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ON THE APPROXIMATION OF QUASIPERIODIC FUNCTIONS WITH DIOPHANTINE FREQUENCIES BY PERIODIC FUNCTIONS

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arXiv 2023年

We present an analysis of the approximation error for a d-dimensional quasiperiodic function f with Diophantine frequencies, approximated by a periodic function with the fundamental domain [0, L1) × [0, L2) × · · · × [0, Ld). When f has a certain regularity, its global behavior can be described by a finite number of Fourier components and has a polynomial decay at infinity. The dominant part of periodic approximation error is bounded by (formula presented), where Lj belongs to the best simultaneous approximation sequence and sj is the number of different irrational elements in j-th dimension component of Fourier frequencies, respectively. Meanwhile, we discuss the approximation rate. Finally, these analytical results are verified by some examples. Copyright © 2023, The Authors. All rights reserved.

关键词： Natural frequencies

Convergence analysis of PM-BDF2 method for quasiperiodic parabolic equations

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arXiv 2024年

作者： Jiang, Kai Li, Meng Zhang, Juan Zhang, Lei Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing and Information Processing Ministry of Education Department of Mathematics and Computational Science Xiangtan University Hunan Xiangtan411105 China School of Mathematical Sciences Institute of Natural Sciences MOE-LSC Shanghai Jiao Tong University Shanghai200240 China

Numerically solving parabolic equations with quasiperiodic coefficients is a significant challenge due to the potential formation of space-filling quasiperiodic structures that lack translational symmetry or decay. In this paper, we introduce a highly accurate numerical method for solving time-dependent quasiperiodic parabolic equations. We discretize the spatial variables using the projection method (PM) and the time variable with the second-order backward differentiation formula (BDF2). We provide a complexity analysis for the resulting PM-BDF2 method. Furthermore, we conduct a detailed convergence analysis, demonstrating that the proposed method exhibits spectral accuracy in space and second-order accuracy in time. Numerical results in both one and two dimensions validate these convergence results, highlighting the PMBDF2 method as a highly efficient algorithm for addressing quasiperiodic parabolic equations. Copyright © 2024, The Authors. All rights reserved.

关键词： Convergence of numerical methods

ACCURATELY RECOVER GLOBAL QUASIPERIODIC SYSTEMS BY FINITE POINTS

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arXiv 2023年

作者： Jiang, Kai Zhou, Qi Zhang, Pingwen Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing and Information Processing Ministry of Education School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan411105 China School of Mathematics and Statistics Wuhan University Wuhan430072 China School of Mathematical Sciences Peking University Beijing100871 China

Quasiperiodic systems, related to irrational numbers, are space-filling structures without decay nor translation invariance. How to accurately recover these systems, especially for non-smooth cases, presents a big challenge in numerical computation. In this paper, we propose a new algorithm, finite points recovery (FPR) method, which is available for both smooth and non-smooth cases, to address this challenge. The FPR method first establishes a homomorphism between the lower-dimensional definition domain of the quasiperiodic function and the higher-dimensional torus, then recovers the global quasiperiodic system by employing interpolation technique with finite points in the definition domain without dimensional lifting. Furthermore, we develop accurate and efficient strategies of selecting finite points according to the arithmetic properties of irrational numbers. The corresponding mathematical theory, convergence analysis, and computational complexity analysis on choosing finite points are presented. Numerical experiments demonstrate the effectiveness and superiority of FPR approach in recovering both smooth quasiperiodic functions and piecewise constant Fibonacci quasicrystals. While existing spectral methods encounter difficulties in accurately recovering non-smooth quasiperiodic *** Codes 11J68, 65D05, 65D15, 68W25 © 2023, CC BY-NC-ND.

关键词： Interpolation

The vehicle audio encryption system based on scrambling thought

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The vehicle audio encryption system based on scrambling thou...

2018 International Conference on Network, Communication, Computer engineering (NCCE 2018)

作者： Peibin Wu Jindong Zhang Yang Liu Dongqi Han Yiding Sun College of Computer Science and Technology Jilin University Key Laboratory of Symbol Computation and Knowledge Engineering of the Ministry of Education Jilin University State Key Laboratory of Automobile Simulation and Control Jilin University

ISBN: (纸本)9781510871076

In this paper,the vehicle audio encryption system based on scrambling thought and applied to vehicle-mounted media is *** pseudo random number is generated each time it is encrypted,and it is substituted into the Fibonacci transform to generate a key with variable *** audio data is divided by the number of pseudo random numbers into the Fibonacci sequence,and is encrypted in permutation *** Fibonacci transform has symmetry and creates an audio file that cannot be parsed and played after the permutation *** the security analysis of encryption algorithm,the audio files of different length are *** key space,anti-attack ability,key sensitivity and other indicators are tested to achieve good *** addition,this algorithm is simple in calculation and can realize the ideal real-time performance in practical application.

