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Nonlinear Extrapolation Estimates of π

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Acta Mathematicae Applicatae Sinica 2024年第1期40卷 91-108页

作者： Wen-qing Xu Sha-sha Wang Da-chuan Xu Beijing Institute for Scientific and Engineering computing Beijing University of TechnologyBeijing100124China department of mathematics and Statistics California State UniversityLong BeachCalifornia90840USA department of mathematics and Physics Shijiazhuang Tiedao UniversityShijiazhuang050043China

The classical Archimedean approximation of π uses the semiperimeter or area of regular polygons inscribed in or circumscribed about a unit circle in R~2 and it is well-known that by using linear combinations of these basic estimates,modern extrapolation techniques can greatly speed up the approximation process.Similarly,when n vertices are randomly selected on the circle,the semiperimeter and area of the corresponding random inscribed and circumscribing polygons are known to converge to π almost surely as n→∞,and by further applying extrapolation processes,faster convergence rates can also be achieved through similar linear combinations of the semiperimeter and area of these random polygons.In this paper,we further develop nonlinear extrapolation methods for approximating π through certain nonlinear functions of the semiperimeter and area of such polygons.We focus on two types of extrapolation estimates of the forms χ_n=S_n~αA_n~β and Y_n(p)=(αS_n~p+βA_n~p)~(1/p) where α+β=1,p≠0,and Sn and An respectively represents the semiperimeter and area of a random n-gon inscribed in the unit circle in R~2,and Xn may be viewed as the limit of Y_n(p) when p→0.By deriving probabilistic asymptotic expansions with carefully controlled error estimates for Xn and Y_n(p),we show that the choice α=4/3,β=-1/3 minimizes the approximation error in both cases,and their distributions are also asymptotically normal.

关键词： Archimedean polygons random polygons approximations ofπ extrapolation processes asymptotic convergence

ONE-PARAMETER FINITE DIFFERENCE METHODS AND THEIR ACCELERATED SCHEMES FOR SPACE-FRACTIONAL SINE-GORDON EQUATIONS WITH DISTRIBUTED DELAY

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Journal of Computational mathematics 2024年第3期42卷 705-734页

作者： Tao Sun Chengjian Zhang Haiwei Sun School of mathematics and Statistics Huazhong University of Science and TechnologyWuhan 430074China Hubei Key Laboratory of Engineering Modeling and Scientific computing Huazhong University of Science and TechnologyWuhan 430074China department of mathematics University of MacaoMacaoChina

This paper deals with numerical methods for solving one-dimensional(1D)and twodimensional(2D)initial-boundary value problems(IBVPs)of space-fractional sine-Gordon equations(SGEs)with distributed delay.For 1D problems,we construct a kind of oneparameter finite difference(OPFD)method.It is shown that,under a suitable condition,the proposed method is convergent with second order accuracy both in time and space.In implementation,the preconditioned conjugate gradient(PCG)method with the Strang circulant preconditioner is carried out to improve the computational efficiency of the OPFD method.For 2D problems,we develop another kind of OPFD method.For such a method,two classes of accelerated schemes are suggested,one is alternative direction implicit(ADI)scheme and the other is ADI-PCG scheme.In particular,we prove that ADI scheme can arrive at second-order accuracy in time and space.With some numerical experiments,the computational effectiveness and accuracy of the methods are further verified.Moreover,for the suggested methods,a numerical comparison in computational efficiency is presented.

关键词： Fractional sine-Gordon equation with distributed delay One-parameter finite difference methods Convergence analysis ADI scheme PCG method

Energy Stable Nodal DG Methods for Maxwell’s Equations of Mixed-Order Form in Nonlinear Optical Media

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Communications on Applied mathematics and Computation 2024年第1期6卷 30-63页

作者： Maohui Lyu Vrushali A.Bokil Yingda Cheng Fengyan Li State Key Laboratory of Scientific and Engineering computing(LSEC) Academy of Mathematics and System SciencesChinese Academy of SciencesBeijing100190China department of mathematics College of ScienceOregon State UniversityCorvallisOR97331USA department of mathematics Michigan State UniversityEast LansingMI48824USA department of Computational mathematics Science and EngineeringMichigan State UniversityEast LansingMI48824USA department of Mathematical Sciences Rensselaer Polytechnic InstituteTroyNY12180USA

