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Integrable discretization of soliton equations via bilinear method and Backlund transformation

Science China mathematics 2015年第2期58卷 279-296页

作者： ZHANG Ying Nan CHANG Xiang Ke HU Juan HU Xing Biao TAM Hon-Wah Institute of Computational Mathematics and Scientific Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences School of Mathematical Sciences University of Chinese Academy of Sciences Department of Applied Mathematics Zhejiang University of Technology Department of Computer Science Hong Kong Baptist University

We present a systematic procedure to derive discrete analogues of integrable PDEs via Hirota’s bilinear *** approach is mainly based on the compatibility between an integrable system and its B¨acklund *** apply this procedure to several equations,including the extended Korteweg-deVries(Kd V)equation,the extended Kadomtsev-Petviashvili(KP)equation,the extended Boussinesq equation,the extended Sawada-Kotera(SK)equation and the extended Ito equation,and obtain their associated semidiscrete *** the continuum limit,these differential-difference systems converge to their corresponding smooth *** these new integrable systems,their B¨acklund transformations and Lax pairs are derived.

关键词： integrable discretization bilinear method Backlund transformation

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Some mathematical aspects of Anderson localization:boundary effect,multimodality,and bifurcation

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Communications in Theoretical Physics 2022年第11期74卷 45-64页

作者： Chen Jia Ziqi Liu Zhimin Zhang Applied and Computational Mathematics Division Beijing Computational Science Research CenterBeijing 100193China Department of Mathematical Sciences Tsinghua UniversityBeijing 100084China Department of Mathematics Wayne State UniversityDetroitMichigan 48202United States of America

Anderson localization is a famous wave phenomenon that describes the absence of diffusion of waves in a disordered *** we generalize the landscape theory of Anderson localization to general elliptic operators and complex boundary conditions using a probabilistic approach,and further investigate some mathematical aspects of Anderson localization that are rarely discussed ***,we observe that under the Neumann boundary condition,the low energy quantum states are localized on the boundary of the domain with high *** provide a detailed explanation of this phenomenon using the concept of extended subregions and obtain an analytical expression of this probability in the one-dimensional ***,we find that the quantum states may be localized in multiple different subregions with high probability in the one-dimensional case and we derive an explicit expression of this probability for various boundary ***,we examine a bifurcation phenomenon of the localization subregion as the strength of disorder *** critical threshold of bifurcation is analytically computed based on a toy model and the dependence of the critical threshold on model parameters is analyzed.

关键词： landscape spectrum eigenvalue eigenmode eigenfunction elliptic operator Schr?dinger operator confinement

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Phaseless Imaging by Reverse Time Migration: Acoustic Waves

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Numerical mathematics(Theory,Methods and Applications) 2017年第1期10卷 1-21页

作者： Zhiming Chen Guanghui Huang LSEC Institute of Computational MathematicsAcademy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing 100190China School of Mathematical Sciences University of Chinese Academy of SciencesBeijing 100049China The Rice Inversion Project Department of Computational and Applied MathematicsRice UniversityHoustonTX 77005-1892USA

We propose a reliable direct imaging method based on the reverse time migration for finding extended obstacles with phaseless total field *** prove that the imaging resolution of the method is essentially the same as the imaging results using the scattering data with full phase information when the measurement is far away from the *** imaginary part of the cross-correlation imaging functional always peaks on the boundary of the *** experiments are included to illustrate the powerful imaging quality.

关键词： Reverse time migration phaseless data inverse scattering extended obstacle

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The Illinois resource information system: Early innovations in geographic information system design

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Cartography and Geographic Information Science 2006年第2期33卷 97-114页

作者： Armstrong, Marc P. Department of Geography Program in Applied Mathematical and Computational Sciences The University of Iowa 316 Jessup Hall Iowa City IA 52242 United States

The Illinois Resource Information System (IRIS) was designed and implemented in the early 1970s by computer scientists at the University of Illinois. These researchers did not continue to work in the emerging field of Geographic Information Systems (GIS), and as a consequence, IRIS innovations were not reported in the GIS literature. IRIS operated across the then-new ARPANet, making it the first networked GIS. The system also had an interactive query language with a rich syntax. Retrievals in IRIS were optimized to promote high levels of interactivity. The development of IRIS had elements in common with the first supercomputer (ILLIAC IV), which was designed at the University of Illinois during that time period. A student who worked on the IRIS project also produced the first doctoral dissertation with "Geographic Information Systems" in its title.

