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Analysis of multivariate Gegenbauer approximation in the hypercube

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arXiv 2018年

作者： Wang, Haiyong Zhang, Lun School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China School of Mathematical Sciences Fudan University Shanghai200433 China

In this paper, we are concerned with multivariate Gegenbauer approximation of analytic functions defined in the d-dimensional hypercube. Two new and sharper bounds for the coefficients of multivariate Gegenbauer expansion are presented based on two different extensions of the Bernstein ellipse. We then establish an explicit error bound for the multivariate Gegenbauer approximation associated with an q ball index set in the uniform norm. As an application, we extend our arguments to obtain some new tight bounds for the coefficients of tensorized Legendre expansions in the context of polynomial approximation of parameterized *** Codes 41A10, 41A63, 41A25 Copyright © 2018, The Authors. All rights reserved.

关键词： Geometry

A spectral penalty method for two-sided fractional differential equations with general boundary conditions

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arXiv 2018年

作者： Wang, Nan Mao, Zhiping Huang, Chengming Karniadakis, George Em School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China Division of Applied Mathematics Brown University ProvidenceRI02912 United States Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China

We consider spectral approximations to the conservative form of the two-sided Riemann-Liouville (R-L) and Caputo fractional differential equations (FDEs) with nonhomogeneous Dirichlet (fractional and classical, respectively) and Neumann (fractional) boundary conditions. In particular, we develop a spectral penalty method (SPM) by using the Jacobi poly-fractonomial approximation for the conservative R-L FDEs while using the polynomial approximation for the conservative Caputo FDEs. We establish the well-posedness of the corresponding weak problems and analyze sufficient conditions for the coercivity of the SPM for different types of fractional boundary value problems. This analysis allows us to estimate the proper values of the penalty parameters at boundary points. We present several numerical examples to verify the theory and demonstrate the high accuracy of SPM, both for stationary and time dependent FDEs. Moreover, we compare the results against a Petrov-Galerkin spectral tau method (PGS-τ, an extension of [23]) and demonstrate the superior accuracy of SPM for all cases considered. Copyright © 2018, The Authors. All rights reserved.

关键词： Constrained optimization

A rigorous computer aided estimation for gelfond exponent of weighted thue-morse sequences

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arXiv 2018年

作者： ZHANG, YIWEI YIN, KE WU, WANQUAN School of Mathematics and Statistics Center for Mathematical Science Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology 1037 Luoyu Road Wuhan430074 China School of Computer Science and Technology Huazhong University of Science and Technology 1037 Luoyu Road Wuhan430074 China

In this paper, we will provide a mathematically rigorous computer aided estimation for the exact values and robustness for Gelfond exponent of weighted Thue-Morse sequences. This result improves previous discussions on Gelfond exponent by Gelfond, Devenport, Mauduit, Rivat, Sarközy and Fan et. al.37C40, 90C27 Copyright © 2018, The Authors. All rights reserved.

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A note on the run length function for intermittent maps

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arXiv 2018年

作者： Cui, Hongfei Fang, Lulu Zhang, Yiwei Wuhan Institute of Physics and Mathematics Chinese Academy of Sciences Wuhan430071 China Sun Yat-sen university Guangzhou510275 China School of Mathematics and Statistics Center for Mathematical Sciences Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Sciences and Technology Wuhan430074 China

We study the run length function for intermittent maps. In particular, we show that the longest consecutive zero digits (resp. one digits) having a time window of polynomial (resp. logarithmic) length. Our proof is relatively elementary in the sense that it only relies on the classical Borel-Cantelli lemma and the polynomial decay of intermittent maps. Our results are compensational to the Erd?os-Rényi law obtained by Denker and Nicol in [8]. Copyright © 2018, The Authors. All rights reserved.

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Modelling brain-wide neuronal morphology via rooted Cayley trees

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arXiv 2018年

作者： Lin, Congping Huang, Yuanfei Quan, Tingwei Zhang, Yiwei Center for Mathematical Sciences Huazhong University of Science and Technology Wuhan China Hubei Key Lab of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan China Britton Chance Center for Biomedical Photonics Wuhan National Laboratory for Optoelectronics-Huazhong University of Science and Technology Wuhan China MoE Key Laboratory for Biomedical Photonics Collaborative Innovation Center for Biomedical Engineering School of Engineering Sciences Huazhong University of Science and Technology Wuhan China

Neuronal morphology is an essential element for brain activity and function. We take advantage of current availability of brain-wide neuron digital reconstructions of the Pyramidal cells from a mouse brain, and analyze several emergent features of brain-wide neuronal morphology. We observe that axonal trees are self-affine while dendritic trees are self-similar. We also show that tree size appear to be random, independent of the number of dendrites within single neurons. Moreover, we consider inhomogeneous branching model which stochastically generates rooted 3-Cayley trees for the brain-wide neuron topology. Based on estimated order-dependent branching probability from actual axonal and dendritic trees, our inhomogeneous model quantitatively captures a number of topological features including size and shape of both axons and dendrites. This sheds lights on a universal mechanism behind the topological formation of brain-wide axonal and dendritic trees. Copyright © 2018, The Authors. All rights reserved.

