**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.