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A Dual Yamabe Flow and Related Integral Flows

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Chinese Annals of mathematics,Series B 2024年第3期45卷 319-348页

作者： Jingang XIONG school of mathematical sciences Laboratory of Mathematics and Complex SystemsMinistry of EducationBeijing Normal UniversityBeijing 100875China

The author studies a family of nonlinear integral flows that involve Riesz potentials on Riemannian manifolds. In the Hardy-Littlewood-Sobolev (HLS for short)subcritical regime, he presents a precise blow-up profile exhibited by the flows. In the HLS critical regime, by introducing a dual Q curvature he demonstrates the concentrationcompactness phenomenon. If, in addition, the integral kernel matches with the Green’s function of a conformally invariant elliptic operator, this critical flow can be considered as a dual Yamabe flow. Convergence is then established on the unit spheres, which is also valid on certain locally conformally flat manifolds.

关键词： Hardy-Littlewood-Sobolev functional Dual Q curvature Integral flow

MAXIMAL FUNCTION CHARACTERIZATIONS OF HARDY SPACES ASSOCIATED WITH BOTH NON-NEGATIVE SELF-ADJOINT OPERATORS SATISFYING GAUSSIAN ESTIMATES AND BALL QUASI-BANACH FUNCTION SPACES

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Acta Mathematica Scientia 2024年第2期44卷 484-514页

作者：林孝盛杨大春杨四辈袁文 laboratory of mathematics and complex systems(Ministry of Education of China) School of Mathematical SciencesBeijing Normal UniversityBeijing100875China school of mathematics and Statistics Gansu Key Laboratory of Applied Mathematics and Complex SystemsLanzhou UniversityLanzhou730000China

Assume that L is a non-negative self-adjoint operator on L^(2)(ℝ^(n))with its heat kernels satisfying the so-called Gaussian upper bound estimate and that X is a ball quasi-Banach function space onℝ^(n) satisfying some mild *** HX,L(ℝ^(n))be the Hardy space associated with both X and L,which is defined by the Lusin area function related to the semigroup generated by *** this article,the authors establish various maximal function characterizations of the Hardy space HX,L(ℝ^(n))and then apply these characterizations to obtain the solvability of the related Cauchy *** results have a wide range of generality and,in particular,the specific spaces X to which these results can be applied include the weighted space,the variable space,the mixed-norm space,the Orlicz space,the Orlicz-slice space,and the Morrey ***,the obtained maximal function characterizations of the mixed-norm Hardy space,the Orlicz-slice Hardy space,and the Morrey-Hardy space associated with L are completely new.

关键词： Hardy space ball quasi-Banach function space Gaussian upper bound estimate non-negative self-adjoint operator maximal function

An Example of Embedded Singular Continuous Spectrum for Discrete Schrödinger Operators

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Acta Mathematica Sinica,English Series 2024年第8期40卷 1837-1849页

作者： Zheng Qi FU Xiong LI laboratory of mathematics and complex systems(Ministry of Education) School of Mathematical SciencesBeijing Normal UniversityBeijing 100875P.R.China

We present an example of a potential such that the corresponding discrete Schrödinger operator has singular continuous spectrum embedded in the absolutely continuous spectrum.

关键词： Sparse potentials singular continuous spectrum Prüfer variables

Recursive Projected Filter Algorithm with Binary-Valued Observations

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Journal of systems Science & complexity 2024年第5期37卷 1832-1860页

作者： ZHANG Xiaoyan WANG Ying XUE Wenchao ZHAO Yanlong Key laboratory of systems and Control Institute of Systems Science Academy of Mathematics and Systems Science Chinese Academy of Sciences school of mathematical sciences University of Chinese Academy of Sciences

This paper focuses on the state estimate for a class of systems with both process noise and measurement noise under binary-valued observations, in which the Gaussian assumption on the predicted density of the state is not required. A recursive projected filter algorithm with time-varying thresholds is constructed to estimate the state under binary-valued observations. The time-varying thresholds are designed as the prediction value of the measurement, which can provide more information about the system state. The convergence property is established with some suitable stability,boundedness and observability conditions. In particular, the estimation error between state and estimate is proved to be asymptotically bounded in the mean-square sense, whose upper bound is related to the variance of process noise. Finally, the theoretical results are demonstrated via numerical examples of first-order and high-order systems.

关键词： Binary-valued observations convergence recursive filter time-varying thresholds

Sharp bilinear decomposition for products of both anisotropic Hardy spaces and their dual spaces with its applications to endpoint boundedness of commutators

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Science China mathematics 2024年第9期67卷 2091-2152页

作者： Jun Liu Dachun Yang Mingdong Zhang school of mathematics China University of Mining and TechnologyXuzhou 221116China laboratory of mathematics and complex systems(Ministry of Education of China) School of Mathematical SciencesBeijing Normal UniversityBeijing 100875China

