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On Strichartz Estimates for Many-Body Schrödinger Equation in the Periodic Setting

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SSRN

SSRN 2023年

作者： Yu, Xueying Zhao, Zehua Zheng, Jiqiang Department of Mathematics University of Washington C138 Padelford Hall Box 354350 SeattleWA98195 United States Department of Mathematics and Statistics Beijing Institute of Technology Beijing China MIIT Key Laboratory of Mathematical Theory and Computation in Information Security Beijing China Institute of Applied Physics and Computational Mathematics Beijing100088 China

In this paper, we prove Strichartz estimates for N-body Schrödinger equations in the periodic setting, specifically on tori T d , where d ≥ 3. The results hold for both rational and irrational tori, and for small interacting potentials in a certain sense. Our work is based on the standard Strichartz estimate for Schr¨odinger operators on periodic domains, as developed by Bourgain-Demeter [2], and can also be extended to the waveguide case using Barron’s recent result [1]. This result can be regarded as a periodic analogue of Hong [20]. © 2023, The Authors. All rights reserved.

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ON THE DECAY PROPERTY OF THE CUBIC FOURTH-ORDER SCHRÖDINGER EQUATION

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arXiv 2022年

作者： Yu, Xueying Yue, Haitian Zhao, Zehua Department of Mathematics University of Washington C138 Padelford Hall Box 354350 SeattleWA98195 United States Institute of Mathematical Sciences ShanghaiTech University Pudong Shanghai China Department of Mathematics and Statistics Beijing Institute of Technology MIIT Key Laboratory of Mathematical Theory and Computation in Information Security Beijing China

In this short paper, we prove that the solution of the cubic fourth-order Schrödinger equation (4NLS) on d (5 ≤ d ≤ 8) enjoys the same (pointwise) decay property as its linear solution does. This result is proved via a bootstrap argument based on the corresponding global result Pausader [24]. This result can be extended to more general dispersive equations (including some more 4NLS models) with scattering asymptotics. Copyright © 2022, The Authors. All rights reserved.

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Input-to-state stabilization for a 2 × 2 hyperbolic system cascaded with an ODE

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Input-to-state stabilization for a 2 × 2 hyperbolic system ...

Youth Academic Annual Conference of Chinese Association of Automation (YAC)

作者： Han-Wen Zhang Jun-Min Wang Jing Wang School of Automation and Software Engineering Shanxi University Taiyuan Shanxi China School of Mathematics and Statistics & Key Laboratory of Mathematical Theory and Computation in Information Security Beijing Institute of Technology Beijing China Delft Institute of Applied Mathematics Delft University of Technology Mekelweg 4 Delft Netherlands

ISBN: (纸本)9781665465373

The paper deals with the input-to-state stabilization for the 2×2 system of first-order hyperbolic equations, which convect in opposite directions cascaded with an unstable ODE equation. First, an inverse backstepping transformation is introduced to obtain a target system. Then, by active disturbance rejection control (ADRC) method, the disturbance is estimated via a disturbance estimator with time-varying gain. When the unmatched disturbances are absent, the disturbance estimator is exponentially convergent to the matched disturbance. Furthermore, in order to reject the matched disturbance and obtain the input-to-state stability of the system, the controller is proposed by using the disturbance estimator. Finally, numerical simulations are presented to validate theoretical results.

关键词： Robust control Backstepping Automation Numerical simulation Control systems mathematical models Stability analysis

Observer Design for a Reaction-Diffusion Equation with Delayed Boundary Measurement

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Observer Design for a Reaction-Diffusion Equation with Delay...

International Conference on Control and Robots (ICCR)

作者： Jing Zhang Wen Kang School of Automation and Electrical Engineering University of Science and Technology Beijing Beijing China Shunde Innovation School University of Science and Technology Beijing Foshan China MIIT Key Laboratory of Mathematical Theory and Computation in Information Security School of Mathematics and Statistics Beijing Institute of Technology Beijing China

This paper addresses the problem of observer design for a reaction-diffusion equation with delayed boundary measurement. The predictor method is employed for the delay compensation. The reaction coefficient of the plant is considered, which may lead to the instability. To obtain the stability result, the backstepping transformation is introduced to determine the observer gains. Linear matrix inequalities (LMIs) conditions are derived to guarantee the exponential stability of the error system. The numerical simulation is provided to validate the proposed results.

关键词： Backstepping Observers Numerical simulation mathematical models Stability analysis Delays Linear matrix inequalities

ON DECAYING PROPERTIES OF NONLINEAR SCHRÖDINGER EQUATIONS

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arXiv 2022年

作者： Fan, Chenjie Staffilani, Gigliola Zhao, Zehua Academy of Mathematics and Systems Science Hua Loo-Keng Key Laboratory of Mathematics Chinese Academy of Sciences Beijing China Department of Mathematics Massachusetts Institute of Technology CambridgeMA02139 United States Department of Mathematics and Statistics Beijing Institute of Technology Beijing China MIIT Key Laboratory of Mathematical Theory and Computation in Information Security Beijing China

In this paper we discuss quantitative (pointwise) decay estimates for solutions to the 3D cubic defocusing Nonlinear Schrödinger equation with various initial data, deterministic and random. We show that nonlinear solutions enjoy the same decay rate as the linear ones. The regularity assumption on the initial data is much lower than in previous results (see [20] and the references therein) and moreover we quantify the decay, which is another novelty of this work. Furthermore, we show that the (physical) randomization of the initial data can be used to replace the L1-data assumption (see [23] for the necessity of the L1-data assumption). At last, we note that this method can be also applied to derive decay estimates for other nonlinear dispersive equations. Copyright © 2022, The Authors. All rights reserved.

