COLLEGE OF SCIENCE
Department of Mathematics
Degrees Offered: B.S., M.S., Ph.D.
TO Webpage:
Chair: Chen, Kung-yu
The Department
This department was founded in 1958 and since 1966 has consisted of the mathematics and the mathematical statistics sections. The master's program was established in 1969 with the addition of the Ph.D. program in 1990.
The primary goal of the department is to provide students with the opportunities to understand basic concepts of mathematics, to explore various areas and to broaden their views in mathematics and statistics. The department offers both need-based and merit-based scholarships.
The department has its own computer laboratory which houses 100 personal computers with a local-area network completed in 1994. Mathematics/statistics software packages such as SAS, MATHEMATICA, and MAPLE are available for teaching and research. The Tamkang Journal of Mathematics, an internationally known quarterly, was first published by the department in 1970.
Faculty
Professors
Chang, Whei-ching ; Chang, Yue-cune ; Chen, Kung-yu ; Chen, Shun-yi ; Chen, Tien-wen ; Cheng, Wei-hou ; Chyan, Chuan-jen ; Hu, Shou-jen ; Hu, Thak-yin ; Kau , Chin-mei ; Lin, Chien-tai ; Liu, Fon-che ; Tam, Bit-shun ; Tseng, Shio-jenn ; Wang, Hsiao-lan ; Yang, Gou-sheng
Associate Professors
Huang, Yih-huei ; Lee, Wu-yen ); Shieh, Chung-tsun ; Wang, Kui-jang ; Wu, Hsiu-fen ;Wu, Jyh-shyang ; Wu, Meng-nien ; Yu, Cherng-yih
Assistant Professors
Wen, Chi-chung ; Yang, Ting-hui
Degree Requirements
The Department of Mathematics offers two programs at both the graduate and undergraduate levels, namely the Mathematics Program and Statistics Program.
- Requirements for a degree of B.Sc. in Mathematics:
Successful completion of 134 credits of courses, including 85 credits of required courses and 42 credits of elective mathematic courses. - Requirements for a degree of B.Sc. in Statistics:
Successful completion of 134 credits of courses, including 89 credits of required courses and 40 credits of elective Statistics courses. - Requirements for a Master's degree in Mathematics and Statistics:
Successful completion of 24 credits of the required courses. Students are also required to submit a written master's thesis completed under the supervision of a faculty member, and pass an Oral Examination. - Requirements for a degree of Ph.D. in Science:
Successful completion of 30 credits of courses. Students are required to pass one qualifying examination within the first five semesters and the second qualifying examination within seven semesters, publish at least one research paper in any journal listed in Science Citation Index, and submit a written doctoral dissertation completed under the supervision of a faculty member, and pass an Oral Examination.
Course Descriptions
Undergraduate Courses
Mathematics Section
E0767 Numerical Analysis (3/3) Interpolating polynomials, Newton's method, fixed point iteration, numerical differentiation and integration, Euler's method, Runge-Kutta method, Gaussian elimination with pivoting, power method, Householder transformation, QR algorithm, least square approximation, orthogonal functions.
M0517 Statistics (0/3) Fundamental concepts of statistics including estimation, testing of hypotheses and applications.
S0024 Analysis I (3/3) Various topics in real analysis, including measure, measurable functions, integrable functions, the Lebesque spaces, modes of convergence, decomposition of measures, and generation of measures.
S0027 Analysis II (3/3) Further studies on various topics in real analysis.
S0051 Algebra (3/3) Basic algebra structures, including groups, rings and algebraic field extensions.
S0090 Vector Analysis (0/3) Tangent, normal and binomial vector, curvature, orthogonal curvilinear coordinates, Laplacian, line integral, conservative fields, potential function, oriented surface, Green's theorem, divergence theorem, Stoke's theorem.
S0132 Topology (3/3) Essentials in point set topology, including the concept of topological spaces, connectedness, compactness, countability axioms, separation axioms.
S0210 Advanced Calculus (4/4) The number systems, topological structures of Rn, continuous functions, differentiable functions of one variable, Riemann-Stieltjes integrals, sequences and series of functions, differentiation on Rn, inverse and implicit function theorems, integration on Rn.
S0155 Modern Algebra (3/3) Further studies in the structures of groups, rings, fields and Galois theory.
S0252 Fundamentals of Mathematics (2/2) Introduction to basic notion of set theory: topics include axioms of set, relations, partially ordered sets, natural numbers, finite and infinite sets and logic.
S0277 Combinatorics (3/3) Enumeration, generating functions, recurrence relations, graph theory and networks.
S0284 Geometry (3/3) Study of curves and surfaces including first and second fundamental forms, Gaussian map, Gauss-Bonnet theorem, geodesics.