Let a:=(a_(1),...,a_(n))2[1,∞)^(n),p∈(0,1),andα:=1/*** any x∈R^(n)and t∈[0,∞),letΦ_(p)(x,t):={t/1+(t[x]_(a)^(ν))^(1-p)if να■N,t/1+(t[x]_(a)^(ν))^(1-p)[log(e+|x|a)]^(p)if να∈N,let where [·]a:=1+|·|a,|·|a denotes the anisotropic quasi-homogeneous norm with respect to a,and ν:=a_(1)+…+a_(n).Let H_(a)^(p)(R^(n)),L_(a)^(a)(R^(n)),and H_(a)^(Φ_(p))(R^(n))be,respectively,the anisotropic Hardy space,the anisotropic Campanato space,and the anisotropic Musielak-Orlicz Hardy space associated with Φ_(p) on R^(n).In this article,via first establishing the wavelet characterization of anisotropic Campanato spaces,we prove that for any f∈H_(a)^(p)(R^(n))and g∈L_(a)^(a)(R^(n)),the product of f and g can be decomposed into S(f,g)+T(f,g) in the sense of tempered distributions,where S is a bilinear operator bounded from H_(a)^(p)(R^(n))*L_(a)^(a)(R^(Φ_(p))) to L^(1)(R^(n)) and T is a bilinear operator bounded from H_(a)^(p)(R^(n))*L_(a)^(a)(R^(n)) to H_(a)^(Φ_(p))(R^(n)) .Moreover,this bilinear decomposition is sharp in the dual sense that any y■H_(a)^(Φ_(p))(R^(n)) that fits into the above bilinear decomposition should satisfy(L^(1)(R^(n))+y)*=(L^(1)(R^(n)+H_(a)^(Φ_(p))(R^(n))*.As applications,for any non-constant b∈L_(a)^(a)(R^(n)) and any sublinear operator T satisfying some mild bounded assumptions,we find the largest subspace of H_(a)^(p)(R^(n)),denoted by H_(a,b)^(p)(R^(n)),such that the commutator [b,T] is bounded from H_(a,b)^(p)(R^(n))to L^(1)(R^(n)).In addition,when T is an anisotropic CalderónZygmund operator,the boundedness of [b,T] from H_(a,b)^(p)(R^(n))to L^(1)(R^(n))(or to H_(a)^(1)(R^(n)) is also *** key of their proofs is the wavelet characterization of function spaces under consideration.

关键词： anisotropic Euclidean space bilinear decomposition Hardy space Campanato space Musielak-Orlicz Hardy space commutator wavelet

F_(p) as a Discrete Metric Space

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Algebra Colloquium 2024年第1期31卷 165-180页

作者： Zezhou Zhang school of mathematical sciences MOE Key laboratory of mathematics and complex systems Beijing Normal University Beijing 100875China

We define a metric that makes the algebraic closure of a finite field F_(p) into a UDBG(uniformly discrete with bounded geometry)metric *** metric stems from algebraic properties of F_(p).From this perspective,for F_(p)we explore common research themes in metric spaces,reveal how peculiar properties naturally arise,and present it as a new type of example for certain well-studied questions.

关键词： algebraic closure of finite field discrete metric space non-expanding Falner

THE PERSISTENCE OF SOLUTIONS IN A NONLOCAL PREDATOR-PREY SYSTEM WITH A SHIFTING HABITAT

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Acta Mathematica Scientia 2024年第3期44卷 1096-1114页

作者：赵敏袁荣 school of Science Tianjin Chengjian UniversityTianjin300384China laboratory of mathematics and complex systems(Ministry of Education) School of Mathematical SciencesBeijing Normal UniversityBeijing100875China

In this paper,we mainly study the propagation properties of a nonlocal dispersal predator-prey system in a shifting *** is known that Choi et al.[J Differ Equ,2021,302:807-853]studied the persistence or extinction of the prey and of the predator separately in various moving *** particular,they achieved a complete picture in the local diffusion ***,the question of the persistence of the prey and of the predator in some intermediate moving frames in the nonlocal diffusion case was left open in Choi et al.'s *** using some a prior estimates,the Arzelà-Ascoli theorem and a diagonal extraction process,we can extend and improve the main results of Choi et *** achieve a complete picture in the nonlocal diffusion case.

关键词： predator-prey system persistence nonlocal dispersal shifting environment

THE ABSENCE OF SINGULAR CONTINUOUS SPECTRUM FOR PERTURBED JACOBI OPERATORS

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Acta Mathematica Scientia 2024年第2期44卷 515-531页

作者：傅钲淇黎雄 Department of Basic sciences Naval Submarine AcademyQingdao266199China laboratory of mathematics and complex systems(Ministry of Education) School of Mathematical SciencesBeijing Normal UniversityBeijing100875China

This paper is mainly about the spectral properties of a class of Jacobi operators(H_(c,b)u)(n)=c_(n)u(n+1)+c_(n-1)u(n-1)+b_(n)u(n),.where∣c_(n)−1∣=O(n^(−α))and b_(n)=O(n^(−1)).We will show that,forα≥1,the singula... 详细信息

关键词： Jacobi operator singular continuous spectrum Prüfer variables

Distributed Least Squares Algorithm of Continuous-Time Stochastic Regression Model Based on Sampled Data

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Journal of systems Science & complexity 2024年第2期37卷 609-628页

作者： ZHU Xinghua GAN Die LIU Zhixin The Key laboratory of systems and Control Academy of Mathematics and Systems ScienceChinese Academy of Sciencesand School of Mathematical SciencesUniversity of Chinese Academy of SciencesBeijing 100190China Zhongguancun laboratory Beijing 100094China

In this paper,the authors consider the distributed adaptive identification problem over sensor networks using sampled data,where the dynamics of each sensor is described by a stochastic differential *** minimizing a local objective function at sampling time instants,the authors propose an online distributed least squares algorithm based on sampled data.A cooperative non-persistent excitation condition is introduced,under which the convergence results of the proposed algorithm are established by properly choosing the sampling time *** upper bound on the accumulative regret of the adaptive predictor can also be ***,the authors demonstrate the cooperative effect of multiple sensors in the estimation of unknown parameters by computer simulations.

关键词： Cooperative excitation condition distributed least squares regret sampled data stochastic differential equation