递归神经网络
课程名称（英文）：Recurrent Neural Networks
授课内容：神经网络是受人脑功能的启发而发展起来的，并试图去模拟某些生物系统。神经网络是由大量的元素，即神经元相互联接而成。神经元的输入由神经元输出的适当加权及偏置项组成。神经元是由适当的函数，也称激活函数来描述。探索和揭示“万物之灵”的人类的高度发达大脑的奥秘是当代科学面临的重大的研究课题之一。 借鉴生物脑神经的研究而发展起来的非生物信息处理方式——人工神经网络是目前最活跃的信息处理科学研究领域之一。人工神经网络通常分为两类；一类是前馈神经网络（即没有反馈的神经网络）；另一类是递归神经网络（即有反馈的神经网络）。由于递归神经网络引入反馈，所以它是一个非线性动力学系统。递归神经网络的动力学行为可理解为状态的变换或迁移过程，可用常微分方程、泛函微分方程、偏微分方程和微分流形等研究。递归神经网络的信息处理功能体现在其动力特征中，包括平衡态、周期过程、振动性、吸引子和混沌等。本课程讲授的内容：
一、绪论：递归神经网络的基本概念。
二、递归神经网络理论基础。介绍一些递归神经网络模型。
三、模拟Hopfield神经网络的定性分析。
四、参数摄动的定性影响。
五、时间延迟的定性影响。
六、联想记忆的的一些综合方法。
七、互连约束的影响。
八、大规模神经网络理论和方法。
九、递归神经网络微分流形方法。
应用数学__________
代 码：________070104___________
研究生课程简介
一、 硕士学位基础课
非线性泛函分析
开课院系：数学系
课程编号：100K0001
课程英文名称：nonlinear functional analysis
授课内容：本课主要讲授无穷维赋范线性空间中非线性算子的连续性、有界性、全连续性等属性；非线性算子的Frechet微分与Gateaux微分等概念; 拓扑度理论及其应用；变分原理等。
本课程课内周学时为4学时，3学分。
Introduce to the course contents:
1. nonlinear operators in
infinity dimensional normed linear spaces
2. Frechet-derivative and Gateaux-derivative of nonlinear
operators.
3. topology degree theory
and its applications
4. Morse theory and its
applications
Class hours: 4 per week;
Credits: 3
课程编号：100K0005
课程名称（中文）：一般拓扑学
课程名称（英文）：General Topology
授课内容：本课程讲授一般拓扑学中的基本概念、基本理论和基本方法。通过本课程的学习，使学生以一般的观点总结和提高在本科阶段学过的有关概念、理论和方法。同时，为数学学科各研究方向的学生进一步学习拓扑、几何、泛函和微分方程等课程提供基础理论支持。
学习本课程，要求学生理解拓扑空间的基本概念；掌握常见的拓扑空间类(如紧空间、连通空间、正规空间、仿紧空间等)及若干拓扑不变性；学会熟练地构造拓扑的多种方法；掌握一些重要定理，如Urysohn引理、Tychonoff定理等；理解和掌握拓扑学中重要的思想方法和证明技巧。通过本课程的学习，培养和提高学生的抽象思维和逻辑推理能力。
本课程课内周学时为4学时，3学分。
Introduce to the course contents:
This course is to introduce the
students the fundamental concepts, theories and methods of general topology,
and make the students summarize and improve their known contents in their
undergraduate study from a general viewpoint. Furthermore, it offers theoretic
support to further studying of Topology, Geometry, functional analysis, the
theory of complex functions, differential equation, etc.
By this course, the students
should understand the concept of topological space, grasp general topological
spaces, such as compact space, connected space, normal space and paracompact
space etc and know the approaches to construct topological space. Furthermore,
the students should have the basic ideas and basic ratiocination skills for the
important theorems such as Urysohn lemma and Tychonoff Theorem etc.
The contents of this course can
be summarized as follows:
1. Topological spaces and its
examples
2. The methods constructing
topologies on a universe
3. Some important types of
topological spaces, such as compact spaces, connected spaces, normal spaces,
paracompact space, etc.
4. countability axioms
5. Some of important theorems including
Urysohn lemma and Tychonoff theorem.
Class hours: 4 per week;
Credits: 3
课程编号：100K0032
课程名称（中文）：常微分方程几何理论
课程名称（英文）：Geometric
Theory of Ordinary Differential Equations
授课内容：常微分方程几何理论是现代动力系统理论的基础。通过本课程的教学使学生 掌握如下知识：Liapunov稳定性、Lagrange原理、向量场的Poincare指数、平面向量场的指数及其基本性质、Bendixson的指数公式、Poincare-Birkhoff扭转定理、Poincare的最后几何定理、Poincare-Birkhoff扭转定理的改进、极限点集、P-式回复运动、环面上的P-式回复运动、准极小集、 B-式回复运动、回复时间的间隔、极小集、极小集的存在性、概周期运动、概周期运动的Liapunov稳定性、Poincare-Bendixson的平面环域定理、Poincare-Bendixson定理的推广、Seifert猜测、遗留问题、拟周期运动，了解混沌的基本知识、Peano现象对混沌的影响、混沌流的定义、混沌的必要条件、 混沌集的几何结构、混沌与分形、混沌流的一个存在定理、没有几何病态的混沌集．从而使学生为学习相关课程打下基础。
本课程课内周学时为4学时，3学分。
Introduce to the course contents:
Geometric Theory of Ordinary
Differential Equations is a base of modern dynamical systems. This course
intent to let students acknowledge following contents:
Liapunov stability, Lagrange
principle, Poincare index of vector fields, Index of planary vector fields ,
Bendixion formula for index, Poincare-Birkhoff
twist theorem, Poincare,s last geometric theorem, improvement of Poincare-Birkhoff
twist theorem, limit set, P-type
recurrent motion, quisi-minimal set,
B-type recurrent motion, the interval of recurrent, minimal set and its
existence, almost periodic motion, Liapunov stability of almost periodic
motion, Poincare-Bendixson theorem, Seifert’s conjiecture, some remained open
problems, quisi-periodic motion, elements of Chaos, necessary condition for
chaos, geometric structure of chaos set, chaos and fractals, etc.