S0317 Differential Equations (3/0) Ordinary differential equations, first order differential equations, higher order linear differential equations, system of linear differential equations, Laplace transforms, series method.
S0325 Calculus (4/4) Limits, differentiation and integration of functions of one variable, infinite series, functions of several variables, partial derivatives, multiple integrals.
S0336 Computer Applications in Mathematics (3/3) The use of computer and software packages in solving problems in mathematics.
S0439 Linear Algebra (3/3) Vector spaces, linear transformations, matrices, eigenvalues and eigenvectors, Jordan and rational canonical forms, inner product spaces.
S0450 Probability Theory (3/0) Basic concepts in probability, discrete and continuous random variables, expectation, bivariate probability distributions and functions of random variables , sampling distributions.
S0579 Complex Analysis (3/3) Analytic functions, complex integration, Cauchy's theorem, sequence and series of analytic functions, conformal mappings, and analytic continuation.
S0616 Linear Algebra II (3/3) Further studies of various topics in linear algebra.
Data Science and Mathematical Statistics Section
M0115 Multivariate Analysis (3/3) Multivariate normal distribution, Hotelling's test, MANOVA, Factor analysis.
M0153 Operation Research (3/3) Linear programming, the simplex algorithm, sensitivity analysis, transportation, assignment, transshipment problems, network models, integer programming, game theory, queuing theory, inventory models.
M0202 Quality Control (3/3) Importance of quality control, early history, Deming's philosophy, process thinking, improving a process, the seven basic tools, control charts for means, ranges, individuals, proportions and counts.
M1043 Survival Analysis (3/3) Special features of survival data, survival function, KM estimate, Cox's PH model and its assumption, general stratified Cox procedure, extension of Cox's PH model.
S0210 Advanced Calculus (4/4) The number systems, topological structures of Rn, continuous functions, differentiable functions of one variable, Riemann-Stieltjes integrals, sequences and series of functions, differentiation on Rn, inverse and implicit function theorems, integration on Rn.
S0250 Applied Statistical Software (2/2) Introduction to data input, output and programming using SAS and S-plus.
M0264 Time Series (0/3) Single variable time series models, estimation, ARIMA models, model building and forecasting, seasonal models.
S0061 Reliability Analysis (3/3) Reliability concepts, and statistical analysis of censored data, degradation data and accelerated life tests.
S0266 Introduction to Statistics (2/2) This is the first course in statistics which covers the basic concepts of statistics and its uses in daily life.
S0295 Nonparametric Statistics (3/3) This course introduces nonparametric methods and related theories.
S0325 Calculus (4/4) Limits, differentiation and integration of functions of one variable, infinite series, functions of several variables, partial derivatives, multiple integrals.
S0364 Computer Applications in Statistics (3/3) Advanced programming of SAS including SAS/connect, SAS/graph, SAS/AF and SAS/insight.
S0408 Experimental Design (3/3) One-way and two-way classification, Latin squares, factorial designs.
S0423 Mathematical Statistics I (4/4) Some probability concepts, random variables and their distribution, moments of random variables, characteristic function, moment generating functions. Stochastic independence, limit theorem, transformations of random variables and random vectors, order statistics, point estimation, testing bypothesis, confidence intervals, Quadratic forms.
S0424 Mathematical Statistics II (3/3) Further studies of various topics in mathematical statistics.
S0439 Linear Algebra (3/3) Vector spaces, linear transformations, matrices, eigenvalues and eigenvectors, Jordan and rational canonical forms, inner product spaces.
S0458 Stochastic Process (3/0) Poisson process, Markov chains, and applications.
S0487 Discrete Mathematics (3/3) Counting, logic, mathematical induction, relations, finite state machines, generating functions, recurrence relations and graph theory.
S0487 Discrete Mathematics (3/3) Fundamental mathematics, generating functions, recurrence relations, graph theory, networks and Boolean algebra.
S0722 Clinical Trials (3/3) Planning and design, basic design consideration, randomization and blinding, sample size determination, efficacy and safety evaluations.
S0733 Queuing Theory (3/3) Birth-death models, M/M/1 system, M/M/2 systems, M/G/1 system, G/M/1 system, networks of queues, transient solutions.
Master's Program
Mathematics
S0079 Abelian Groups (3/3) Ulm's Theorem and various structure theorems, homological methods and recent results.
S0024 Analysis (3/3) Measure, Lebesgue measure, Lebesgue integral, LP-spaces, integration on product spaces, complex measure.
S0046 Algebraic Topology (3/3) Singular homology theory, cohomology ring and duality in manifolds.
S0051 Algebra (3/3) Groups and rings; free, projective and injective modules; Hom and tensor product, field extensions and Galois Theory.
S0187 Matrix Theory (3/3) Similarity, diagonalization, unitary equivalence, normal matrices, Jordan canonical forms, variational characterizations of eigenvalues of Hermitian matrices, matrix norms, location of eigenvalues, nonnegative matrices.
