拓扑学

Other semirings arise naturally in such diverse areas of mathematics as combinatorics, functional analysis, topology, graph theory, Euclidean geometry, probability theory, commutative and noncommutative ring theory, optimization theory, discrete event dynamical systems, automata theory, formal language theory and the mathematical modeling of quantum physics and parallel computation systems.

在许多不同的数学领域存在着其它形式的半环，比如组合学，泛函分析，拓扑学，图论，欧式几何，概率论，交换和非交换环，优化论，离散动力系统，自动化理论，形式语言理论，量子物理的数据模型和并行计算系统。

Graph theory and topology was researched to establish graphic interpreter, boolean multiplication ⊙ of a matrix is inferred and defined, and simple sub-graphs are disconnected from complex graph by the multiplication ⊙; parameterization of graph is realized through a sequence of sub-graph entities linked by constraints.

研究了图形翻译器的图论和拓扑学理论基础，推出并定义了矩阵布尔乘运算〓，使用矩阵运算〓将复杂的图形拆分成一系列简单子图；以约束关联的子图实体序列实现图形的参数化。

According to the viewpoint of combination tope, any convex polyhedron can be composed of some simplexes, and both convex polyhedron and simplex are important concepts of programming theory, so it's necessary to find their volumes in some calculations.

由组合拓扑学可知，任意凸多面体均为若干个单纯形的某种组合，两者均为规划论中最重要的基本概念之一，因而在一些计算中对其求积是不可少的。

After the founding of the new China, as social stability and national scientific establishment improved, the diffusion entered a new phrase, with Poincare's groundbreaking works in qualitative theory of differential equations and topology being well spread and developed.

建国后，随着社会的稳定和国家科技体制的逐渐完备，庞加莱学说在中国的传播进入新的阶段，他在微分方程定性理论、拓扑学等领域的开拓性工作都得到了很好的传播和发展。

Topology is traditionally decomposed into three parts: General topology, Algebraic topology and Differential topology.

习惯上拓扑学被分成点集拓扑、代数拓扑和微分拓扑三部分。

In mathematics, Chinese scholars have achieved important results in fields such as number theory, theorem-proving by computer, differential geometry, topology, complex analysis, probability and mathematical statistics, PDE, functional analysis, numerical analysis and control theory and so on.

在数学方面，据我所知，中国数学家在诸如数论、定理机器证明、微分几何、拓扑学、复分析、概率论与数理统计、偏微分方程、泛函分析、数值分析和控制论等领域里也都取得了具有国际水平的成果。

This paper investigates topological properties of the general gradient system, and demonstrates by results of differential topology that the boundedness of the general gradient system is equivalent to the boundedness of the potential energy boundary surface. Moreover, a general gradient system is bounded if, and only if the system has both sink point and source point.

该文研究了广义梯度系统的拓扑性质，应用微分拓扑学原理证明，广义梯度系统的完全稳定性与势能界面的有界性是等价的；广义梯度系统完全稳定当且仅当该系统既有渊点又有源点。

The curriculum comprises the fundamental concepts of point set topology, differential manifold and differential dynamics, followed by the conceptions and theoretical methods of abrupt change, bifurcation, chaos and fractal. The analysis and compare combined with the practice in power system make it easy for readers to learn the basic methods of nonlinear dynamics and write high quality papers.

本课程主要内容包括点集拓扑学、微分流形和微分动力系统的基本概念，后面部分主要涉及到突变与分叉、混沌、分形和分维概念和理论方法，并结合电力系统实际例子加以分析与比较，使读者很快掌握非线性动力学的基本方法，能够写出高质量的文章和学术论文。

Subsequent chapters develop measure theory, point-set topology, Fourier series for the Lebesgue integral, and the basics of Banach and Hilbert spaces

制定措施以后各章的理论，点集拓扑学，傅立叶的勒贝格积分系列，与Banach和Hilbert空间的基本知识

Our studies mainly focus on the free groups with finite ranks . Free groups with finite ranks are the fundamental subjects of combinatorial group theory, and they have central importance for the studies of both topology and algebra.

我们研究比较多的是有限秩的自由群，有限秩自由群是组合群论的中心课题，是一个在拓扑学及代数学上都具有重要意义的研究对象，获取有关这些自由群的结构的信息的一个主要方法是研究其上的各种自同构作用。

On the other hand, the more important thing is because the urban housing is a kind of heterogeneity products.

另一方面，更重要的是由于城市住房是一种异质性产品。

Climate histogram is the fall that collects place measure calm value, cent serves as cross axle for a few equal interval, the area that the frequency that the value appears according to place is accumulated and becomes will be determined inside each interval, discharge the graph that rise with post, also be called histogram.

气候直方图是将所收集的降水量测定值，分为几个相等的区间作为横轴，并将各区间内所测定值依所出现的次数累积而成的面积，用柱子排起来的图形，也叫做柱状图。