Junior Year
40312 Numerical Linear Algebra
Matrix decomposition, least-squares method, eigenvalue problem, iterative methods.
40317 Experimental Mathematics
Bezier curves, Splines, Markov Chains, Chaos and Fractals, Computer Graphics, The Arnold Cat, The Singular Value Decomposition.
40321 40324 Complex Variable (I)(II)
1.Complex variables 2.Analytic functions 3.Power series 4.Cauchy integral formula 5.Residue and its applications 6.Conformal mappings 7.Riemann mapping theorem 8.Harmonic functions.
40327 40328 Algebraic Coding Theory (I)(II)
The object of this course is to present only the fundamentals of the algebraic coding theory. It is not the intention of this course to give a categorical survey of important results in coding theory. The material covered may be regarded as a first course on the subject. 1.Linear Codes 2.Hamming Codes 3.Cyclic Codes 4.BCH Codes 5.Reed-Solomon Codes 6.Quadratic Residue Codes
40331 Topology
Upper and lower bounds. Finite and infinite sets. Open sets and Closed sets on R. The nested intervals theorem. Open sets and closed sets in metric spaces. Interior, closure, and boundary. Continuous functions. Complete metric spaces. Open sets and closed sets in topological spaces. Basis and subbasis. Subspaces. Connected spaces. Path-connected spaces. Compact spaces. The Cantor set. Product spaces. Quotient spaces.
40332 Applied Analysis
Extends concepts and techniques of linear algebra and develops further applications. Introduction to normed and Hilbert spaces; projections and bounded operators; emphasis on matrix and function space applications.
40336 Topic in Analysis
This course is selected from the following topics by two year rotated alternatively,
1.Euclidean space, Elementary classical analysis 2.Topological space, compact and connectness 3.Metric space, completeness 4.Measure space 5.Measurable function, and integrations 6.Normed space, Banach space, Lp space 7.Linear operator and Linear functional 8.Hilber space 9.Fourier series and Fourier transform 10.Gateux and Frechet derivative 11.Convex set and convex function 12.Introduction to optimization analysis 13.Topological group and Fourier analysis
40343 Stochastic Processes
The prerequisites for this course are a previous course covering calculus, probability, and statistics. The course begins with an introduction to random processes and covers Markov chains. Then, an introduction to continuous-time Markov processes will be provided. Some important processes will also be explored, such as Poisson process, Gaussion process, and Wiener process, during the semester. Next, we discuss integration and differentiation of stochastic processes. Finally, we discuss solutions to nonhomogeneous ordinary differential equations having constant coefficients whose right-hand side is either a stochastic process or white noise.
40344 Nonparametric Statistics
This is a one-semester course at the advanced undergraduate level . Students in this class have at least one introductory course in classical statistics. We will discuss one single sample, two independent samples, and two related samples first. When the available data for analysis consist of observations from three or more related samples, these are also interesting topics to pursue. Finally, we would like to introduce the simple linear regression model, but the assumptions underlying parametric inference are suspect.
40345 40346 Mathematical Statistics (I)(II)
The course is designed for one year in mathematical statistics. We roughly divide this course into two parts. In part 1, the concepts of a field and a σ-field, and also the definition of a random variable as a measurable function, are introduced. Other important topics are as follows: the discussion of various discrete and continuous random variables; the establishment of several moments and probability inequalities; the exploration of various moment generating functions; an extension of limit theorems, including all common modes of convergence and their relationship; and also statements of the Laws of Large Numbers and further useful limit theorems. In the second part of this course, we are going to discuss the concept of sufficiency and estimation problems in detail. The principles of unbiasedness, UMVUE, and MLE are also considered. Hypothesis testing, exponential families, linear models, multivariate normal distributions, and quadratic forms are also going to be discussed in this course.
40347 Regression Analysis
The students have had an introductory course in statistical inference. We use calculus and linear algebra to demonstrate some important results. Also, some basic elements of matrix algebra are needed for the multiple regression models. Many interesting topics which will be discussed are as follows: simple regression analysis, polynomial regression, logistic regression, polytomous logistic regression, and generalized linear model. The use of residual analysis and other diagnostics for examining the appropriateness of a regression model is a recurring theme throughout this course. Statistical software, such as Minitab or SAS, is also needed for doing the data analysis.
40349 Statistical Methods
We suggest that the students have more background in statistics before taking this course. We will introduce the types of data with which the course mainly deals. Topics will include sample surveys, controlled experiments, and comparative observational studies, plus the concept of random sampling. Regression analysis, analysis of variance, and time series will be discussed during the semester, especially we will focus on time series analysis.
40350 Mathematics of Finance
Measurement of interest and discount, not accumulated value and present value, annuities, sinking funds, amortization schedules, and determination of yield rates.
40354 Risk Theory
Frequency and severity distributions; individual and collective models, ruin theory, reinsurance, and simulation.