The demand for advanced technicians with expertise in sciences and mathematics is increasing as our society becomes more modernized and informatized. The Department of Mathematics, while focusing on providing education and conducting research in applied mathematics, offers basic courses on mathematic principles and mathematic thinking as well as mathematics, statistics and insurance mathematics courses for scientific computations such as mathematic modeling, numerical analysis and computational mathematics.

**Location** : Rm.611, Ogui Hall

**Tel** : 82-2-940-5220

**Fax** : 82-2-942-0108

**e-mail** : kwmath@gmail.com

**Website** : http://math.kw.ac.kr/

### Course Descriptions

#### College mathematics 1

This course covers basic concepts of limit and continuity, critical points and max/min problems, mean value theorem, the derivative of various functions (geometric interpretation and analytic definition) and its applications, linear approximation, trigonometric functions, hyperbolic functions, inverse functions.

#### College Mathematics 2

This course covers the fundamental concepts of definite integral and its applications, the fundamental theorem of integral calculus, series, power series, ratio test, Taylor's theorem.

#### Set Theory

This course covers elementary logic, set operations, relations and functions, denumerable and nondenumerable sets, cardinal numbers and cardinal arithmetic, the axiom of choice, ordinal numbers and ordinal arithmetic.

#### Number Theory

This course covers arithmetic properties of integers; Congruences, arithmetic functions, diophantine equations, quadratic residues, primitive roots, and algebraic number.

#### Linear Algebra 1

This course is an introduction to linear algebra providing the basic concepts such as properties of matrices, determinants, methods to solve systems of linear equations by using matrices, vector spaces, linearly independent vectors, linearly dependent vectors, bases and dimensions, linear maps, relations between matrices and linear maps.

#### Linear Algebra 2

This course provides more concepts about linear algebra such as eigenvalues, eigenvectors, characteristic polynomials, minimal polynomials of matrices and linear transformations, diagonalization of matrices, inner product spaces, Gram-Schmidt orthogonalization process, Jordan canonical form.

#### Multivariable Calculus

This course introduces students to the fundamental concepts and basic elementary theorems from vector calculus with emphasis on functions of several variables. TOPICS : vector spaces, linear independence, linear transformation, inner product, orthogonality; partial derivatives, differentiability, Taylor's Theorem, multiple integrals, surface area, surface integrals, curves, arclength, curvature, line integrals, vector fields, Green's Theorem, Divergence Theorem, Stokes Theorem.

#### Analysis 1

This course covers supremum and infimum, completeness properties of the real numbers, limits of numerical sequences and series; limits and continuity, properties of continuous functions on closed bounded intervals; the intermediate value theorem; derivatives in one variable.

#### Analysis 2

This course covers sequences and series of functions, power series, uniform convergence, term by term differentiation and integration Riemann integration in one variable; open and closed sets and convergence of sequences in R^n; limits and continuity in several variables, properties of continuous functions on compact sets.

#### Numerical Analysis 1

This course covers Taylor polynomial, the error in the Taylor's polynomial, binary number system errors: definitions, sources, and examples, bisection method, Newtons's method, secant method, fixed point iteration, polynomial interpolation, divided differences, error in polynomial interpolation, spline functions, best approximation problem Chebyshev polynomials, trapezoidal and Simpson rules, Gaussian numerical integration.

#### Numerical Analysis 2

This course covers system of linear equations, Gaussian elimination, LU factorization, error in solving linear systems, least squares data fitting, eigenvalue problem iteration methods, Euler method, convergence of Euler's method, Taylor and Runge-Kutta methods, multistep methods, stability of numerical methods systems of differential equations, introduction to finite element method.

#### Computational Mathematics

This course gives students experiences in using mathematical softwares such as Matlab, Mathematica and Maple to solve problems from various areas of mathematics. It also provides students with tools for the further study of numerical methods or numerical analysis.

#### Introduction to Ordinary Differential Equations

The objective of this course is primarily to introduce the students in disciplines which emphasize methods of explicit solutions and the theory of ordinary differential equations. Topics: Linear differential equations, series solutions of linear differential equations, higher order equations; Laplace transform, systems of differential equations, linear systems, stability of solutions, qualitative behavior of non-linear systems, Lyapunov's second method; boundary value problems and Sturm-Liouville theory.

