue-measurable and satisfiesBasic propertiesGiven integrable functions f (x ), g (x ), and h (x ) denote their Fourier transforms by, , andrespectively. The Fourier transform has the following basic properties (Pinsky 2002).LinearityFor any complex numbers a and b , if h (x ) = aƒ(x ) + bg(x ), thenTranslationFor any real number x 0, if h (x ) = ƒ(x − x 0), thenModulationFor any real number ξ0, if h (x ) = e 2πixξ0ƒ(x ), then.ScalingFor a non-zero real number a , if h (x ) = ƒ(ax ), then. The case a = −1 leads to the time-reversal property, which states: if h (x ) = ƒ(−x ), then.ConjugationIf , thenIn particular, if ƒ is real, then one has the reality conditionAnd ifƒ is purely imaginary, thenConvolutionIf , thenUniform continuity and the Riemann–Lebesgue lemmaThe rectangular function is Lebesgue integrable.The sinc function, which is the Fourier transform of the rectangular function, is bounded andcontinuous, but not Lebesgue integrable.The Fourier transform of an integrable function ƒ is bounded and continuous, but need not be integrable – for example, the Fourier transform of the rectangular function, which is a step function (and hence integrable) is the sinc function, which is not Lebesgue integrable, though it does have an improper integral: one has an analog to thealternating harmonic series, which is a convergent sum but not absolutely convergent.It is not possible in general to write the inverse transform as a Lebesgue integral. However, when both ƒ and are integrable, the following inverse equality holds true for almost every x:Almost everywhere, ƒ is equal to the continuous function given by the right-hand side. If ƒ is given as continuous function on the line, then equality holds for every x.A consequence of the preceding result is that the Fourier transform is injective on L1(R).The Plancherel theorem and Parseval's theoremLet f(x) and g(x) be integrable, and let and be their Fourier transforms. If f(x) and g(x) are also square-integrable, then we have Parseval's theorem (Rudin 1987, p. 187):where the bar denotes complex conjugation.The Plancherel theorem, which is equivalent to Parseval's theorem, states (Rudin 1987, p. 186):The Plancherel theorem makes it possible to define the Fourier transform for functions in L2(R), as described in Generalizations below. The Plancherel theorem has the interpretation in the sciences that the Fourier transform preserves the energy of the original quantity. It should be noted that depending on the author either of these theorems might be referred to as the Plancherel theorem or as Parseval's theorem.See Pontryagin duality for a general formulation of this concept in the context of locally compact abelian groups.Poisson summation formulaThe Poisson summation formula provides a link between the study of Fourier transforms and Fourier Series. Given an integrable function ƒ we can consider the periodic summation of ƒ given by:where the summation is taken over the set of all integers k. The Poisson summation formula relates the Fourier series of to the Fourier transform of ƒ. Specifically it states that the Fourier series of is given by:Convolution theoremThe Fourier transform translates between convolution and multiplication of functions. If ƒ(x) and g(x) are integrablefunctions with Fourier transforms and respectively, then the Fourier transform of the convolution is given by the product of the Fourier transforms and (under other conventions for the definition of theFourier transform a constant factor may appear).This means that if:where ∗ denotes the convolution operation, then:In linear time invariant (LTI) system theory, it is common to interpret g(x) as the impulse response of an LTI systemwith input ƒ(x) and output h(x), since substituting the unit impulse for ƒ(x) yields h(x) = g(x). In this case, represents the frequency response of the system.Conversely, if ƒ(x) can be decomposed as the product of two square integrable functions p(x) and q(x), then theFourier transform of ƒ(x) is given by the convolution of the respective Fourier transforms and .Cross-correlation theoremIn an analogous manner, it can be shown that if h(x) is the cross-correlation of ƒ(x) and g(x):then the Fourier transform of h(x) is:As a special case, the autocorrelation of function ƒ(x) is:for whichEigenfunctionsOne important choice of an orthonormal basis for L2(R) is given by the Hermite functionswhere are the "probabilist's" Hermite polynomials, defined by Hn(x) = (−1)n exp(x2/2) D n exp(−x2/2). Under this convention for the Fourier transform, we have thatIn other words, the Hermite functions form a complete orthonormal system of eigenfunctions for the Fourier transform on L2(R) (Pinsky 2002). However, this choice of eigenfunctions is not unique. There are only four different eigenvalues of the Fourier transform (±1 and ±i) and any linear combination of eigenfunctions with the same eigenvalue gives another eigenfunction. As a consequence of this, it is possible to decompose L2(R) as a directsum of four spaces H0, H1, H2, and H3where the Fourier transform acts on Hksimply by multiplication by i k. Thisapproach to define the Fourier transform is due to N. Wiener (Duoandikoetxea 2001). The choice of Hermite functions is convenient because they are exponentially localized in both frequency and time domains, and thus give rise to the fractional Fourier transform used in time-frequency analysis (Boashash 2003).Fourier transform on Euclidean spaceThe Fourier transform can be in any arbitrary number of dimensions n. As with the one-dimensional case there are many conventions, for an integrable function ƒ(x) this article takes the definition:where x and ξ are n-dimensional vectors, and x·ξ is the dot product of the vectors. The dot product is sometimes written as .All of the basic properties listed above hold for the n-dimensional Fourier transform, as do Plancherel's and Parseval's theorem. When the function is integrable, the Fourier transform is still uniformly continuous and the Riemann–Lebesgue lemma holds. (Stein & Weiss 1971)Uncertainty principleGenerally speaking, the more concentrated f(x) is, the more spread out its Fourier transform must be. In particular, the scaling property of the Fourier transform may be seen as saying: if we "squeeze" a function in x, its Fourier transform "stretches out" in ξ. It is not possible to arbitrarily concentrate both a function and its Fourier transform.The trade-off between the compaction of a function and its Fourier transform can be formalized in the form of an Uncertainty Principle by viewing a function and its Fourier transform as conjugate variables with respect to the symplectic form on the time–frequency domain: from the point of view of the linear canonical transformation, the Fourier transform is rotation by 90° in the time–frequency domain, and preserves the symplectic form.Suppose ƒ(x) is an integrable and square-integrable function. Without loss of generality, assume that ƒ(x) is normalized:It follows from the Plancherel theorem that is also normalized.The spread around x = 0 may be measured by the dispersion about zero (Pinsky 2002) defined byIn probability terms, this is the second moment of about zero.The Uncertainty principle states that, if ƒ(x ) is absolutely continuous and the functions x ·ƒ(x ) and ƒ′(x ) are square integrable, then(Pinsky 2002).The equality is attained only in the case (hence ) where σ > 0is arbitrary and C 1 is such that ƒ is L 2–normalized (Pinsky 2002). In other words, where ƒ is a (normalized) Gaussian function, centered at zero.In fact, this inequality implies that:for any in R (Stein & Shakarchi 2003).In quantum mechanics, the momentum and position wave functions are Fourier transform pairs, to within a factor of Planck's constant. With this constant properly taken into account, the inequality above becomes the statement of the Heisenberg uncertainty principle (Stein & Shakarchi 2003).Spherical harmonicsLet the set of homogeneous harmonic polynomials of degree k on R n be denoted by A k . The set A k consists of the solid spherical harmonics of degree k . The solid spherical harmonics play a similar role in higher dimensions to the Hermite polynomials in dimension one. Specifically, if f (x ) = e −π|x |2P (x ) for some P (x ) in A k , then. Let the set H k be the closure in L 2(R n ) of linear combinations of functions of the form f (|x |)P (x )where P (x ) is in A k . The space L 2(R n ) is then a direct sum of the spaces H k and the Fourier transform maps each space H k to itself and is possible to characterize the action of the Fourier transform on each space H k (Stein & Weiss 1971). Let ƒ(x ) = ƒ0(|x |)P (x ) (with P (x ) in A k ), then whereHere J (n + 2k − 2)/2 denotes the Bessel function of the first kind with order (n + 2k − 2)/2. When k = 0 this gives a useful formula for the Fourier transform of a radial function (Grafakos 2004).Restriction problemsIn higher dimensions it becomes interesting to study restriction problems for the Fourier