Class hours: 4 per week; Credits: 3
课程编号：100K0003
课程名称（中文）：概率论基础
课程名称（英文）：Foundations of Probability Theory
授课内容：概率论是关于随机现象分析的数学分支，其主要内容是对随机变量进行数学分析，即研究那些没有确定规律但却存在统计规律的问题。概率论已被广泛应用于统计、计算机科学、工程等领域，从中得出一些复杂系统潜在事件发生可能性的有关结论。作为统计学的基础，概率论知识对大数据集的量化分析也是十分必要的。
本门课程是建立在测度论基础之上，其讲授对象是研究生。它将为那些进一步学习和研究各种可能性的学生展现较为宽广的视野。为达到此目标，要求学生必须吃透概念，掌握方法，并持之以恒地深入钻研，获取收益。本门课程包括如下内容：
第一章是对初等实变函数的复习。
第二章给出了必需的“测度论和积分”的纲要，集类，概率测度和分布函数也将在这一章给予介绍。
第三章纯粹是介绍概率论的语言和结构，但我们将其内容严格限制在随机变量、数学期望和独立性的范围之内，这是一些关键且在此阶段可行的内容。
第四章可以认为是适用于概率的实函数论之概念和技巧的综合。其中包括各种形式的收敛，条件期望和条件独立性的特性。
第五章介绍大数定理，主要包括简单的极限定理、弱大数定理、级数的收敛、强大数定理。
第六章发展一些主要的分析工具，即付立叶变换。主要介绍特征函数的一般性质、卷积、唯一性和逆转定理及收敛定理。
第七章着手介绍古典概率论的所谓“中心问题”。主要包括林德贝格－费勒定理及中心极限定理的分歧。
本课程课内周学时为4学时，3学分。
Foundations of probability theory
Probability
theory is
the branch of mathematics concerned with analysis of random phenomena. Its subject matter is the mathematical analysis of
random variables, that is, of those empirical phenomena which do not have
deterministic regularity but possess some statistical regularity. Now it is used extensively in
areas such as statistics, computer science and engineering … to draw
conclusions about the likelihood of potential events and the underlying
mechanics of complex systems. As a mathematical foundation for statistics,
probability theory is essential to many quantitative analyses of large sets of
data.
The probability theory is founded on the measure
theory. The course is
designed for graduate students, which will
offer a broad perspective of the open field and prepare the student for various
further study and research of possibilities. To this aim, it needs the students
must acquire knowledge of ideas and practice in methods, and dwell with them
long and deeply enough to reap the benefits. A brief description will now be
given of the course.
A review of elementary real variables is given in chapter
1 and a synopsis of the requisite "measure and integration" is given
in Chapter 2, namely Measure theory, The classes of sets, probability measures
and their distribution functions will be introduced. Chapter 3 is the true
introduction to the language and framework of probability theory, but its
content have been restricted to random variable, expectation and independence.
They are crucial and feasible at this stage. Chapter 4 can be regarded as an
assembly of notions and techniques of real function theory adapted to the usage
of probability. Various modes of convergence will be introduced in this
chapter. The basic properties of conditional expectation and conditional
independence are also discussed. Law of large numbers and random series
theorems are given in Chapter 5. They are simple limit theorems, weak law of
large numbers, convergence of series, strong law of large numbers and
applications. Chapter 6 develops some of the chief analytical weapons, namely
Fourier transform. It includes the general properties of characteristic
function, convolutions, uniqueness, inversion, convergence theorems. Chapter 7
initiates what has been called the "central problem" of classical
probability theory. The important theorems are Lindeberg-Feller theorem and ramifications
of the central limit theorem.
Class hours: 4 per week;
Credits: 3
课程编号：100K0033
课程名称（中文）：微分流形
课程名称（英文）：Differential
manifold
授课内容：随着科学技术的飞速发展，流形、 Stokes定理、 张量、 外微分形式等较深的知识不仅成为数学本身重要的和活跃的研究领域，而且其他学科领域中，如力学及物理学（特别是Einstein的广义相对论及规范场论）中，已获得越来越广泛、 深刻而富有成效的应用。微分流形是三维欧氏空间的光滑曲线（一维微分流形）和光滑曲面（二维微分流形）在高维空间甚至一般拓扑空间中的自然推广。著名的格林公式、高斯公式、斯托克斯公式等在高维流形上，利用外微分，统一为一种形式。微分流形课程的主要内容包括：微分流形、微分形式和外微分、流形上的积分、Stokes定理 Riemann流形等。
本课程课内周学时为4学时，3学分
Introduce to the course contents:
Course content: With the advanced
development of scientific technology, some deep knowledge such as manifold, Stokes
theory, tensor, exterior differential forms etc. not only have been important
and active research areas of mathematics itself, but also have obtained more
and more wide, deep and successful application in other subject areas such as
dynamics and physics (especially Einstein’s general theory of relativity and
gauge field theory). Differential manifold is the nature generalization of the
smooth curve (1-dimensional differential manifold) in 3-Euclidean space and the
smooth surface (2-dimensional differential manifold) in 3-Euclidean space. On
high-dimensional manifold, the famous Green formula, Gauss formula, Stokes
formula etc. are unified to be one form by exterior differential. The main
content of this course is as follows: differential manifold, differential
forms, exterior differential, integral on manifold, Stokes theory, Riemann
manifold etc..
