同胚映射的代数指数、H?lder指数和拟对称指数

给定实轴上的同胚映射,定义了拟对称指数、代数指数和H?lder指数.上述指数刻画了同胚映射的局部特征,同时在拟对称映射和拟共形映射的研究中具有重要作用.探索了三个指数的相互关系并给出了几个实例.     

模和拟共形映射

研究了模和拟共形映射，得到了一个同胚为拟共形映射的几个条件．     

内球性质和拟共形映射

研究了一类具有内球性质区域的几何与分析性质,证明了f(∞)=∞的同胚f:-Rn→-Rn是拟共形映射当且仅当f保持区域的内球性质不变,并获得了该类区域若干有趣的几何性质.     

μ(z)-同胚的紧致性

本文研究μ(z)-同胚的紧致性.当一族μ(z)-同胚的伸张函数同时受控于一控制函数K(z)时,得到该族的紧致性质.利用紧性,得出一列μn(z)-同胚可收敛于一个拟共形映射.另外,还给出了μ(z)同胚在其逆映射为ACL情况下的分解定理.     

拟共形映射和弱Cigar域

设f是R~n到R~n上的同胚,本文证明了f是拟共形映射的充要条件是f将R~n中的任一弱Cigar域映成R~n中的弱Cigar域。     

μ(z)-同胚的紧致性

本文研究μ（ｚ）－同胚的紧致性．当一族μ（ｚ）－同胚的伸张函数同时受控于一控制函数Ｋ（ｚ）时,得到该族的紧致性质．利用紧性,得出一列μｎ（ｚ）－同胚可收敛于一个拟共形映射．另外,还给出了μ（ｚ）－同胚在其逆映射为ＡＣＬ 情况下的分解定理．     

关于单叶函数和万有Teichmuller空间的一些注记

万有Ｔｅｉｃｈｍｕｌｌｅｒ空间Ｔ是所有规范的拟对称同胚组成的集合，在Ｂｅｒｓ嵌入下，Ｔ与△上共形映射的Ｓｃｈｗａｒｚ导数密切相关，本文讨论由规范的对称同胚所 子集Ｔ０的相应性质。     

Notes on Univalent Functions and the Universal Teichmüller Space

万有Ｔｅｉｃｈｍüｌｅｒ空间Ｔ是所有规范的拟对称同胚组成的集合．在Ｂｅｒｓ嵌入下，Ｔ与Δ上共形映射的Ｓｃｈｗａｒｚ导数密切相关．本文讨论由规范的对称同胚所组成的子集Ｔ０的相应性质．     

覆盖近似空间的连续与同胚映射

利用一般映射研究了覆盖近似空间的一些性质,并证明了一些结论.接着定义了覆盖空间的粗糙连续映射及粗糙同胚映射.最后在覆盖粗糙连续映射和覆盖粗糙同胚映射的条件下,研究了两个覆盖近似空间的有关性质,进而在某种程度上为覆盖近似空间的分类提供了理论依据.     

双圆性质曲线和拟共形映射

设 f : R²→R² 是一同胚, f (∝)=∝. 该文证明了f 是拟共形映射的充要条件是f 保持曲线的双圆性质不变.     

Generalized local Morrey spaces and multilinear commutators generated by Marcinkiewicz integrals with rough kernel associated with Schr\"{o}dinger operators and local Campanato functions

Let $L=-\Delta+V\left( x\right) $ be a Schr\"{o}dinger operator, where $\Delta$ is the Laplacian on ${\mathbb{R}^{n}}$, while nonnegative potential $V\left( x\right) $ belongs to the reverse H\"{o}lder class. In this paper, we consider the behavior of multilinear commutators of Marcinkiewicz integrals with rough kernel associated with Schr\"{o}dinger operators on generalized local Morrey spaces.     

Global higher integrability of solutions to subelliptic double obstacle problems

In this paper we consider the double obstacle problems associated with nonlinear subelliptic equation \[X^*A(x,u,Xu)+ B(x,u,Xu)=0, \ \ x\in\Omega,\] where $X=(X_1,\ldots,X_m)$ is a system of smooth vector fields defined in $\mathbb{R}^n$ satisfying H\"{o}rmander"s condition. The global higher integrability for the gradients of the solutions is obtained under a capacitary assumption on the complement of the domain $\Omega$.     

On the images of Sobolev spaces under the heat kernel transform on the Heisenberg group

The aim of this paper is to obtain certain characterizations for the image of a Sobolev space on the Heisenberg group under the heat kernel transform. We give three types of characterizations for the image of a Sobolev space of positive order $H^m(\mathbb {H}^n), m\in \mathbb {N}^n,$ under the heat kernel transform on $\mathbb {H}^n,$ using direct sum and direct integral of Bergmann spaces and certain unitary representations of $\mathbb {H}^n$ which can be realized on the Hilbert space of Hilbert‐Schmidt operators on $L^2(\mathbb {R}^n).$ We also show that the image of Sobolev space of negative order $H^{-s}(\mathbb {H}^n), s(>0) \in \mathbb {R}$ is a direct sum of two weighted Bergman spaces. Finally, we try to obtain some pointwise estimates for the functions in the image of Schwartz class on $\mathbb {H}^n$ under the heat kernel transform.     

