Introduction

MAS 101, MAS 121 and MAS 131 are prerequisites for all the other degree courses. The additional prerequisites are given in the descriptions below. The prerequisites may, in exceptional cases, but not for the chain courses, be replaced with approval of the instructor and consent of the Department. The courses in each of the following groups are considered as chain courses:
a) MAS 101, MAS 102, MAS 103
b) MAS 101, MAS 102, MAS 201
c) MAS 121, MAS 221
d) MAS 171, MAS 271
e) MAS 151, MAS 250, MAS 251
f) MAS 203, MAS 204.
Moreover, all the 200-level courses have as a prerequisite the successful completion of a one-semester foreign-language course.

MAS 101 CALCULUS I
The real-number system. Real functions. Limits of sequences and real functions of one variable. Continuity and differentiation of functions of one variable.

MAS 102 CALCULUS II
(Prerequisite: MAS 101 )
Riemann integration. Applications of differentiation and integration. Sequences and infinite series of real numbers.

MAS 103 CALCULUS III (Prerequisite: MAS 102 )
Functions of several variables. Continuity, differentiation, basic theorems and applications. Vector calculus, directional derivative, gradient. Extremum problems, multiple integration. Differential forms, surface integrals, Stokes theorem.

MAS 121 LINEAR ALGEBRA I
Linear spaces. Linear dependence, basis, and dimension. Matrix theory and determinants. Linear systems and the Gauss elimination method. Inverse matrix. Linear transformations, image, and kernel. Eigenvalues, eigenvectors and matrix diagonalization.

MAS 131 GENERAL MATHEMATICS
Fundamental theory of sets, functions, equivalence relations with examples from geometry, number theory and group theory. Mathematical induction with examples from number theory and topology. Introduction to complex numbers. Introduction to number theory and group theory.

MAS 151 PROBABILITY-STATISTICS I (Prerequisite: MAS 102 )
Probability, random variables, distribution functions. Distributions, independence, expected value. Covariance. Modes of convergence of sequences of random variables, laws of large numbers.

MAS 171 NUMERICAL ANALYSIS I ( Prerequisites: CSC 002, MAS 102 )
Sources, propagation and analysis of errors. Solution of non-linear equations. Solution of systems of linear equations. Polynomial interpolation. Numerical differentiation and integration.

MAS 181 VECTOR CALCULUS (Prerequisite: MAS 102 )
Vector algebra, vector functions. Real functions of two variables, maxima, minima. Double and triple integrals, line integrals. Theorems of Green, Gauss, Stokes.

MAS 201 REAL ANALYSIS (Prerequisite: MAS 102 )
Basic topology, metric spaces, sequences, series, continuity, differentiation and integration. Special functions. Fourier Series.

MAS 203 ORDINARY DIFFERENTIAL EQUATIONS
Existence, uniqueness, smooth dependence of the solutions on parameters. Methods for solving 1st and 2nd degree linear equations. Linear equations with constant coefficients. Method of powerseries: Smooth and singular solutions. Linear systems of ordinary differential equations. Nonlinear equations: phase plane, stability, periodicity, chaos.

MAS 204 PARTIAL DIFFERENTIAL EQUATIONS (Prerequisite: MAS 203 )
First order equations - Characteristics-Second order equations-Elliptic, Hyperbolic and Parabolic equations- Laplace's, Wave and Diffusion equations. Method of separation of variables- Fourier transforms- Boundary value problems- Applications to Mathematical Physics- Similarity solutions.

MAS 205 FUNCTIONAL ANALYSIS (Prerequisite: MAS 201 )
Metric spaces, linear transformations, linear functional, dual spaces. Hahn-Banach theorem. Banach spaces. Fixed point theorem and applications. Banach-Steinhaus theorems. Reflexivity and separability. Hilbert space. Topological vector spaces. Open mapping theorem and closed graph.

MAS 206 COMPLEX ANALYSIS (Prerequisite: MAS 181 )
Complex numbers, holomorphic functions, Cauchy- Riemann equations, harmonic functions. Exponential, trigonometric and logarithmic functions. Integration, Cauchy's theorem, Cauchy's integral formula. Morera's theorem, Liouville's theorem, the fundamental theorem of algebra. The Maximum modulus theorem. Taylor series, Laurent series, calculus of residues. Conformal mapping, linear fractional transformation, the Riemann Mapping theorem.

MAS 221 LINEAR ALGEBRA II
Groups. Rings. Fields. Algebras. Linear mappings. Isomorphisms. The algebra L(V)=Hom(V,V). Linear mappings and matrices. Change of basis matrix. Inner product spaces. Cauchy-Schwarz inequality. Gram-Schmidt algorithm. Positive definite matrices. Eigenvalues and eigenvectors. Cayley-Hamilton theorem. Minimum polynomial. Canonical forms. Invariant subspaces. Jordan canonical form.

MAS 223 NUMBER THEORY
Divisibility theory in the integers, Euclidean algorithms. Primes and their distribution. Congruence modulo n, Fermat's theorem, Quandatic reciprocity. Perfect numbers, representation of integers as sum of squares. Fermat's conjecture. Continued fractions.

MAS 225 GROUP THEORY
Definition of a group. Subgroups. Lagrange's theorem. Examples of groups: Cyclic groups, Dihedral groups, symmetric groups. Abelian groups. Normal subgroups and quotient groups. The homomorphism theorems. The Jordan-Holder theorem. Action of group on sets. Sylow's theorems.

MAS 226 THEORY OF RINGS AND FIELDS
Rings, ideals, integral domains, Euclidean domains. Polynomial rings. Fields, field extensions. Algebraically closed fields, finite fields. Ruler and Compass constructions.

MAS 231 DIFFERENTIAL GEOMETRY I (Prerequisites: MAS 102 , MAS 181 )
Theory of curves in R2 and R3. Curvature, Frenet formulae. Theory of surfaces in R3. First and second fundamental forms. Gauss map, Gauss curvature, isometry. Theorema Egregium (Gauss). Parallel transport, geodesics.

MAS 233 TOPOLOGY (Prerequisite: MAS 201 )
Topological spaces, bases, separation axioms. Metric spaces, complete metric spaces. Baire's theorem. Compact spaces, connected spaces. Product topology, quotient topology. The Fundamental group.

MAS 234 ALGEBRAIC TOPOLOGY (Prerequisites: MAS 225, MAS 233 )
Fundamental group, homotopy, covering spaces. Introduction to homology.

MAS 250 PROBABILITY-STATISTICS II (Prerequisite: MAS 151 )
Conditional distributions and conditional expected values. Moment generating functions and characteristic functions. Limit theorems. Introduction to inferential statistics (estimation, confidence intervals, tests).

MAS 251 PROBABILITY-STATISTICS III (Prerequisite: MAS 250 )
Statistics, estimation (sufficiency, unbiasedness, UMVU estimators, consistency, maximum likelihood estimators) exponential families of distributions, confidence intervals, hypothesis testing, X

MAS 252 STOCHASTIC PROCESSES (Prerequisite: MAS 250 )
Examples of stochastic processes. Random walk. Markov chains, continuous time Markov chains. Poisson processes, birth and death processes, renewal processes, branching processes, applications.

MAS 253 LINEAR MODELS (Prerequisite: MAS 251 )
Normal, t,F,X2 distributions. Linear regression with one independent variable: estimation, hypothesis testing problems and robustness. Multiple regression: estimation, tests and model selection. Single-factor analysis of variance: tests and analysis of factor effects. Two-factor analysis of variance for equal and unequal sample sizes. Single-and two-factor analysis of covariance.