Class hours: 4 per week;
Credits: 3
课程编号：100K0004
课程名称（中文）：近世代数
课程名称（英文）：Modern Algebra
授课内容：近世代数是为数学专业硕士生一年级开设的一门基础课，它主要讲述现代代数学的一些基本概念和理论，目的是让学生了解现代代数学的发展状况，掌握一些基本思想和方法，进一步培养逻辑思维和抽象思维能力，为以后的学习打下基础。这门课的主要内容包括以下五方面：
一、
群论。包括群的有关概念与性质；循环群；正规子群与商群；同态定理；群的直和与分解等。
二、
环论。包括环的有关概念与性质；理想与商环；环同态定理等。
三、
模论。包括模的有关概念与性质；自由模；投射模等。
四、
范畴论。包括范畴的有关概念；自然变换；可表函子与伴随函子；Abel范畴等。
五、
同调论。包括复形与同调模；长正合列；导出函子等。
授课教师可根据课时和学生的实际情况，适当调节所列知识的顺序和内容。
本课程课内周学时为4学时，3学分。
Introduce to the course contents:
Modern algebra is a one-semester course intended for the
first year graduates majoring in mathematics. The main content of the course
includes some basic concepts and theories in modern algebra. By studying this
course students will know the developments of modern algebra, learn its basic
methods and ideas,raise the abilities in logistic thinking and abstract
thinking, and build the foundation for further study.
The content of the course should cover the following chapters:
Chapter 1 Groups.
It includes the related definitions and properties of groups,
cyclic groups, normal subgroups, quotient groups, homomorphism theorems, direct
sums, and decompositions, etc.
Chapter 2 Rings.
It includes the related definitions and properties of rings,
ideas, quotient rings, and homomorphism theorems, etc.
Chapter 3 Modules.
It includes the related definitions and properties of
modules, free modules, and projective modules, etc.
Class hours: 4 per week; Credits: 3
二、硕士学位专业课
课程编号：100K0044
课程名称（中文）：泛函微分方程理论
课程名称（英文）：Theory
of Functional Differential Equations
授课内容：在自然和社会现象中，许多系统的发展趋势或未来状态不仅与现状有关，而且或多或少与过去的发展趋势有关，这类现象称为滞后现象，或遗传效应。从工程技术、物理、力学、控制论、化学反应、生物医学、遗传问题、流行病学、动物与植物的循环系统、社会科学中的各种经济现象中提出的数学模型带有明显的滞后量。例如，医学中的传染病的潜伏期，弹性力学中的滞后效应，特别是自动控制中任何一个含有反馈的系统一般总有时滞，这类时滞的出现是因为需要有限的时间接受信息，做出反应。这类时滞系统无法用微分方程来描述，需要用微分差分方程——一类泛函微分方程来描述。泛函微分方程早在1750年欧拉(Euler)的几何问题中已出现。但系统研究则从20世纪才开始，特别近20年来在极其广泛的应用课题推动下获得了实质性的、全面的进展。泛函微分方程主要包括如下类型的方程：滞后型、中立型、混合型、复杂偏差型。泛函微分方程研究的课题主要有：适定性、稳定性与有界性、周期解、振动性与渐近性、概周期解、吸引子等。
本课程课内周学时为4学时，3学分。
Course number: 100M4201
Course name: Theory of Functional Differential
Equations
Course content: In natural and social
phenomena, the development trends or future state of many systems have
something to do not only with present condition, but also with the past
development trends more or less. This phenomenon is called delay or inheritance.
The mathematical models brought forward from engineering, physics, mechanics,
cybernetics, chemical reaction, biomedicine, heredity, epidemiology,
circular systems of animal and plant and kinds of economic phenomena in social
science have obvious delay arguments. For example, the incubation of infectious
disease in the medical science, delay effect in elasticity, especially
any system with feedback in automatic control always possesses time-delay. The
appearance of this delay is because it needs limited time to accept information
and make reaction. This kind of system can not be described by differential equations (DE) but by differential difference equations
(DDE)——a kind of
functional differential equations (FDE). FDE appeared in Euler geometry
problems as early as 1750. But the system research didn’t begin until 20
century. Especially in recent 20 years, it gains material, all-around progress
through the promotion of most broad application tasks. FDE mainly include the following equations:
Delay type, neutral type, mixed
type, complex deviating argument type. The subjects studied in FDE mainly
contains: well posedness, stability and boundary, periodic solutions,
oscillations and asymptotic, almost periodic solutions, attractors ect..
