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Relation Between the Eventual Continuity and the E-property for Markov-Feller Semigroups

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Acta Mathematicae Applicatae Sinica 2024年第1期40卷 1-16页

作者： Yong Liu Zi-yu Liu lmam School of Mathematical SciencePeking UniversityBeijing100871China

We investigate some relations between two kinds of semigroup regularities, namely the e-property and the eventual continuity, both of which contribute to the ergodicity for Markov processes on Polish *** precisely, we prove that for Markov-Feller semigroup in discrete time and stochastically continuous MarkovFeller semigroup in continuous time, if there exists an ergodic measure whose support has a nonempty interior,then the e-property is satisfied on the interior of the support. In particular, it implies that, restricted on the support of each ergodic measure, the e-property and the eventual continuity are equivalent for the discrete-time and the stochastically continuous continuous-time Markov-Feller semigroups.

关键词： Markov-Feller semigroup ergodicity e-property eventual continuity

EPR-Net: constructing a non-equilibrium potential landscape via a variational force projection formulation

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National Science Review 2024年第7期11卷 135-147页

作者： Yue Zhao Wei Zhang Tiejun Li Center for Data Science Peking University Zuse Institute Berlin Department of Mathematics and Computer Science Freie Universit?t Berlin Laboratory of Mathematics and Applied Mathematics(lmam)and School of Mathematical Sciences Peking University Center for Machine Learning Research Peking University

We present EPR-Net, a novel and effective deep learning approach that tackles a crucial challenge in biophysics: constructing potential landscapes for high-dimensional non-equilibrium steady-state ***-Net leverages a nice mathematical fact that the desired negative potential gradient is simply the orthogonal projection of the driving force of the underlying dynamics in a weighted inner-product ***, our loss function has an intimate connection with the steady entropy production rate(EPR),enabling simultaneous landscape construction and EPR estimation. We introduce an enhanced learning strategy for systems with small noise, and extend our framework to include dimensionality reduction and the state-dependent diffusion coefficient case in a unified fashion. Comparative evaluations on benchmark problems demonstrate the superior accuracy, effectiveness and robustness of EPR-Net compared to existing methods. We apply our approach to challenging biophysical problems, such as an eight-dimensional(8D)limit cycle and a 52. multi-stability problem, which provide accurate solutions and interesting insights on constructed landscapes. With its versatility and power, EPR-Net offers a promising solution for diverse landscape construction problems in biophysics.

关键词： high-dimensional potential landscape non-equilibrium system entropy production rate dimensionality reduction deep learning

Pole assignment in descriptor periodic systems

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Science China Mathematics 2005年第10期48卷 1309-1332页

作者： FAN Hung-Yuan~1 LIN Wen-Wei~1 & XU Shufang~2.1. Department of Mathematics,Tsing Hua University,Hsinchu 30043,China 2. lmam,School of Mathematical Sciences,Peking University,Beijing 100871,China 1. Department of Mathematics Tsing Hua University 30043 Hsinchu China 2. lmam School of Mathematical Sciences Peking University 100871 Beijing China

In this paper a pole assignment problem is considered for the descriptor linear discrete-time periodic systems, which is using the periodic proportional-derivative feedback to modify a given system such that the closed loop system has a specified selfconjugate set of eigenvalues. It is shown that the complete reachability of an open loop periodic system is equivalent to the possibility of assigning an arbitrary set of the eigenvalues to the system by choosing the suitable periodic proportional-derivative feedback.A computational approach is also proposed to solve the problem, which uses the reliable numerical techniques based on the orthogonal transformations. Numerical examples are presented to illustrate the effectiveness of the proposed approach.

关键词： descriptor linear periodic system, pole assignment, proportional and derivative feedback, numerical method, periodic Schur decomposition.

The regular representation and regular covariant representation of crossed products of Woronowicz C^(*)-algebras

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Science China Mathematics 2005年第9期48卷 1245-1259页

作者： ZHANG Xiaoxia~(1,2. & GUO Maozheng~(1,3) 1. School of Mathematical Sciences,Peking University,Beijing 100871,China 2. Department of Mathematics,Yantai University,Yantai 2.4005,China 3. lmam,Peking University,Beijing 100871,China 1. School of Mathematical Sciences Peking University 100871 Beijing China 2. Department of Mathematics Yantai University 264005 Yantai China 3. lmam Peking University 100871 Beijing China

In this paper,it is shown that the regular representation and regular covariant representation of the crossed products A×α G correspond to the twisted multiplicative unitary operators,where A is a Woronowicz C~*-algebra acted upon by a discrete group ***,it is also shown that the regular covariant C~*-algebra is the Woronowicz C~*-algebra which corresponds to a multiplicative ***,an explicit description of the multiplicative unitary operator for C(SU_q(2. )×α (?) is given in terms of those of the Woronowicz C~*-algebra C(SU_q(2. ) and the discrete group G.