transform. The Fourier transform of an integrable function is continuous and the restriction of this function to any set is defined. But for a square-integrable function the Fourier transform could be a general class of square integrable functions. As such, the restriction of the Fourier transform of an L 2(R n ) function cannot be defined on sets of measure 0. It is still an active area of study to understand restriction problems in L p for 1 < p < 2. Surprisingly, it is possible in some cases to define the restriction of a Fourier transform to a set S , provided S has non-zero curvature. The case when S is the unit sphere in R n is of particular interest. In this case the Tomas-Stein restriction theorem states that the restriction of the Fourier transform to the unit sphere in R n is a bounded operator on L p provided 1 ≤ p ≤ (2n + 2) / (n + 3).One notable difference between the Fourier transform in 1 dimension versus higher dimensions concerns the partial sum operator. Consider an increasing collection of measurable sets E R indexed by R ∈ (0,∞): such as balls of radius R centered at the origin, or cubes of side 2R . For a given integrable function ƒ, consider the function ƒR defined by:Suppose in addition that ƒ is in L p (R n ). For n = 1 and 1 < p < ∞, if one takes E R = (−R, R), then ƒR converges to ƒ in L p as R tends to infinity, by the boundedness of the Hilbert transform. Naively one may hope the same holds true forn > 1. In the case that ERis taken to be a cube with side length R, then convergence still holds. Another naturalcandidate is the Euclidean ball ER= {ξ : |ξ| < R}. In order for this partial sum operator to converge, it is necessary that the multiplier for the unit ball be bounded in L p(R n). For n ≥ 2 it is a celebrated theorem of Charles Fefferman that the multiplier for the unit ball is never bounded unless p = 2 (Duoandikoetxea 2001). In fact, when p≠ 2, thisshows that not only may ƒR fail to converge to ƒ in L p, but for some functions ƒ ∈ L p(R n), ƒRis not even an element ofL p.GeneralizationsFourier transform on other function spacesIt is possible to extend the definition of the Fourier transform to other spaces of functions. Since compactly supported smooth functions are integrable and dense in L2(R), the Plancherel theorem allows us to extend the definition of the Fourier transform to general functions in L2(R) by continuity arguments. Further : L2(R) →L2(R) is a unitary operator (Stein & Weiss 1971, Thm. 2.3). Many of the properties remain the same for the Fourier transform. The Hausdorff–Young inequality can be used to extend the definition of the Fourier transform to include functions in L p(R) for 1 ≤ p≤ 2. Unfortunately, further extensions become more technical. The Fourier transform of functions in L p for the range 2 < p < ∞ requires the study of distributions (Katznelson 1976). In fact, it can be shown that there are functions in L p with p>2 so that the Fourier transform is not defined as a function (Stein & Weiss 1971).Fourier–Stieltjes transformThe Fourier transform of a finite Borel measure μ on R n is given by (Pinsky 2002):This transform continues to enjoy many of the properties of the Fourier transform of integrable functions. One notable difference is that the Riemann–Lebesgue lemma fails for measures (Katznelson 1976). In the case that dμ = ƒ(x) dx, then the formula above reduces to the usual definition for the Fourier transform of ƒ. In the case that μ is the probability distribution associated to a random variable X, the Fourier-Stieltjes transform is closely related to the characteristic function, but the typical conventions in probability theory take e ix·ξ instead of e−2πix·ξ (Pinsky 2002). In the case when the distribution has a probability density function this definition reduces to the Fourier transform applied to the probability density function, again with a different choice of constants.The Fourier transform may be used to give a characterization of continuous measures. Bochner's theorem characterizes which functions may arise as the Fourier–Stieltjes transform of a measure (Katznelson 1976). Furthermore, the Dirac delta function is not a function but it is a finite Borel measure. Its Fourier transform is a constant function (whose specific value depends upon the form of the Fourier transform used).Tempered distributionsThe Fourier transform maps the space of Schwartz functions to itself, and gives a homeomorphism of the space to itself (Stein & Weiss 1971). Because of this it is possible to define the Fourier transform of tempered distributions. These include all the integrable functions mentioned above, as well as well-behaved functions of polynomial growth and distributions of compact support, and have the added advantage that the Fourier transform of any tempered distribution is again a tempered distribution.The following two facts provide some motivation for the definition of the Fourier transform of a distribution. First let ƒ and g be integrable functions, and let and be their Fourier transforms respectively. Then the Fourier transform obeys the following multiplication formula (Stein & Weiss 1971),Secondly, every integrable function ƒ defines a distribution Tƒby the relationfor all Schwartz functions φ.In fact, given a distribution T, we define the Fourier transform by the relationfor all Schwartz functions φ.It follows thatDistributions can be differentiated and the above mentioned compatibility of the Fourier transform with differentiation and convolution remains true for tempered distributions.Locally compact abelian groupsThe Fourier transform may be generalized to any locally compact abelian group. A locally compact abelian group is an abelian group which is at the same time a locally compact Hausdorff topological space so that the group operations are continuous. If G is a locally compact abelian group, it has a translation invariant measure μ, called Haar measure. For a locally compact abelian group G it is possible to place a topology on the set of characters so that is also a locally compact abelian group. For a function ƒ in L1(G) it is possible to define the Fourier transform by (Katznelson 1976):Locally compact Hausdorff spaceThe Fourier transform may be generalized to any locally compact Hausdorff space, which recovers the topology but loses the group structure.Given a locally compact Hausdorff topological space X, the space A=C(X) of continuous complex-valued functions on X which vanish at infinity is in a natural way a commutative C*-algebra, via pointwise addition, multiplication, complex conjugation, and with norm as the uniform norm. Conversely, the characters of this algebra A, denoted are naturally a topological space, and can be identified with evaluation at a point of x, and one has an isometric isomorphism In the case where X=R is the real line, this is exactly the Fourier transform. Non-abelian groupsThe Fourier transform can also be defined for functions on a non-abelian group, provided that the group is compact. Unlike the Fourier transform on an abelian group, which is scalar-valued, the Fourier transform on a non-abelian group is operator-valued (Hewitt & Ross 1971, Chapter 8). The Fourier transform on compact groups is a major tool in representation theory (Knapp 2001) and non-commutative harmonic analysis.Let G be a compact Hausdorff topological group. Let Σ denote the collection of all isomorphism classes of finite-dimensional irreducible unitary representations, along with a definite choice of representation U(σ) on theHilbert space Hσ of finite dimension dσfor each σ ∈ Σ. If μ is a finite Borel measure on G, then the Fourier–Stieltjestransform of μ is the operator on Hσdefined bywhere is the complex-conjugate representation of U(σ) acting on Hσ. As in the abelian case, if μ is absolutely continuous with respect to the left-invariant probability measure λ on G, then it is represented asfor some ƒ ∈ L 1(λ). In this case, one identifies the Fourier transform of ƒ with the Fourier –Stieltjes transform of μ.The mapping defines an isomorphism between the Banach space M (G ) of finite Borel measures (see rca space) and a closed subspace of the Banach space C ∞(Σ) consisting of all sequences E = (E σ) indexed by Σ of (bounded) linear operators E σ : H σ → H σ for which the normis finite. The "convolution theorem" asserts that, furthermore, this isomorphism of Banach spaces is in fact an isomorphism of C * algebras into a subspace of C ∞(Σ), in which M (G ) is equipped with the product given by convolution of measures and C ∞(Σ) the product given by multiplication of operators in each index σ.The Peter-Weyl theorem holds, and a version of the Fourier inversion formula (Plancherel's theorem) follows: if ƒ ∈ L 2(G ), thenwhere the summation is understood as convergent in the L 2 sense.The generalization of the Fourier transform to the noncommutative situation has also in part contributed to the development of noncommutative geometry. In this context, a categorical