Class
hours: 4 per week; Credits: 3
课程编号：100K0015
课程名称(中文)：模糊数学基础
课程名称(英文)：A Foundation of
Fuzzy Mathematics
授课内容： 1965年美国控制论专家L.A.Zadeh发表第一篇模糊集文章以来，标志模糊数学和模糊系统的诞生，几十年来，在广大科学工作者的努力之下，模糊数学的理论和应用发展迅速，成为一门备受关注的新兴的热点学科。
模糊数学基础课着眼于学科的基础理论，目的是比较系统地介绍理论研究和应用研究的基础知识，为进一步学习打下牢固的基础，属于应用数学硕士研究生学位基础课。
它的内容主要包括：模糊集的基本概念，集合套、模糊化理论，一致性问题，各种形式的分解定理、表现定理、扩张原理，模糊关系、模糊变换、模糊数等理论，此外包括L型模糊集，模糊系统的基本理论及模糊测度与积分等。
学习该课程需要有一定的数学基础，特别要求有较扎实的近世代数、集合论、函数论方面的基本功。
本课程课内周学时为4学时，3学分。
A foundation of fuzzy mathematics:
Fuzzy mathematics is originated
in 1965 when L.A.Zadeh published the
first article about fuzzy mathemaics. After that it experiences rapid development. Now, it has proven its
usefulness in both theoretical and applied application.
This course aims at introducing
students the fundamental theory on fuzzy mathematics and the necessary math basis for further study and research.
Its contents are summarized as
follows:
1. Triangular norm
2. residuated lattice
3. A primer on fuzzy sets.
4. Decomposition theorems
5. Representation theorems
6. Lifting principle
7. Fuzzy relation
Note that this course needs some
preliminary courses about Abstract algebra, Set theory and theory of fuctions.
Class hour: 4 per week; Credit:
3
课程编号：100K0006
课程名称(中文)：L-拓扑学导引
课程名称(英文)：Introduction to L-Topology
授课内容： Fuzzy拓扑学(即模糊拓扑学)的历史从1968年C.L. Chang提出Fuzzy拓扑空间概念的第一篇论文算起，至今仅有三十多年的历史。层次结构的特点是它不同于一般拓扑学(General Topology)的特有风格，与完备格代数结构的紧密联系又赋予了它新的创造力。本课程主要介绍以刘应明院士创立的重域以及后来由王国俊教授创立且更具一般性的远域为工具建立起来的Fuzzy拓扑学理论。主要包括Fuzzy拓扑空间的基本理论、诱导空间理论、连通性、可数性、分离性和紧性理论等。本课程仅要求学生具有一般拓扑学的初步知识，但希望学生有较好的数学修养。
本课程课内周学时4学时，3学分。
Introduce to the course
contents:
A fresh start of L-topology was
made in 1968, when a concept of L-topological space was proposed by C. L. Chang
in his paper. The characteristic of stratified structure made it very different
from general topology. It is its close relation to algebraic structure of
complete lattices that L-topology gains rapid developments.
This couse is to introduce the Fuzzy
topology theory, which is based on Liu Yingming’s coincidencent
neighborhood or more general Wang Guojun’s remote neighborhood concept.
The contents of this course are
as follows:
1. Fundamental concepts
2. Induced L-topological spaces
3. Connected L-topological
spaces
4. Countability of L-topological
spaces
5. Separation axioms in
L-topology
6. Different compactness in
L-topology
The requisites are the general
topology and lattice theory, and good mathematical understanding and
ratiocination ability.
Class hour: 4 per week; Credit:
3
课程编号： 100K0035
课程名称(中文)：多元统计分析
课程英文名称： Multivariate Statistical Analysis
授课内容：多元统计分析是数理统计学中内容十分丰富的一个分支，它研究各种多元统计方法的数学机理。多元统计方法处理多元数据——包含多个指标的观测数据。现在，各个领域的研究者们面临的数据几乎都是多元数据，并且，常常是包含几十、几百个变量、成千上万个观测的超大容量数据。如何有效的简化数据结构、如何快速地从纷繁复杂的数据集中提取有用信息，正是多元统计方法的主要研究内容。最近几十年来，随着计算机技术的进步，多元统计方法在自然科学和社会科学的许多分支学科中得到了广泛应用，发展十分迅速。
多元统计方法是各种多元数据分析处理统计方法的集成。其中有些经典方法的理论和应用已比较完善，也有些方法至今还在发展和完善之中，并且不断有一些新的方法出现。按其科学研究的目标分，多元统计方法可大致如下归类：
1、
数据结构简化方法：这类方法的主要目标是以少数变量代表原有的多个变量，要求有用信息的损失尽量小，并且能有更好的现实解释。这类方法包括主成分分析、因子分析、典型相关分析等等。
2、
分类与判别方法：这类方法的目标是根据数据对观测对象或变量进行分组或分类判别，包括聚类分析方法和判别分析方法，每种方法之下又有不同的方法，其统计处理思想各不相同。这两类方法也是两种重要的模式识别技术。
3、
变量间的依赖关系研究方法：变量间的依赖关系是人们最感兴趣的一方面，一个实际的推动力是各种预测问题。这类方法包括多元回归分析技术、多重回归、偏最小二成法等等。
4、
假设的构造与检验：假设检验是统计学的基本问题，相对于一元统计，多元的假设检验要复杂的多。多元方差分析是这种检验最有代表性的一个。
本课程主要介绍上述四方面的多元统计方法及其相关理论与应用。多元正态分布的有关理论以及多元抽样分布是不少多元统计分析技术的基础，本课程在介绍各种分析方法之前先来介绍这些理论。矩阵代数和数理统计基础知识是学习多元统计分析课程的必要前提，若有必要，在一开始学习时会复习梳理这两方面的相关知识。
本课程课内周学时为4学时，3学分。
Introduce to the course contents:
Multivariate statistical analysis has
experienced pronounced growth over the last few decades. The primary objective of such multivariate
methods is to analyze multivariate data that consist of a lot of variables
measured on large numbers of experimental units. Almost all data collected by
today’s researchers can be classified as multivariate data. Multivariate
methods are extremely useful for helping researchers make sense of those large,
complicated, and complex multivariate data sets.