关键词： Nonlinear equations

A Highly Efficient Adaptive-Sieving-Based Algorithm for the High-Dimensional Rank Lasso Problem

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arXiv 2022年

作者： Bai, Xiaoning Li, Qingna School of Mathematics and Statistics Beijing Institute of Technology No. 5 Zhongguancun South Street Haidian District Beijing100081 China School of Mathematics and Statistics Beijing Key Laboratory on MCAACI Key Laboratory of Mathematical Theory and Computation in Information Security Beijing Institute of Technology No. 5 Zhongguancun South Street Haidian District Beijing100081 China

The high-dimensional rank lasso (hdr lasso) model is an efficient approach to deal with high-dimensional data analysis. It was proposed as a tuning-free robust approach for the high-dimensional regression and was demonstrated to enjoy several statistical advantages over other approaches. The hdr lasso problem is essentially an L1-regularized optimization problem whose loss function is Jaeckel’s dispersion function with Wilcoxon scores. Due to the nondifferentiability of the above loss function, many classical algorithms for lasso-type problems are unable to solve this model. In this paper, inspired by the adaptive sieving strategy for the exclusive lasso problem [1], we propose an adaptive-sieving-based algorithm to solve the hdr lasso problem. The proposed algorithm makes full use of the sparsity of the solution. In each iteration, a subproblem with the same form as the original model is solved, but in a much smaller size. We apply the proximal point algorithm to solve the subproblem, which fully takes advantage of the two nonsmooth terms. Extensive numerical results demonstrate that the proposed algorithm (AS-PPA) is robust for different types of noises, which verifies the attractive statistical property as shown in [2]. Moreover, AS-PPA is also highly efficient, especially for the case of high-dimensional features, compared with other *** Codes 90C06 (Primary), 90C25 (Secondary) © 2022, CC BY.

关键词： Constrained optimization

A Facial Reduction Approach for the Single Source Localization Problem

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arXiv 2021年

作者： Shi, He Li, Qingna School of Mathematics and Statistics Beijing Institute of Technology Beijing100081 China School of Mathematics and Statistics Beijing Key Laboratory on MCAACI Key Laboratory of Mathematical Theory and Computation in Information Security Beijing Institute of Technology Beijing100081 China

The single source localization problem (SSLP) appears in several fields such as signal processing and global positioning systems. The optimization problem of SSLP is nonconvex and difficult to find its globally optimal solution. It can be reformulated as a rank constrained Euclidean distance matrix (EDM) completion problem with a number of equality constraints. In this paper, we propose a facial reduction approach to solve such an EDM completion problem. For the constraints of fixed distances between sensors, we reduce them to a face of the EDM cone and derive the closed formulation of the face. We prove constraint nondegeneracy for each feasible point of the resulting EDM optimization problem without a rank constraint, which guarantees the quadratic convergence of semismooth Newton’s method. To tackle the nonconvex rank constraint, we apply the majorized penalty approach developed by Zhou et al. (IEEE Trans Signal Process 66(3):4331-4346, 2018). Numerical results verify the fast speed of the proposed approach while giving comparable quality of solutions as other methods. Copyright © 2021, The Authors. All rights reserved.

关键词： Numerical methods

ON WELL-POSEDNESS RESULTS FOR THE CUBIC-QUINTIC NLS ON T3

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arXiv 2023年

作者： Luo, Yongming Yu, Xueying Yue, Haitian Zhao, Zehua Institut für Wissenschaftliches Rechnen Technische Universität Dresden Zellescher Weg 25 Dresden01069 Germany Department of Mathematics University of Washington C138 Padelford Hall Box 354350 SeattleWA98195 United States Institute of Mathematical Sciences ShanghaiTech University Pudong Shanghai China Department of Mathematics and Statistics Beijing Institute of Technology Beijing China MIIT Key Laboratory of Mathematical Theory and Computation in Information Security Beijing China

We consider the periodic cubic-quintic nonlinear Schrödinger equation (CQNLS) (i∂t + ∆)u = µ1|u|2u + µ2|u|4u on the three-dimensional torus T3 with µ1, µ2 ∈ R \ {0}. As a first result, we establish the small data well-posedness of (CQNLS) for arbitrarily given µ1 and µ2. By adapting the crucial perturbation arguments in [33] to the periodic setting, we also prove that (CQNLS) is always globally well-posed in H1(T3) in the case µ2 > 0. Copyright © 2023, The Authors. All rights reserved.

关键词： Nonlinear equations

The e-positivity and schur positivity of the chromatic symmetric functions of some trees

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arXiv 2021年

作者： Wang, David G.L. Wang, Monica M.Y. School of Mathematics and Statistics Beijing Key Laboratory on MCAACI Beijing Institute of Technology Key Laboratory of Mathematical Theory and Computation in Information Security Ministry of Industry and Information Technology Beijing102400 China School of Mathematics and Statistics Beijing Institute of Technology Beijing102400 China

We investigate the e-positivity and Schur positivity of the chromatic symmetric functions of some spider graphs with three legs. We obtain the positivity classification of all broom graphs and that of most double broom graphs. The methods involve extracting particular e-coefficients of the chromatic symmetric function of these graphs with the aid of Orellana and Scott’s triple-deletion property, and using the combinatorial formula of Schur coefficients by examining certain special rim hook tabloids. We also propose some conjectures on the e-positivity and Schur positivity of trees. © 2021, CC0.

关键词： Graphic methods