关键词： Geographic information systems

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FUNDAMENTAL-SOLUTIONS IN PERTURBED SHEAR-FLOW

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FLUID DYNAMICS RESEARCH 1994年第3-4期13卷 167-195页

作者： CRIMINALE, WO SMITH, FIP [a]Department of Applied Mathematics University of Washington Seattle Washington USA [b]Department of Mathematical Computational Sciences University of St. Andrews. St. Andrews Scotland UK

In this paper infinite plane Couette flow in a viscous incompressible fluid is considered subject to general three-dimensional perturbations and the equations of motion are linearized. Furthermore, initial-value problems are posed and a set of closed-form solutions are obtained for a variety of conditions, such as the system under the influence of: (i) a mass source;(ii) an external force;or (iii) initial vorticity. The result is a knowledge of both the early transient dynamics and the near spatial field behavior, as well as the state after a long time and the far field behavior. It is shown that the solutions can be considered as fundamental (in the sense that source-sink solutions are regarded fundamental for irrotational motion) and therefore are useful in analyzing other boundary-value, initial-value problems where the basic flow can be synthesized from piece-wise linear (constant shear) variations. To this end a generalized Green's function for the system is determined.

关键词： Viscous flow

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A class of intrinsic schemes for orthogonal integration

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SIAM Journal on Numerical Analysis 2002年第6期40卷 2069-2084页

作者： Celledoni, Elena Owren, Brynjulf Dept. of Computational Engineering SINTEF Applied Mathematics N-7465 Trondheim Norway Department of Mathematical Sciences NTNU N-7491 Trondheim Norway

Numerical integration of ODEs on the orthogonal Stiefel manifold is considered. Points on this manifold are represented as n × k matrices with orthonormal columns, of particular interest is the case when n k. Mainly two requirements are imposed on the integration schemes. First, they should have arithmetic complexity of order nk2. Second, they should be intrinsic in the sense that they require only the ODE vector field to be defined on the Stiefel manifold, as opposed to, for instance, projection methods. The design of the methods makes use of retractions maps. Two algorithms are proposed, one where the retraction map is based on the QR decomposition of a matrix, and one where it is based on the polar decomposition. Numerical experiments show that the new methods are superior to standard Lie group methods with respect to arithmetic complexity, and may be more reliable than projection methods, owing to their intrinsic nature.

关键词： Integration

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STABILITY ANALYSIS FOR NONLINEAR SCHRODINGER EQUATIONS WITH NONLINEAR ABSORBING BOUNDARY CONDITIONS

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Journal of computational mathematics 2017年第1期35卷 1-18页

作者： Jiwei Zhang Zhenli Xu Xiaonan Wu Desheng Wang Applied and Computational Mathematics Division Beijing Computational Science Research Center China Department of Mathematics Institute of Natural Sciences and MOE Key Lab of Scientific and Engineering Computing Shanghai Jiao Tong University Shanghai 200240 China Department of Mathematics Hong Kong Baptist University Kowloon Hong Kong China Division of Mathematical Sciences School of Physical and Mathematical Sciences Nanyang Technological University Singapore 637371 Singapore

Local absorbing boundary conditions （LABCs） for nonlinear Schr6dinger equations have been constructed in papers [PRE 78（2008） 026709; and PRE 79 （2009） 046711] using the so-called unified approach. In this paper, we present stability analysis for the reduced problem with LABCs on the bounded computational domain by the energy estimate, and discuss a class of modified versions of LABCs. To prove the stability analysis of the re- duced problem, we turn to the technique of some auxiliary variables which reduces the higher-order derivatives in LABCs into a family of equations with lower-order derivatives. Furthermore, we extend the strategy to the stability analysis of two-dimensional problems by carefully dealing with the LABCs at corners. Numerical examples are given to demon- strate the effectiveness of our boundary conditions and validate the theoretical analysis.

关键词： Nonlinear SchrSdinger equations Energy estimates Absorbing boundaryconditions.

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Artificial intelligence for modelling infectious disease epidemics

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Nature 2025年第8051期638卷 623-635页