Multivariate analysis are often concerned
with finding relationships among (1) the response variables, (2) the
experimental units, and (3) both response variables and experimental units. Some
multivariate methods are classified as “variable-directed” techniques which are
primarily concerned with relationships that might exist among the response variables,
such as principal components analysis (PCA), factor analysis (FA), regression
analysis, and canonical correlation analysis. Some other multivariate methods
are classified as “individual-directed” techniques which are primarily
concerned with relationships that might exist among the experimental units
and/or individuals being measured, some examples of this type of technique are
discriminate analysis (DA), cluster analysis (CA), and multivariate analysis of
variance (MANOVA).
The underlying theme behind most multivariate
techniques is simplification. Many techniques tend to be exploratory in nature
rather than confirmatory. That is, these methods tend to motivate hypotheses
rather than test them. Multivariate techniques are often useful for exploring
data in attempt to learn if there is worthwhile and valuable information
contained in data.
In
this course, most of the ordinary multivariate methods will be introduced,
including their theory and applications. The theory of multivariate normal
distribution and linear statistical models is the base of many multivariate
methods. They will be introduced before the multivariate methods.
The basic knowledge of matrix algebra and
mathematical statistics is necessary for the studying of this course. We will
review the basic results of matrix algebra and mathematical statistics in the beginning
of the course.
Class
hour: 4 per week; Credit: 3
课程编号: 100K0034
课程名称（中文）：动力系统
课程名称（英文）：Dynamical
Systems
授课内容：在非线性系统中，非线性动力学研究是引人注目的问题。在非线性动力学问题中最古老的，人们最感兴趣的两类问题：一是天体力学问题，二是流体湍流问题。前者是有限维动力学问题，后者是无穷维动力学问题，所谓维数是指在给定的瞬时内描述系统结构所必须的参数，除流体湍流问题是无穷维动力系统外，近年来，在科学技术问题中涌现了大量的无穷维动力系统问题。例如：等离子物理、激光、非线性光学、燃烧、数理经济、机器人等都存在无穷维动力系统问题。因此，开展无穷维动力系统研究是时代的需要，是国际学术热点问题，具有辉煌的发展前景，本课程主要内容如下：
不变集和吸引子的一般性结果、Sobolev空间理论简介、耗散发展方程的吸引子、耗散波动方程的吸引子、吸引子的Liapunov指数与Hausdorff维数与分形维数、某些物理系统的吸引子的维数估计、不稳定流形、Liapunov泛函和维数下界估计、锥和压缩性质、惯性流形、非自伴情况下惯性流形和缓流形、吸引子和惯形流形的逼近。
本课程课内周学时为4学时，3学分。
Introduce to the course contents:
Course content: In nonlinear systems, the research
of nonlinear dynamic is an attractive problem, in which there are two oldest
and most interesting problems— celestial mechanics and turbulent flow. The
former is finite-dimensional dynamical systems, and the later is infinite-dimensional
dynamical systems. The so-called dimension is the necessary parameter in the
description of the system structure within the certain instant. In recent years, lots of infinite-dimensional dynamical systems appear in scientific technical
areas except for turbulent flow problems. For instance, there are infinite-dimensional
dynamical systems problems in plasma physics, laser, nonlinear optics, burning,
mathematical economics, robot etc. As a result, the development of infinite-dimensional dynamical systems research
is the needs of the age, as well as the hotspot of the international learning,
and it has the brilliant developing foreground. The main content of the course
is as follows:
The
universal result of invariant sets and attractors, the introduction of Sobolev
space theory, attractors in dissipative evolutionary equations, attractors in
dissipative wave equations, Liapunov index, Hausdorff dimension, fractal
dimension of attractors, the estimation of dimension for attractors in some
physical systems, instable flow; Liapunov functional and the estimation of
dimension lower bound, cone and contraction property, inertial manifold, inertial manifold and slow
manifold under non-self-adjoint cases, the approximation for attractors and
inertial manifold.
Class hour: 4 per week; Credit:
3
课程编号：100K0016
课程名称（中文）：偏微分方程现代理论
课程英文名称：Modern Theory of Partial Differential Equations
授课内容：本课程主要介绍Sobolev 空间Hs(W), 并在Hs(W)中讨论偏微分方程理论，包括椭圆、抛物及双曲方程。
1．研究Sobolev
空间Hs(W)的各种性质，诸如：完备性，可分性，自反性，光滑逼近理论，内插和延拓定理，空间嵌入、紧嵌入及迹定理等。
2.讨论椭圆、抛物及双曲方程在合适条件下解的存在性及正则性问题，以椭圆方程为主。其中将用到变分法、半群方法、Fourier变换方法、Galerkin方法、特征函数展开等。
本课程课内周学时为4学时，3学分。
Introduce to the course contents:
The course introduces the
Sobolev space Hs(W) and the partial differential equations in Hs(W), and the main course is the following:
1. introduce the space Hs(W) and its main characteristic: completeness,
reflexivity, imbedding theory etc.
2. the existence of
solutions for partial differential equations under suitable conditions, and the
regular estimate, where we will introduce variational methods, semi-group
methods, Fourier transformation, Galerkin methods etc.
