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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

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

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

Simulation of macrosegregation in a 36-t steel ingot using a multiphase model

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International Journal of Minerals,Metallurgy and Materials 2020年第2期27卷 200-209页

作者： Zhuo Chen Hou-fa Shen Key Laboratory for Advanced Material Processing Technology Ministry of Education School of Material Science and Engineering Tsinghua University

Macrosegregation is the major defect in large steel ingots caused by solute partitioning and melt convection during *** this study,a three-phase(liquid,columnar dendrites,and equiaxed grains)model is proposed to simulate macrosegregation in a 36-t steel ingot.A supplementary set of conservation equations are employed in the model such that two types of equiaxed grains,either settling or adhering to the solid shell,are well *** predicted concentration agrees quantitatively with the experimental value.A negative segregation cone was located at the bottom owing to the grain settlement and solute-enriched melt leaving from the mushy *** interdendritic liquid flow was carefully analyzed,and the formation of A-type segregations in the mid-height of the ingot is *** segregation was observed near the riser neck due to the specific relationship between flow direction and temperature ***,the as-cast macrostructure of the ingot is presented,including the grain size distribution and columnar–equiaxed transition.

关键词： macrosegregation multiphase model transportation solidification steel ingot

Comparison of some properties of star forming galaxies and active galactic nuclei between two BOSS galaxy samples from SDSS DR9

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Research in Astronomy and Astrophysics 2014年第4期14卷 401-410页

作者： Xin-fa Deng School of Science Nanchang University

Using the LOWZ and CMASS samples of the ninth data release （DR9） from the SDSS-III Baryon Oscillation Spectroscopic Survey （BOSS）, I investigate properties of star forming galaxies and active galactic nuclei （AGNs）. The CMASS sample seriously suffers from the radial selection effect, even within the redshift 0.44 〈 z 〈 0.6, which will likely lead to statistical conclusions in the CMASS sample being less robust. In the LOWZ sample, the fraction of star-forming galaxies is nearly constant from the least dense regime to the densest regime; the AGN fraction is also insensitive to the local environment. In addition, I note that in the LOWZ sample, the distributions of stellar mass and stellar velocity dispersion for star forming galaxies and AGNs are nearly the same.

关键词： galaxies: active- galaxies: statistics

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

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

Laplacian Speech Model and Soft Decision Based MMSE Estimator for Noise Power Spectral Density in Speech Enhancement

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Chinese Journal of Electronics 2018年第6期27卷 1214-1220页

作者： ou Shifeng SONG Peng GAO Ying School of Science and Technology for Opto-electronic Information Yantai University School of Computer and Control Engineering Yantai University

The estimation of noise Power spectral density(PSD) is a very crucial issue for speech enhancement as a result of its significant effect on the quality and intelligibility of the enhanced speech. Most of the existing estimators for noise PSD try to employ Gaussian speech priors, which, however, have been proven inconsistent with the reality. We derived an effective solution to this problem of estimating noise PSD in the Minimum mean square error(MMSE) sense when the speech component is modeled by a Laplacian distribution. Meanwhile, the soft decision technique instead of the hard Voice activity detection(VAD) is evolved into our algorithm, which can automatically makes the estimation unbiased without requiring a bias compensation. The performance of the proposed method is tested by several objective and subjective measures under various stationary and nonstationary noise environments. The results confirm that our method achieves good performance for all the noise conditions and Signalnoise-ratio(SNR) settings.

关键词： Noise PSD estimation Speech enhancement Laplacian speech model Soft decision

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