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Unique continuation properties for one dimensional higher order Schrödinger equations

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arXiv 2019年

作者： Huang, Tianxiao Huang, Shanlin Zheng, Quan Sun Yat-sen University Guangdong Zhuhai519082 China School of Mathematics and Statistics Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Hubei Wuhan430074 China School of Mathematics and Statistics Huazhong University of Science and Technology Hubei Wuhan430074 China

We study two types of unique continuation properties for the higher order Schrödinger equation with potential i∂tu = (−∆x)mu + V(t, x)u, (t, x) ∈ 1+n, 2 ≤ m ∈ N+. The first one says if u has certain exponential decay at two times, then u ≡ 0, and this result is sharp by constructing critical non-trivial solutions. The second one says if u ≡ 0 in an arbitrary half-space of 1+n, then u ≡ 0 identically. The uniqueness theorems are given when n = 1, but we also prove partial results when n ∈ N+ for their own interests. Possibility or obstacles to proving these unique continuation properties in higher spatial dimensions are also discussed. Copyright © 2019, The Authors. All rights reserved.

关键词： Geometry

Jacobi polynomials on the Bernstein ellipse

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arXiv 2017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 School of Mathematical Sciences Shanghai Key Laboratory for Contemporary Applied Mathe-matics Fudan University Shanghai200433

In this paper, we are concerned with Jacobi polynomials P(α,β)n(x) on the Bernstein ellipse with motivation mainly coming from recent studies of convergence rate of spectral interpolation. An explicit representation of P(α,β)n(x) is derived in the variable of parametrization. This formula further allows us to show that the maximum value of |P(α,β)n(z)| over the Bernstein ellipse is attained at one of the endpoints of the major axis if α+β ≥ -1. For the minimum value, we are able to show that for a large class of Gegenbauer polynomials (i.e., α = β), it is attained at two endpoints of the minor axis. These results particularly extend those previously known only for some special cases. Moreover, we obtain a more refined asymptotic estimate for Jacobi polynomials on the Bernstein ellipse.65N35, 65D05, 41A05, 41A25 Copyright © 2017, The Authors. All rights reserved.

关键词： Polynomials

The role of slow manifolds in parameter estimation for a multiscale stochastic system with α-stable Lévy noise

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arXiv 2020年

作者： Chao, Ying Wei, Pingyuan Duan, Jinqiao School of Mathematics and Statistics Center for Mathematical Sciences 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

This work is about parameter estimation for a fast-slow stochastic system with non-Gaussian α-stable Lévy noise. When the observations are only available for slow components, a system parameter is estimated and the accuracy for this estimation is quantified by p-moment with p ∈ (1, α), with the help of a reduced system through random slow manifold approximation. This method provides an advantage in computational complexity and cost, due to the dimension reduction in stochastic systems. To numerically illustrate this method, and to corroborate that the parameter estimator based on the reduced slow system is a good approximation for the true parameter value of the original system, a prototypical example is present. Copyright © 2020, The Authors. All rights reserved.

关键词： Parameter estimation

The Onsager-Machlup Function as Lagrangian for the Most Probable Path of a Jump-Diffusion Process

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arXiv 2018年

作者： Chao, Ying Duan, Jinqiao School of Mathematics and Statistics Center for Mathematical Sciences 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

This work is devoted to deriving the Onsager-Machlup function for a class of stochastic dynamical systems under (non-Gaussian) Lévy noise as well as (Gaussian) Brownian noise, and examining the corresponding most probable paths. This Onsager-Machlup function is the Lagrangian giving the most probable path connecting metastable states for jump-diffusion processes. This is done by applying the Girsanov transformation for measures induced by jump-diffusion processes. Moreover, we have found this Lagrangian function is consistent with the result in the special case of diffusion processes. Finally, we apply this new Onsager-Machlup function to investigate dynamical behaviors analytically and numerically in several examples. These include the transitions from one metastable state to another metastable state in a double-well system, with numerical experiments illustrating most probable transition paths for various noise parameters. Copyright © 2018, The Authors. All rights reserved.

