切测度的几何分析

本文研究了Radon测度μ的切测度,给出了切测度的Rectifiability.利用对偶性原理及Blow-up 技巧证明了连续函数的微切集的存在性.另外,本文给出了切测度的平坦性特征.作为切测度的应用, 证明了Marstrand定理.     

类切饼集的测度维数

本文考虑了满足一定条件的类切饼集,主要利用密度定理,获得其Hausdorff测度和填充测度是等价的.同时,给出这类类切饼集维数和测度的统计解释.     

连续函数微切集的存在性定理

该文研究了在Hausdorff 度量及分布意义下连续函数之微切集的存在性问题, 证明了连续的典型函数具有丰富的(万有)微切集结构. 这一结果推广了Z. Buczolich^[3]的相关结论.     

Banach空间上集值测度与模糊数测度的扩张

在Banach空间上,给出集值测度的扩张定理并借助集测度的扩张给出了模糊数测度的扩张定理。     

Cookie-Cutter集上的Gibbs测度

Cookie-Cutter集不仅是动力系统中重要的研究对象,而且是分形中一类重要的集合.而支撑在其上的Gibbs测度对计算分形维数和热力学机制的熵起关键性作用.本文借助于定理2.1构造了支撑在其上的Gibbs测度,并用遍历性证明了该测度的唯一性.     

类切饼集上测度的局部维数存在的充要条件

本文研究了类切饼集上有限Borel正则测度的局部维数性质.利用中值定理和有界畸变原理,获得了一类类切饼集上测度的局部维数存在的一个充要条件,所获结果包含了已有结果.     

集值测度的延拓定理

讨论了集值测度的若干基本性质，在此基础上，研究了集值测度凸性，紧性及其靠拢定理，特别是到紧凸值集值测度的延拓定理。     

扩散 Cantor 集的最小Hausdorff 测度

In this paper, we construct a scattered Cantor set having the value 1/2 of log2/log3- dimensional Hausdorff measure. Combining a theorem of Lee and Baek, we can see the value 21 is the minimal Hausdorff measure of the scattered Cantor sets, and our result solves a conjecture of Lee and Baek.     

数值函数于集值测度的积分的极限定理

本文给出了实值函数关于集值测度的积分序列的极限定理。     

集值测度的收敛性及其与Castaing向量表示中测度的收敛性之间的关系

本文研究了集值测度的收敛性. 利用极限理论的方法,在X有RNP,且M(∪)s-lim inf_(n→∞) M_n条件下,获得了集值测度的收敛性与其 Castaing表示中向量测度收敛性的关系,建立了集值测度列与集值测度选择列的收敛性,从而对进一步研究集值测度的一些件质提供了理论依据.     

QUALITATIVE METHOD IN ANALYSIS OF A P-L-L WITH TANGENT CHARACTERISTIC AND FREQUENCY MODULATION INPUT

In analysis of p-L-L with tangent characteristic and frequency modulation input, we have obtained the following two types of the phase looked loop equation. \[\begin{array}{l} \frac{{{\partial ^2}\varphi }}{{\partial {t^2}}} + \alpha \frac{{d\varphi }}{{dt}} + \gamma \tan \varphi = {\beta _1} + {\beta _2}(\cos {\Omega _M}t + {\Omega _M}\sin {\Omega _M}t){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (I)\\frac{{{\partial ^2}\varphi }}{{\partial {t^2}}} + (\alpha + \eta {\sec ^2}\varphi )\frac{{d\varphi }}{{dt}} + \gamma \tan \varphi = {\beta _1} + {\beta _2}(\cos {\Omega _M}t - {\Omega _M}\sin {\Omega _M}t){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (II) \(\alpha > 0,\gamma > 0,\eta > 0,{\beta _1} > 0,{\beta _2} > 0,{\Omega _M} > 0) \end{array}\] In this paper, our aim is to explain the usual qualitative method and Lyapunov's function method, by which the existence of a periodic solution of (I), (II) is established. In addition, we especially point out: How is to construct the Lyapunovas function for the nonlinear and nonairtoiiomous system? This is a very important problem.     