#### Topology 1

This course covers the fundamental concepts of point-set or general topology; topological spaces, basic open sets, subspaces and continuity, homeomorphisms, product spaces, connected spaces, and so on - are covered rigorously but at an elementary level. The student is required to explain each problem logically.

#### Topology 2

This course covers the basic topics; connectedness, compactness, separation properties, metric spaces, and so on. In particular, Topology 1 is a prerequisite.

#### Modern Algebra 1

This course provides the important concepts of group theory such as the definition of groups, subgroups, cyclic groups, Lagrange theorem, normal subgroups, factor groups, isomorphism theorems, direct sums of groups, orbits, Sylow theorem.

#### Modern Algebra 2

This course provides the basic concepts of ring theory such as the definition of rings, subrings, ideals, integral domains, the relations between maximal ideals and prime ideals, factor rings, polynomial rings, Euclidean domain, principal ideal domain, unique factorization domain.

#### Complex Analysis 1

This course covers the geometry of complex numbers, Cauchy-Riemann equations, harmonic functions, mapping properties of logarithm, exponential, and other functions, contour integrals, the Cauchy-Goursat Theorem, Cauchy integral formulas.

#### Complex Analysis 2

This course covers Morera's Theorem, maximum modulus principle, Liouville's theorem, linear fractional transformations as mappings, preservation of angles, harmonic conjugates, applications of conformal mapping to physical problems.

#### Actuarial Mathematics

This course covers interests, mortality table, life insurance, life annuity, net premium, liability reserve, office's premium.

#### Mathematical Statistics 1

This course will offer essentially all the distribution theory, estimation and tests of statistical hypotheses, expectation, random variables, multiple random variables.

#### Mathematical Statistics 2

Many of the topics of this course are estimation and tests of statistical hypotheses including nonparametric methods, sufficient statistics, Rao-Cramer inequality, and robust estimation after measures of the quality of estimators. Multiple comparisons and the analysis of variance, multi-variable normal distributions will be provided.

#### Introduction to Partial Differential Equations

The objective of this course is to provide students with the techniques necessary for the formulation and solutions of partial differential equations and prepare students for further study in partial differential equations. Topics: Derivations of partial differential equations; classification of linear partial differential equations; separation of variables applied to the heat equation, the wave equation and Laplace's equations in various geometries; Sturm-Liouville theory for second order ordinary differential equations; Bessel functions; Fourier series; Existence and uniqueness; Fourier transforms.

#### Topics in Algebra

This course covers several concepts not in the course of Modern Algebra 1,2 such as the extension fields, algebraic extensions, Galois theorem, modules. Besides, this course provides various concepts needed in algebra course.

#### Topics in Topology

We study the application of general topology and introduce some special topics. Topology 1 and Topology 2 are prerequisites.

#### Topics in Applied Mathematics

This course deals with mathematical methodology and many kinds of subjects in applied mathematics.

#### Differential Geometry

This course covers vectors, vector functions of a real variable, concept of a curve, curvature and torsion, frame fields, Euclidean geometry. Vector functions of a vector variable, concept of a surface, fundamental forms, tensor analysis, Riemannian geometry.

#### History of Mathematics

This course covers the historical developments of the various fields of mathematics, such as numerical systems, Euclidean and non-Euclidean geometry, analytic geometry, calculus and analysis, algebra, probability, set theory, topology, mathematical logic and philosophy.

#### Real Analysis

This course covers Caratheodory extension theorem, Lebesgue measure on the real line, general measure theory, convergence theorems, Lusin's theorem, Egorov's theorem, Lp-spaces, Fubini's theorem, functions of bounded variation, absolutely continuous functions, Lebesgue differentiation theorem.

#### Mathematical Modeling and Scientific Computing

This course will be centered around examples, arising from many application areas of applied mathematics, of how one carries through all the steps : problem → mathematical formulation → theoretical analysis → numerical solution. The goals of the course include

- provide illustrating examples of major modeling tasks of various problem arising from many application areas of Applied Mathematics,

- convoy practical approaches of how to approach problems not originally stated in mathematical form, and

- introduce a number of key techniques from analysis and numerics (only at the points where these are needed).

#### Probability Theory

This course is intended for introduction on probability theory and random processes. Markov chains, queueing theory and sequences of independent identically distributed random variables will be provided. Random processed analysis and processing of random signals will be also provided.

#### Selected topics in Mathematics

This course deals with mathematical subjects selected from the contemporary mathematical interests.