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Undergraduate Program Mandatory Basic Courses MAS Calculus I This course deals with differentiation and integration of one variable real-valued functions, emphasizing basic concepts and applications.

The topics are: differentiation and integration of trigonometric functions, logarithmic functions, hyperbolic functions and their inverse functions, improper integral and its convergence tests, polar coordinates, infinite series and their convergence tests, Taylor series, and power series. MAS Calculus II This course deals with differentiation and integration of multivariable real-valued functions, emphasizing basic concepts and applications. The topics include matrices, determinants, characteristic equations, eigenvalues, eigenvectors, inner product spaces, orthogonalization, diagonalization of square matrices and quadratic forms.

The topics include ordinary linear differential equations, Laplace transform, systems of differential equations and some partial differential equations. MAS Applied Mathematical Analysis This course introduces Fourier series, Fourier transform, differentiation and integration of complex variable functions, power series for complex variable functions, and residue theorem.

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Topics include independence of events and random variables, various probability distributions, expectation, conditional expectation, the law of large numbers, the central limit the orem, tests of hypothesis, the analysis of variance, and regression. Elective general MAS College Mathematics Designed for students who are not ready for Calculus I MAS , this course deals with differentiation and integration of one variable real-valued functions, emphasizing basic concepts and applications.

Students who have taken this course are allowed to register for MASZ. Topics include congruence equations, arithmetic functions, residues, quadratic residues, continued fractions, algebraic properties of quadratic fields, the prime number theorem, diophantine approximation, diophantine equation, and applications to cryptography. MAS Linear Algebra This course gives students the opportunity to manipulate the concepts of linear algebra and to develop an intuitive understanding of their geometric meanings.

Topics include unitary and hermitian mappings, eigenvalues and eigenvectors, spectral decomposition, triangulation and the Jordan normal form, and multilinear algebra. MAS Analysis I This course provides sophomores in mathematics with a thorough background in mathematical analysis. Topics include real number system, sequences, open sets, closed sets, connected sets, compact sets, limits and continuity of functions, differentiation, differentiation of multivariable functions, the mean value theorem, the intermediate value theorem, Riemann integration, sequences and series of functions. Topics include series of functions, uniform continuity, double series, uniform convergence, differentiation of sequences and series of functions, integration of sequences and series of functions, special functions, Hilbert space, Fourier series, orthogonality, completeness, transformations, the inverse function theorem, the implicit function theorem, vector analysis, multiple integration, line integration, and some basic concepts of differentiable manifolds.

MAS Applied Mathematics and Modeling This course introduces problem-oriented mathematics with case studies for real world problems. MAS Computational Geometry and Computer Graphics This course introduces mathematical methods and theories to describe curves and surfaces in space, and deals with applications to computer-aided design and computer graphics.

MAS Discrete Mathematics This course introduces discrete objects, such as permutations, combinations, networks, and graphs. Topics include enumeration, partially ordered sets, generating functions, graphs, trees, and algorithms. MAS Introduction to Differential Geometry This course is an introduction to the differential geometry of curves and surfaces in 3-dimensional space.

Topics include local theory of curves, Gauss maps and the curvature of surfaces, intrinsic geometry, and the global geometry of surfaces. MAS Topology This course studies basic general topological properties and concepts, including topologies, open sets, closed sets, compactness and connectedness, separation axioms for Hausdorff spaces, regular spaces and normal spaces, and countability.

Basic properties of metric spaces and various metrization theorems are studied as well. MAS Ordinary Differential Equations and Dynamical systems We study solutions to various ordinary differential equations and geometric properties. Topics include the Poincare-Bendixon theorem, modelling, dynamical systems, and applications. Topics include conditional probability and independence, expectation, various random variables and distributions, the law of large numbers, the central limit theorem, martingale theory, the Poisson process, Markov chains, Brownian motion, and stationary random processes.

The course also includes inverse transform methods and rejection methods for simulation. MAS Mathematical Statistics This course covers basic theories for statistical methodologies and applications to engineering and applied sciences.

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Topics include basic theories of probability, random variables, probability distributions and their inter-relationship, average and variance, variable transformations, sampling distributions, estimation and hypothesis testing, the law of large numbers, two-dimensional distributions, decision making, linear models, and nonparametric methods. MAS Matrix Computation and Application Coming from the application of linear algebra, this course introduces the classification of matrices, according to their properties, together with the theory of matrix computations and computational algorithms.