generalization of the Fourier transform to noncommutative groups is Tannaka-Krein duality, which replaces the group of characters with the category of representations. However, this loses the connection with harmonic functions.AlternativesIn signal processing terms, a function (of time) is a representation of a signal with perfect time resolution, but no frequency information, while the Fourier transform has perfect frequency resolution, but no time information: the magnitude of the Fourier transform at a point is how much frequency content there is, but location is only given by phase (argument of the Fourier transform at a point), and standing waves are not localized in time – a sine wave continues out to infinity, without decaying. This limits the usefulness of the Fourier transform for analyzing signals that are localized in time, notably transients, or any signal of finite extent.As alternatives to the Fourier transform, in time-frequency analysis, one uses time-frequency transforms or time-frequency distributions to represent signals in a form that has some time information and some frequency information – by the uncertainty principle, there is a trade-off between these. These can be generalizations of the Fourier transform, such as the short-time Fourier transform or fractional Fourier transform, or can use different functions to represent signals, as in wavelet transforms and chirplet transforms, with the wavelet analog of the (continuous) Fourier transform being the continuous wavelet transform. (Boashash 2003). For a variable time and frequency resolution, the De Groot Fourier Transform can be considered.Applications Analysis of differential equationsFourier transforms and the closely related Laplace transforms are widely used in solving differential equations. The Fourier transform is compatible with differentiation in the following sense: if f (x ) is a differentiable function withFourier transform , then the Fourier transform of its derivative is given by . This can be used to transform differential equations into algebraic equations. Note that this technique only applies to problems whose domain is the whole set of real numbers. By extending the Fourier transform to functions of several variables partial differential equations with domain R n can also be translated into algebraic equations.。
fourier analysis an introduction
fourier analysis an introductionFourier Analysis: An IntroductionIntroduction:Fourier analysis is a mathematical tool used to analyze periodic functions and signals. It is named after Jean-Baptiste Joseph Fourier, a French mathematician, who developed the concept in the early 19th century. Fourier analysis is widely used in various fields, including physics, engineering, computer science, and signal processing. In this article, we will take a step-by-step approach to understand the basics of Fourier analysis and its applications.1. Periodic Functions:To understand Fourier analysis, let's first define periodic functions.A periodic function is a function that repeats itself after a certain period. In other words, for any value of x, f(x) = f(x + T), where T is the period. Some common examples of periodic functions include sine and cosine functions.2. Fourier Series:The Fourier series is a mathematical representation of a periodic function as an infinite sum of sinusoidal functions. It allows us todecompose a complex periodic function into simpler sinusoidal components. The formula for the trigonometric Fourier series for a periodic function f(x) with period T is given by:f(x) = a0 + Σ[an*cos(nωx) + bn*sin(nωx)]where a0 is the DC component, an and bn are the Fourier coefficients, n is an integer, and ω= 2π/T.3. Fourier Coefficients:The Fourier coefficients an and bn determine the amplitude of each component in the Fourier series. To calculate these coefficients, we can use the formulas:an = (2/T) * ∫[f(x)*cos(nωx)] dxbn = (2/T) * ∫[f(x)*sin(nωx)] dxwhere ∫represents integration over one period.4. Frequency Spectrum:The frequency spectrum is a graphical representation of the Fourier series, showing the amplitude of each sinusoidal component as afunction of frequency. The frequency spectrum helps us understand the distribution of frequency components in the original signal. The DC component (a0), representing the average value of the signal, is located at zero frequency.5. Fast Fourier Transform (FFT):Computing the Fourier coefficients directly using integrals can be computationally expensive for large data sets. The Fast Fourier Transform (FFT) is an algorithm used to efficiently compute discrete Fourier transforms. The FFT reduces the computational complexity from O(n^2) to O(n log n), making it practical for real-time processing of signals.6. Applications of Fourier Analysis:Fourier analysis finds numerous applications in various fields. Some of the key areas where Fourier analysis is used include:- Signal Processing: Fourier analysis is used to extract various frequency components from signals, enabling filtering, noise reduction, and compression.- Image Processing: The Fourier Transform is used to analyze and enhance images, such as in image compression algorithms.- Communication Systems: Fourier analysis helps in the modulation, demodulation, and encoding of signals in various communication systems.- Quantum Mechanics: Fourier analysis is used to describe and analyze wave functions in quantum mechanics.- Music and Audio Processing: Fourier analysis is used for audio compression, synthesis, and analysis of music signals.Conclusion:Fourier analysis is a powerful mathematical tool for analyzing periodic functions and signals. Its ability to decompose complex signals into simple sinusoidal components allows us to gain insights into the frequency content of a signal and enables various applications such as signal processing, image processing, communication systems, and more. Understanding the basics of Fourier analysis provides a solid foundation for exploring its advanced concepts and applications in different fields.。
Mathematisches Forschungsinstitut Oberwolfach Report No. 482006 Combinatorics, Probability
Combinatorics, Probability and Computing
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Workshop: Combinatorics, Probability and Computing Table of Contents
J´ ozsef Balogh (joint with B´ ela Bollob´ as and Robert Morris) Majority bootstrap percolation on the hypercube . . . . . . . . . . . . . . . . . . . . . . 2879 Christian Borgs (joint with J.T. Chayes, L. Lov´ asz, V.T. S´ os and K. Vesztergombi) Convergent sequences of dense graphs II. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2882 Peter Bro Miltersen The two probe conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2887 Jennifer Chayes (joint with C. Borgs, L. Lov´ asz, V.T. S´ os and K. Vesztergombi) Convergent sequences of dense graphs I. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2888 Amin Coja-Oghlan (joint with Michael Behrisch and Mihyun Kang) Local limit theorems for the giant component . . . . . . . . . . . . . . . . . . . . . . . . 2892 Benjamin Doerr (joint with Tobias Friedrich) Deterministic random walks on the two-dimensional grid . . . . . . . . . . . . . 2896 Uriel Feige (joint with Jeong Han Kim and Eran Ofek) Witnesses for non-satisfiability of dense random 3CNF formulas . . . . . . 2897 Ehud Friedgut (joint with Irit Dinur) Intersecting families are essentially contained in juntas . . . . . . . . . . . . . . . 2898 Alan Frieze (joint with Jon Kleinberg, R. Ravi and Warren Debany) Line-of-sight networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2902 Stefanie Gerke (joint with Paul Balister and B´ ela Bollob´ as) Connectivity of random addable graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2906 Svante Janson The probability that a random multigraph is simple . . . . . . . . . . . . . . . . . . 2907 Peter Keevash (joint with Dhruv Mubayi, Benny Sudakov and Jacques Verstra¨ ete) Rainbow Tur´ an problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2910 Yoshiharu Kohayakawa (joint with Vojtˇ ech R¨ odl and Eduardo Tengan) The size-Ramsey number of short subdivisions . . . . . . . . . . . . . . . . . . . . . . . 2913 Michael Krivelevich (joint with Dan Hefetz and Tibor Szab´ o) Hamilton cycles in expanding and highly connected graphs . . . . . . . . . . . . 2916 Nati Linial (joint with Adi Shraibman) Complexity measures of sign matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2917
university and their function
•
• (P3)It enables man to construct an intellectual vision of a new world , and it preserves the zest of life by the suggestion of satisfying purposes. • People with imagination will be able to form a new outlook which is different from that of people without imagination. Imagination is capable of preserving people’s enthusiasm for life because it can show people that life has many purposes which can be pleasing.
