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arXiv 2021年

作者： Hu, Jun Key Laboratory of Mathematical Theory and Computation in Information Security School of Mathematics and Statistics Beijing Institute of Technology Beijing100081 China

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ON DECAYING PROPERTIES OF NONLINEAR SCHRÖDINGER EQUATIONS

arXiv

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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

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

arXiv

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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

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Optimization Models and Interpretations for Three Types of Adversarial Perturbations against Support Vector Machines

arXiv

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arXiv 2022年

作者： Su, Wen Li, Qingna Cui, Chunfeng 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 Ministry of Education School of Mathematical Sciences Beihang University 100191 China

Adversarial perturbations have drawn great attentions in various deep neural networks. Most of them are computed by iterations and cannot be interpreted very well. In contrast, little attentions are paid to basic machine learning models such as support vector machines. In this paper, we investigate the optimization models and the interpretations for three types of adversarial perturbations against support vector machines, including sample-adversarial perturbations (sAP), class-universal adversarial perturbations (cuAP) as well as universal adversarial perturbations (uAP). For linear binary/multi classification support vector machines (SVMs), we derive the explicit solutions for sAP, cuAP and uAP (binary case), and approximate solution for uAP of multi-classification. We also obtain the upper bound of fooling rate for uAP. Such results not only increase the interpretability of the three adversarial perturbations, but also provide great convenience in computation since iterative process can be avoided. Numerical results show that our method is fast and effective in calculating three types of adversarial perturbations. Copyright © 2022, The Authors. All rights reserved.

关键词： Support vector machines

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Corrigendum to “Trace forms on the cyclotomic Hecke algebras and cocenters of the cyclotomic Schur algebras” [J. Pure Appl. Algebra 227(4) (2023) 107281]

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Journal of Pure and Applied Algebra 2025年第6期229卷

作者： Zhekun He Jun Hu Huang Lin School of Mathematics and Statistics Beijing Institute of Technology Beijing 100081 PR China MIIT Key Laboratory of Mathematical Theory and Computation in Information Security School of Mathematics and Statistics Beijing Institute of Technology Beijing 100081 PR China

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Two cycle-chord graphs are e-positive

arXiv

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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 Beijing102400 China MIIT Key Laboratory of Mathematical Theory and Computation in Information Security Beijing102400 China School of Mathematics and Statistics Beijing Institute of Technology Beijing102400 China

We prove Gebhard and Sagan's (e)-positivity of the line graphs of tadpoles in noncommuting variables. This implies the e-positivity of these line graphs. We then extend this (e)-positivity result to that of certain cycle-chord graphs, and derive the bivariate generating function of all cycle-chord *** Codes 05E05 05A15 05C15. © 2021, CC0.

关键词： Graphic methods

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Switching Control of 2-D Kuramoto-Sivashinsky Equation under Averaged Measurements

Switching Control of 2-D Kuramoto-Sivashinsky Equation under...

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Industrial Cyber-Physical Systems (ICPS)

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

This paper deals with the switching control design of Kuramoto-Sivashinsky equation (KSE) in two-dimensional (2-D) rectangular domain $\Omega$ . It is supposed that $\Omega$ is divided into $N$ subdomains and the discrete-time averaged measurements are available. Under the distributed sensors and actuators in space, the stabilization of 2-D KSE by switching is studied where only one pair of actuator and sensor is active during a certain interval of time (or only one pair of actuator and sensor is employed to move in the spatial domain). In the latter case, the moving time of actuator-sensor pair is considered as an additional switching between the open-loop system and the closed-loop switched system. Sufficient conditions are established in terms of linear matrix inequalities (LMIs) to ensure regional stability of the closed-loop system.

关键词： Switched systems Actuators Sufficient conditions Control design Two dimensional displays Switches Cyber-physical systems

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LONG TIME BEHAVIOR of STOCHASTIC NLS with A SMALL MULTIPLICATIVE NOISE

arXiv

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arXiv 2021年

作者： Fan, Chenjie Xu, Weijun Zhao, Zehua Academy of Mathematics and Systems Science and Hua Loo-Keng Key Laboratory of Mathematics Chinese Academy of Sciences Beijing China Beijing International Center for Mathematical Research Peking University China Department of Mathematics and Statistics Beijing Institute of Technology MIIT Key Laboratory of Mathematical Theory and Computation in Information Security Beijing China

We prove the global space-time bound for the mass critical nonlinear Schrödinger equation perturbed by a small multiplicative noise in dimension three. The associated scattering behavior are also obtained. We also prove a global Strichartz space-time bound for the linear stochastic model, which is new itself and serves a prototype model for the nonlinear case. The proof combines techniques from [16], [19] as well as local smoothing estimates for linear Schrödinger operators. Copyright © 2021, The Authors. All rights reserved.

关键词： Phase noise

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

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

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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

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

arXiv

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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

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