MAS Introduction to Numerical Analysis This course discusses some of the central problems that arise in applications of mathematics and the development of constructive methods for the numerical solution of the problems. It also provides elementary numerical tools for scientific computation. Topics include computing error analysis, algorithms, Gaussian elimination, Cholesky decomposition, error bounds, ill-conditioned problem, eigenvalues, Jacobi rotation and eigenvalue estimates, the power method, solution of nonlinear equations, the OR algorithm, interpolation, numerical integrations, and how to solve differential equations.

The concepts of financial terms will be explained and stochastic methods on how the financial market products are priced will also be introduced. Through this course, students are expected to learn how probability, statistics, and applied mathematics are used in financial markets. Topics include convex sets, convex functions, separation theorem, Karush-Kuhn-Tucker theorem, Brouwer fixed point theorem, Ky-Fan inequality, and Nash equilibrium.

MAS Introduction to Cryptography This course introduces classical cryptosystems, symmetric cryptosystems, DES, AES, public key cryptosystems, digital signature, communication protocols, and information theory. MAS Introduction to Algebraic Geometry This course introduces basic concepts in algebraic geometry and related theorems. Various problems are treated. MAS Analysis on Manifolds This course covers the elementary theory of functions of several variables and introduces basic concepts of differentiable manifolds and differential forms. It considers elementary concepts in differential geometry such as manifolds, curvature and geodesics. The exterior derivative and integrals of differential forms are defined on Euclidean spaces and generalized to differentiable manifolds. The course includes applications to surface theory.

MAS Combinatorial Topology This course introduces some basic algebraic topological concepts using combinatorial methods according to triangulations. Topics include simplicial complexes and triangulations of spaces, homotopy and fundamental groups, classification of surfaces, covering spaces, simplicial homology of surfaces, and the Euler-Poincare formula.

MAS Matrix Groups This course provides an introduction to basic Lie group theory at the concrete level of matrix groups. Topics include general linear groups and subgroups, Lie algebras, exponentials and logarithms, maximal tori, Spin groups, and the Weyl group. MAS Introduction to Partial Differential Equations This course introduces solutions and properties of first and second order linear partial differential equations, as well as solutions of first order nonlinear partial differential equations. Laplace equations, heat equations, wave equations, and methods for boundary value problems are introduced.

MAS Fourier Analysis and Applications Basic properties of Fourier series and Fourier transforms will be treated with applications to differential equations and signal processing. MAS Linear Models This course covers methods of linear regression analysis, and major topics include simple linear regression analysis, multiple regression analysis, goodness-of-fit test, model building and model selection methods, regression analysis with incomplete data, and non-least-squares estimation.

MAS Statistical Methods with Computer This course introduces data analysis methods using computer statistical program packages Minitab, SAS, SPLUS, etc , and the main goal of the course is to teach and train students for effective analysis methods over a variety of data types and analysis purposes.

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It cover the topics of the definition of random processes, second moment theory, special descriptions of random processes, linear transforms, signal detection and estimation, and Gaussian processes. MAS Theory and Application of Transforms Transform theories for continuous and discrete signals are widely applied to many engineering problems.

This course covers the theories and applications of various types of transform methods. We cover topics such as complex variables, contour integrals, Laplace transforms, Fourier transforms, and Z transforms. MAS Mathematical Mechanics This course introduces some of the important concepts of physics such as classical mechanics, statistical mechanics, and quantum mechanics to help students apply them in the study of mathematics.

The constant of proportionality between the viscous stress tensor and the velocity gradient is known as the viscosity. A simple equation to describe incompressible Newtonian fluid behaviour is. For a Newtonian fluid, the viscosity, by definition, depends only on temperature and pressure , not on the forces acting upon it. If the fluid is incompressible the equation governing the viscous stress in Cartesian coordinates is. If the fluid is not incompressible the general form for the viscous stress in a Newtonian fluid is. If a fluid does not obey this relation, it is termed a non-Newtonian fluid , of which there are several types.

Non-Newtonian fluids can be either plastic, Bingham plastic, pseudoplastic, dilatant, thixotropic, rheopectic, viscoelastic. In some applications another rough broad division among fluids is made: ideal and non-ideal fluids. An Ideal fluid is non-viscous and offers no resistance whatsoever to a shearing force.

An ideal fluid really does not exist, but in some calculations, the assumption is justifiable. One example of this is the flow far from solid surfaces. In many cases the viscous effects are concentrated near the solid boundaries such as in boundary layers while in regions of the flow field far away from the boundaries the viscous effects can be neglected and the fluid there is treated as it were inviscid ideal flow.