几何代数在定理证明中的消元与化简算法

在符号计算中,最困难的一个地方是中间计算过程的表达式快速膨胀.基于不变量代数的符号几何计算为解决这个困难提供了可能.比如,利用零几何代数证明欧氏几何定理时,就可以给出很短的证明,甚至是单项式证明.中间的证明过程里有很多地方涉及到消元,展开,化简等问题.从程序实现的角度出发,在充分利用零几何代数计算特点的基础上,给出用于机器证明的消元、化简算法.     

THE JUMPING PHENOMENON OF THE DIMENSIONS OF COHOMOLOGY GROUPS OF TANGENT SHEAF

Let X be a compact complex manifold.Consider a small deformation π :X→B of X,the dimensions of the cohomology groups of tangent sheaf H q(X_t,T_X_t) may vary under this deformation.This article studies such phenomena by studying the obstructions to deform a class in H q(X,T X) with parameter t and gets a formula for the obstructions.     

ON THE PLANE SECTIONS OF THE TANGENT SURFACE OF A SPACE CURVE

<正> 1.Introduction.Let the tangent surface of an analytic curve Cin ordinary space be cut by a plane through a tangent at an ordinarypoint O of C,then the plane section thus obtained has a point of in-flexion at O,provided that the plane is not osculating to C at O.Thisis a well-known fact.Prof.Bompiani has enriched the projectivedifferential geometry of a plane curve at the neighbourhood of a point ofinflexion by introducing various covariant osculants.Prof.Su hasgiven some remarkable properties of the loci of the cusp-tangent t ofthe seven-point cusped cubic of the plane section of the tangent surfacemade by a variable plane n through a tangent of C at O.Prof.Lanehas investigated the loci of the osculants and various points and linesintroduced by Prof.Bompiani.The object of this note is to obtain someproperties of other loci related to the same problem as considered byProf.Su.     

Wirtinger不等式的一个几何应用

本文研究了著名的Minkowski混合面积不等式.利用平面凸集的支持函数的性质和分析中著名的Wirtinger不等式,得到了Minkowski混合面积不等式的一个简化证明.     

THE STRUCTURE OF THE TANGENT CONE: AN INTERESTING BIJECTION#

ABSTRACT

Let X = Spec(R) be a reduced equidimensional algebraic variety over an algebraically closed field k. Let Y = Spec(R/𝔮) be a codimension one ordinary multiple subvariety, where 𝔮 is a prime ideal of height 1 of R. If U is a nonempty open subset of Y and 𝔪 a closed point of U, we denote by A ? R _𝔪 its local ring in X, by 𝔭 the extension of 𝔮 in A, and by K the algebraic closure of the residue field k(𝔭).

Then there exists a bijection γ^𝔪:Proj(G _𝔭(A) ?  _A/𝔭 k) → Proj(G(A _𝔭) ?  _k(𝔭)K) such that for every subset Σ of Proj(G _𝔭(A) ?  _A/𝔭 k), the Hilbert function of Σ coincides with the Hilbert function of γ^𝔪(Σ). We examine some applications. We study the structure of the tangent cone at a closed point of a codimension one ordinary multiple subvariety.     

EXISTENCE OF AREA MINIMIZING TANGENT CONES OF INTEGRAL CURRENTS WITH PRESCRIBED MEAN CURVATURE

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GEOMETRIC QUANTIZATION OF THE THREE-DIMENSIONAL HARMONIC OSCILLATOR

ABSTRACT

In this article we apply the geometric quantization method of Rund to the case of the three-dimensional harmonic oscillator and show that the eigenvalue spectrum thus obtained coincides precisely with the energy eigenvalue spectrum that is prescribed by standard quantum mechanics.     

多目标规划的ακ-较多有效解的几何特性

In this paper, with the help of properties of P-linear below bounded points, tangent cones, cone k-interior points, projective cones, the geometric char-acterizations of ακ-major efficient solutions and ακ-major optimality solutions of multiobjective programming are obtained.