作者： Kraemer, Moritz U. G. Tsui, Joseph L.-H. Chang, Serina Y. Lytras, Spyros Khurana, Mark P. Vanderslott, Samantha Bajaj, Sumali Scheidwasser, Neil Curran-Sebastian, Jacob Liam Semenova, Elizaveta Zhang, Mengyan Unwin, H. Juliette T. Watson, Oliver J. Mills, Cathal Dasgupta, Abhishek Ferretti, Luca Scarpino, Samuel V. Koua, Etien Morgan, Oliver Tegally, Houriiyah Paquet, Ulrich Moutsianas, Loukas Fraser, Christophe Ferguson, Neil M. Topol, Eric J. Duchêne, David A. Stadler, Tanja Kingori, Patricia Parker, Michael J. Dominici, Francesca Shadbolt, Nigel Suchard, Marc A. Ratmann, Oliver Flaxman, Seth Holmes, Edward C. Gomez-Rodriguez, Manuel Schölkopf, Bernhard Donnelly, Christl A. Pybus, Oliver G. Cauchemez, Simon Bhatt, Samir Pandemic Sciences Institute University of Oxford Oxford United Kingdom Department of Biology University of Oxford Oxford United Kingdom Department of Electrical Engineering and Computer Science University of California Berkeley Berkeley CA United States UCSF UC Berkeley Joint Program in Computational Precision Health Berkeley CA United States Division of Systems Virology Department of Microbiology and Immunology The Institute of Medical Science The University of Tokyo Tokyo Japan Section of Epidemiology Department of Public Health University of Copenhagen Copenhagen Denmark Oxford Vaccine Group University of Oxford and NIHR Oxford Biomedical Research Centre Oxford United Kingdom Department of Epidemiology and Biostatistics Imperial College London London United Kingdom Department of Computer Science University of Oxford Oxford United Kingdom School of Mathematics University of Bristol Bristol United Kingdom MRC Centre for Global Infectious Disease Analysis School of Public Health Imperial College London London United Kingdom Department of Statistics University of Oxford Oxford United Kingdom Doctoral Training Centre University of Oxford Oxford United Kingdom Institute for Experiential AI Northeastern University MA Boston Thailand Santa Fe Institute Santa Fe NM United States World Health Organization Regional Office for Africa Brazzaville Congo WHO Hub for Pandemic and Epidemic Intelligence Health Emergencies Programme World Health Organization Berlin Germany Centre for Epidemic Response and Innovation (CERI) School for Data Science and Computational Thinking Stellenbosch University Stellenbosch South Africa African Institute for Mathematical Sciences (AIMS) South Africa Muizenberg Cape Town South Africa Genomics England London United Kingdom Scripps Research La Jolla CA United States Department of Biosystems Science and Engineering ETH Zürich Basel Switzerland Swiss Institute of Bioinformatics Lausanne Switzerland The Ethox Centre Nuffield

Infectious disease threats to individual and public health are numerous, varied and frequently unexpected. Artificial intelligence (AI) and related technologies, which are already supporting human decision making in economics, medicine and social science, have the potential to transform the scope and power of infectious disease epidemiology. Here we consider the application to infectious disease modelling of AI systems that combine machine learning, computational statistics, information retrieval and data science. We first outline how recent advances in AI can accelerate breakthroughs in answering key epidemiological questions and we discuss specific AI methods that can be applied to routinely collected infectious disease surveillance data. Second, we elaborate on the social context of AI for infectious disease epidemiology, including issues such as explainability, safety, accountability and ethics. Finally, we summarize some limitations of AI applications in this field and provide recommendations for how infectious disease epidemiology can harness most effectively current and future developments in AI. © Springer Nature Limited 2025.

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Singular limits for the Navier-Stokes-Poisson equations of the viscous plasma with the strong density boundary layer

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Science China mathematics 2023年第7期66卷 1495-1528页

作者： Qiangchang Ju Tao Luo Xin Xu Institute of Applied Physics and Computational Mathematics Beijing 100088China Department of Mathematics City University of Hong KongHong KongChina School of Mathematical Sciences and College of Oceanic and Atmospheric Sciences Ocean University of ChinaQingdao 266100China

The quasi-neutral limit of the Navier-Stokes-Poisson system modeling a viscous plasma with vanishing viscosity coefficients in the half-space■is rigorously proved under a Navier-slip boundary condition for velocity and the Dirichlet boundary condition for electric *** is achieved by establishing the nonlinear stability of the approximation solutions involving the strong boundary layer in density and electric potential,which comes from the breakdown of the quasi-neutrality near the boundary,and dealing with the difficulty of the interaction of this strong boundary layer with the weak boundary layer of the velocity field.

关键词： Navier-Stokes-Poisson equations interaction of strong and weak boundary layers singular limit

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Domains of mobility for a planar body moving among obstacles

Domains of mobility for a planar body moving among obstacles

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ASME 1996 Design Engineering Technical Conferences and Computers in Engineering Conference, DETC-CIE 1996

作者： Haug, Edward J. Adkins, Frederick A. Coroian, Dan I. Department of Mechanical Engineering Program in Applied Mathematical and Computational Sciences The University of Iowa United States

ISBN: (纸本)9780791897591

A formulation is presented for defining domains of mobility for a planar convex body moving with three degrees-of-freedom among convex planar obstacles. Applications included are determination of areas of a factory floor or material storage facility in which objects can be manipulated without impacting fixed obstacles. Mobility of the moving body is defined to encompass (1) dextrous mobility of the body;i.e., points that can be reached by a reference point on the body and at which the body can be rotated through its full range of admissible orientations without penetrating any stationary obstacle, and (2) limited mobility of the body;i.e., points that can be reached by the reference point and at which the body does not penetrate any stationary obstacle, for some admissible orientation. Analytical criteria for points on boundaries of domains of mobility are derived and numerical methods suitable for mapping these boundaries are summarized. An elementary example involving a moving and a stationary ellipse, with and without orientation restrictions, is solved analytically to illustrate the method. A more general application with one moving body and three stationary obstacles is solved numerically. © 1996 American Society of Mechanical Engineers (ASME). All rights reserved.

关键词： Numerical methods

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