Beurling-Ahlfors延拓的伸张函数

本文研究了实轴上的保向同胚映照到上半平面的Beurling-Ahlfors延拓的性质.对上半平面及实轴附近的伸张函数分别作了估计,获得了一个最佳估计,改进了已有的结果.     

μ(z) -同胚的边界性质

该文讨论μ(z) -同胚的边界性质.给出了一个充分条件,使得上半平面的μ(z) -自同胚可以拓扑地向边界延拓. 用μ(z) -同胚的伸张函数估计了ρ -函数.     

伸张函数增长阶的估计

本文建立了Beurling-Ahlfors扩张的伸张函数的一个新的估计式,并对当ρ-函数在递减函数控制下伸张函数的增长阶进行了估计,改进了已有的结果,得到估计式: 当ρ(y/2)≥2时,D(x,y)≤2ρ(y/2); 当1≤ρ(y/2)<2时,D(x,y)2≤ρ(y/2)+1/2.     

半平面中解析函数的积分表示

本文证明了半平面中满足某些限制增长条件的解析函数可以用加权Blaschke乘积和在半平面边界上的积分之和表示出来,这一结果改进了在半平面为指数型解析函数的经典结果.     

广义Beurling-Ahlfors定理

建立了具有可积球伸张H(z,f)的,上半平面到自身的ID-同胚w=f(z)的Beurling-Ahlfors定理,研究了广义Beurling-Ahlfors扩张的等距问题.     

关于Beurling和Ahlfors的一个定理的证明

<正> 实轴到自身的保持定向的同胚μ叫作ρ拟对称的，如果它满足下列条件：■对一切实数x及t≠0.1956年A.Beurling与L.Ahlfors证明了一个重要定理:对于任意给定的ρ拟对称函数μ,总存在一个上半平面到自身的K拟共形映照,以μ为边界对应,而K有下述估计式:     

拟对称函数增长阶的估值

设Ｇ１与Ｇ２是由光滑Ｔｏｒｄａｎ闭曲线界成的区域,ｆ为Ｇ１到Ｇ２的μ（ｚ）－同胚,当ｆ的平均伸长地数函数控制时,则ｆ可拓扑地延拓到边界,记边界函数为ｈ,本文引进了由ｈ生成的拟对称函数ρｈ,利用模理论及极值长度方法,我们估计了拟对称函数的增长阶,得到了一个双向不等式。     

全平面上随机Dirichlet级数的值分布

对全平面上ρ(0ρ+∞)级随机Dirichlet级数,在一定条件下,在宽度为吾的任何闭水平带形内几乎必然有一个没有例外小函数的ρ级Borel线.     

半平面中有限阶解析函数的因子分解

与经典有限阶整函数的Hadamard因子分解定理和半平面中属于Hardy空间的解析函数的内外函数的因子分解类似,对右半平面中有限阶ρ解析函数f,可以分解为三个解析函数G,eQ和eg的乘积GeQeg,其中G是一个加权Blaschke乘积,Q是一个次数不超过ρ的多项式以及eg是一个加权外函数,log|G|,ReQ和Reg-log|f|在右半平面的边界恒为零．     

边界过原点的任意半平面中的Hilbert边值问题

给出了边界过原点的任意半平面中的Hilbert边值问题的提法,定义了函数的一种对称扩张,并利用这种对称扩张将此Hilbert边值问题转化为无穷直线上的Riemann边值问题,得到了该问题的一般解和可解性定理.     

关于L-fuzzy拓扑分离性的弱同胚不变性质

本文研究了一种新的弱同胚不变性质，它更具有一般性；证明了目前文献中所论述的各种分离性都是弱同胚不变性质，并且否定回答了王国俊提出的两个分离性问题。     

全局线性化及同胚函数的■连续性

本文证明了微分方程全局线性化中的同胚函数及其逆同胚函数都具有连续性，同时给出了具体的估计式．     

The Heegaard Genus of Amalgamated 3-Manifolds

Let M and M′ be simple 3-manifolds, each with connected boundary of genus at least two. Suppose that Mand M′ are glued via a homeomorphism between their boundaries. Then we show that, provided the gluing homeomorphism is ‘sufficiently complicated’, the Heegaard genus of the amalgamated manifold is completely determined by the Heegaard genus of Mand M′ and the genus of their common boundary. Here, a homeomorphism is ‘sufficiently complicated’ if it is the composition of a homeomorphism from the boundary ofM to some surface S, followed by a sufficiently high power of a pseudo-Anosov onS, followed by a homeomorphism to the boundary of M′. The proof uses the hyperbolic geometry of the amalgamated manifold, generalised Heegaard splittings and minimal surfaces.     