S0238 Partial Differential Equations (3/3) First-order equations, principles for higher-order equations, Fourier methods, the differential equations of physics and engineering
S0277 Combinatorial Mathematics (3/3) Introduction to enumerative combinatorics, graph theory and combinatorial designs.
S0320 Differential Geometry (3/3) Euclidean geometry, geometry of surfaces in Euclidean space, Riemannian geometry.
S0402 Graph Theory (3/3) Planar graphs, graphs coloring domination, independence, chromatic numbers and networks.
S0566 Ordinary Differential Equation (3/3) Existence and uniqueness, continuation. autonomous and nonautonomous system. Poincaré-Bendixson theorem, linear and linearization, Poincaré map, stability near equilibia and periodic orbit, bifurcation diagram, congugacy, structurally stable.
S0573 Special Topics in Analysis (2/2) Selected special topics in mathematical analysis.
S0598 Combinatorial Design (3/3) Orthogonal Latin squares, symmetric designs, Steiner systems and tournament designs.
S0602 Special Topics in Algebra (3/3) Various topics in algebra, such as homological algebra, representations of finite groups and characters.
S0631 Fractal Geometry (3/3) Hausdoff measure and dimension, alternative definitions of dimension, techniques for calculating dimensions.
S0632 Hyperspace Theory (3/3) Various topologies on spaces whose elements are certain subsets of a given underlying space are studied.
S0686 Commutative Algebra (3/3) Various topics in commutative rings, including Noetherian, Artinian rings and modules, localization, primary decomposition, Hilbert Nullstellensatz, integral extensions and valuations, analysis of Dedekind domains.
Mathematical Statistics
M0115 Multivariate Analysis (3/3) Multivariate normal distribution, inferences about multivariate means and linear models, principal components, factor analysis, discrimination and classification, clustering.
M0202 Quality Control (3/3) Importance of quality control, early history, Deming's philosophy, process thinking, improving a process, the seven basic tools, control charts for means, ranges, individuals, proportions and counts, design of experiments, factorial, fictional factorial and screening designs.
S0061 Reliability Analysis (3/3) Censoring and statistical methods, life table and graphs, inference procedures for distributions of exponential, Weibull, extreme-value and other models, parametric regression models, proportional hazards and related regression models, nonparametric methods, goodness-of-fit tests.
S0075 Statistical Application in Biology (3/3) Generalized linear model, categorical data analysis, survival analysis, nonparametric methods, with applications in various areas of biostatistics.
S0231 Advanced Mathematical Statistics (3/3) Probability theory, transformations and expectations, common families of distributions, multiple random variables, properties of a random sample, principles of data reduction, point estimation, hypothesis testing, interval estimation, decision theory.
S0233 Advanced Probability (3/3) Topics includes random walks, probability theory, random variables independence, expectation, convergence, limit theorems, conditional expectation, Martingales.
S0264 Time Series (3/3) Autocorrelation function, stationary models, nonstationary models, seasonal models, transfer function models, intervention models.
S0269 Statistical Methods (3/3) Regression analysis, analysis of frequencies variable, introduction to time series data, CR and RCB designs, nest design, factorial experiment.
S0295 Nonparametric Statistics (3/3) This course introduces the important theoretical foundations of nonparametric statistics, both classical and current.
S0408 Experimental Designs (3/3) Factorial treatment designs, random and mixed models, complete block designs, incomplete block designs, fractional factorial designs, split-plot designs, repeated measure designs, cross-over designs.
S0441 Linear Statistical Models (3/3) This course covers general linear model, generalized linear model, with basic concepts, theorems, and applications.
Ph.D. Program
E1197 Dynamic Systems (3/3) Diffeomorphisms and flows, stable manifold, center manifold, normal form, versal deformation.
S0137 Functional Analysis (3/3) Topological vector spaces, local convexity, completeness, convexity, duality, Banach algebras, Gelfand-Naimark theory, the spectral theorem.
S0427 Number Theory (3/3) Algebraic integers, quadratic and cyclotomic fields, class-group and class-number, p-adic numbers, Zeta and L-functions.
S0590 Nonlinear Functional Analysis (3/3) Basic problems of the theory of non-expansive mappings in Banach spaces, fixed point theorems and convergence of successive approximations.
S0591 Linear Integral Equations (3/3) Basic existence theorem, integral equations with L2 kernels, applications to partial differential equations, Fourier transforms, the Fredholm theory.
S0593 Smooth Dynamic Systems (3/3) Diffeomorphisms, flows, invariant manifold, transversality, generic properties, structural stability.
S0594 Nonparametric Regression (3/3) Theorems, methods, and applications of kernel regression procedure.
