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Schauder Estimates for Elliptic and Parabolic Equations

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Chinese Annals of Mathematics,Series B 2006年第6期27卷 637-642页

作者： Xu-Jia WANG centre for Mathematics and its applications Australian National University Canberra ACT 0200 Australia

In this note the author gives an elementary and simple proof for the Schauder estimates for elliptic and parabolic equations. The proof also applies to nonlinear equations.

关键词： Schauder estimate Elliptic equation Parabolic equation

Strong stability of Nash equilibria in load balancing games

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Science China Mathematics 2014年第7期57卷 1361-1374页

作者： CHEN Bo LI SongSong ZHANG YuZhong centre for Discrete Mathematics and its applications Warwick Business SchoolUniversity of Warwick School of Management Qufu Normal University

We study strong stability of Nash equilibria in load balancing games of m(m 2)identical servers,in which every job chooses one of the m servers and each job wishes to minimize its cost,given by the workload of the server it chooses.A Nash equilibrium(NE)is a strategy profile that is resilient to unilateral *** an NE in such a game is ***,an NE assignment is not stable against coordinated deviations of several jobs,while a strong Nash equilibrium(SNE)*** study how well an NE approximates an *** any job assignment in a load balancing game,the improvement ratio(IR)of a deviation of a job is defined as the ratio between the pre-and post-deviation *** NE is said to be aρ-approximate SNE(ρ1)if there is no coalition of jobs such that each job of the coalition will have an IR more thanρfrom coordinated deviations of the *** it is already known that NEs are the same as SNEs in the 2-server load balancing game,we prove that,in the m-server load balancing game for any given m 3,any NE is a(5/4)-approximate SNE,which together with the lower bound already established in the literature yields a tight approximation *** closes the final gap in the literature on the study of approximation of general NEs to SNEs in load balancing *** establish our upper bound,we make a novel use of a graph-theoretic tool.

关键词： load balancing game Nash equilibrium strong Nash equilibrium approximate strong Nash equilibrium

The Christoffel problem by the fundamental solution of the Laplace equation

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Science China Mathematics 2021年第7期64卷 1599-1612页

作者： Qi-Rui Li Dongrui Wan Xu-Jia Wang School of Mathematical Sciences Zhejiang UniversityHangzhou 310027China centre for Mathematics and its applications The Australian National UniversityCanberraACT 2601Australia College of Mathematics and Statistics Shenzhen UniversityShenzhen 518060China

The Christoffel problem is equivalent to the existence of convex solutions to the Laplace equation on the unit sphere S^(n).Necessary and sufficient conditions have been found by Firey(1967)and Berg(1969),by using the Green function of the Laplacian on the *** the Christoffel problem as the Laplace equation on the entire space R^(n+1),we observe that the second derivatives of the solution can be given by the fundamental solutions of the Laplace *** we find new and simpler necessary and sufficient conditions for the solvability of the Christoffel *** also study the Lp extension of the Christoffel problem and provide sufficient conditions for the problem,for the case p≥2.

关键词： Christoffel problem Laplace equation fundamental solution

On Self-Adjointness of the Product of Two Second-Order Differential Operators

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Acta Mathematica Sinica,English Series 1999年第3期15卷 375-383,385-386页

作者： D. E. Edmunds Department of Mathematics Inner Mongolia University Huhhot 010021 P. R. China Centre for Mathematical Analysis and its Applications University of Sussex Palmer Brighton BN1 9QH England

In this paper, the problem of self-adjointness of the product of two differential operators is considered. A number of results concerning self-adjointness of the product L2L1 of two second-order self-adjoint differential operators are obtained by using the general construction theory of self-adjoint extensions of ordinary differential operators.

关键词： Self-adjoint operator Differential operator Product of two differential operators

Global regularity of optimal mappings in non-convex domains

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Science China Mathematics 2019年第11期62卷 2057-2072页

作者： Shibing Chen Jiakun Liu Xu-Jia Wang School of Mathematical Sciences University of Science and Technology of ChinaHefei 230026China School of Mathematics and Applied Statistics University of WollongongWollongongNSW 2522Australia centre for Mathematics and its applications The Australian National UniversityCanberraACT 0200Australia

In this paper, we establish a global regularity result for the optimal transport problem with the quadratic cost, where the domains may not be convex. This result is obtained by a perturbation argument,using a recent ... 详细信息

关键词： Monge-Ampère equation optimal transportation global regularity

Mapping of quantitative trait loci using the skew-normal distribution

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Journal of Zhejiang University-Science B(Biomedicine & Biotechnology) 2007年第11期8卷 792-801页