关键词： Lagrange multipliers

Spiking hierarchy in an adaptive exponential integrate-and-fire network synchronization

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arXiv 2021年

作者： Wu, Xiaoyue Lin, Congping Zhang, Yiwei School of Mathematics and Statistics Huazhong University of Sciences and Technology Wuhan China Center for Mathematical Sciences Huazhong University of Science and Technology Wuhan China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan China

Neuronal network synchronization has received wide interests. Network connection structure is known to play a key role in its synchronization. In the present manuscript, we study the influence of initial membrane potentials together with network topology on bursting synchronization, in particular the sequential spiking order in stabilized inter bursts. We find a hierarchical phenomenon on spiking order. We grade neurons into different layers: Primary neurons are those initiate the spike and lower-layer neurons are then determined via network connections successively. This constructs a directed graph to indicate spiking propagation among different layers of neurons. Neurons in upper layers spike earlier than those in lower layers. More interestingly, we find that among the same layer, neuron spiking order is mainly associated with stimuli they receive from upper layer via network connections;more stimuli leads to earlier spiking. Furthermore, we find that with effectively the same stimuli from upper layer, neurons with less connections spike earlier. © 2021, CC BY.

关键词： Neurons

A remark on the central model method for the weak Palis conjecture of higher dimensional singular flows

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arXiv 2018年

作者： Xiao, Qianying Zhang, Yiwei School of Mathematical Sciences Peking University Beijing100871 China School of Mathematics and Statistics Center for Mathematical Sciences Hubei Key Laboratory of Engineering Modeling and Scientific Computing HuaZhong University of Science and Technology 1037 Luoyu Road Wuhan430074 China

For a generic vector field robustly without horseshoes, and an aperiodic chain recurrent class with singularities whose saddle values have different signs, the extended rescaled Poincaré map is associated with a central model. We estimate such central model and show it must have chain recurrent cen-tral segments over the singularities. This obstructs the application of central model to create horseshoes, and indicates that, differing from C1 diffeomor-phisms, solo using central model method is insufficient as a strategy to prove weak Palis conjecture for higher dimensional ( 4) singular flows. Our compu-tation is actually based on simplified way of addressing blowup construction. As a byproduct, we are applicable to directly compute the extended rescaled Poincaré map upto second order derivatives, which we believe has its inde-pendent interests. Copyright © 2018, The Authors. All rights reserved.

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A parameter estimator based on Smoluchowski-Kramers approximation 1

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arXiv 2018年

作者： He, Ziying Duan, Jinqiao Cheng, Xiujun Center for Mathematical Sciences School of Mathematics and Statistics 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 devise a simplified parameter estimator for a second order stochastic differential equation by a first order system based on the Smoluchowski-Kramers approximation. We establish the consistency of the estimator by using Γ-convergence theory. We further illustrate our estimation method by an experimentally studied movement model of a colloidal particle immersed in water under conservative force and constant diffusion. Copyright © 2018, The Authors. All rights reserved.

关键词： Differential equations

Characterization of the most probable transition paths of stochastic dynamical systems with stable Lévy noise

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arXiv 2018年

作者： Huang, Yuanfei Chao, Ying Yuan, Shenglan Duan, Jinqiao School of Mathematics and Statistics Center for Mathematical Sciences Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China Department of Applied Mathematics Illinois Institute of Technology ChicagoIL60616 United States

This work is devoted to the investigation of the most probable transition path for stochastic dynamical systems driven by either symmetric α-stable Lévy motion (0 Copyright © 2018, The Authors. All rights r... 详细信息

关键词： Brownian movement

STATIONARY FLUCTUATIONS FOR THE FACILITATED EXCLUSION PROCESS

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arXiv 2023年

作者： Erignoux, Clément Zhao, Linjie Inria Univ. Lille CNRS UMR 8524 - Laboratoire Paul Painlevé LilleF-59000 France School of Mathematics and Statistics Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China

We derive the stationary fluctuations for the Facilitated Exclusion Process (FEP) in one dimension in the symmetric, weakly asymmetric and asymmetric cases. Our proof relies on the mapping between the FEP and the zero-range process, and extends the strategy in [10], where hydrodynamic limits were derived for the FEP, to its stationary fluctuations. Our results thus exploit works on the zero-range process’s fluctuations [17, 19], but we also provide a direct proof in the symmetric case, for which we derive a sharp estimate on the equivalence of ensembles for the FEP’s stationary states. © 2023, CC BY.

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