• Alfred North Whitehead (1861– 1947) was a British mathematician, logician and philosopher best known for his work in mathematical logic and the philosophy of science. In collaboration with Bertrand Russell, he authored the landmark three-volume Principia Mathematica《数学原理》 (1910, 1912, 1913) and contributed significantly to twentieth-century logic, philosophy of science and metaphysics形而上学.
(2008)Dimensionality reduction： A comparative review
L.J.P. van der Maaten ∗ , E.O. Postma, H.J. van den Herik
MICC, Maastricht University, P.O. Box 616, 6200 MD Maastricht, The Netherlands.
22 February 2008
the number of techniques and tasks that are addressed). Motivated by the lack of a systematic comparison of dimensionality reduction techniques, this paper presents a comparative study of the most important linear dimensionality reduction technique (PCA), and twelve frontranked nonlinear dimensionality reduction techniques. The aims of the paper are (1) to investigate to what extent novel nonlinear dimensionality reduction techniques outperform the traditional PCA on real-world datasets and (2) to identify the inherent weaknesses of the twelve nonlinear dimenisonality reduction techniques. The investigation is performed by both a theoretical and an empirical evaluation of the dimensionality reduction techniques. The identification is performed by a careful analysis of the empirical results on specifically designed artificial datasets and on the real-world datasets. Next to PCA, the paper investigates the following twelve nonlinear techniques: (1) multidimensional scaling, (2) Isomap, (3) Maximum Variance Unfolding, (4) Kernel PCA, (5) diffusion maps, (6) multilayer autoencoders, (7) Locally Linear Embedding, (8) Laplacian Eigenmaps, (9) Hessian LLE, (10) Local Tangent Space Analysis, (11) Locally Linear Coordination, and (12) manifold charting. Although our comparative review includes the most important nonlinear techniques for dimensionality reduction, it is not exhaustive. In the appendix, we list other important (nonlinear) dimensionality reduction techniques that are not included in our comparative review. There, we briefly explain why these techniques are not included. The outline of the remainder of this paper is as follows. In Section 2, we give a formal definition of dimensionality reduction. Section 3 briefly discusses the most important linear technique for dimensionality reduction (PCA). Subsequently, Section 4 describes and discusses the selected twelve nonlinear techniques for dimensionality reduction. Section 5 lists all techniques by theoretical characteristics. Then, in Section 6, we present an empirical comparison of twelve techniques for dimensionality reduction on five artificial datasets and five natural datasets. Section 7 discusses the results of the experiments; moreover, it identifies weaknesses and points of improvement of the selected nonlinear techniques. Section 8 provides our conclusions. Our main conclusion is that the focus of the research community should shift towards nonlocal techniques for dimensionality reduction with objective functions that can be optimized well in practice (such as PCA, Kernel PCA, and autoencoders).
含导函数Stieltjes积分边界条件下二阶问题的正解
应用泛函分析学报Vol.22, No.4Dec., 2020第22卷第4期2020年12月ACTA ANALYSIS FUNCTIONALIS APPLICATA DOI ： 10.12012/1009-1327(2020)04-0193-14文献标识码：A含导函数Stieltjes 积分边界条件下二阶问题的正解计倩,张国伟(东北大学理学院数学系，沈阳110819)摘 要 本文研究了一类含导函数Stieltjes 积分边值条件下二阶边值问题的正解.由 于边值条件中带有导数，导致讨论过程与已有文献不同，并且给出相应的格林函数.应 用不动点指数理论证明非线性项/(x,y,z)关于x , y 有超(次)线性增长情形下方程正 解的存在性.通过两个具体例子进行说明理论结果的有效性，例子中边值条件包含积分 型与多点型的形式.关键词 正解；不动点指数;锥中图分类号O175.14; O177.91Positive Solutions for Second Order Problems under Stieltjes Integral Boundary Conditions with DerivativeJI Qian, ZHANG Guowei(Department of Mathematics, College of Science, Northeastern University, Shenyang 110819, China)Abstract In this paper, we study positive solutions for a class of second order prob lems under Stieltjes integral boundary conditions with derivative. Due to the derivative in the boundary conditions, the procedure of discussing is different from one in previ ous literature, and Green's function corresponding to the problem is given. The fixed point index theory is applied to prove the existence of positive solutions when the non linear term /(x, y, z) has superlinear or sublinear growth on x and y. The validity of the theoretical results is illustrated by two concrete examples, in which the boundary conditions include the forms of integral and multi-point types.Keywords positive solution; fixed point index; cone收稿日期：2020-09-13作者简介：计倩(1994-),女辽宁本溪人硕士研究生，研究方向：非线性泛函分析，E-mail: ******************.194应用泛函分析学报第22卷Chinese Library Classification O175.14;O177.911引言近些年来，非线性微分方程边值问题在科学研究和工程技术等领域中都具有重要的应用，并受到诸多学者的广泛关注，取得了许多研究成果二阶非线性常微分方程边值问题的正解存在性及多解性成为一个重要研究领域，文献[3]利用锥上不动点指数方法讨论了边值条件中带有Stieltjes积分的方程—u〃(t)=/(t,u(t),G[0,1],au(0)—bu z(0)=a[u],cu(1)+du'(1)=0[u]正解的存在性，但是在Stieltjes积分中不含未知函数的导数Stieltjes积分中含有未知函数导数的边值问题也有一些研究结果文献[5]研究方程—u''(t)=g(t)f(t,u(t)),t G[0,1],u(0)=a[u],u'(1)=0[u]+入[u']正解的存在性，其中a[u]=/u(t)d A(t),0[u]=/u(t)d B(t),入[u']=/u'(t)dA(t),丿0Jo JoA,B和A为界变差函数但是非线性项函数不含有未知函数的导数受此启发，本文用锥上不动点指数方法讨论如下含导函数Stieltjes积分边界条件下二阶边值问题正解的存在性：—u''(t)=f(t,u(t),u'(t)),t G[0,1],(1)u'(0)=a[u],u(1)=0[u]—A[u'].关于非线性项和Stieltjes积分形式边值条件含未知函数导数的工作可见文献[9,10].本文讨论方程(1)的内容和所使用的方法与[9,10]不相同.2预备知识定义c”0,1]空间的范数为||u||ci=max{||训c,||u'||c}.首先我们假设，(C1)f:[0,1]x R+x R t R+是连续函数其中R+=[0,Q.(C2)A(1)-A(s)、0,V s G[0,1].由于方程⑴边值条件中含有入[u'],类似于[5]中Webb所使用的方法,我们需要给出相应的Green函数.弓|理1在(C i)的条件下，考虑当a[u]=0[u]=0,即—u''(t)=f(t,u(t),u'(t)),t G[0,1],(2)u'(0)=0,u(1)+入[u']=0第4期计倩等：含导函数Stieltjes 积分边界条件下二阶问题的正解195时,⑵在C 1[0,1]中的解由如下定义(Hu )(t ) = / (A(1) — A(s ))f (s,u (s ),u '(s ))d s + [ (t, s )f (s,u (s ),u '(s )) d s丿0丿0:=/ kH (t, s )f (s,u (s ),u '(s ))d s J0的算子H 不动点给出，其中⑶{1 — s, 0 < t < s < 1,1 — t, 0 < s < t < 1,⑷I A (1) — A(s ) + 1 — s, 0 < t < s < 1, k H (t, s )= <(A (1) — A(s ) + 1— t, 0 < s < t < 1.⑸证明 首先,对—u ''(t ) = f (t,u (t ),u '(t ))在[0,t 和[t, 1]求两次积分,并利用 ⑵中的边值条件就可得到(3)式•其次，(Hu )(t ) = / (A(1) — A (s ))f (s,u (s ),u '(s ))d s + [ k o(t, s )f (s,u (s ),u '(s )) d s丿o丿o =[(A(1) — A(s ))f (s,u (s ),u '(s ))d s + [ (1 — t )f (s,u (s ),u '(s ))d s +丿0丿0/ (1 — s)f(s,u(s),u '(s))ds,[f (s, u (s ), u '(s )) d s + (1 — t )f (t, u (t ), u '(t )) — (1 — t )f (t, u (t ), u '(t ))0(Hu)''(t) = —f (t,u(t),u '(t)),显然(Hu )(1) = J 0X (A(1) — A(s ))f (s,u (s ),u '(s ))d s ,而tf (s, u (s ), u '(s )) ds)dA(t )f (s, u (s ), u '(s )) dA(t )d s = f (A (1) — A(s ))f (s, u (s ), u '(s )) d s = (Hu )(1),丿0所以(Hu )''(t ) = —f (t, u (t ), u '(t )), (Hu )'(0) = 0, (Hu )(1) + 入[(Hu )'] = 0•由此可见，u 是 H 不 动点.弓I 理2如果(C 2)满足，则存在非负函数①h (s ) = A(1) — A(s ) + 1 — s,使得V t, s e [0,1]有(1 一 t )① H (s ) < k H (t, s ) < ① H (s ).容易证明在C 1 [0,1]中,BVP(1)有解当且仅当如下的积分方程u (t ) =(t 一 dA(t ) — 1)a [u ] + 0[u ] + (Hu )(t )I f(s,u(s),u '(s))ds,丿0dA(t ) — t) a [u ]-入[(Hu )']f (s, u (s ), u '(s )) d s dA(t )dA(t)) 0[u ] + (Hu )(t ) := (Tu)(t)⑹196应用泛函分析学报第22卷存在解，其中a [u ] = 0[u ] — a [u ]•记2是a [u ]对应的有界变差函数，并且假设(C 3) K a (s ) := / k n (t, s )d A (t ) > 0,K b (s ) ：= / k n (t, s )d B (t ) > 0, V s G [0,1].丿0 丿0令 y (t ) = 1 — t + fl dA(t ), d (t ) = t — fl dA(t ).再假设(C 4)0 < S [7] < 1,0[y ] > 0, 0 < 0[d ] < 1, a [d ] > 0, D := (1 — d [7])(1 — 0[d ]) — d [J ]0[7] > 0. 定义算子S 如下(Su)(t)：=丄 d - dA (t)—t + 人"A") (1 一 00]) / K A (s)f (s,u (s ),u z (s ))d s + 丘0] / K b (s )f (s,u (s ),u z (s ))d s_ 丿0 丿0 .0[Y ] / K A (s )f (s,u (s ),u '(s ))ds + (1 — a[Y ]) / k b (s)f (s,u(s ),u '(s ))ds 一 丿0 丿0 _t + - D+ k n (t, s )f (s,u (s ),u z (s ))d s丿0/ ks (t, s )f (s, u (s ), u z (s )) d s,丿0即(Su )(t ) = / k s (t, s )f (t,u (t ),u '(t ))d s.丿0⑺显然k s (t,s ) =1 — t +D 0 dA(t ) [(1 — 0[d ])K A (s ) + a [d ]K B (s )] +-~『叮人")[0[Y ]K A (s) + (1 — a[Y ])K B (s)] + k n (t, s).由上述条件可以很容易的看出(Su )(t ) > 0.并且dk s (t, s )~dt -=D [(1 — 00])K A (s ) + a[J]K B (s)] + D [0[Y ]K A (s) + (1 — a[Y ])k b (s)]—以仔 s )—1 1< D [(1 — 0[d ])K A(s ) + a [d ]K B(s )] + 万[0[y ]K a (s ) + (1 — S [y ])K b (s )] + 1 :=巫(s ).(8)⑼引理3假设满足(C 2)〜(C 4),则存在非负函数Q(s )和v(t) = min {t, 1—讣使得V t, s G [0,1],v (t )Q(s ) < k s (t, s ) < $(s ),其中 Q(s ) = D [(1 — 0Q ])K a (s ) + &[d ]K B (s )] + D [0[y ]K a (s ) + (1 —殆])恥(s )] + $n (s ).我们定义如下两个锥和三个线性算子：P = {u G C 1[0,1] : u(t) > 0, V t G [0,1]},(10)(L i u )(t )K = {u G P : u (t ) > v(t) ||u||C , V t G [0, 1]},[K s (t, s)(a 2u(s) + C 2)d s, @2,C 2为正常数),(11)(12)第4期计倩等：含导函数Stieltjes 积分边界条件下二阶问题的正解197(13)(14)(02u )(s ) = / K s (t, s )u (t )d t, u G C [0,1],Jo (L g u )(t ) = f K s (t, s )u (s )d s, u G C [0,1].JoV x,y G X ，若x — y G P ，记为x A y 或者y Y x ,称为由锥P 导出的半序.弓I 理4假设(C i )〜(4)都成立，那么S : P t K 和L i : C [0,1] t C [0,1]均是全连续算 子并且 L i (P ) U K, (i =1, 2, 3).证明 当(7), (8)和(C i )〜(C 4)都成立，则当u G P 时，有(Su )(t ) > 0.由(C i )条件可 以得到S : P T C i [0,1]是连续的算子取锥P 上的有界集合F ，则存在一个数M > 0,使得 ||训。
法国数学家拉格朗日著作《解析函数论》英文名