Mappings by the solutions of second-order elliptic equations

The properties of mappings by the solutions of second-order elliptic partial differential equations in the plane are studied. We obtain conditions on a function, continuous on the unit circle, that are sufficient for the solution of the Dirichlet problem in the open unit disk for the given equation with the given boundary function to be a homeomorphism between the open unit disk and a Jordan simply connected domain. The properties of the zeros of the solutions of the given equations are also studied. In particular, an analog of the main theorem of algebra is proved for polynomial solutions.     

Triple positive solutions to third order three-point BVP with increasing homeomorphism and positive homomorphism

In this paper, we investigate the existence of triple positive solutions for nonlinear differential equations boundary value problems with increasing homeomorphism and positive homomorphism operator. By using fixed point theorems in cones, we establish results on the existence of three positive solutions with suitable growth conditions imposed on the nonlinear term. As applications, two examples are given to demonstrate our result. The conclusions in this paper essentially extend and improve the known results.     

Circle decompositions of surfaces

We determine which connected surfaces can be partitioned into topological circles. There are exactly seven such surfaces up to homeomorphism: those of finite type, of Euler characteristic zero, and with compact boundary components. As a byproduct, we get that any circle decomposition of a surface is upper semicontinuous.     

A topological version of the Poincaré-Birkhoff theorem with two fixed points

The main result of this paper gives a topological property satisfied by any homeomorphism of the annulus \mathbb A = \mathbb S¹ ×[-1, 1]{\mathbb {A} = \mathbb {S}^1 \times [-1, 1]} isotopic to the identity and with at most one fixed point. This generalizes the classical Poincaré-Birkhoff theorem because this property certainly does not hold for an area preserving homeomorphism h of \mathbb A{\mathbb {A}} with the usual boundary twist condition. We also have two corollaries of this result. The first one shows in particular that the boundary twist assumption may be weakened by demanding that the homeomorphism h has a lift H to the strip [(\mathbbA)\tilde] = \mathbbR ×[-1, 1]{\tilde{\mathbb{A}} = \mathbb{R} \times [-1, 1]} possessing both a forward orbit unbounded on the right and a forward orbit unbounded on the left. As a second corollary we get a new proof of a version of the Conley–Zehnder theorem in \mathbb A{\mathbb {A}} : if a homeomorphism of \mathbb A{\mathbb {A}} isotopic to the identity preserves the area and has mean rotation zero, then it possesses two fixed points.     

O-minimal Hauptvermutung for polyhedra I

Milnor discovered two compact polyhedra which are homeomorphic but not PL homeomorphic (a counterexample to the Hauptvermutung). He constructed the homeomorphism by a finite procedure repeated infinitely often. Informally, we call a procedure constructive if it consists of an explicit procedure that is repeated only finitely many times. In this sense, Milnor did not give a constructive procedure to define the homeomorphism between the two polyhedra. In the case where the homeomorphism is semialgebraic, the author and Yokoi proved that the polyhedra in R ⁿ are PL homeomorphic. In that article, the required PL homeomorphism was not constructively defined from the given homeomorphism. In the present paper we obtain the PL homeomorphism by a constructive procedure starting from the homeomorphism. We prove in fact that for any ordered field R equipped with any o-minimal structure, two definably homeomorphic compact polyhedra in R ⁿ are PL homeomorphic (the o-minimal Hauptvermutung theorem 1.1). Together with the fact that any compact definable set is definably homeomorphic to a compact polyhedron we can say that o-minimal topology is “tame”.     

Limit Behaviour of Solutions to a Class of Equivalued Surface Boundary Value Problems for Parabolic Equations

In this paper, we discuss the limit behaviour of solutions for a class of equivalued surface boundary value problems for parabolic equations. When the equivalued surface boundary \overline{\Gamma}^\varepsilon_1 shrinks to a fixed point on boundary \Gamma_1, only homogeneous Neumann boundary conditions or Neumann boundary conditions with Dirac function appear on \Gamma_1.