?? - ????????

????: ?????
????: 070102C03
????: C
? ?: 70
? ?: 4
????: ????????,????
????:
???????????????????????????????????????????,?????????????????????
????????: ????????????????
????:
??????????????????????????????????,???????????????????????L-
S??????????????????????????????????????????????,????????????????????????????
?????????
????:
???:??????????
§1. ?????????
§2. ????
§3. ?????
???:?????????????
§1. ???????????
§2. ?????
???:?????????????
§1. ????????
§2. ??????????
§3. ??????????????
§4. ????????????
§5. ????????????
§6. ????????????
???:????
§1. ?????
§2. ????
§3. 0-1?????????????
§4. ????
???:??????
§1. ????????????
§2. ??????????
§3. Berry-Esseen ???
§4. ????????????Poisson????
§5 ?????????????
????: ????
???????:
1. Billingsley, P., Convergence of Probability Measures, Wiley, New York,
1968.
2. Feller, W., An Introduction to Probability Theory and Its Applications,
Wiley, New York, 1971.
3. Laha, R.G. and Rohatgi, V.K., Probability Theory. Wiley, New York, 1979.
4. Loeve, M., Probability Theory, Springer-Verlag, 1977.
????:
????: ?? (20%),????(20%),????(60%)
?????: ??? Title: Foundations of Probability Theory
Course Number: 070102C03
Course Type: C
Session: 70
Credit: 4
Designed for: Graduate students majoring in Probability, Statistics and/or
Applied Mathematics.
Objectives:
This course is designed to provide the graduate students with a solid
background and understanding of the basic results and methods in
probability measures and probability limit theory needed in more advanced
research in related subjects.
Prerequisites:
A working knowledge of real analysis and basic probability theory is
assumed.
Major Contents:
This course introduces basic concepts, tools and properties of modern
probability theory in measure-theoretic foundations, including measure and
probability, measurable functions and rand om variables, mathematical
expectation and L-S integral, basic probability inequalities, convergence
concepts, characteristic functions, infinitely divisible distributions,
conditional probability, conditional expectation, martingales and its
applications. The course also discusses the weak law of large numbers, the
strong law of large numbers, the law of the iterated logarithm and the
central limit theorems under several cases.
Main Chapters:
Chapter 1 Basic Concepts of Probability and Measure
1. Probability Spaces and Random Variables
2. Mathematical Expectation
3. Convergence Concepts
Chapter 2 Conditioning and Martingales
2.1 Conditional Probability and Conditional Expectation
2.2 Martingales and its Applications
Chapter 3 Distribution and Characteristic Functions
3.1 Weak convergence
3.2 Definitions and Properties of Characteristic Functions
3.3 The Inversion and Uniqueness Theorems, Continuity Theorem
3.4 Weak convergence and Tightness on a Metric Space
3.5 Definitions and Properties of Infinitely Divisible Distributions
3.6 Infinitely Divisible Characteristic Functions and its Applications
Chapter 4 The Laws of Large Numbers
4.1 The Weak Law of Large Numbers
4.2 Basic Techniques
4.3 0-1 Law and The Strong Law of Large Numbers
4.4 The Law of the Iterated Logarithm
Chapter 5 The Central Limit Theorems
5.1 The Bounded Variance Case
2. The General Central Limit Theorem
3. The Berry-Esseen inequality
5.4 Normal, Degenerate, and Poisson Convergence
5.5 The Central Limit Theorem for Infinite Variance Case
Ways of Instruction: Lectures
Textbooks and References:
1. Billingsley, P., Convergence of Probability Measures, Wiley, New York,
1968.
2. Feller, W., An Introduction to Probability Theory and Its Applications,
Wiley, New York, 1971.
3. Laha, R.G. and Rohatgi, V.K., Probability Theory. Wiley, New York, 1979.
4. Loeve, M., Probability Theory, Springer-Verlag, 1977.
Instructor:
Couse Evaluation: The course grade will be determined by Homework (20%),
Midterm exam (20%), Final exam (60%).
Program Designer: Wang Lihong
????:????
????:0701B0200
????:B
??:80
??:4
????:??????
????:
???????????????????,?????????????????????????????????????????
????????:?????????????,????????????
????:
????????????????,????????????????????????????????,???????????????,???????
????????,??????????????????????????????????,????????????????????????????????
????????,????????,?:Artin?,Noether?,???,????????,?????????
????:
1. : ??
1. ????????
2. ?????????
3. ???????
4. ?????
5. ????
6. Hom??,?????
7. ??
8. ???
2. ?
2.1 ???R-???
2.2 Artin??Noether?
2.3 ???
2.4 ???,???
2.5 ?????,???
2.6 ?????????
2.7 ??????
????:????
???????:
??:Nathan Jacobson, Basic Algebra II, W. H. Freeman and Campany, 1980.