作者： FERNANDES Elisabete PACHECO António PENHA-GONALVES Carlos centre for Mathematics and its applications IST-Technical University of Lisbon1049-001 LisboaPortugal Department of Statistics and Operational ResearchFaculty of SciencesUniversity of Lisbon1749-016 LisboaPortugal Gulbenkian Institute of ScienceP-2781-901 OeirasPortugal Department of Mathematics and centre for Mathematics and its applications IST-Technical University of Lisbon1049-001 LisboaPortugal Gulbenkian Institute of Science P-2781-901 OeirasPortugal

In standard interval mapping (IM) of quantitative trait loci (QTL), the QTL effect is described by a normal mixture model. When this assumption of normality is violated, the most commonly adopted strategy is to use the previous model after data transformation. However, an appropriate transformation may not exist or may be difficult to find. Also this approach can raise interpretation issues. An interesting alternative is to consider a skew-normal mixture model in standard IM, and the resulting method is here denoted as skew-normal IM. This flexible model that includes the usual symmetric normal distribution as a special case is important, allowing continuous variation from normality to non-normality. In this paper we briefly introduce the main peculiarities of the skew-normal distribution. The maximum likelihood estimates of parameters of the skew-normal distribution are obtained by the expectation-maximization (EM) algorithm. The proposed model is illustrated with real data from an intercross experiment that shows a significant departure from the normality assumption. The performance of the skew-normal IM is assessed via stochastic simulation. The results indicate that the skew-normal IM has higher power for QTL detection and better precision of QTL location as compared to standard IM and nonparametric IM.

关键词： Interval mapping (IM) Quantitative trait loci (QTL) Skew-normal distribution Expectation-maximization (EM)algorithm

BERNSTEIN—JORGENS THEOREM FOR A FOURTH ORDER PARTIAL DIFFERENTIAL EQUATION

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Journal of Partial Differential Equations 2002年第1期15卷 78-88页

作者： NeilS.Trudinger 汪学家等 centre for Mathematics and its applications The Australian National University Canberra ACT 0200 Australia

We introduce a metric, conformal to the affine metric, on a convex graph, and consider the Euler equation of the volume functional. We establish a priori estimates for solutions and prove a Bernstein-Jorgens type resu... 详细信息

关键词：解估计四阶偏微分方程 Bernstein定理

A boundary expansion of solutions to nonlinear singular elliptic equations

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Science China Mathematics 2022年第1期65卷 9-30页

作者： Huaiyu Jian Jian Lu Xu-Jia Wang Department of Mathematical Sciences Tsinghua UniversityBeijing 100084China centre for Mathematics and its applications The Australian National UniversityCanberraACT 0200Australia South China Research Center for Applied Mathematics and Interdisciplinary Studies South China Normal UniversityGuangzhou 510631China

In this paper we establish an asymptotic expansion near the boundary for solutions to the Dirichlet problem of elliptic equations with singularities near the *** expansion formula shows the singularity profile of solutions at the *** deal with both linear and nonlinear elliptic equations,including fully nonlinear elliptic equations and equations of Monge-Ampère type.

关键词： boundary asymptotic expansion singular nonlinear elliptic equations Dirichlet problem

Quantum Fibrations:Quantum Computation on an Arbitrary Topological Space

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Acta Mathematica Sinica,English Series 2024年第11期40卷 2693-2718页

作者： Kazuki IKEDA Co-design Center for Quantum Advantage(C2QA)&Center For Nuclear Theory Department of Physics and AstronomyStony Brook UniversityStony BrookUSA Department of Mathematics and Statistics&centre for Quantum Topology and its applications(quanTA) University of SaskatchewanSaskatoonCanada

Using operator algebras,we extend the theory of quantum computation on a graph to a theory of computation on an arbitrary topological *** computation is usually implemented on finite discrete sets,and the purpose of this study is to extend this to theories on arbitrary *** conventional theory of quantum computers can be viewed as a simplified algebraic geometry theory in which the action of SU(2)is defined on each point of a discrete *** this study,we extend this in general as a theory of quantum fibrations in which the action of the von Neumann algebra is defined on an arbitrary topological *** quantum channel is then naturally extended as a net of von Neumann *** allows for a more mathematically rigorous discussion of general theories,including physics and chemistry,which are defined on sets that are not necessarily discrete,from the perspective of quantum computer science.

关键词： Quantum computation operator algebra von Neumann algebra fibration quantum computational chemistry complexity theory