法国数学家拉格朗日著作《解析函数论》英文名Analysis of Functions by French mathematician LagrangeAnalysis of Functions, also known as Mémoire sur larésolution des équations numériques, is a groundbreaking work by French mathematician Joseph-Louis Lagrange. This seminal work, published in 1809, laid the foundation for the field of complex analysis and played a pivotal role in shaping modern mathematics.Lagrange's work in Analysis of Functions focused on the study of functions of a complex variable and their properties. He developed new methods for solving equations involving complex numbers, uncovering fundamental principles that would later become the basis of complex analysis. In particular, Lagrange's work on power series and their convergence properties was a major contribution to the understanding of complex functions.One of the key concepts introduced in Analysis of Functions is the concept of a holomorphic function, which is a complex function that is differentiable at every point in its domain. Lagrange's study of holomorphic functions and their propertieshelped lay the groundwork for the development of the theory of analytic functions, a central area of study in complex analysis.Analysis of Functions also includes Lagrange's work on the theory of residues, which are complex numbers associated with singularities of a complex function. Lagrange developed new techniques for calculating residues and applying them to the evaluation of complex integrals, a key tool in the study of complex functions.In addition to his mathematical contributions, Lagrange's Analysis of Functions had a significant impact on the development of mathematics as a whole. His work inspired future generations of mathematicians to explore the rich and diverse field of complex analysis, leading to further advancements in the study of functions of a complex variable.Overall, Analysis of Functions by Joseph-Louis Lagrange is a seminal work in the field of complex analysis that has had a lasting impact on the development of modern mathematics. Lagrange's innovative methods and profound insights continue to influence mathematicians to this day, making his work an essential reference for anyone studying the theory of functions of a complex variable.。
两种“内在价值”理论之争——重审摩尔对布伦塔诺的批评
第10卷第6期2019年12月哲学分析Philosophical AnalysisVol.10，No.6Dec.，2019两种“内在价值”理论之争——重审摩尔对布伦塔诺的批评周宇郝亿春摘 要：在“是否承认内在价值”的问题上，摩尔属于布伦塔诺少有的同盟者，不过在一些重要方面摩尔却对后者提出了严厉批评：在“内在善好”概念上，摩尔批评布伦塔诺的说明方式没有清晰认识到“内在善好”仅仅取决于相关对象的内在本质而与情感无关；在偏爱原则上，摩尔批评布伦塔诺混淆了正当性的程度差异和正当的偏爱情感，而没有认识到前者才是决定价值排序的本质要素；在总量原则上，摩尔批评布伦塔诺没有恰当认识到在整体和部分的内在价值之间的复杂关系，甚至会导致恶劣的道德后果。
通过重新审视上述批评可知，布伦塔诺不仅可以在很大程度上回应摩尔对其的批评，而且在“内在价值”理论上，他提供了一种与摩尔不同甚至更加精致的理论形态。
重审及回应摩尔对布伦塔诺的批评也有助于弄清“内在价值”这种“实事”本身。
关键词：布伦塔诺；摩尔；内在价值；偏爱原则；总量原则中图分类号：B82 文献标志码：A 文章编号：2095-0047（2019）06-0079-11摩尔在《伦理学原理》的序言中不无惊喜地指出：“我发现在布伦塔诺《道德认识的源头》一书中的一些见解，远比我所熟知的其他伦理学家都更为接近我自己的观点。
”a不过，摩尔在对身处遥远德语区的布伦塔诺报以盛赞的同时，也清晰地认识到他们之间的理论分野。
虽然摩尔对布伦塔诺在伦理学上的“客观主义”理论及其对“自身善”和“手段善”所做出的正确区分不吝褒扬b，但他还是批评了布伦塔 作者简介： 周宇：中山大学哲学系博士研究生；郝亿春：中山大学哲学系教授。
a G. E. Moore，Principia Ethica，Cambridge：Cambridge University Press，1993，p.36；也可参见摩尔：《伦理学原理》，长河译，北京：商务印书馆1983年版，第3—4页。
两类偏微分方程的数学问题研究 英语
两类偏微分方程的数学问题研究英语Two types of partial differential equations, theelliptic and hyperbolic equations, have been the focus of mathematical research for many years. These equations arisein many fields, including physics, engineering, and computer science. Understanding these equations and solving the mathematical problems that arise from them is crucial for advancing our understanding of the natural world and for developing new technologies. In this article, we will explore the two types of partial differential equations and the mathematical problems that arise from them.Step 1: Understanding elliptic equationsElliptic equations are characterized by their smoothness and their tendency to model equilibrium situations, such as the steady state solution of a heat equation. These equations are often described as being “well-posed” because they have unique solutions that depend continuously on the data. Mathematical problems that arise from elliptic equations often involve finding the existence and uniqueness of solutions, the regularity of solutions, and the decay or stability of solutions. Some common techniques used to solve these problems include the use of maximum principles, variational methods, and the Hardy-Littlewood-Sobolev inequality.Step 2: Understanding hyperbolic equationsHyperbolic equations, on the other hand, are characterized by their wave-like behavior and their tendency to model dynamic situations, such as the propagation of wavesin water or sound in air. These equations are often described as being “ill-posed” because they do not have unique solutions that depend continuously on the data. Mathematical problems that arise from hyperbolic equationsoften involvethe stability of solutions, the sharpness of estimates, andthe existence of solutions in large time intervals. Some common techniques used to solve these problems include theuse of energy methods, entropy inequalities, and theStrichartz estimates.Step 3: Bridging the gap between elliptic and hyperbolic equationsWhile elliptic and hyperbolic equations are quitedifferent in nature, they are also connected in interesting ways. For example, elliptic equations can arise as the limitof hyperbolic equations as the speed of propagation becomes infinitely fast. Additionally, some problems that arise from hyperbolic equations can be reformulated as elliptic problems, allowing for the use of techniques developed for elliptic equations. Bridging the gap between elliptic and hyperbolic equations is an active area of research, and has led to the development of new techniques and insights into both types of equations.In conclusion, the study of elliptic and hyperbolic equations is an important area of mathematical research, with far-reaching applications in physics, engineering, and computer science. By understanding the mathematical problems that arise from these equations, and the techniques used to solve them, we can gain a deeper understanding of the natural world and develop new technologies to improve our lives.。
用数学方法证明埃氏筛法
用数学方法证明埃氏筛法The Sieve of Eratosthenes is a mathematical method used to find all prime numbers up to a certain limit. It works by marking the multiples of each prime number starting from 2, and then eliminating the composite numbers, leaving only the prime numbers. This method is not only efficient, but also elegant in its simplicity.埃氏筛法是一种用来找到所有小于等于某个限制的素数的数学方法。
它通过标记从2开始的每个素数的倍数，然后消除复合数，留下只有素数。
这种方法不仅高效，而且在其简单性上也是优雅的。
To prove the effectiveness of the Sieve of Eratosthenes, we can consider the underlying principles behind it. The method relies on the fundamental theorem of arithmetic, which states that every integer greater than 1 can be uniquely expressed as a product of prime numbers. By marking off the multiples of prime numbers, we are essentially eliminating the composite numbers and retaining only the prime numbers.要证明埃氏筛法的有效性，我们可以考虑背后的原理。
Eulid辗转相除法与二部图的一个对应
3. 正文
3.1. 一道组合最值问题
有若干正整数，它们和为 m ⋅ n ，且既可以分为和相等的 m 个组，又可以分为和相等的 n 个组，求这
些正整数个数的最小值 f (m, n) 。 经过对一些较小的 m 和 n 的试验我们猜测 f (m, n) 可能是 m + n − (m, n) 。我们的证明如下：
DOI: 10.12677/pm.2021.112030
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理论ቤተ መጻሕፍቲ ባይዱ学
Eulid辗转相除法与二部图的一个对应
金长松，陈 贺 东北大学秦皇岛分校数学与统计学院，河北 秦皇岛
收稿日期：2021年1月2日；录用日期：2021年2月2日；发布日期：2021年2月9日
摘要
本文对一道组合最值问题的更一般情况进行了猜测和证明，利用了图论方法证明满足要求的值不小于目 标值，再利用归纳构造证明目标值是可行的。以此得到了Eulid辗转相除法与二部图的一个对应。
∑ ∑ x|(m,n)
m
+ x
n
−1
=
m + n − 1 ≥ m + n − (m, n)
x|(m,n) x
DOI: 10.12677/pm.2021.112030
220
理论数学
金长松，陈贺
由于图 G 的连线方法，如果两个点所代表的组都含有 f (m, n) ≥ m + n − (m, n) 中的某个数，则它们之
的点，其所在组里的正整数必定出现两次，否则一定可以再加进一个新的点与图 T 连通，与 G'的最大性
矛盾。对这些数求和，得到 ms = nt，于是其中点的个数为 s + t =m + n ，这里 x | (m, n) 因为图 G'是连通
含Pell与Pell-Lucas数之积的斜循环矩阵的行列式及其性质
−
ζ
2 n
−
8n
−
4
32
.
iv) 令
n
∑ Sn = ζ i ai = ζ1a + ζ 2a2 + + ζ nan .
(12)
i =1
由(1)和(12)以及下面给出的公式
DOI: 10.12677/aam.2020.912247
2132
应用数学进展
李笑丽 等
可得
a2 Sn= ζ1a3 + ζ 2a4 + + ζ nan+2 , 6aS=n 6ζ1a2 + 6ζ 2a3 + + 6ζ nan+1 .
Open Access
1. 引言
循环矩阵在编码、预处理器、通信、信号处理、图像处理、解决 Toeplitz 矩阵问题等领域中起着重 要的作用[1] [2] [3]。近年来，矩阵理论工作者对包含著名数列的循环矩阵进行了一系列的研究，如循环 矩阵的行列式、范数、逆矩阵及扩展式等。如[4]中通过构造变换矩阵给出具有 Fibonacci 和 Lucas 数的斜 循环的行列式，以及利用左斜循环矩阵和斜循环矩阵之间的关系求得左斜循环矩阵的行列式；[5]中使用 多项式的逆因式分解给出涉及 Perrin，Padovan，Tribonacci 和广义 Lucas 数的这些矩阵的确切行列式；[6] 是由获得的新公式来计算 H 矩阵的行列式。文献[7]利用 Pell 和 Pell-Lucas 数的一些性质来获得斜循环量 和左斜循环量矩阵的行列式；[8]是关于求 Fibonacci 数和 Lucas 数的循环矩阵的范数；[9] [10] [11]均是先 证明矩阵的可逆性，求出矩阵的行列式进一步写出其逆矩阵；[12]用 Tribonacci 数和广义 Lucas 数来表示 循环矩阵和左循环矩阵的扩展式的上下界。文献[13]给出了以 Fibonacci 数和 Lucas 数之积为元素的斜循 环矩阵的行列式、逆矩阵和范数等；郑和顺[10]研究了广义 Lucas 斜循环矩阵的精确行列式和逆矩阵； Bozkurt [11]给出了带有 Pell 和 Pell-Lucas 数列的经典循环矩阵的行列式和逆矩阵。[14]研究了包含 Pell 和 Pell-Lucas 数之和的斜循环矩阵行列式、范数等。受上述研究的启发，本文将对以 Pell 与 Pell-Lucas 数之积为元素的斜循环矩阵、左斜循环矩阵的行列式、范数及扩展式的上下界进行研究。
A binary algorithm for the Jacobi symbol
R(u; v) = R(u=2; v=2) and v is odd, and when
uloisg2o(dud=2an+dvv=2is)
evelong. 2(u
+
v).
A
similar
inequality
holds
when
u
is
even
Finally, assume both u and v are odd. Then if u v, the algorithm performs a subtraction step
particularly e cient implementation on binary digital computers, where division by 2 can be done
quickly by using a \shift". Although it is not surprising that it is possible to adapt Stein's algorithm
found on symbolic algebra systems such as Mathematica and Maple.
