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Bilateral Hardy-type Inequalities

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Acta Mathematica Sinica,English Series 2013年第1期29卷 1-32页

作者： mu fa chen School of Mathematical Sciences Beijing Normal University Laboratory of Mathematics and Complex Systems Beijing Normal University Ministry of Education

This paper studies the Hardy-type inequalities on the intervals （may be infinite） with two weights, either vaaishing at two endpoiats of the interval or having mean zero. For the first type of inequalities, in terms of new isoperimetric constants, the factor of upper and lower bounds becomes smaller than the known ones. The second type of the inequalities is motivated from probability theory and is new ia the analytic context. The proofs are now rather elementary. Similar improvements are made for Nash inequality, Sobolev-type inequality, and the logarithmic Sobolev inequality on the intervals.

关键词： Hardy-type inequality vanishing at two endpoints mean zero splitting technique normedlinear space Nash inequality logarithmic Sobolev inequality

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

Hermitizable,isospectral complex matrices or differential operators

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Frontiers of Mathematics in China 2018年第6期13卷 1267-1311页

作者： mu-fa chen School of Mathematical Sciences Beijing Normal UniversityLaboratory of Mathematics and Complex Systems (Beijing Normal University)Ministry of EducationBeijing 100875China

The main purpose of the paper is looking for a larger class of matrices which have real spectrum. The first well-known class having this property is the symmetric one, then is the Hermite one. This paper introduces a new class, called Hermitizable matrices. The closely related isospectral problem, not only for matrices but also for differential operators is also studied. The paper provides a way to describe the discrete spectrum, at least for tridiagonal matrices or one-dimensional differential operators. Especially, an unexpected result in the paper says that each Hermitizable matrix is isospectral to a birth-death type matrix (having positive sub-diagonal elements, in the irreducible case for instance). Besides, new efficient algorithms are proposed for computing the maximal eigenpairs of these class of matrices.

关键词： Real spectrum symmetrizable Hermitizable isospectral matrix differential operator

Efficient initials for computing maximal eigenpair

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Frontiers of Mathematics in China 2016年第6期11卷 1379-1418页

作者： mu-fa chen School of Mathematical Sciences Beijing Normal University Laboratory of Mathematics and Complex Systems (Beijing Normal University) Ministry of Education Beijing 100875 China

This paper introduces some efficient initials for a well-known algorithm （an inverse iteration） for computing the maximal eigenpair of a class of real matrices. The initials not only avoid the collapse of the algorithm but are also unexpectedly efficient. The initials presented here are based on our analytic estimates of the maximal eigenvalue and a mimic of its eigenvector for many years of accumulation in the study of stochastic stability speed. In parallel, the same problem for computing the next to the maximal eigenpair is also studied.

关键词： Perron-Frobenius theorem power iteration Rayleigh quotient iteration efficient initial tridiagonal matrix Q-matrix

Spectral Gap and Logarithmic Sobolev Constant for Continuous Spin Systems

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Acta Mathematica Sinica,English Series 2008年第5期24卷 705-736页

作者： mu fa chen School of Mathematical Sciences Beijing Normal University Beijing 100875 P. R. China

The aim of this paper is to study the spectral gap and the logarithmic Sobolev constant for continuous spin systems. A simple but general result for estimating the spectral gap＇of finite dimensional systems is given by Theorem 1.1, in terms of the spectral gap for one-dimensional marginals. The study of this topic provides us a chance, and it is indeed another aim of the paper, to justify the power of the results obtained previously. The exact order in dimension one （Proposition 1.4）, and then the precise leading order and the explicit positive regions of the spectral gap and the logarithmic Sobolev constant for two typical infinite-dimensional models are presented （Theorems 6.2 and 6.3）. Since we are interested in explicit estimates, the computations become quite involved. A long section （Section 4） is devoted to the study of the spectral gap in dimension one.

关键词： spectral gap logarithmic Sobolev constant spin system principal eigenvalue

Efficient algorithm for principal eigenpair of discrete p-Laplacian

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Frontiers of Mathematics in China 2018年第3期13卷 509-524页

作者： mu-fa chen School of Mathematical Sciences Beijing Normal University Laboratory of Mathematics and Complex Systems (Beijing Normal University) Ministry of Education Beijing 100875 China

This paper is a continuation of the author＇s previous papers [Front. Math. China, 2016, 11（6）： 1379-1418; 2017, 12（5）： 1023-1043], where the linear case was studied. A shifted inverse iteration algorithm is introduced, as an acceleration of the inverse iteration which is often used in the non-linear context （the p-Laplacian operators for instance）. Even though the algorithm is formally similar to the Rayleigh quotient iteration which is well-known in the linear situation, but they are essentially different. The point is that the standard Rayleigh quotient cannot be used as a shift in the non-linear setup. We have to employ a different quantity which has been obtained only recently. As a surprised gift, the explicit formulas for the algorithm restricted to the linear case （p = 2） is obtained, which improves the author＇s approximating procedure for the leading eigenvalues in different context, appeared in a group of publications. The paper begins with p-Laplacian, and is closed by the non-linear operators corresponding to the well-known Hardy-type inequalities.

关键词： Discrete p-Laplacian principal eigenpair shifted inverse iteration approximating procedure

Global algorithms for maximal eigenpair

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Frontiers of Mathematics in China 2017年第5期12卷 1023-1043页

作者： mu-fa chen School of Mathematical Sciences Beijing Normal University Laboratory of Mathematics andComplex Systems (Beijing Normal University) Ministry of Education Beijing 100875 China

This paper is a continuation of our previous work [Front. Math. China, 2016, 11（6）： 1379-1418] where an efficient algorithm for computing the maximal eigenpalr was introduced first for tridiagonal matrices and then extended to the irreducible matrices with nonnegative off-diagonal elements. This paper introduces mainly two global algorithms for computing the maximal eigenpair in a rather general setup, including even a class of real （with some negative off-diagonal elements） or complex matrices.

关键词： Maximal eigenpair shifted inverse iteration global algorithm

The Optimal Constant in Hardy-type Inequalities

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Acta Mathematica Sinica,English Series 2015年第5期31卷 731-754页

作者： mu-fa chen School of Mathematical Sciences Beijing Normal University

To estimate the optimal constant in Hardy-type inequalities, some variational formulas and approximating procedures are introduced. The known basic estimates are improved considerably. The results are illustrated by typical examples. It is shown that the sharp factor is meaningful for each finite interval and a classical sharp model is re-examined.

关键词： Hardy-type inequality optimal constant variational formulas approximating procedure

Improved global algorithms for maximal eigenpair

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Frontiers of Mathematics in China 2019年第6期14卷 1077-1116页

作者： mu-fa chen Yue-Shuang LI School of Mathematical Sciences Beijing Normal UniversityLaboratory of Mathematics and Complex Systems(Beijing Normal University)Ministry of EducationBeijing 100875China

This paper is a continuation of our previous paper[***,2017,12(5):10231043]where global algorithms for computing the maximal cigcnpair were introduced in a rather general *** efficiency of the global algorithms is improved in this paper in terms of a good use of power iteration and two quasi-symmetric ***,the new algorithms are applied to Hua’s economic optimization model.

关键词： Maximal eigenpair global algorithm power iteration shifted inverse iteration quasi-symmetrization