In this work,we develop energy stable numerical methods to simulate electromagnetic waves propagating in optical media where the media responses include the linear Lorentz dispersion,the instantaneous nonlinear cubic Kerr response,and the nonlinear delayed Raman molecular vibrational response.Unlike the first-order PDE-ODE governing equations considered previously in Bokil et al.(J Comput Phys 350:420–452,2017)and Lyu et al.(J Sci Comput 89:1–42,2021),a model of mixed-order form is adopted here that consists of the first-order PDE part for Maxwell’s equations coupled with the second-order ODE part(i.e.,the auxiliary differential equations)modeling the linear and nonlinear dispersion in the material.The main contribution is a new numerical strategy to treat the Kerr and Raman nonlinearities to achieve provable energy stability property within a second-order temporal discretization.A nodal discontinuous Galerkin(DG)method is further applied in space for efficiently handling nonlinear terms at the algebraic level,while preserving the energy stability and achieving high-order accuracy.Indeed with d_(E)as the number of the components of the electric field,only a d_(E)×d_(E)nonlinear algebraic system needs to be solved at each interpolation node,and more importantly,all these small nonlinear systems are completely decoupled over one time step,rendering very high parallel efficiency.We evaluate the proposed schemes by comparing them with the methods in Bokil et al.(2017)and Lyu et al.(2021)(implemented in nodal form)regarding the accuracy,computational efficiency,and energy stability,by a parallel scalability study,and also through the simulations of the soliton-like wave propagation in one dimension,as well as the spatial-soliton propagation and two-beam interactions modeled by the two-dimensional transverse electric(TE)mode of the equations.

关键词： Maxwell’s equations Kerr and Raman Discontinuous Galerkin method Energy stability

Secure and Reliable Routing in the Internet of Vehicles Network:AODV-RL with BHA Attack Defense

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Computer Modeling in Engineering & Sciences 2024年第4期139卷 633-659页

作者： Nadeem Ahmed Khalid Mohammadani Ali Kashif Bashir Marwan Omar Angel Jones Fayaz Hassan School of Electronic Science Beijing University of Post and TelecommunicationBeijing100086China department of Computer Science Huanggang Normal UniversityHuanggangChina department of computing and mathematics Manchester Metropolitan UniversityManchesterUK Woxsen School of Business Woxsen UniversityHyderabad502345India department of Computer Science and mathematics Lebanese American UniversityBeirutLebanon Information Technology and Management Illinois Institute of TechnologyChicagoUSA College of computing Capital Technology UniversityLaurelUSA

Wireless technology is transforming the future of transportation through the development of the Internet of Vehicles(IoV).However,intricate security challenges are intertwinedwith technological progress:Vehicular ad hoc Networks(VANETs),a core component of IoV,face security issues,particularly the Black Hole Attack(BHA).This malicious attack disrupts the seamless flow of data and threatens the network’s overall reliability;also,BHA strategically disrupts communication pathways by dropping data packets from legitimate nodes altogether.Recognizing the importance of this challenge,we have introduced a new solution called ad hoc On-Demand Distance Vector-Reputation-based mechanism Local Outlier Factor(AODV-RL).The significance of AODVRL lies in its unique approach:it verifies and confirms the trustworthiness of network components,providing robust protection against BHA.An additional safety layer is established by implementing the Local Outlier Factor(LOF),which detects and addresses abnormal network behaviors.Rigorous testing of our solution has revealed its remarkable ability to enhance communication in VANETs.Specifically,Our experimental results achieve message delivery ratios of up to 94.25%andminimal packet loss ratios of just 0.297%.Based on our experimental results,the proposedmechanismsignificantly improves VANET communication reliability and security.These results promise a more secure and dependable future for IoV,capable of transforming transportation safety and efficiency.

关键词： Black hole attack IoV vehicular ad hoc network AODV routing protocol

Transformer-Aided Deep Double Dueling Spatial-Temporal Q-Network for Spatial Crowdsourcing Analysis

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Computer Modeling in Engineering & Sciences 2024年第4期139卷 893-909页

作者： Yu Li Mingxiao Li Dongyang Ou Junjie Guo Fangyuan Pan department of computing Hangzhou Dianzi UniversityHangzhou310018China

With the rapid development ofmobile Internet,spatial crowdsourcing has becomemore andmore popular.Spatial crowdsourcing consists of many different types of applications,such as spatial crowd-sensing services.In terms of spatial crowd-sensing,it collects and analyzes traffic sensing data from clients like vehicles and traffic lights to construct intelligent traffic prediction models.Besides collecting sensing data,spatial crowdsourcing also includes spatial delivery services like DiDi and Uber.Appropriate task assignment and worker selection dominate the service quality for spatial crowdsourcing applications.Previous research conducted task assignments via traditional matching approaches or using simple network models.However,advanced mining methods are lacking to explore the relationship between workers,task publishers,and the spatio-temporal attributes in tasks.Therefore,in this paper,we propose a Deep Double Dueling Spatial-temporal Q Network(D3SQN)to adaptively learn the spatialtemporal relationship between task,task publishers,and workers in a dynamic environment to achieve optimal allocation.Specifically,D3SQNis revised through reinforcement learning by adding a spatial-temporal transformer that can estimate the expected state values and action advantages so as to improve the accuracy of task assignments.Extensive experiments are conducted over real data collected fromDiDi and ELM,and the simulation results verify the effectiveness of our proposed models.

