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Global well-posedness for the 2. critical dissipative quasi-geostrophic equation

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Science China Mathematics 2007年第4期50卷 485-494页

作者： Zhi-fei ZHANG school of mathematical sciences, peking university, {2., China school of mathematical sciences Peking University Beijing 100871 China

We study the initial value problem for the 2. critical dissipative quasi-geostrophic equation. We prove the global existence for small data in the scale invariant Besov spaces ? p,1 2.p , 1 ? p ? ∞. In particular, for p = ∞, our result does not impose any regularity on the initial data. Our proofs are based on an exponential decay estimate of the semigroup $e^{{ - t\kappa ( - \Delta )}^{\alpha}} $ and the use of space-time Besov spaces.

关键词： quasi-geostrophic equation Besov space Littlewood-Paley decomposition well-posedness 35Q35

Totally geodesic submanifolds of maximal rank in symmetric spaces

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Science China Mathematics 2004年第2期47卷 264-271页

作者： ZHU Fuhai & LIANG Keschool of mathematical sciences and lmam, {2., beijing 100871, China College of mathematical sciences and LPMC, Nankai university, Tianjin 300071, China 1.school of mathematical sciences and lmam Peking University 100871 Beijing China 2.College of Mathematical Sciences and LPMC Nankai University 300071 Tianjin China

We gave a complete list of totally geodesic submanifolds of maximal rank in symmetric spaces of noncompact type. The compact cases can be obtained by the duality.

关键词： totally geodesic submanifold, Lie triple system, maximal rank.

Maslov-type index and brake orbits in nonlinear Hamiltonian systems

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Science China Mathematics 2007年第6期50卷 761-772页

作者： Duan-zhi ZHANG school of mathematical sciences, peking university, {2., China 1. school of mathematical sciences Peking University Beijing 100871 China

In this paper, we study the Maslov-type index theory for linear Hamiltonian systems with brake orbits boundary value conditions and its applications to the existence of multiple brake orbits of nonlinear Hamiltonian s... 详细信息

关键词： brake orbit Maslov-type index relative Morse index dual variational method 58E05 70H05 34C2.

Adjoint method for the optimum planning of industrial pollutant sources

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Science China Earth sciences 2005年第8期48卷 1270-1279页

作者： LIU Feng1,2. HU Fei2.& ZHU Jiang2.1. College of Environmental sciences, peking university, {2., China 2. LAPC, Institute of Atmospheric Physics, Chinese Academy of sciences, beijing 10002., China 1. College of Environmental sciences Peking University 100871 Beijing China 2. LAPC Institute of Atmospheric Physics Chinese Academy of Sciences 100029 Beijing China

The optimum planning of industrial pollutant sources, which optimizes the economic object without violating environmental constraints, is an important and hard task to be conquered. In this paper, an adjoint method is developed to solve the problem. The penalty function is in-troduced to deal with the environmental inequality constraints, and Lagrange function is con-structed to derive the adjoint equation and the gradient of the object function. In this means, the gradient of the object function can be calculated by solving the adjoint equation, and the infor-mation from the gradient is used to make the object function descend and approach to an optimal solution after some iterations. A two-dimensional, simplified model is used for numerical experi-ments. The theoretical derivations are verified by the results of the experiments. Furthermore, the adjoint method is shown to be of excellent convergence and efficiency, which is adaptive to the fast development of air quality numerical models and super computers.

关键词： optimum planning, industrial pollutant sources, adjoint equation, penalty function, gradient.

Finite Groups in Which Sylow Normalizers Admit Supplements

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Acta Mathematica Sinica,English Series 2007年第3期23卷 417-422页

作者： Tian Ze LI school of mathematical sciences,peking university,{2.,P.R.China school of mathematical sciences Peking University Beijing 100871 P. R.China

For a group G and a subgroup M of G,we say that a subgroup A of G is a supplement toM in G,if G=*** prove that a finite group in which every Sylow normalizer admits a nilpotentsupplement is *** particular we confirm a... 详细信息

关键词： Sylow normalizer Supplement Wreath product

OBLIQUE DERIVATIVE PROBLEMS FOR SECOND ORDER NONLINEAR MIXED EQUATIONS WITH DEGENERATE LINE

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Acta Mathematica Scientia 2008年第3期28卷 604-612页

作者：闻国椿 school of mathematical sciences Peking University

The present article deals with oblique derivative problems for some nonlinear mixed equations with parabolic degeneracy, which include the Tricomi problem as a special case. First, the formulation of the problems for the equations is given; next, the representation and estimates of solutions for the above problems are obtained; finally, the existence of solutions for the problems is proved by the successive iteration and the compactness principle of solutions of the problems. In this article, the author uses the complex method, namely, the complex functions in the elliptic domain and the hyperbolic complex functions in hyperbolic domain are used.

关键词： Oblique derivative problems nonlinear mixed equations parabolic degeneracy

r-matrix and algebraic-geometric symplectic map

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Chinese Science Bulletin 1999年第2期44卷 114-118页

作者： QIAO ZhijunInstitute of Mathematics and school of mathematical sciences, peking university, {2., China Institute of Mathematics and school of mathematical sciences Peking University Beijing China

A new Lax matrix is introduced for the integrable symplectic map (ISM), and the non-dynamical(i. e. constant) r-matrix of ISM is obtained. Moreover, an effective approach is systematically presented to construct the explicit solution(here, the explicit solution means algebraic-geometric solution expressed by the Riemann-Theta function) of a soliton system or nonlinear evolution equation from Lax matrix, r-matrix, and the theory of nonlinearization through taking the Toda lattice as an example. The given algebraic-geometric solution of the Toda lattice is almost-periodic and includes the periodic and finite-band solution.

关键词： symplectic map r-matrix algebraic-geometric solution.

SOME BOUNDARY VALUE PROBLEMS FOR NONLINEAR DEGENERATE ELLIPTIC EQUATIONS OF SECOND ORDER

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Acta Mathematica Scientia 2007年第3期27卷 663-672页

作者：闻国椿 school of mathematical sciences Peking University

The present article deals with some boundary value problems for nonlinear elliptic equations with degenerate rank 0 including the oblique derivative problem. Firstly the formulation and estimates of solutions of the oblique derivative problem are given, and then by the above estimates and the method of parameter extension, the existence of solutions of the above problem is proved. In this article, the complex analytic method is used, namely the corresponding problem for degenerate elliptic complex equations of first order is firstly discussed, afterwards the above problem for the degenerate elliptic equations of second order is solved.

关键词： Boundary value problems nonlinear elliptic equations parabolic degeneracy

New feature extraction in gene expression data for tumor classification

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Progress in Natural Science:Materials International 2005年第9期15卷 861-864页

作者： HE Renya, CHENG Qiansheng**, WU Lianwen and YUAN Kehong(lmam, school of mathematical sciences, Institute of Molecular Medicine, {2., beijing 100871, China) lmam School of Mathematical Sciences Institute of Molecular Medicine Peking University Beijing 100871 China

Using gene expression data to discriminate tumor from the normal ones is apowerful method. However, it is sometimes difficult because the gene expression data are in highdimension and the object number of the data sets is very small. The key technique is to find a newgene expression profiling that can provide understanding and insight into tumor related cellularprocesses. In this paper, we propose a new feature extraction method based on variance to the centerof the class and employ the supportvector machine to recognize the gene data either normal ortumor. Two tumor data sets are used to demonstrate the effectiveness of our methods. The resultsshow that the performance has been significantly improved.

关键词： tumor classification support vector machine (SVM) bioinformatics featureextraction gene expression