Several e cient algorithms modeled on Euclid's algorithm for computing the greatest common di-
and replaces u with (u ? v)=2. Hence we have
R(u; v) = 1 + R((u ? v)=2; v) 1 + log2((u ? v)=2 + v) = 1 + log2((u + v)=2) = log2(u + v):
欧拉方程的正保护
On positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes qXiangxiong Zhang a ,Chi-Wang Shu b ,*a Department of Mathematics,Brown University,Providence,RI 02912,United StatesbDivision of Applied Mathematics,Brown University,Providence,RI 02912,United Statesa r t i c l e i n f o Article history:Received 4March 2010Received in revised form 6July 2010Accepted 13August 2010Available online 18August 2010Keywords:Hyperbolic conservation laws Discontinuous Galerkin method Positivity preserving High order accuracyCompressible Euler equations Gas dynamicsFinite volume schemeEssentially non-oscillatory scheme Weighted essentially non-oscillatory schemea b s t r a c tWe construct uniformly high order accurate discontinuous Galerkin (DG)schemes which preserve positivity of density and pressure for Euler equations of compressible gas dynam-ics.The same framework also applies to high order accurate finite volume (e.g.essentially non-oscillatory (ENO)or weighted ENO (WENO))schemes.Motivated by Perthame and Shu (1996)[20]and Zhang and Shu (2010)[26],a general framework,for arbitrary order of accuracy,is established to construct a positivity preserving limiter for the finite volume and DG methods with first order Euler forward time discretization solving one-dimen-sional compressible Euler equations.The limiter can be proven to maintain high order accuracy and is easy to implement.Strong stability preserving (SSP)high order time dis-cretizations will keep the positivity property.Following the idea in Zhang and Shu (2010)[26],we extend this framework to higher dimensions on rectangular meshes in a straightforward way.Numerical tests for the third order DG method are reported to dem-onstrate the effectiveness of the methods.Ó2010Elsevier Inc.All rights reserved.1.IntroductionIn this paper we are interested in constructing high order accurate schemes for solving hyperbolic conservation law sys-tems.For scalar conservation laws,the entropy solution is total variation diminishing (TVD),which is a desired property for numerical solutions.While traditional TVD schemes (e.g.[10])measure the total variation by that of the cell averages or grid values,leading to a necessary degeneracy of accuracy to first order at smooth extrema [18],genuinely high order TVD schemes can be constructed for one-dimensional scalar conservation laws [21,27]by measuring the total variation of the reconstruction polynomials.For multi-dimensional scalar conservation laws,it is difficult to enforce the TVD property for a high order scheme,however it is reasonable to insist on a strict maximum principle,which is satisfied by the entropy solu-tion.Genuinely high order accurate finite volume and discontinuous Galerkin (DG)schemes which satisfy a strict maximum principle have been constructed recently in [26].For hyperbolic conservation law systems,the entropy solutions in general satisfy neither the TVD property nor the max-imum principle.In this paper we are mainly interested in the Euler equations for the perfect gas,the one-dimensional ver-sion being given by0021-9991/$-see front matter Ó2010Elsevier Inc.All rights reserved.doi:10.1016/j.jcp.2010.08.016qResearch supported by AFOSR Grant FA9550-09-1-0126and NSF Grant DMS-0809086.*Corresponding author.Tel.:+14018632549;fax:+14018631355.E-mail addresses:zhangxx@ (X.Zhang),shu@ (C.-W.Shu).w t þf ðw Þx ¼0;t P 0;x 2R ;ð1:1Þw ¼qm E0B @1C A ;f ðw Þ¼m q u 2þp ðE þp ÞuB @1CAð1:2Þwithm ¼q u ;E ¼12q u 2þq e ;p ¼ðc À1Þq e ;where q is the density,u is the velocity,m is the momentum,E is the total energy,p is the pressure,e is the internal energy,and c >1is a constant (c =1.4for the air).The speed of sound is given by c ¼ffiffiffiffiffiffiffiffiffiffiffic p =q p and the three eigenvalues of the Jaco-bian f 0(w )are u Àc ,u and u +c .Physically,the density q and the pressure p should both be positive.We are interested in positivity-preserving high order schemes,which maintain the positivity of density and pressure at time level n +1,provided that they are positive at time level n .The techniques developed in this paper can be considered as generalizations of the maximum-principle-satisfying limiters in [26]and the positivity-preserving schemes in [20].We remark that failure of pre-serving positivity of density or pressure may cause blow-ups of the numerical algorithm,for example,for low density prob-lems in computing blast waves,and low pressure problems in the computing high Mach number astrophysical jets [9].We also remark that most commonly used high order numerical schemes for solving hyperbolic conservation law systems,including,among others,the Runge–Kutta discontinuous Galerkin (RKDG)method with a total variation bounded (TVB)lim-iter [2,4],the essentially non-oscillatory (ENO)finite volume and finite difference schemes [11,25],and the weighted ENO (WENO)finite volume and finite difference schemes [17,13],do not in general satisfy the positivity property for Euler equa-tions automatically.We now consider the Euler equations (1.1)in more detail.Let p ðw Þ¼ðc À1ÞE À1m 2qbe the pressure function.It can be easily verified that p is a concave function of w =(q ,m ,E )T if q P 0.For w 1=(q 1,m 1,E 1)T and w 2=(q 2,m 2,E 2)T ,Jensen’s inequality implies,for 06s 61,p s w 1þð1Às Þw 2ðÞP sp w 1ðÞþð1Às Þp w 2ðÞif q 1P 0;q 2P 0:ð1:3ÞDefine the set of admissible states byG ¼w ¼q m E 0B @1C Aq >0and p ¼ðc À1ÞE À12m 2q >08><>:9>=>;;then G is a convex set.If the density or pressure becomes negative,the system (1.1)will be non-hyperbolic and thus the ini-tial value problem will be ill-posed.We are interested in schemes for (1.1)producing the numerical solutions in the admissible set G .We start with a first order finite volume schemew n þ1j¼w n j Àk h w n j ;w n j þ1 Àh w n j À1;w n j h i;ð1:4Þwhere h (Á,Á)is a numerical flux,n refers to the time step and j to the spatial cell (we assume uniform mesh size only for sim-plicity),and k ¼D tD xis the ratio of time and space mesh sizes.w n j is the approximation to the cell average of the exact solution v (x ,t )in the cell I j ¼x j À1;x j þ1h i,or the point value of the exact solution v (x ,t )at x j ,at time level n .The scheme (1.4)and itsnumerical flux h (Á,Á)are called positivity preserving,if the numerical solution w n j being in the set G for all j implies the solu-tion w n þ1jbeing also in the set G .This is usually achieved under a standard CFL condition k kðj u j þc Þk 16a 0:ð1:5ÞExamples of positivity preserving fluxes include the Godunov flux [6],the Lax–Friedrichs flux [20],the Boltzmann type flux[19],and the Harten–Lax–van Leer flux [12].We now consider a general high order finite volume scheme,or the scheme satisfied by the cell averages of a DG method solving (1.1),which has the following formw n þ1j ¼w n j Àk h w Àj þ1;w þj þ12Àh w Àj À1;w þj À12h i;ð1:6Þwhere h is a positivity preserving flux under the CFL condition (1.5),w n j is the approximation to the cell average of the exact solution v (x ,t )in the cell I j ¼x j À12;x j þ12h iat time level n ,and w Àj þ12;w þj þ12are the high order approximations of the point values v x j þ1;t nwithin the cells I j and I j +1respectively.These values are either reconstructed from the cell averages w n j in a finitevolume method or read directly from the evolved polynomials in a DG method.We assume that there is a polynomial vectorX.Zhang,C.-W.Shu /Journal of Computational Physics 229(2010)8918–89348919q j (x )=(q j (x ),m j (x ),E j (x ))T (either reconstructed in a finite volume method or evolved in a DG method)with degree k ,wherek P 1,defined on I j such that w n j is the cell average of q j (x )on I j ,w þj À12¼q j x j À12and w Àj þ12¼q j x j þ12.A general framework to construct a high order positivity preserving finite volume scheme for the Euler equations wasintroduced in [20],in which a sufficient condition for the solution w n þ1jof (1.6)to be in the set G is that,all the nodal values w Æj þ12and w n þ1j Àa w Àj þ12þw þj À1 are in the set G under the CFL conditionk kðj u j þc Þk 16a a 0;where a 2(0,1]is a constant.Strong stability preserving (SSP)high order Runge–Kutta [25]and multi-step [24]time discret-ization will keep the positivity since G is convex.It is reasonable to require and easy to enforce the positivity of the point values w Æj þ1.However,it is more difficult to en-force the positivity of w n þ1j Àa w Àj þ12þw þj À12without destroying accuracy for an arbitrary high order scheme.We refer to Perthame and Shu [20]for more discussions on this point.In this paper,we provide a similar sufficient condition,whichis however much easier to enforce.We need the N -point Legendre Gauss–Lobatto quadrature rule on the intervalI j ¼x j À12;x j þ12h i,which is exact for the integral of polynomials of degree up to 2N À3.We would need to choose N such that2N À3P k .Denote these quadrature points on I j asS j ¼x j À12¼b x 1j ;b x 2j ;...;b x N À1j ;b x N j¼x j þ12n o:ð1:7ÞWe will prove that a sufficient condition for w n þ1j2G is simply q j b x a j2G for a =1,2,...,N ,under a suitable CFL condition.The same type of the linear scaling limiter used in [26]can enforce this sufficient condition without destroying accuracy.This limiter is also very easy to implement.Furthermore,we provide a straightforward extension of this result to arbitrary high order two-dimensional schemes on rectangular meshes.The main conclusion of this paper is,by adding a positivity preserving limiter which will be specified later to a high order accurate finite volume scheme or a discontinuous Galerkin scheme solving one or multi-dimensional Euler equations,with the time evolution by a SSP Runge–Kutta or multi-step method,we obtain a uniformly high order accurate scheme preserving the positivity in the sense that the density and pressure of the cell averages are always positive if they are positive initially.The paper is organized as follows:we first prove the positivity result for schemes in one space dimension in Section 2.In Section 3,we show a straightforward extension to two space dimensions on rectangular meshes.In Section 4,numerical tests for the third order DG method will be shown.Concluding remarks are given in Section 5.2.Positivity-preserving high order schemes in one dimension 2.1.A sufficient conditionWe consider the first order Euler forward time discretization (1.6);higher order time discretization will be discussed la-ter.Let b w a be the Legendre Gauss–Lobatto quadrature weights for the interval À12;12ÂÃsuch that P Na ¼1b w a ¼1,with2N À3P k .Motivated by the approach in [20,26],our first result isTheorem 2.1.For a finite volume scheme or the scheme satisfied by the cell averages of a DG method (1.6),if q j b x a j2G for all j and a ,then w n þ1j 2G under the CFL conditionk kðj u j þc Þk 16b w1a 0:ð2:1ÞProof.The exactness of the quadrature rule for polynomials of degree k impliesw n j¼1ZI jq j ðx Þdx ¼XN a ¼1b w a q j b x a j :By adding and subtracting h w þj À12;w Àj þ12,the scheme (1.6)becomes w n þ1j¼XN a ¼1b w a q j b x a jÀk h w Àj þ12;w þj þ1 Àh w þj À1;w Àj þ12þh w þj À1;w Àj þ12Àh w Àj À12;w þj À1h i ¼XN À1a ¼2b wa q jb x ajþb w N w Àj þ12Àk w N h w Àj þ12;w þj þ12 Àh w þj À12;w Àj þ12 h iþb w 1w þj À12Àk b w 1h w þj À12;w Àj þ12 Àh w Àj À12;w þj À12 h i ¼XN À1a ¼2b w a q j b x a jþb w N H N þb w 1H 1;8920X.Zhang,C.-W.Shu /Journal of Computational Physics 229(2010)8918–8934whereH1¼wþjÀ1Àkw1h wþjÀ1;wÀjþ12Àh wÀjÀ12;wþjÀ1h i;ð2:2ÞH N¼wÀjþ12Àkw Nh wÀjþ12;wþjþ1Àh wþjÀ1;wÀjþ12h i:ð2:3ÞNotice that(2.2)and(2.3)are both of the type(1.4),and b w1¼b w N,therefore H1and H2are in the set G under the CFLcondition(2.1).Now,it is easy to conclude that w nþ1jis in G,since it is a convex combination of elements in G.hRemark2.2.Here we only discuss Euler forward.Strong stability preserving high order Runge–Kutta[25]and multi-step [24]time discretization will keep the validity of Theorem2.1since G is convex.Remark2.3.From the proof of Theorem2.1,we can see that any type of quadrature rule will work as long as the quadrature points include the two cell ends and the quadrature is exact for polynomials of degree k.It would appear that there is a pos-sibility to achieve a larger CFL number if we canfind a better quadrature in the sense that b w1is larger.However,for k=2,3, we have verified that the Gauss–Lobatto quadrature is the best choice.Remark2.4.For the Lax–Friedrichsfluxhðu;vÞ¼1½fðuÞþfðvÞÀa0ðvÀuÞ ;where a0=k(j u j+c)k1,the CFL condition(1.5)was proven for k a0612in[20]by proving the numerical solution of thefirstorder Lax–Friedrichs scheme to be the cell average of the exact solution.Here,we prove that(1.5)for the Lax–Friedrichsflux can be relaxed to k a061.The Lax–Friedrichs scheme can be written asw nþ1j ¼w njÀk h w nj;w njþ1Àh w njÀ1;w njh i¼ð1Àk a0Þw njþk a0w njþ1À1f w njþ1!þk a0w njÀ1þ1f w njÀ1!:Assume w nj ;w njÀ1and w njþ1are in the set G,we want to show w nþ1j2G under the CFL k a061.Notice that w nþ1jis a convexcombination of the three vectors w nj ;w njþ1À1a0f w njþ1and w njÀ1þ1a0f w njÀ1,we only need to show w njÀ1þ1a0f w njÀ1andw njþ1À1a0f w njþ1are in the set G.It is easy to check that thefirst components of the both vectors are positive.The only non-trivial part is to check the positivity of the‘‘pressure”.For simplicity,we drop the subscripts and superscripts,i.e.,we provewÆ10fðwÞ2G if w2G.Let p¼1c EÀ1m2qand u=m/q.By a direct calculation,we havep wÆ1a0fðwÞ¼pð1Æua0Þq;ð1Æua0ÞmÆ1a0p;ð1Æua0ÞEÆua0pT"#¼1ÀpqcÀ12ða0ÆuÞ2!1Æua0pTherefore,p wÆ1a0fðwÞ>0()pqcÀ12ða0ÆuÞ<1()cpq<2ccÀ1ða0ÆuÞ2()ffiffiffiffiffiffiffic pqr<ffiffiffiffiffiffiffiffiffiffiffiffi2ccÀ1sða0ÆuÞ:Since c¼ffiffiffiffiffiffiffiffiffiffiffic p=qpand a0=k(j u j+c)k1,we have p wÆ1fðwÞ>0.The CFL condition(2.1)using the Lax–Friedrichsflux and the Gauss–Lobatto quadrature points for k=2,3,4,5are listed in Table2.1.We note that these conditions are comparable with and only slightly more restrictive than the standard CFL conditions for linear stability of discontinuous Galerkin methods[5].Table2.1The CFL condition(2.1)of Lax–Friedrichsflux for26k65and the Gauss–Lobatto quadrature points onÀ12;1 2ÂÃ.k CFL Quadrature points onÀ12;1 2ÂÃ2k a0616À12;0;12ÈÉ3k a0616À12;0;12ÈÉ4k a06112À12;À1ffiffiffiffi20p;1ffiffiffiffi20p;12n o5k a06112À12;À1ffiffiffiffi20p;1ffiffiffiffi20p;12n oX.Zhang,C.-W.Shu/Journal of Computational Physics229(2010)8918–893489212.2.A limiter to enforce the sufficient conditionGiven the vector of approximation polynomials q j (x )=(q j (x ),m j (x ),E j (x ))T ,either reconstructed for a finite volume scheme or evolved for a DG scheme,with its cell average w n j ¼q n j ;m n j ;E n jT 2G ,we would like to modify q j (x )into ~q j ðx Þsuch that it satisfiesAccuracy:For smooth solutions,the limiter does not destroy accuracyk ~qj ðx ÞÀq j ðx Þk ¼O ðD x k þ1Þ8x 2I j ;where kÁk denotes the Euclidean norm.Positivity:~q j b x a j2G for a =1,2,...,N . Conservativity:1xZI j~q j ðx Þdx ¼w n j :Definep n j¼ðc À1ÞE n j À12m n j 2=q n j .Then q n j >0and p nj >0for all j .Assume there exists a small number e >0such thatq n j P e and p nj P e for all j .For example,we can take e =10À13in the computation.The first step is to limit the density.Replace q j (x )byb q j ðx Þ¼h 1q j ðx ÞÀq n jþq n j ;ð2:4Þwhereh 1¼min q n j Àe q n jÀq min ;1();q min ¼min a q j b x a j :ð2:5ÞThen the cell average of b q j ðx Þover I j is still q n jand b q j b x a jP e for all a .The accuracy of b q j ðx Þcan be proven following the same lines as in [26].The second step is to enforce the positivity of the pressure.We need to introduce some notations.Letb q j ðx Þ¼ðb q j ðx Þ;m j ðx Þ;E j ðx ÞÞT and b q a j denote b q j b x a j.DefineG e¼w ¼q m E 1C A 0B @q P e and p ¼ðc À1ÞE À12m 2q P e 8><>:9>=>;;ð2:6Þ@G e ¼w ¼q m E 1C A 0B @q P eand p ¼ðc À1ÞE À1m 2q ¼e 8><>:9>=>;;ð2:7Þands aðt Þ¼ð1Àt Þw n jþt b q j b x a j;06t 61:ð2:8ÞG e is a convex set thanks to (1.3).o G e in (2.7)is a surface which contains part of the boundary of G e .s a (t )in (2.8)is the straight line passing through the two points w n j and b q j b x a j.If b q a j lies outside of G e ,namely p b q a j<e ,then there exists an intersection point of the straight line s a (t )and the surface @G e .Let s a e denote this intersection,then s a e ¼s a t a e ÀÁfor some t a e 2½0;1 satisfying p s a t a eÀÁÀÁ¼e .We will abuse the notation and let s a e ¼b q a j if p ðb q a j Þ2G e .So we haves ae ¼s a t a e ÀÁ;if p b q a j<e ;b q a j ;if p b q a jP e :8><>:ð2:9ÞWe consider the following new vector of polynomials~q j ðx Þ¼h 2b q j ðx ÞÀw n jþw n j ;ð2:10Þ8922X.Zhang,C.-W.Shu /Journal of Computational Physics 229(2010)8918–8934withh 2¼min a ¼1;2;...;N s a e Àw n jb q a j Àw n j:ð2:11ÞIt is easy to see that the cell average of ~q j ðx Þover I j is w n j .Next we would like to show the following lemma.Lemma 2.5.The ~q j ðx Þdefined in (2.10)and (2.11)satisfies ~q j b x a j2G e &G for all a .Proof.Notice that ~q j ðx Þis actually a convex combination of b q j ðx Þand w n j ,so the density of ~q j b x a jis no less than e .For the same reason,p ð~q j b x a j ÞP e if p b q j b x a jP e .If p b q j b x a j<e ,then p s a e ÀÁ¼e and ~qj b x aj¼h 2b q j b x a j Àw n j þw n j ¼h 2a e t a e b q j b x a j Àw n j þw n j h i þ1Àh 2a e w n j ¼h 2a e s a e þ1Àh 2a ew n j :Notice that s a e Àw n jb q a jÀw n j¼t a e .Therefore,(2.11)implies h 26t a e .So ~q j b x a j is a convex combination of s a e and w n j ,and thus p ~q j b x a j P e .h Finally,we need to show the limiter (2.10)and (2.11)does not destroy accuracy when q j (x )approximates a smooth solu-tion.Define d (z ,G )=min w 2G k z Àw k .Assume the exact solution v (x ,t n )is smooth and d (v (x ,t n ),G e )P M ,"x ,for some constant M >0.It suffices to show h 2=1+O (D x k +1).If h 2<1,then h 2¼s b e Àw n j =b q b j Àw n jfor some b where s b e is the intersection of thestraight line and the surface.Since w n j is a (k +1)th order approximation to the cell average of v (x ,t n ),we have d w n j ;G eP M þO ðD x k þ1ÞP M2if D x is smallenough.We can also assume the overshoot s b e Àb q b j ¼O ðD x k þ1Þsince b q b j is a (k +1)th order approximation to a point in G e .Thus,j 1Àh 2j ¼1Àh 2¼1Às b e Àw n j b q b j Àw n j ¼s b e Àb q b j b q bj Àw n j6s b e Àb q b jd w n j ;Ge ¼O ðD x k þ1Þ;where the third equality is due to the fact