Möbius Homogeneous Hypersurfaces with Three Distinct Principal Curvatures in Sn+1

Let x : M~n→ S~(n+1) be an immersed hypersurface in the(n + 1)-dimensional sphere S~(n+1). If, for any points p, q ∈ Mn, there exists a Mbius transformation φ :S~(n+1)→ S~(n+1) such that φox(Mn~) = x(M~n) and φ ox(p) = x(q), then the hypersurface is called a Mbius homogeneous hypersurface. In this paper, the Mbius homogeneous hypersurfaces with three distinct principal curvatures are classified completely up to a Mbius transformation.     

$R^n$空间中的Cauchy积分公式

In this note p(D) = Dm+ b1Dm 1+···+ bmis a polynomial Dirac operator in R~n, where D =nj=1ej xjis a standard Dirac operator in Rn, bjare the complex constant coefficients. In this note we discuss all decompositions of p(D) according to its coefficients bj,and obtain the corresponding explicit Cauchy integral formulae of f which are the solution of p(D)f = 0.     

Fractional Hamiltonian systems with positive semi-definite matrix

We study the existence of solutions for the following fractional Hamiltonian systems $$ \left\{ \begin{array}{ll} - _tD^{\alpha}_{\infty}(_{-\infty}D^{\alpha}_{t}u(t))-\lambda L(t)u(t)+\nabla W(t,u(t))=0,\\[0.1cm] u\in H^{\alpha}(\mathbb{R},\mathbb{R}^n), \end{array} \right. ~~~~~~~~~~~~~~~~~(FHS)_\lambda $$ where $\alpha\in (1/2,1)$, $t\in \mathbb{R}$, $u\in \mathbb{R}^n$, $\lambda>0$ is a parameter, $L\in C(\mathbb{R},\mathbb{R}^{n^2})$ is a symmetric matrix, $W\in C^1(\mathbb{R} \times \mathbb{R}^n,\mathbb{R})$. Assuming that $L(t)$ is a positive semi-definite symmetric matrix, that is, $L(t)\equiv 0$ is allowed to occur in some finite interval $T$ of $\mathbb{R}$, $W(t,u)$ satisfies some superquadratic conditions weaker than Ambrosetti-Rabinowitz condition, we show that (FHS)$_\lambda$ has a solution which vanishes on $\mathbb{R}\setminus T$ as $\lambda \to \infty$, and converges to some $\tilde{u}\in H^{\alpha}(\R, \R^n)$. Here, $\tilde{u}\in E_{0}^{\alpha}$ is a solution of the Dirichlet BVP for fractional systems on the finite interval $T$. Our results are new and improve recent results in the literature even in the case $\alpha =1$.     

Transverse homoclinic orbit bifurcated from a homoclinic manifold by the higher order melnikov integrals

Consider an autonomous ordinary differential equation in $\mathbb{R}^n$ that has a $d$ dimensional homoclinic solution manifold $W^H$. Suppose the homoclinic manifold can be locally parametrized by $(\alpha,\theta) \in \mathbb{R}^{d-1}\times \mathbb{R}$. We study the bifurcation of the homoclinic solution manifold $W^H$ under periodic perturbations. Using exponential dichotomies and Lyapunov-Schmidt reduction, we obtain the higher order Melnikov function. For a fixed $(\alpha_0,\theta_0)$ on $W^H$, if the Melnikov function have a simple zeros, then the perturbed system can have transverse homoclinic solutions near $W^H$.     

Required Number of Iterations for Sparse Signal Recovery via Orthogonal Least Squares

In countless applications, we need to reconstruct a $K$-sparse signal $\mathbf{x}\in\mathbb{R}^n$ from noisy measurements $\mathbf{y}=\mathbf{\Phi}\mathbf{x}+\mathbf{v}$, where $\mathbf{\Phi}\in\mathbb{R}^{m\times n}$ is a sensing matrix and $\mathbf{v}\in\mathbb{R}^m$ is a noise vector. Orthogonal least squares (OLS), which selects at each step the column that results in the most significant decrease in the residual power, is one of the most popular sparse recovery algorithms. In this paper, we investigate the number of iterations required for recovering $\mathbf{x}$ with the OLS algorithm. We show that OLS provides a stable reconstruction of all $K$-sparse signals $\mathbf{x}$ in $\lceil2.8K\rceil$ iterations provided that $\mathbf{\Phi}$ satisfies the restricted isometry property (RIP). Our result provides a better recovery bound and fewer number of required iterations than those proposed by Foucart in 2013.     

Radix representations, self-affine tiles, and multivariable wavelets

We investigate the connection between radix representations for and self-affine tilings of . We apply our results to show that Haar-like multivariable wavelets exist for all dilation matrices that are sufficiently large.

     

The Coefficient Inequalities for a Class of Holomorphic Mappings in Several Complex Variable

The authors establish the coefficient inequalities for a class of holomorphic mappings on the unit ball in a complex Banach space or on the unit polydisk in ■ⁿ,which are natural extensions to higher dimensions of some Fekete and Szeg? inequalities for subclasses of the normalized univalent functions in the unit disk.