关键词： Historical behavior analysis spatial crowdsourcing deep double dueling Q-networks

New Results on Equivalence of Multivariate Polynomial Matrices

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Journal of Systems Science & Complexity 2024年

作者： GUAN Jiancheng LIU Jinwang ZHENG Licui WU Tao LIU Jie School of mathematics and computing Science Hunan University of Science and Technology

This paper investigates equivalence of square multivariate polynomial matrices with the determinant being some power of a univariate irreducible polynomial.The authors give a necessary and sufficient condition for this equivalence.And the authors present an algorithm to reduce a class of multivariate polynomial matrices to their Smith forms.

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Derivatives of Intersection Local Time for Two Independent Symmetricα-stable Processes

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Acta Mathematica Sinica,English Series 2024年第5期40卷 1273-1292页

作者： Huan ZHOU Guang Jun SHEN Qian YU department of mathematics Anhui Normal UniversityWuhu 241002P.R.China department of mathematics Nanjing University of Aeronautics and AstronauticsNanjing 211106P.R.China Key Laboratory of Mathematical Modelling and High Performance computing of Air Vehicles(NUAA) MIITNanjing 211106P.R.China

In this paper,we consider the derivatives of intersection local time for two independent d-dimensional symmetricα-stable processes X^(α) and X^(α)with respective indices α and α.We first study the sufficient condition for the existence of the derivatives,which makes us obtain the exponential integrability and H?lder continuity.Then we show that this condition is also necessary for the existence of derivatives of intersection local time at the origin.Moreover,we also study the power variation of the derivatives.

关键词： Symmetric stable processes intersection local time exponential integrability power variation

A Survey on Chinese Sign Language Recognition:From Traditional Methods to Artificial Intelligence

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Computer Modeling in Engineering & Sciences 2024年第7期140卷 1-40页

作者： Xianwei Jiang Yanqiong Zhang Juan Lei Yudong Zhang School of mathematics and Information Science Nanjing Normal University of Special EducationNanjing210038China School of computing and Mathematical Sciences University of LeicesterLeicesterLE17RHUK department of Information Technology Faculty of Computing and Information TechnologyKing Abdulaziz UniversityJeddah21589Saudi Arabia

Research on Chinese Sign Language(CSL)provides convenience and support for individuals with hearing impairments to communicate and integrate into society.This article reviews the relevant literature on Chinese Sign Language Recognition(CSLR)in the past 20 years.Hidden Markov Models(HMM),Support Vector Machines(SVM),and Dynamic Time Warping(DTW)were found to be the most commonly employed technologies among traditional identificationmethods.Benefiting from the rapid development of computer vision and artificial intelligence technology,Convolutional Neural Networks(CNN),3D-CNN,YOLO,Capsule Network(CapsNet)and various deep neural networks have sprung up.Deep Neural Networks(DNNs)and their derived models are integral tomodern artificial intelligence recognitionmethods.In addition,technologies thatwerewidely used in the early days have also been integrated and applied to specific hybrid models and customized identification methods.Sign language data collection includes acquiring data from data gloves,data sensors(such as Kinect,LeapMotion,etc.),and high-definition photography.Meanwhile,facial expression recognition,complex background processing,and 3D sign language recognition have also attracted research interests among scholars.Due to the uniqueness and complexity of Chinese sign language,accuracy,robustness,real-time performance,and user independence are significant challenges for future sign language recognition research.Additionally,suitable datasets and evaluation criteria are also worth pursuing.

关键词： Chinese Sign Language Recognition deep neural networks artificial intelligence transfer learning hybrid network models

Moving collocation methods for time fractional differential equations and simulation of blowup

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Science China mathematics 2011年第3期54卷 611-622页

作者： MA JingTang1 & JIANG YingJun2 1School of Economic mathematics, Southwestern University of Finance and Economics, Chengdu 611130, China 2department of mathematics and Scientific computing, Changsha University of Science and Technology, Changsha 410076, China School of Economic mathematics Southwestern University of Finance and Economics Chengdu China department of mathematics and Scientific computing Changsha University of Science and Technology Changsha China

A moving collocation method is proposed and implemented to solve time fractional differential equations. The method is derived by writing the fractional differential equation into a form of time difference equation. The method is stable and has a third-order convergence in space and first-order convergence in time for either linear or nonlinear equations. In addition, the method is used to simulate the blowup in the nonlinear equations.

关键词：分数微分方程运动时间爆破模拟搭配方法非线性方程组分数阶微分方程三阶收敛差分方程