????:1. Saunders Mac Lane, Categories for the Working Mathematician, 2nd
Edition, Graduate Texts in Mathematics, Vol. 5, Springer-Verlag,
1998.
2. Frank W. Anderson and Kent R. Fuller, Rings and Categories of
Modules, 2nd Edition, Graduate Texts in Mathematics, Vol. 13,
Springer-Verlag, 1992.
????:???
????:??
?????:???
Title: Basic Algebra
Course Number: 0701B0200
Course Type B
Session: 80
Credit: 4
Designed for: All specialities in Department of Mathematics
Objectives:
This is an extension of an undergraduate course-abstract algebra. The
aim of this course is to wish the students to understand and master the
fundamental concepts and methods in the theory of categories and the
fundamental properties of some important classes of modules.
Prerequisites:
The students should know the concepts and properties related to groups
and rings very well, and know the definition and fundamental properties of
modules.
Major Contents:
The theory of categories is an important branch of algebra. This theory
has produced essentially many ideals and methods, which are contributing to
an overall understanding of mathematics. Studying the theory of categories
has given rise to a vast amount of new points of view and to the posing of
new questions, which are not only themselves of interest in the theory of
categories, but which have revealed new avenues for investigation in
various concrete categories. This analysis arises in the particular case of
module categories, which have given, in their turn, the motivation for the
development of categories. In this course, we mainly relate the concepts
and fundamental properties of general categories and functors, and
introduce the definitions and fundamental properties of some important
classes of modules, such as Artinian modules, Noetherian modules,
projective modules, injective modules and flat modules and so on.
Main Chapters:
Chapter 1 Categories
1. Definition and examples of categories
2. Some basic categorical concepts
3. Functors and natural transformations
4. Equivalence of categories
5. Products and coproducts
6. The hom functors. Representable functors
7. Universals
8. Adjoints
Chapter 2 Modules
2.1 The categories R-mod and mod-R
2.2 Artinian and Noetherian modules
2.3 Projective modules
2.4 Injective modules. Injective hull
2.5 Tensor product of modules. Flat modules
2.6 Generators and progenerators
2.7 Equivalence of categories of modules
Ways of Instruction: Classroom teaching
Textbooks and References:
Textbook: Nathan Jacobson, Basic Algebra II, W. H. Freeman and Campany,
1980
References: 1. Saunders Mac Lane, Categories for the Working Mathematician,
2nd Edition, Graduate Texts in Mathematics, Vol. 5, Springer-
Verlag, 1998
2. Frank W. Anderson and Kent R. Fuller, Rings and Categories of
Modules, 2nd Edition, Graduate Texts in Mathematics, Vol. 13,
Springer-Verlag, 1992
Instructor: Zhaoyong Huang
Course Evaluation: Written examination
Program Designer: Zhaoyong Huang
????:????
????:070101X09
????:X
??:80
??:4
????:????
????:??????????????????????
????????:???????
????:
????????????????????????????,?,?????,???,???,?????
?????Hom????????,?????????,??????,??????,??(??)??,??????????????????????????
???Hom?????????????Ext ? Tor?????????,????????(??,??)??????????,????Hilbert
????,Serre??????????
????:
??? ????
1) ?????
2) ??,??????
3) ??
??? ????????
1) ?????
2) ??????
3) ????
4) ???? (??)
5) ???? (??)
??? ??
1) ????
2) ????
??? Ext? Tor
1) Ext ? ??
2) Tor ? ?
??? ????
1) ????
2) ????
3) ????
4) ????
5) Hilbert ????
6) Serre??
7) ?????
????:????
???????:
1. J. J. Rotman, An introduction to homological algebra, Academic Press,
1979.
2. C. A. Weibel, An introduction to homological algebra, Cambridge
University Press, 1994.
3. H. Cartan and S. Eilenberg, Homological Algebra, Princeton University
Press.
4. M. Scott Osborne, Basic Homological Algebra, Springer-Verlag, 2000
5. E. E. Enochs and O. M. G. Jenda, Relative homological algebra, Walter de
Gruyter-Berlin-New York, 2000.
6. ???,????, ?????, 1998?
7. ???,????, ???????, 1998?
????:???, ???,???
????:????
?????:??? Title: Homological Algebra
Course Number: 070101X09
Course Type: X
Session: 80
Credit: 4
Designed for: Pure Mathematics
Objectives: Basic concepts, principles and methods in homological algebra
are presented
Prerequisites: Students have learned the graduate algebra course
Major Contents:
The aim of this course is to present basic materials in Homological
Algebra. Chapter 1 gives basic concepts such as categories and factors,
projective, injective and flat modules. Chapter 2 deals with Hom and tensor
product functors. The material in this chapter includes sums and products,
exactness of functors, adjoint isomorphism Theorem, direct limits
(pushouts) and inverse limits (pullbacks). Chapter 3 introduces homology
functors and