Class hour: 4 per week; Credit:
3
三、硕士专业选修课
课程编号：100K0011
课程名称（中文）：代数拓扑基础
课程名称（英文）：Basic Concepts of Algebraic Topology
授课内容： 本课程介绍代数拓扑学的基本概念及理论，使学生掌握代数拓扑学的意义，方法及应用。主要内容：单纯同调理论，包括几何复形和多面体，单纯同调群及其拓扑不变性，单纯逼近，正合序列及Lefschetz不动点定理；奇异同调理论，包括奇异同调群，Mayer-Vietoris同调序列，相对同调，同调的公理；同伦理论，包括基本群，覆盖空间及其分类，高阶同伦群等。
本课程课内周学时为4学时, 3学分。
Introduce to the course
contents:
The aim of this course is to introduce students the
basic concepts and ideas in algebraic topology and make students grasp the
essence, methods and applications of algebraic topology.
The main topics are as follows:
l
Simplicial
homology: simplicial complex and polyhedron, simplicial group and its
topological invariance, simplicial approximation, exact sequence, Lefschetz
fixed-point theorem.
l
Singular
homology: singular homology group, Mayer-Vietoris sequences, relative homology
and axioms for homology, etc.
l
Cellular
homology: cellular homology group, theorem for cellular homology and its
computation.
l
Homotopy theory:
fundamental group, covering space and classification of covering spaces, higher
homotopy groups, etc.
Class hours: 4 per week;
Credits: 3
课程编号：100K0014
课程名称(中文)：格值拓扑学的现代理论
课程名称(英文)：Modern Theory of Fuzzy Topology
授课内容：学习本课程要求学生具备拓扑学(general
topology)、模糊集合论和剩余格的基本知识。受模糊数学思想的影响，格上拓扑学在多值逻辑环境下可以从多种背景下引入， 例如，Ulrich Höhle 和Alexander
Šostak 的格理论途径、Ying Mingsheng 的逻辑途径和张德学范畴途径等等。该课程主要包括下列内容：
1、Fuzzy拓扑理论背景
2、Fuzzifying
拓扑学
3、L-fuzzy
拓扑学
4、多值拓扑学
本课程课内周学时为4学时，3学分。
Introduce to the course contents:
Since the inception of fuzzy set
theory, there has been several different approaches for fuzzy topology in the
literature, such as lattice theoretic approach proposed by Ulrich Höhle and
Alexander Šostak, logic approach proposed by Ying Mingsheng, enriched category approach
proposed by Zhang Dexue, and others. This course is intended to present a
systematic exposition of these different approaches. Students are required to
be familiar with the fundamental notions of topology, fuzzy set theory and
residuated lattices.
The contents are listed as
follows:
1. A primer on fuzzy topology
2. Fuzzifying topology
3. L-fuzzy topology
4. Many valued topology
Class hour: 4 per week; Credit:
3
课程编号：100K0041
课程名称（中文）：随机微分方程
课程名称（英文）：Stochastic differential equations
授课内容：随着科学技术的飞速发展, 随机微分方程广泛应用于系统科学、工程控制、生态学、金融学等领域。随机微分方程起源于马氏过程的构造，起始于Kolmogorov的分析方法和Feller的半群方法。由于Brown运动几乎所有的轨道无处可微，因而不能用一般的方法定义积分。Ito首先给出了积分的定义，并以随机微分方程的解表达扩散过程。Ito定义的随机微积分不仅适用于扩散过程，而且适用于一类广泛的随机过程——鞅。随机微分已经成为研究随机过程的重要工具。随机微分方程主要内容包括：测度论、Markov过程和扩散过程 、Winner过程和白噪声、 随机积分和随机微分、 随机微分方程解的适定性和渐近性、 随机微分方程的应用等。
本课程课内周学时为4学时，3学分。
Introduce to the course contents:
Course content: With the advanced
development of scientific technology, the stochastic differential equations is
widely used in the areas such as systems science, engineering control, ecology,
finance etc.. The stochastic differential equations is observed in the
construction of Markov process and originated from Kolmogorov analytical method
and Feller semi-group method. Since Almost all the orbits differentiable
nowhere because of Brownian motion, the common methods of defining integral fail.
Ito first gave the definition of integral, and expressed the diffusion
processes by the solutions of stochastic differential equations. That
stochastic calculus is suitable for both diffusion processes and a class of
wide stochastic process——Martingale. The stochastic differential
equations have become an important tool of the study of stochastic process. The
main content of this course is as follows: measure theory, Markov process and
diffusion process, Winner process and white noise, stochastic integral and
stochastic differentiation, the well-posedness and asymptotic behavior of
solutions of stochastic differential equations, the application of stochastic
differential equations etc..
.