关键词： multiplicative unitary operator, Woronowicz C*-algebra, crossed product.

KINETIC FLUX VECTOR SPLITTING FOR THE EULER EQUATIONS WITH GENERAL PRESSURE LAWS

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Journal of Computational Mathematics 2004年第4期22卷 622-632页

作者： Hua-zhongTang lmam SchoolofMathematicalSciencesPekingUniversityBeijing100871China

This paper attempts to develop kinetic flux vector splitting (KFVS) for the Euler equations with general pressure laws. It is well known that the gas distribution function for the local equilibrium state plays an important role in the construction of the gas-kinetic schemes. To recover the Euler equations with a general equation of state (EOS), a new local equilibrium distribution is introduced with two parameters of temperature approximation decided uniquely by macroscopic variables. Utilizing the well-known connection that the Euler equations of motion are the moments of the Boltzmann equation whenever the velocity distribution function is a local equilibrium state, a class of high resolution MUSCL-type KFVS schemes are presented to approximate the Euler equations of gas dynamics with a general EOS. The schemes are finally applied to several test problems for a general EOS. In comparison with the exact solutions, our schemes give correct location and more accurate resolution of discontinuities. The extension of our idea to multidimensional case is natural.

关键词： General equation of state KFVS method The Euler equations

Uniqueness and local stability for the inverse scattering problem of determining the cavity

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Science China Mathematics 2005年第8期48卷 1113-1123页

作者： FENG Lixin & MA Fuming lmam, School of Mathematical Sciences, Peking University, Beijing 100871, China School of Mathematical Sciences, Heilongjiang University, Harbin 150080, China School of Mathematical Sciences, Jilin University, Changchun 130012. China 1. lmam School of Mathematical Sciences Peking University 100871 Beijing China 2. School of Mathematical Sciences Heilongjiang University 150080 Harbin China 3. School of Mathematical Sciences Jilin University 130012 Changchun China

Considering a time-harmonic electromagnetic plane wave incident on an arbitrarily shaped open cavity embedded in infinite ground plane, the physical process is modelled by Maxwell's equations. We investigate the inverse problem of determining the shape of the open cavity from the information of the measured scattered field. Results on the uniqueness and the local stability of the inverse problem in the 2.dimensional TM (transverse magnetic) polarization are proved in this paper.

关键词： uniqueness, stability, Maxwell's equations, inverse scattering problem.

A Hybrid Procedure for Finding Real Points on a Real Algebraic Set

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Journal of Systems Science & Complexity 2019年第1期32卷 185-204页

作者： WANG Yu XIA Bican lmam & School of Mathematical Sciences Peking University

Motivated by the idea of Shen, et al.'s work, which proposed a hybrid procedure for real root isolation of polynomial equations based on homotopy continuation methods and interval analysis,this paper presents a hybrid procedure to compute sample points on each connected component of a real algebraic set by combining a special homotopy method and interval analysis with a better estimate on initial intervals. For a real algebraic set given by a polynomial system, the new method ?rst constructs a square polynomial system which represents the sample points, and then solve this system by a special homotopy continuation method introduced recently by Wang, et al.(2.17). For each root returned by the homotopy continuation method, which is a complex approximation of some(complex/real) root of the polynomial system, interval analysis is used to verify whether it is an approximation of a real root and ?nally get real points on the given real algebraic set. A new estimate on initial intervals is presented which helps compute smaller initial intervals before performing interval iteration and thus saves computation. Experiments show that the new method works pretty well on tested examples.

关键词： Homotopy continuation interval arithmetic polynomial system real variety

Mackey Theory for p-adic Lie Groups

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Acta Mathematica Sinica,English Series 2006年第2期22卷 507-514页

作者： Bing Yong HSIE lmam Department of Mathematics Peking University

This paper gives a p-adic analogue of the Mackey theory, which relates representations of a group of type G - H × t A to systems of imprimitivity.

关键词： Cosmooth projection valued measure System of imprimitivity Cosmooth system of imprimitivity Smooth representation

p-adic Modular Symbols and ∧-adic Modular Forms

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Acta Mathematica Sinica,English Series 2006年第2期22卷 331-338页

作者： Fu Zheng WANG lmam&Department of Mathematics.Peking University

A construction of A-adic modular forms from p-adic modular symbols is described. It shows that each A linear map satisfying some certain conditions from the module of p-adic modular symbols to A corresponds to a A-adi... 详细信息

关键词： Modular symbol Manin symbol A-adic modular form