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Jacobian-determinant Equation in the Critical Besov Spaces

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Acta Mathematica Sinica,English Series 2017年第5期33卷 691-704页

作者： xiang Rong zhu Department of Mathematics Zhejiang Normal University Jinhua 321004 P. R. China

Let Ω be a bounded domain in R;with smooth boundary. Here we consider the following Jacobian-determinant equation det u（x）=f（x）,x∈Ω;u（x）=x,x∈?Ω where f is a function on Ω with min;f = δ > 0 and Ωf（x）dx = |Ω|. We prove that if f ∈B;（Ω） for some p∈（n,∞）, then there exists a solution u ∈ B;（Ω）C;（Ω） to this equation. On the other hand, we give a simple example such that u ∈ C;（R;, R;） while detu does not lie in B;（R;） for any p<∞.

关键词： Jacobian-determinant Besov space elliptic system Neumann problem

The Existence of Weak Solutions to the 2D Euler Equations with Initial Vorticity in BMO

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Acta Mathematica Sinica,English Series 2007年第10期23卷 1881-1888页

作者： xiang Rong zhu Department of Mathematics Zhejiang University Hangzhou 310027 P. R. China

We shall prove that there exists a weak solution to the 2D Euler equations with initial vorticity in BMO and initial velocity in L^∞.

关键词： Euler equation weak solution BMO

Maximal operators of pseudo-differential operators with rough symbols

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Applied Mathematics(A Journal of Chinese Universities) 2024年第1期39卷 174-180页

作者： Ramla Benhamoud zhu xiang-rong School of Mathematical Sciences Zhejiang Normal UniversityJinhua 321004China

Consider a pseudo-differential operator T_(a)f(x)=∫_(R^(n))e^(ix,ζ)a(x,ζ)f(ζ)dζwhere the symbol a is in the rough Hormander class L^(∞)S_(ρ)^(m)with m∈R andρ∈[0,1].In this note,when 1≤p≤2,if n(ρ-1)/p and a∈L^(∞)S_(ρ)^(m),then for any f∈S(R^(n))and x∈R^(n),we prove that M(T_(a)f)(x)≤C(M(|f|^(p))(x))^(1/p) where M is the Hardy-Littlewood maximal *** theorem improves the known results and the bound on m is sharp,in the sense that n(ρ-1)/p can not be replaced by a larger constant.

关键词： pseudo-differential operator rough Hormander class H-L maximal operator

原发性骨质疏松潜在生物标志物的生物信息学分析

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中国组织工程研究 2025年第8期29卷 1741-1750页

作者：赵嘉诚任诗齐祝秦刘佳佳朱翔杨洋南通大学附属医院创伤中心江苏省南通市226001 南通大学医学院江苏省南通市226001 南通大学附属医院手外中心江苏省南通市226001

背景:原发性骨质疏松症的发病率高,但发病机制尚不完全清楚,目前尚缺乏有效的早期筛查指标和治疗方案。目的:通过综合生物信息学分析,进一步探索原发性骨质疏松症的发生机制。方法:原发性骨质疏松症数据来自公共基因表达数据库,筛选差... 详细信息

背景:原发性骨质疏松症的发病率高,但发病机制尚不完全清楚,目前尚缺乏有效的早期筛查指标和治疗方案。目的:通过综合生物信息学分析,进一步探索原发性骨质疏松症的发生机制。方法:原发性骨质疏松症数据来自公共基因表达数据库,筛选差异基因分别进行GO功能和KEGG通路富集分析。此外,将差异基因进行蛋白质-蛋白质相互作用网络确定原发性骨质疏松症相关核心基因,并通过最小绝对收缩和选择运算算法来识别并验证原发性骨质疏松症相关的生物标志物,并分别进行免疫细胞相关分析、基因富集分析以及药物标靶网络分析。最后将生物标志物行qPCR实验验证。结果与结论:①该研究中共获得126个差异基因以及前列腺素、表皮生长因子受体、丝裂原活化蛋白激酶3、转化生长因子B1和视网膜母细胞瘤基因1等5个生物标志物。②GO分析表明差异基因主要富集在细胞对氧化应激的反应以及自噬的调节等方面;KEGG分析显示主要集中在自噬以及细胞衰老等通路当中。③生物标志物的免疫分析发现,前列腺素,视网膜母细胞瘤基因1、丝裂原活化蛋白激酶3与免疫细胞密切相关。④基因富集分析表明,生物标志物与免疫相关途径有关。⑤药物标靶网络分析显示5个生物标志物与原发性骨质疏松症药物相关。⑥qPCR结果表明,前列腺素、表皮生长因子受体、丝裂原活化蛋白激酶3和转化生长因子B1在原发性骨质疏松症样本中,与对照样本相比表达显著上升(P<0.001),而视网膜母细胞瘤基因1在原发性骨质疏松症样本中,与对照样本相比表达显著下降(P<0.001)。⑦总之,该研究筛选并验证了5个原发性骨质疏松潜在生物标志物,为进一步深入探究原发性骨质疏松症的发病机制、早期筛查诊断及靶向治疗提供了参考依据。