Class hour: 4 per week; Credit:
3
课程编号：100K0040
课程名称（中文）：随机过程
课程名称（英文）：Stochastic process
授课内容：随机过程是概率理论的重要内容。随机过程不只研究一个或有限个随机变量，而是研究无限多个随机变量。在最简单的情况，随机过程等同于一列随机变量序列（如马尔科夫链）。随机过程另一种类型是定义在空间某个区域上的随机场。随机过程的研究方法之一是将其作为一个或几个确定自变量（或称为“输入”）的函数，函数值（或称为“输出”）为具有确定概率分布的随机变量。尽管相应于各个时刻（或点，在随机场的情况下）随机变量可以是完全独立的，但在一般情况下，这些变量更多地呈现出具有复杂的统计相关性。
这门课程主要集中于随机过程的基本概念、基础理论的教学并为进一步学习和研究随机过程理论的学生提供基本的入门知识。本门课程包括五章：
１．
第一章 引论：随机过程的基本概念及应用，有限维分布族和随机过程的基本类型。
２．
第二章 马尔科夫过程：马尔科夫链，纯不连续马尔科夫过程，扩散过程。
３．
第三章 二阶矩过程和随机分析：预备知识，随机分析，正态过程，伊藤随机积分和伊藤随机微分方程。
４．
第四章 平稳过程：平稳过程和协方差函数，平稳过程和协方差函数的谱分解，线性系统中的平稳过程，自回归滑动和过程。
５．
时间序列分析：时间序列的预测与滤波，线性模型中的均值估计，自回归模型的拟合。
本课程课内周学时为4学时，3学分。
Introduce to the course contents:
Stochastic process is the important content of probability
theory. Instead of dealing only with one random variable or finite variables, a
stochastic process concerns with infinite variables. In the simplest possible
case, It amounts to a sequence of random variables known as a time series (for
example, see Markov chain). Another basic type of a stochastic process is a
random field, whose domain is a region of space, in other words, a random
function whose arguments are drawn from a range of continuously changing
values. One approach to stochastic processes treats them as function of one or
several deterministic arguments (or called 'inputs') whose values (or 'outputs')
are random variables which have certain probability distributions. Although the
random values of a stochastic process at different times may be independent
random variables, in most commonly considered situations they exhibit
complicated statistical correlations.
The course is focusing on basic concepts and basic
theory of stochastic processes and provides basic access knowledge to further
study and research stochastic processes. There are five chapters.
1. Introduction: The basic concepts of stochastic processes,
some applications, family of finite dimensional distribution functions and the
basic types of stochastic processes.
2. Markov processes: Markov chains, purely
discontinuous Markov process, diffusion process.
3. Second moment stochastic process and random
analysis: Random analysis, Normal process, Ito Kiyosi stochastic integrals and
stochastic differential Equations.
4. Stationary process: stationary process and covariance
function, stationary process and spectral decomposition of covariance function,
stationary process in linear system, ARMA process.
5. Time series analysis: Forecasting and filtering of
time series, average value estimation of linear model and fitting of ARMA
model.
Class hour: 4 per week; Credit:
3
课程编号：100K0037
课程名称（中文）：非线性偏微分方程专题
课程名称（英文）：Topics on Nonlinear
Partial Differential Equations
授课内容：非线性偏微分方程是现代数学体系中的重要内容。现代科学与技术中许多数学模型都可归结为非线性偏微分方程。通过本课程的教学使学生掌握非线性偏微分方程理论基础，了解现代非线性偏微分方程若干研究领域，比如哈密顿-雅可比方程的粘性解理论，非线性偏微分方程的平均化理论，有助于研究生选择研究课题。
Introduce to the course contents:
Nonlinear partial differential equations is very important
contents of modern mathematics.
Nonlinear partial differential equations are often arise from modern
sciences and technology as mathematical model. This course focus on some resent
topics such as: Viscosity solutions of Hamilton-Jacobi equations,
Homogenization theory, p-Laplacian equations etc. This course will be helpful
for graduate students to decide the topics for their theses.
Class hours: 4 per week;
Credits: 3
课程编号：100K0039
课程名称（中文）：时滞偏微分方程专题
课程名称（英文）：Special Topics Partial Differential Equations with Delays
授课内容：时滞微分方程与偏微分方程的互相渗透自然产生时滞偏微分方程，人们把含有时滞的偏微分方程称为偏泛函微分方程，时滞偏微分方程起源于生物学、化学、物理学，工程技术等诸学科。20世纪60年代开始研究，到70年代才逐渐活跃起来，时滞偏微分方程无论在理论上，还是在应用上都富有发展的前景，特别自80年代以来，众多学者利用半群理论，线性、非线性泛函分析理论，辛几何辛拓扑理论，借助于常微分方程，偏微分方程，泛函微分方程的方法研究时滞偏微分方程，在适定性、上解下解、爆炸性、行波解、稳定性、有界性、孤立子等方面取得了重大进展。而最近几年又对时滞偏微分方程的吸引域、吸引盆、吸引子、吸引子的Hausdoff维数估计等进行研究，正在成为国际研究的热点，这是因为时滞偏微分方程解的长期动力行为是由吸引子决定的，因此，对吸引子的研究就显得更加重要，时滞偏微分方程是国际上方兴未艾的研究领域，需要较多的数学工具，难度很大，但这正是赶超国际先进水平的好时机，因此，开设这门课程对学生未来的发展有很强的促进作用。
本课程课内周学时为4学时，3学分。
Introduce to the course contents:
Course content: The infiltration between
differential equations with delays and partial differential equations produces
partial differential equations with delays naturally, and it is called partial
functional differential equations. Partial differential equations originates
from biology, chemistry, physics, engineer technique etc. Its study started
from 1960s, and did not develop until 1970s. The partial differential equations
have developing foreground either in theory or in application. Especially
since 1980s, many scholars have made use of semi-group theory, linear or
nonlinear functional analysis theory, symplectic geometry and symplectic
topology, as well as the methods in ODE, PDE, FDE to study the partial
differential equations with delays, and have made great progress in well
posedness, upper and lower solutions, exploration，travelling
wave solutions, stability, boundedness, soliton etc.. In recent years the study in attraction region,
attraction basin, attractors and estimates of
Hausdorff dimension of attractors of partial differential equations with delays gradually became the
hotspot in international researches. The reason is that the long-term dynamical
behavior of solution to partial
differential equations with delays is determined by attractors, and thus the
study of attractors becomes more important. Partial differential equations with delays is in the ascendant in
international areas; it needs many mathematical tools，and
has great difficulty, but it is a good opportunity to overtake the
international advanced level, and therefore，opening
the course has the
Class hour: 4 per week; Credit: 3
课程编号：100K0038
课程名称（中文）：哈密顿动力学专题
课程名称（英文）：Topics on Hamiltonian Dynamics
授课内容：哈密顿动力学是现代数学得重要内容，在现代物理理论中有重要应用。通过本课程的教学使学生掌握哈密顿动力系统的基础知识，了解现代哈密顿系统理论中的摄动理论、Aubry-Mather理论，有助于研究生选择研究课题。
本课程课内周学时为4学时，3学分。
Introduce to the course contents:
Hamiltonian
dynamics is active joint field of modern mathematics and physics. This course
help students to hold basic knowledge of Hamiltonian dynamical systems. And
then we focus on some special topics such as: perturbation theory, KAM thery,
Aubry-Mather, weak KAM theory, and some connection with nonlinear partial
differential equations etc. So this course will benefit for students to decide
the topics for their thesis.