关键词： vehicle media audio encryption scrambling thought security analysis

A posteriori error estimation for an interior penalty virtual element method of Kirchhoff plates

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arXiv 2024年

作者： Feng, Fang Hu, Yuming Yu, Yue Zhao, Jikun School of Mathematics and Statistics Nanjing University of Science and Technology Nanjing China Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education National Center for Applied Mathematics in Hunan School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan411105 China School of Mathematics and Statistics Zhengzhou University Zhengzhou450001 China

In this paper, we develop a residual-type a posteriori error estimation for an interior penalty virtual element method (IPVEM) for the Kirchhoff plate bending problem. Building on the work in [18], we adopt a modified discrete variational formulation that incorporates the H1-elliptic projector in the jump and average terms. This allows us to simplify the numerical implementation by including the H1-elliptic projector in the computable error estimators. We derive the reliability and efficiency of the a posteriori error bound by constructing an enriching operator and establishing some related error estimates that align with C0-continuous interior penalty finite element methods. As observed in the a priori analysis, the interior penalty virtual elements exhibit similar behaviors to C0continuous elements despite its discontinuity. This observation extends to the a posteriori estimate since we do not need to account for the jumps of the function itself in the discrete scheme and the error estimators. As an outcome of the error estimator, an adaptive VEM is introduced by means of the mesh refinement strategy with the one-hanging-node rule. Numerical results from several benchmark tests confirm the robustness of the proposed error estimators and show the efficiency of the resulting adaptive VEM. Copyright © 2024, The Authors. All rights reserved.

关键词： Discrete element methods

Improved stability criteria for time-varying delayed T–S fuzzy systems via delay partitioning approach

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Fuzzy Sets and Systems 2011年第1期185卷 83-94页

作者： Jiyao An Guilin Wen The Key Laboratory of Advanced Design and Simulation Techniques for Special Equipment (Ministry of Education) the State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body Hunan University Changsha 410082 PR China College of Information Science and Engineering Hunan University Changsha 410082 PR China

This paper focuses on the stability analysis for uncertain Takagi–Sugeno (T–S) fuzzy systems with interval time-varying delay. The uncertainties of system parameter matrices are assumed to be time-varying and norm-bounded. Some new Lyapunov–Krasovskii functionals (LKFs) are constructed by nonuniformly dividing the whole delay interval into multiple segments and choosing different Lyapunov functionals to different segments in the LKFs. By employing these LKFs, some new delay-derivative-dependent stability criteria are established for the nominal and uncertain T–S fuzzy systems in a convex way. These stability criteria are derived that depend on both the upper and lower bounds of the time derivative of the delay. By employing the new delay partitioning approach, the obtained stability criteria are stated in terms of linear matrix inequality (LMI). They are equivalent or less conservative while involving less decision variables than the existing results. Finally, numerical examples are given to illustrate the effectiveness and reduced conservatism of the proposed results.

关键词： Uncertain system T–S fuzzy system Delay-derivative-dependent stability Interval time-varying delay Linear matrix inequality (LMI)

A multigrid-reduction-in-time solver with a new two-level convergence for unsteady fractional Laplacian problems

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arXiv 2019年

作者： Yue, Xiaoqiang Pan, Kejia Zhou, Jie Weng, Zhifeng Shu, Shi Tang, Juan Key Laboratory of Intelligent Computing & Information Processing of Ministry of Education Hunan Key Laboratory for Computation and Simulation in Science and Engineering School of Mathematics and Computational Science Xiangtan University Xiangtan411105 China School of Mathematics and Statistics HNP-LAMA Central South University Changsha410083 China School of Mathematical Sciences Fujian Province University Key Laboratory of Computation Science Huaqiao University Quanzhou362021 China

The multigrid-reduction-in-time (MGRIT) technique has proven to be successful in achieving higher run-time speedup by exploiting parallelism in time. The goal of this article is to develop and analyze a MGRIT algorithm, using FCF-relaxation with time-dependent time-grid propagators, to seek the finite element approximations of unsteady fractional Laplacian problems. The multigrid with line smoother proposed in [L. Chen, R. H. Nochetto, et al., Math. Comp. 85 (2016) 2583–2607] is chosen to be the spatial solver. Motivated by [B. S. Southworth, SIAM J. Matrix Anal. Appl. 40 (2019) 564–608], we provide a new temporal eigenvalue approximation property and then deduce a generalized two-level convergence theory which removes the previous unitary diagonalization assumption on the fine and coarse time-grid propagators required in [X. Q. Yue, S. Shu, et al., Comput. Math. Appl. 78 (2019) 3471–3484]. Numerical computations are included to confirm the theoretical predictions and demonstrate the sharpness of the derived convergence upper bound. Copyright © 2019, The Authors. All rights reserved.

关键词： Finite element method

A Facile and Efficient Synthesis of Spiropyrrolidine Derivatives Via [3+2] Cycloaddition

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A Facile and Efficient Synthesis of Spiropyrrolidine Derivat...

中国化学会第六届有机化学学术会议

作者： YI,Pin-Gui LI,Xiao-Fang YU,Xian-Yong Wang,Tao Key Laboratory of Theoretical Chemistry and Molecular Simulation of Ministry of Education Hunan Province College Key Laboratory of QSAR/QSPR School of Chemistry and Chemical Engineering Hunan University of Science and Technology Xiangtan 411201 China

Synthesis of N-Substituted Dihydropyrimidinones Via an Unexpected Reaction of Thiazolo[3,2-a]Pyrimidine Derivatives and Nitrile Oxide

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Synthesis of N-Substituted Dihydropyrimidinones Via an Unexp...

中国化学会第六届有机化学学术会议

作者： ZENG,Rong-Jin LI,Xiao-Fang YU,Xian-Yong Key Laboratory of Theoretical Chemistry and Molecular Simulation of Ministry of Education Hunan Province College Key Laboratory of QSAR/QSPR School of Chemistry and Chemical Engineering Hunan University of Science and Technology Xiangtan 411201 China