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variational formulas of Poincare-Type Inequalities in Banach Spaces of Functions on the Line

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Acta Mathematica Sinica,English Series 2002年第3期18卷 417-436页

作者： Mu Fa CHEN Department of Mathematics. Beijing Normal. University. Beijing 100875. P. R. China Department of Mathematics Beijing Normal University Beijing P. R. China

Motivated from the study on logarithmic Sobolev. Nash and other functional inequalities, the variational formulas for Poincare inequalities are extended to a large class of Banach (Orlicz) spaces of functions on the line. Explicit criteria for the inequalities to hold and explicit estimates for the optimal constants in the inequalities are presented. As a typical application, the logarithmic Sobolev constant is carefully examined.

关键词： variational formula Poincare inequality Logarithmic Sobolev inequality Orliez space

variational formulas of Poincaré-type Inequalities for Birth-Death Processes

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Acta Mathematica Sinica,English Series 2003年第4期19卷 625-644页

作者： MuFaCHEN DepartmentofMathematics BeijingNormalUniversityBeijing100875P.R.China

In author's one previous paper, the same topic was studied for onedimensional diffusions. As a continuation, this paper studies the discrete case, that is thebirth-death processes. The explicit criteria for the inequalities, the variational formulas andexplicit bounds of the corresponding constants in the inequalities are presented. As typicalapplications, the Nash inequalities and logarithmic Sobolev inequalities are examined.

关键词： variational formula Poincare inequality Nash inequality logarithmicSobolev inequality orlicz space birth-death process

variational formulas and approximation theorems for the first eigenvalue in dimension one

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Science China Mathematics 2001年第4期44卷 409-418页

作者：陈木法 Department of Mathematics Beijing Normal University Beijing China

Some complete variational formulas and approximation theorems for the first eigenvalue of elliptic operators in dimension one or a class of Markov chains are presented.

关键词： first eigenvalue variational formula elliptic operator birth-death process

New point view of spectral gap in functional spaces for birth-death processes

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Frontiers of Mathematics in China 2014年第3期9卷 523-535页

作者： Yutao MA Yonghua MAO School of Mathematical Sciences Key Laboratory on Mathematics and Complex SystemBeijing Normal University Beijing 100875 China

Constructing some proper functional spaces, we obtain the corresponding norm for the operator （-.L）^-1, and then, via spectral theory, we revisit two variational formulas of the spectral gap, given by M. F. Chen [Fr... 详细信息

关键词： Birth-death processes Dirichlet first eigenvalue variational formula spectral theory duality

variational Principles for Asymptotic Variance of General Markov Processes

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Acta Mathematica Sinica,English Series 2023年第1期39卷 107-118页

作者： Lu Jing HUANG Yong Hua MAO Tao WANG School of Mathematics and Statistics Fujian Normal UniversityFuzhou 350007P.R.China School of Mathematical Sciences Beijing Normal UniversityLaboratory of Mathematics and Complex SystemsMinistry of EducationBeijing 100875P.R.China

A variational formula for the asymptotic variance of general Markov processes is *** application,we get an upper bound of the mean exit time of reversible Markov processes,and some comparison theorems between the reve... 详细信息

关键词： Markov process asymptotic variance variational formula the mean exit time comparison theorem semi-Dirichlet form

Mixed principal eigenvalues in dimension one

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Frontiers of Mathematics in China 2013年第2期8卷 317-343页

作者： Mu-Fa CHEN Lingdi WANG Yuhui ZHANG School of Mathematical Sciences Beijing Normal University Laboratory of Mathematics andComplex Systems (Beijing Normal University) Ministry of Education Beijing 100875 China

This is one of a series of papers exploring the stability speed of one-dimensional stochastic processes. The present paper emphasizes on the principal eigenvalues of elliptic operators. The eigenvalue is just the best constant in the L2-Poincare inequality and describes the decay rate of the corresponding diffusion process. We present some variational formulas for the mixed principal eigenvalues of the operators. As applications of these formulas, we obtain case by case explicit estimates, a criterion for positivity, and an approximating procedure for the eigenvalue.

关键词： Eigenvalue variational formula explicit estimate positivity criterion approximating procedure

Lower bounds of principal eigenvalue in dimension one

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Frontiers of Mathematics in China 2012年第4期7卷 645-668页

作者： Mu-Fa CHEN School of Mathematical Sciences Beijing Normal UniversityLaboratory of Mathematics andComplex Systems(Beijing Normal University)Ministry of EducationBeijing 100875China

For the principal eigenvalue with bilateral Dirichlet boundary condition, the so-called basic estimates were originally obtained by capacitary method. The Neumann case （i.e., the ergodic case） is even harder, and was deduced from the Dirichlet one plus a use of duality and the coupling method. In this paper, an alternative and more direct proof for the basic estimates is presented. The estimates in the Dirichlet case are then improved by a typical application of a recent variational formula. As a dual of the Dirichlet case, the refine problem for bilateral Neumann boundary condition is also treated. The paper starts with the continuous case （one-dimensional diffusions） and ends at the discrete one （birth-death processes）. Possible generalization of the results studied here is discussed at the end of the paper

关键词： Principal eigenvalue lower estimate variational formula one- dimensional diffusion birth-death process

The Polar φ-Brunn-Minkowski Inequalities

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Wuhan University Journal of Natural Sciences 2020年第4期25卷 293-300页

作者： WEI Hongyu School of Mathematical Sciences Chongqing Normal UniversityChongqing 401331China

A general framework of polar Brunn-Minkowski theory that unifies the Orlicz Brunn-Minkowski inequality, the Lp Brunn-Minkowski inequality for p ∈(0,1) and the logBrunn-Minkowski inequality all for polar bodies is provided. It is shown that this general polar φ Brunn-Minkowski inequality is equivalent to a general polar φ Minkowski mixed volume inequality.

关键词： polar body variational formula general Brunn-Minkowski theory Minkowski mixed volume inequality Brunn-Minkowski inequality

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MEG反源问题的一种数值方法

高等学校计算数学学报 2012年第3期34卷 238-251页

作者：王红艳马富明黑龙江大学数学科学学院吉林大学数学研究所

1引言所谓MEG(脑磁图)反源问题,就是根据头外电磁场数据确定脑内神经电流源的活动情况.此类反问题在生物医学中有着广泛的应用,其最经典的一个应用即利用脑磁图仪器确定癫痫病人脑部病灶的位置.癫痫是由脑部神经元群阵发性异常放电所致... 详细信息

1引言所谓MEG(脑磁图)反源问题,就是根据头外电磁场数据确定脑内神经电流源的活动情况.此类反问题在生物医学中有着广泛的应用,其最经典的一个应用即利用脑磁图仪器确定癫痫病人脑部病灶的位置.癫痫是由脑部神经元群阵发性异常放电所致,这些神经元群的体积一般都很小,因而可以被近似看作是偶极子源[1,6].本文即是考虑由此衍生的反

关键词： invese source problem harmonic electromagnetic field dipole variational formula text function