Class hours: 4 per week; Credits: 3
课程编号：100K0036
课程名称（中文）递归神经网络
课程名称（英文）：Recurrent Neural Networks
授课内容：神经网络是受人脑功能的启发而发展起来的，并试图去模拟某些生物系统。神经网络是由大量的元素，即神经元相互联接而成。神经元的输入由神经元输出的适当加权及偏置项组成。神经元是由适当的函数，也称激活函数来描述。探索和揭示“万物之灵”的人类的高度发达大脑的奥秘是当代科学面临的重大的研究课题之一。 借鉴生物脑神经的研究而发展起来的非生物信息处理方式——人工神经网络是目前最活跃的信息处理科学研究领域之一。人工神经网络通常分为两类；一类是前馈神经网络（即没有反馈的神经网络）；另一类是递归神经网络（即有反馈的神经网络）。由于递归神经网络引入反馈，所以它是一个非线性动力学系统。递归神经网络的动力学行为可理解为状态的变换或迁移过程，可用常微分方程、泛函微分方程、偏微分方程和微分流形等研究。递归神经网络的信息处理功能体现在其动力特征中，包括平衡态、周期过程、振动性、吸引子和混沌等。本课程讲授的内容：
一、绪论：递归神经网络的基本概念。
二、递归神经网络理论基础。介绍一些递归神经网络模型。
三、模拟Hopfield神经网络的定性分析。
四、参数摄动的定性影响。
五、时间延迟的定性影响。
六、联想记忆的的一些综合方法。
七、互连约束的影响。
八、大规模神经网络理论和方法。
九、递归神经网络微分流形方法。
本课程课内周学时为4学时，3学分。
Introduce to the course contents:
Course content: Neural networks（NN）is
developed according to the enlightenment of the function of the human brain，and tries to simulate some biological system. NN is
composed by the joint of many elements, or neurons. The neurons input is
composed by the proper weighting to neurons output and bias terms. The neurons
are described by a proper function i.e. activation function. The exploration
and unveiling of the secret of the human’s high developed brain is one of the
important research tasks in modern science. The non-biology information
processing mode developed by the use of the brain nerve research——artificial neural networks（ANN） is one of the most active information processing
scientific research. ANN is usually divided into two kinds: one is the
feed-forward neural networks (i.e. no feedback neural networks), the other is
recurrent neural networks (RNN) (i.e. feedback neural networks). Since RNN
inducts feedback, it is a nonlinear dynamical system. The dynamical behavior of
RNN can be read as state transition or transferring process, and can be studied
by ODE, FDE, PDE and differentiable manifolds. The information processing
function of RNN is realized through dynamic characteristics, including
equilibrium state, periodic process oscillations, attractors and chaos etc..
The content of the course is as follows：
1. Introduction: the basic
concept of RNN.
2. RNN theoretical basis with
the introduction of some Recurrent Neural Networks models.
3. The qualitative analysis
of Hopfield neural networks.
4. The qualitative influence
of parameter perturbation.
5. The qualitative influence of
time delay.
6. Some synthesis methods of
associative memory
7. The influence of
interconnect confinement.
8. Great scale NN theories
and methods.
9. Differentiable manifolds
methods in RNN.
Class hour: 4 per week; Credit: 3
课程编号：100K0019
课程名称(中文)：现代格伦
课程名称(英文)：Modern theory of lattices
授课内容：本课程的目的完整介绍格论的若干最新进展, 并为后续课程及研究生的文献阅读提供必要的基础和知识预备。本课程主要介绍：
1. 偏序集与格
2. Galois 对
3. Heyting代数
4. Locales与拓扑空间
5. 特殊化序与拓扑
6. 连续格的一般理论
本课程需学生有一般拓扑学(General Topology)的基本知识和较好的数学修养。
本课程课内周学时为4学时，3学分。
Introduce to the course contents:
The
purpose of this course is to completely present the latest theory on lattices,
and offer necessary math base for further reference reading and research.
The
contents are listed as follows:
1. Partial ordered sets and
lattice
2. Galois connections
3. Heyting algebra
4. Locales and topological
spaces
5. Specialization ordering and
topology
6. Continuous lattices
The prerequisites for
understanding this course are good command of general topology and algebra.
Class hour: 4 per week; Credit:
3
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