关键词： Brain

A wong-zakai approximation for random slow manifolds with application to parameter estimation

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arXiv 2018年

作者： He, Ziying Zhang, Xinyong Jiang, Tao Liu, Xianming Center for Mathematical Sciences Huazhong University of Sciences and Technology Wuhan430074 China School of Mathematics and Statistics Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Sciences and Technology Wuhan430074 China Department of Mathematical Sciences Tsinghua University Beijing100084 China Collaborative Innovation Center of China Pilot Reform Exploration and Assessment Hubei Sub-Center Hubei University of Economics Wuhan430205 China

We study a Wong-Zakai approximation for the random slow manifold of a slow-fast stochastic dynamical system. We first deduce the existence of the random slow manifold about an approximation system driven by an integrated Ornstein-Uhlenbeck (O-U) process. Then we compute the slow manifold of the approximation system, in order to gain insights of the long time dynamics of the original stochastic system. By restricting this approximation system to its slow manifold, we thus get a reduced slow random system. This reduced slow random system is used to accurately estimate a system parameter of the original system. An example is presented to illustrate this *** Codes Primary: 37L55, 35R60, Secondary: 60H15, 58J65 Copyright © 2018, The Authors. All rights reserved.

关键词： Parameter estimation

A stochastic pitchfork bifurcation in most probable phase portraits

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arXiv 2018年

作者： Wang, Hui Chen, Xiaoli Duan, Jinqiao Center for Mathematical Sciences School of Mathematics and Statistics Hubei Key Laboratory for Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China Department of Applied Mathematics Illinois Institute of Technology ChicagoIL60616 United States Center for Mathematical Sciences Huazhong University of Science and Technology Wuhan430074 China

We study stochastic bifurcation for a system under multiplicative stable Lévy noise (an important class of non-Gaussian noise), by examining the qualitative changes of equilibrium states in its most probable phase portraits. We have found some peculiar bifurcation phenomena in contrast to the deterministic counterpart: (i) When the non-Gaussianity parameter in Lévy noise varies, there is either one, two or none backward pitchfork type bifurcations;(ii) When a parameter in the vector field varies, there are two or three forward pitchfork bifurcations;(iii) The non-Gaussian Lévy noise clearly leads to fundamentally more complex bifurcation scenarios, since in the special case of Gaussian noise, there is only one pitchfork bifurcation which is reminiscent of the deterministic situation. Copyright © 2018, The Authors. All rights reserved.

关键词： Gaussian noise (electronic)

Maxwell-Stefan theory based lattice Boltzmann model for diffusion in multicomponent mixtures

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arXiv 2018年

作者： Chai, Zhenhua Guo, Xiuya Wang, Lei Shi, Baochang School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China School of Mathematics and Physics China University of Geosciences Wuhan430074 China School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China

The phenomena of diffusion in multicomponent (more than two components) mixtures are very universal in both science and engineering, and from mathematical point of view, they are usually described by the Maxwell-Stefan (MS) based continuum equations. In this paper, we propose a multiple-relaxation-time lattice Boltzmann (LB) model for the mass diffusion in multicomponent mixtures, and also perform a Chapman-Enskog analysis to show that the MS based continuum equations can be correctly recovered from the developed LB model. In addition, considering the fact that the MS based continuum equations are just a diffusion type of partial differential equations, we can also adopt much simpler lattice structures to reduce the computational cost of present LB model. We then conduct some simulations to test this model, and find that the results are in good agreement with some available works. Besides, the reverse diffusion, osmotic diffusion and diffusion barrier phenomena are also captured. Finally, compared to the kinetic theory based LB models for multicomponent gas diffusion, the present model does not include any complicated interpolations, and its collision process can be still implemented locally. Therefore, the advantages of single-component LB method can also be preserved in present LB model. Copyright © 2018, The Authors. All rights reserved.

关键词： Mixtures

A role of random slow manifolds in detecting stochastic bifurcation

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arXiv 2018年

作者： He, Ziying Cai, Rui Duan, Jinqiao Liu, Xianming Center for Mathematical Sciences Huazhong University of Sciences and Technology Wuhan430074 China School of Mathematics and Statistics Huazhong University of Sciences and Technology Wuhan430074 China Hubei Key Laboratory for Engineering Modeling and Scientific Computing Huazhong University of Sciences and Technology Wuhan430074 China Department of Applied Mathematics Illinois Institute of Technology ChicagoIL60616 United States

We consider the relation for the stochastic equilibrium states between the reduced system on a random slow manifold and the original system. This provides a theoretical basis for the reduction about sophisticated detailed models by the random slow manifold without significant damage to the overall qualitative properties. Based on this result, we reveal a role of random slow manifolds in detecting stochastic bifurcation by an example. The example exhibits a stochastic bifurcation phenomenon and possesses a random slow manifold that carries the stochastic bifurcation information of the system. Specifically, the lower dimensional reduced system on the random slow manifold retains the stochastic bifurcation phenomenon from the original *** Codes 60H10, 58J65 Copyright © 2018, The Authors. All rights reserved.

关键词： Perturbation techniques