that b q b j ;s b e and w n j lie on the same line.Therefore,the limiting process (2.4),(2.5),(2.10)and (2.11)returns ~qj ðx Þsatisfying the accuracy,positivity and conservativity.2.3.Implementation for the DG methodAt time level n ,assuming the DG polynomial in cell I j is q j (x )=(q j (x ),m j (x ),E j (x ))T with degree k ,and the cell average of q j (x )is w n j ¼q n j ;m n j ;E nj T2G ,then the algorithm flowchart of our algorithm for the Euler forward is Set up a small number e ¼min j 10À13;q n j ;p w n jn o . In each cell,modify the density first:evaluate min a ¼1;...;N q j b x a j and get b q j ðx Þby (2.4)and (2.5),set b q j ðx Þ¼b q j ðx Þ;m j ðx Þ;E j ðx ÞÀÁT. Then modify the pressure:let b q a j denote b q j x a j ,for each a ,if p b q a j<e ,then solve the following quadratic equation for t a e ,p 1Àt a e ÀÁw nj þt a e b q j b x aj h i ¼e :ð2:12ÞIf p b q a jP e ,then set t a e ¼1.h 2in (2.11)is mathematically equivalent to h 2¼min a ¼1;...;N t a e .Get ~q j ðx Þby (2.10). Use ~qj ðx Þinstead of q j (x )in the DG scheme with Euler forward in time under the CFL condition (2.1).h For SSP high order time discretizations,we need to use the limiter in each stage for a Runge–Kutta method or in each stepfor a multi-step method.Remark 2.6.The implementation for a finite volume method is similar,but it will be a little bit more complicated for WENO since there are only nodal values but no polynomials in each cell after WENO reconstruction.One way to implement the limiter is to construct polynomials using the nodal values and cell averages,see [26]for details.We are also exploring other,simpler ways to implement this positivity preserving limiter for WENO finite volume schemes.These implementation details and numerical tests will be reported elsewhere.X.Zhang,C.-W.Shu /Journal of Computational Physics 229(2010)8918–89348923Remark 2.7.Theoretically,there is a complication regarding the CFL condition (2.1)for a Runge–Kutta time discretization.Consider the third order SSP Runge–Kutta method.To enforce (2.1)rigorously,we need to get an accurate estimation of k (j u j +c )k 1for all the three stages based only on the numerical solution at time level n ,which is highly nontrivial mathe-matically.In practice,we can simply multiply a factor,for example 2to 3,to the quantity k (j u j +c )k 1of w n ,as an estimation for all the stages.Although this is a rough estimation,it works well for us to choose a time step satisfying (2.1)in all the examples in Section 4.To be more efficient,we could implement this more stringent CFL condition only when a preliminary calculation to the next time step produces negative density or pressure.This complication does not exist if we use a SSP multi-step time discretization.3.Positivity-preserving high order schemes in two dimensions 3.1.A sufficient conditionIn this section we extend our result to finite volume or DG schemes of (k +1)th order accuracy on rectangular meshes solving two-dimensional Euler equationsw t þf ðw Þx þg ðw Þy ¼0;t P 0;ðx ;y Þ2R 2;ð3:1Þw ¼qm n EB B B @1C C C A;f ðw Þ¼mq u 2þp q u v ðE þp ÞuB B B @1C CC A;g ðw Þ¼nq u v q v 2þp ðE þp ÞvB B B @1C CC Að3:2Þwithm ¼q u ;n ¼q v ;E ¼12q u 2þ12q v 2þq e ;p ¼ðc À1Þq e ;where q is the density,u is the velocity in x direction,v is the velocity in y direction,m and n are the momenta,E is the totalenergy,p is the pressure,e is the internal energy.The speed of sound is given by c ¼ffiffiffiffiffiffiffiffiffiffiffic p =q p .The eigenvalues of the Jacobian f 0(w )are u Àc ,u ,u and u +c and the eigenvalues of the Jacobian g 0(w )are v Àc ,v ,v and v +c .The pressure function p is still concave with respect to w if q P 0and the set of admissible statesG ¼w ¼q m n E1C C C A 0B B B @q >0and p ¼ðc À1ÞE À12m 2q À12n 2q >08>>><>>>:9>>>=>>>;is still convex.For simplicity we assume we have a uniform rectangular mesh.At time level n ,we have a vector of approximation poly-nomials of degree k ,q ij (x ,y )=(q ij (x ,y ),m ij (x ,y ),n ij (x ,y ),E ij (x ,y ))T with the cell average w n ij ¼q n ij ;m n ij ;n n ij ;E nij Ton the (i ,j )cell x i À1;x i þ1h i Ây j À1;y j þ1h i .Letw þi À12;j ðy Þ;w Ài þ12;j ðy Þ;w þi ;j À12ðx Þ;w Ài ;j þ12ðx Þdenote the traces of q ij (x ,y )on the four edges respectively,see Fig.3.1.All of the traces are vectors of single variable polynomials of degree k.8924X.Zhang,C.-W.Shu /Journal of Computational Physics 229(2010)8918–8934。
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ANALYSIS OF THE BINARY EUCLIDEAN ALGORITHMRICHARD P.BRENTAbstractThe classical Euclidean algorithm forfinding the greatest common divisor of two positive integers has been exhaustively analyzed since the time of Gauss.The theory of binary Euclidean algorithms is less well-developed.We analyze the“right-shift”binary Euclidean algorithm of Silver and Terzian[11]and Stein[12].In particular,we show that the expected number of iterations for uniformly distributed inputs in{1,2,3,...,N}is asymptotic to K log2N as N→∞,where K 0.706.We introduce another binary Euclidean algorithm,the“left-shift”algorithm,and consider its expected behaviour on random inputs.The expected number of iterations for the left-shift algorithm is slightly greater than for the right-shift algorithm,but the left-shift algorithm is worth considering for use on a computer with a“normalize”instruction,as then the left-shifting loop may be replaced by a single instruction.Either of the binary algorithms could be implemented in hardware(or microcode)with approximately the same expense as integer division.CommentsOnly the Abstract is given here.The full paper appeared as[2].Binary Euclidean algorithms were later applied to give linear-time systolic algorithms for integer GCD computation[6,8,7,1]. The polynomial GCD problem[5]is simpler because of the lack of carries.The probabilistic assumptions of[2]were given a rigorous foundation by Vall´e e[13,14].Minor ErrataIn the definition of D0(x)on the last line of page326D0(x)=0should be replaced by D0(x)=1.In equation(6.3)on page342,the term−x2(1+x)should be replaced by−12(1+x).Major ErrataSome of the results are incorrect.For example,(3.1),(3.29),(3.34),(3.35)are wrong(though a close approximation to the truth).Further details are given in[3,4].See also[10,§4.5.2].1991Mathematics Subject Classification.Primary68Q25;Secondary65Y10,68Q35.Key words and phrases.Euclidean algorithm,binary Euclidean algorithm,greatest common divisor,GCD, continued fraction,left shift,right shift,systolic algorithm.This research was supported in part by the National Science Foundation under Grant MCS75-222-55and the Office of Naval Research under Contract N0014-76-C-0370,NR044-422.The author thanks Frank de Hoog and Don Knuth for their assistance and encouragement.Copyright c 1976,Academic Press,Inc.Abstract and Comments c 1993–1999,R.P.Brent.rpb037a typeset using A M S-L A T E X.2RPB037A,BRENTAcknowledgementsThanks to Don Knuth,Philippe Flajolet and Brigitte Vall´e e for their assistance in correcting and extending my1976results[2].References[1]A.W.Bojanczyk and R.P.Brent,“A systolic algorithm for extended GCD computation”,Comput.Math.Applic.14(1987),233–238.rpb096.[2]R.P.Brent,“Analysis of the binary Euclidean algorithm”,in New Directions and Recent Results in Algorithmsand Complexity(edited by J.F.Traub),Academic Press,New York,1976,321–355.MR54#14417,55#11701;Zbl363.00013,373.68040.Also appeared as a Technical Report,Department of Computer Science,Carnegie-Mellon University(June1976),35pp.Extended abstract appeared in SIGSAM Bulletin(May1976).rpb037.[3]R.P.Brent,“Twenty years’analysis of the binary Euclidean algorithm”,Proc.Symposium in Celebration ofthe Work of C.A.R.Hoare,Oxford,Sept.1999,to appear.rpb183.[4]R.P.Brent,Further analysis of the Binary Euclidean algorithm,Tech.Report,Oxford University ComputingLaboratory,to appear.rpb183tr[5]R.P.Brent and H.T.Kung,“Systolic VLSI arrays for polynomial GCD computation”,IEEE Trans.onComputers C–33(1984),731–736.rpb073.[6]R.P.Brent and H.T.Kung,“A systolic VLSI array for integer GCD computation”,in ARITH-7,Proc.Seventh Symposium on Computer Arithmetic(edited by K.Hwang),IEEE/CS Press,1985.rpb077.[7]R.P.Brent and H.T.Kung,“Systolic VLSI arrays for linear-time GCD computation”,in VLSI83(editedby F.Anceau and E.J.Aas),North-Holland,Amsterdam,1983,145–154.rpb082.[8]R.P.Brent,H.T.Kung and F.T.Luk,“Some linear-time algorithms for systolic arrays”,in InformationProcessing83(edited by R.E.A.Mason),North-Holland,Amsterdam,1983,865–876.rpb079.[9]D.E.Knuth,The Art of Computer Programming,Volume2,Addison-Wesley,Menlo Park,first edition,1969.[10]D.E.Knuth,The Art of Computer Programming,Volume2,Addison-Wesley,Menlo Park,third edition,1997.[11]R.Silver and J.Terzian,unpublished,1962.See Knuth[9,§4.5.2–4.5.3].[12]J.Stein,“Computational problems associated with Racah algebra”,put.Phys.1(1967),397–405.[13]B.Vall´e e,The complete analysis of the binary Euclidean algorithm,Proc.ANTS’98,Lecture Notes in Com-puter Science1423,Springer-Verlag,1998,77–94.[14]B.Vall´e e,Dynamics of the binary Euclidean algorithm:functional analysis and operators,manuscript,Feb.1998(to appear in Algorithmica)..unicaen.fr/~brigitte/Publications/bin-gcd.ps Australian National University and Carnegie-Mellon University。