关键词：原发性骨质疏松生物标志物生物信息学药物标靶网络蛋白质-蛋白质相互作用网络

On the Small Time Asymptotics of the Dynamical Φ_(1)^(4)Model

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Acta Mathematica Sinica,English Series 2021年第3期37卷 436-446页

作者： Bing Guang CHEN xiang Chan zhu Academy of Mathematics and Systems Science Chinese Academy of ScienceBeijing 100190P.R.China Department of Mathematics University of BielefeldD-33615 BielefeldGermany

In this paper,we establish a small time large deviation principle(small time asymptotics)for the dynamicalΦ_(1)^(4)model,which not only involves study of the space-time white noise with intensity√ε,but also the inv... 详细信息

关键词： Φ_(1)^(4)model space-time white noise small time asymptotics large deviation

Large Deviation Principle for the Two-dimensional Stochastic Navier-Stokes Equations with Anisotropic Viscosity

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Acta Mathematicae Applicatae Sinica 2023年第3期39卷 511-549页

作者： Bing-guang CHEN xiang-chan zhu Academy of Mathematics and Systems Science Chinese Academy of SciencesBeijing 100190China Department of Mathematics University of BielefeldD-33615 BielefeldGermany

In this paper we establish the large deviation principle for the the two-dimensional stochastic Navier-Stokes equations with anisotropic viscosity both for small noise and for short *** proof for large deviation principle is based on the weak convergence *** small time asymptotics we use the exponential equivalence to prove the result.

关键词： large deviation principle stochastic Navier-Stokes equations anisotropic viscosity small time asymptotics weak convergence approach

Weak type(1, 1) of the Riesz transform on some direct product manifolds with exponential volume growth

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Science China Mathematics 2024年第7期67卷 1599-1622页

作者： Hong-Quan Li Jie-xiang zhu School of Mathematical Sciences Fudan UniversityShanghai200433China Center for Applied Mathematics Tianjin UniversityTianjin300072China

In this paper, we are concerned with the Riesz transform on the direct product manifold H^(n)× M,where H^(n) is the n-dimensional real hyperbolic space, and M is a connected complete non-compact Riemannian manifold satisfying the volume doubling property and generalized Gaussian or sub-Gaussian upper estimates for the heat kernel. We establish its weak type(1, 1) property. In addition, we obtain the weak type(1, 1) of the heat maximal operator in the same setting. Our arguments also work for a large class of direct product manifolds with exponential volume growth. Particularly, we provide a simpler proof of weak type(1, 1) boundedness of some operators considered in the work of Li et al.(2016).

关键词： Riesz transform heat kernel direct product real hyperbolic space

Integrability of the general product Hardy operators on the product Hardy spaces

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Applied Mathematics(A Journal of Chinese Universities) 2012年第2期27卷 225-233页

作者： zhu xiang-rong CHEN Jie-cheng Department of Mathematics Zhejiang Normal UniversityJinhua 321004China

In this paper,we prove that the general product Hardy operators are bounded from the product Hardy space H1n（Rm1×···×Rmn） to L1（R（∑ni=1mi）.

关键词： product Hardy operator product Hardy space atom decomposition.

Spectrum and Spectral Singularities of a Quadratic Pencil of a Schr?dinger Operator with Boundary Conditions Dependent on the Eigenparameter

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Acta Mathematica Sinica,English Series 2023年第11期39卷 2164-2180页

作者： xiang zhu Zhao Wen ZHENG Kun LI School of Mathematical Sciences Qufu Normal UniversityQufu 273165P.R.China College of Mathematics and Systems Science Guangdong Polytechnic Normal UniversityGuangzhou 510665P.R.China

In this paper,we consider the following quadratic pencil of Schr?dinger operators L(λ)generated in L~2(R~+)by the equation−y″+[p(x)+2λq(x)]y=λ^(2)y,x∈R^(+)=[0,+∞)with the boundary condition y′(0)/y(0)=β_(1)λ+β0-α_(1)λ+α0,where p(x)and q(x)are complex valued functions andα_(0),α_(1),β_(0),β_(1)are complex numbers withα_(0)β_(1)-α_(1)β_(0)≠*** is proved that L(λ)has a finite number of eigenvalues and spectral singularities,and each of them is of a finite multiplicity,if the conditions p(x),q′(x)∈AC(R^(+)),limx→∞[|p(x)|+|q(x)|+|q′(x)|]=0 and sup 0≤x0.

关键词： Schrodinger operator Strum-Liouville problems finite spectrum eigenparameter dependence