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Static Theory for Planar Ferromagnets and Antiferromagnets

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Acta Mathematica Sinica,English Series 2001年第4期17卷 541-580页

作者： Feng Bo HANG Fang Hua LIN courant institute, 251 Mercer Street. New York. NY 10012, USA Dept. of Math, Princeton University, Fine Hall. Washington Road, Princeton, NJ 08544 USA. courant institute New York USA

Here we generalize the "BBH"-asymptotic analysis to a simplified mathematical model for tho planar ferromaguets and antiferromagnets. To develop such a static theory is a necessary step for a rigorous mathematical jus... 详细信息

关键词： Ginzburg-Landau-type equations Vortices Minimizing harmonic maps Gradient cstimate Radial solutions Stability Quantization

Nodal Sets and Doubling Conditions in Elliptic Homogenization

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Acta Mathematica Sinica,English Series 2019年第6期35卷 815-831页

作者： Fanghua Lin Zhongwei Shen courant institute of Mathematical Sciences Department of Mathematics University of Kentucky

This paper is concerned with uniform measure estimates for nodal sets of solutions in elliptic homogenization. We consider a family of second-order elliptic operators {L_ε} in divergence form with rapidly oscillating and periodic coefficients. We show that the(d-1)-dimensional Hausdorff measures of the nodal sets of solutions to L_ε(u_ε) = 0 in a ball in Rdare bounded uniformly in ε > 0. The proof relies on a uniform doubling condition and approximation of u_ε by solutions of the homogenized equation.

关键词： Nodal sets homogenization doubling condition

Mapping Problems,Fundamental Groups and Defect Measures

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Acta Mathematica Sinica,English Series 1999年第1期15卷 25-52页

作者： Fanghua Lincourant institute of Mathematics,New York University,New York,NY 10012,USA courant institute of Mathematics New York University New York USA

We study all the possible weak limits of a minimizing sequence,for p-energy functionals, consisting of continuous maps between Riemannian manifolds subject to a Dirichlet boundary condition or a homotopy *** show that if p is not an integer,then any such weak limit is a strong limit and,in particular,a stationary p-harmonic map which is C1,α continuous away from a closed subset of the Hausdorff dimension ≤n-[p]-*** p is an integer,then any such weak limit is a weakly p-harmonic map along with a（n-p）-rectifiable Radon measure μ.Moreover,the limiting map is C1,α continuous away from a closed subset ∑=spt μ∪S with Hn-p（S）=***,we discuss the possible varifolds type theory for Sobolev mappings.

关键词： Defect measure Harmonic mapping Generalized varifold Rectifiability

A generic approach to the dynamical interpretation of ocean-atmosphere processes

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Acta Oceanologica Sinica 2008年第z1期27卷 74-92页

作者： X. San Liang courant institute of Mathematical Sciences New York New York 10012 USA

This paper summarizes the recent development of a portable self-contained system to unravel the intricate multiscale dynamical processes from real oceanic flows, which are in nature highly nonlinear and intermittent in space and time. Of particular focus are the interactions among largescale, mesoscale, and submesoscale *** firsu introduce the concept of scale window, and an orthogonal subspace decomposition technigue called multiscale window transform (MWT). Established on MWT is a rigorous formalism of multiscale transport, perfect transfer, and multiscale conversion, which makes a new methodology, multiscale energy and vorticity analysis (MS-EVA). A direct application of the MS-EVA is the development of a novel localized instability analysis, generalizing the classical notion of hydrodynamic instability to finite amplitude processes on irregularly variable domains. The theory is consistent with the analytical solutions of Eady's model and Kuo's model, the benchmark models of baroclinic instability and barotropic instability; it is further validated with a vortex shedding control problem. We have put it to application with a variety of complicated real ocean problems, which would be otherwise very difficult, if not impossible, to tackle. Briefly shown in this paper include the dynamical studies of a highly variable open ocean front, and a complex coastal ocean circulation. In the former, it is found that underlying the frontal meandering is a convective instability followed by an absolute instability, and correspondingly a rapid spatially amplifying mode locked into a temporally growing mode; in the latter, we see a real ocean example of how upwelling can be driven by winds through nonlinear instability, and how winds may excite the ocean via an avenue which is distinctly different from the classical paradigms. This system is mathematically rigorous, physically robust, and practically straightforward.

关键词： Multiscale energy and vorticity analysis multiscale window transform multiscale transport perfect transfer finite-amplitude hydrodynamic instability analysis mean-eddy-turbulence interaction

Stable Stationary Harmonic Maps to Spheres

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Acta Mathematica Sinica,English Series 2006年第2期22卷 319-330页

作者： Fang Hua LIN Chang You WANG courant institute of Mathematical Sciences New York University Department of Mathematics Univrsity of Kentucky

For k ≥ 3, we establish new estimate on Hausdorff dimensions of the singular set of stable-stationary harmonic maps to the sphere S^k. We show that the singular set of stable-stationary harmonic maps from B5 to 83 is the union of finitely many isolated singular points and finitely many HSlder continuous curves. We also discuss the minimization problem among continuous maps from B^n to S^2.

关键词： stable stationary harmonic map Hausdorff dimension rectifiablity

Geek Talents:Who are the Top Experts on GitHub and Stack Overflow?

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Computers, Materials & Continua 2019年第8期61卷 465-479页

作者： Yijun Tian Waii Ng Jialiang Cao Suzanne McIntosh New York University Courant Institute of Mathematical SciencesNew York10012USA

In the field of Computer Science,software developers need to use a wide array of social collaborative platforms for learning and *** most popular ones are GitHub and Stack *** platforms only support search queries to extract relevant repository information from GitHub,or questions and answers from Stack *** ignores the valuable coder-related part-who are the top experts(geek talents)in a specific area?This information is important to companies,open source projects,and to those who want to learn from an expert role ***,how to find the right developers is quite a crucial yet challenging *** of the current works mainly focus on recommending experts in a particular software engineering task and ignore the relationship between developers within different *** this paper,we propose a novel technique that automatically identifies geek talents from GitHub,Stack Overflow,and across both *** results show that our work performs well at recommending proper developers in diverse areas.

关键词： Developer recommendation collaborative filtering stack overflow GitHub

Boundary Harnack principle on nodal domains Dedicated to Professor Jiaxing Hong on the Occasion of His 80th Birthday

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Science China Mathematics 2022年第12期65卷 2441-2458页

作者： Fanghua Lin Zhengjiang Lin courant institute of Mathematical Sciences New York UniversityNewYorkNY10012USA

We study some geometric and potential theoretic properties of nodal domains of solutions to certain uniformly elliptic equations. In particular, we establish corkscrew conditions, Carleson type estimates and boundary ... 详细信息

关键词： boundary Harnack principle nodal domain modified Harnack chain Carleson type estimate frequency bound

Large m asymptotics for minimal partitions of the Dirichlet eigenvalue

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Science China Mathematics 2022年第1期65卷 1-8页

作者： Zhiyuan Geng Fanghua Lin courant institute of Mathematical Sciences New York UniversityNew YorkNY 10012USA

In this paper,we study large m asymptotics of the l 1 minimal m-partition problem for the Dirichlet *** any smooth domainΩ⊂R^(n)such that|Ω|=1,we prove that the limit lim_(m→∞)l^(1)_(m)(Ω)=c 0 exists,and the constant c 0 is independent of the shape ofΩ.Here,l^(1)_(m)(Ω)denotes the minimal value of the normalized sum of the first Laplacian eigenvalues for any m-partition ofΩ.

关键词： Dirichlet eigenvalue l1 minimal partition problem large m asymptotics

Non-gaussian Test Models for Prediction and State Estimation with Model Errors

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Chinese Annals of Mathematics,Series B 2013年第1期34卷 29-64页

作者： Michal BRANICKI Nan CHEN Andrew J.MAJDA Department of Mathematics and Center for Atmosphere Ocean Science Courant Institute of Mathematical SciencesNew York University

Turbulent dynamical systems involve dynamics with both a large dimensional phase space and a large number of positive Lyapunov exponents. Such systems are ubiqui- tous in applications in contemporary science and engineering where the statistical ensemble prediction and the real time filtering/state estimation are needed despite the underlying complexity of the system. Statistically exactly solvable test models have a crucial role to provide firm mathematical underpinning or new algorithms for vastly more complex scien- tific phenomena. Here, a class of statistically exactly solvable non-Gaussian test models is introduced, where a generalized Feynman-Ka~ formulation reduces the exact behavior of conditional statistical moments to the solution to inhomogeneous Fokker-Planck equations modified by linear lower order coupling and source terms. This procedure is applied to a test model with hidden instabilities and is combined with information theory to address two important issues in the contemporary statistical prediction of turbulent dynamical systems： the coarse-grained ensemble prediction in a perfect model and the improving long range forecasting in imperfect models. The models discussed here should be use- ful for many other applications and algorithms for the real time prediction and the state estimation.

关键词： Prediction Model error Information theory Feynman-Kac framework Fokker planck Turbulent dynamical systems