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These are the Graduate Courses of the Institute of Mathematics.

 

COURSE NO. DESCRIPTION SYLLABUS
Math 201

Concepts and Techniques in Abstract Algebra
Groups, rings and homomorphisms.

Prerequisite: Math 109 or COI

Credit: 3 u.

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Math 202.1

Analysis I
Real numbers; sequences of real numbers and limits; continuity of functions; derivatives; Riemann integral.

Prerequisite: COI

Credit: 3 u.

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Math 202.2

Analysis II
n-dimensional Euclidean space; functions of several variables; partial derivatives; multiple integrals; complex numbers; complex-valued functions and their derivatives

Prerequisite: Math 202.1

Credit: 3 u.

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Math 203

Matrices and Applications
Linear systems of equations and matrices; matrix operations; determinants; vector spaces and linear transformations; eigenvalues and eigenvectors.

Prerequisite: COI

Credit: 3 u.

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Math 204

Classical and Modern Geometry
Finite geometries, euclidean and non-euclidean geometries,projective geometry, geometric transformations

Prerequisite: COI

Credit: 3 u.

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Math 205

Concepts and Techniques in Probability and Statistics
Descriptive statistics, probability and probability distributions, sampling theory, estimation and test of hypothesis, linear correlation and regression analysis.

Prerequisite: COI

Credit: 3 u.

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Math 208

History and Development of the Fundamental Concepts of Mathematics
The Development of Mathematics: a historical over; The Nature of Mathematics, Issues and Aspects of Mathematics.

Prerequisite: COI

Credit: 3 u.

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Math 209.1

Selected Topics in Applied Mathematics

Prerequisite: Math 201

Credit: 3 u.

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Math 209.2

Selected Topics in Discrete Mathematics

Prerequisite: COI

Credit: 3 u. ; topic to be specified for record purposes

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Math 210.1

Modern Algebra I
Semi-groups and groups; rings; fields; groups with operators; selected topics.

Prerequisite: COI

Credit: 3 u.

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Math 210.2

Modern College Algebra II
Continuation of Math 210.1

Prerequisite: Math 210.1

Credit: 3 u.

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Math 211

Linear Algebra
Vector spaces; linear mappings, theorem o Hamilton-Cayley; modules over principal ideal domains; Jordan cononical forms; rational canonical forms; bilinear forms; inner products; law of inertia; spectral theorem; multi-linear forms; tensors products.

Prerequisite: 114 or Math 110.2 or COI

Credit: 3 u.

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Math 212

Theory of Groups
Definitions and examples; normal subgroups and homomorphisms; Abelian groups; Sylow theorems; composition series and solvable groups.

Prerequisite: COI

Credit: 3 u.

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Math 213

Theory of Rings
Representation and structure of rings; Ideal Theory.

Prerequisite: COI

Credit: 3 u.

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Math 214

Theory of Matrices

Prerequisite: COI

Credit: 3 u.

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Math 216

Lie Groups and Lie Algebras
Classical matrix Lie groups, Lie algebras of Lie groups, nilpotent and solvable algebras, semi-simple algebras, representations.

Prerequisite: Math 210.1

Credit: 3 u.

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Math 217

Theory of Numbers
Linear consequences; Euler's and Wilson's theorem; quadratic residues; quadratic reciprocity law; Jacobi's and Kroneeker's symbols; Polian equations; positive binary and temary quadratic forms; theory of the sums of two and three squares.

Prerequisite: COI

Credit: 3 u.

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Math 218

Theory of Algebraic Numbers
Algebraic number fields; algebraic integrals; basic and discriminants ideals, fundamental theorem on the decomposition of ideals; ideal classes; Minkowski's theorem; the class formula; units; Fermat's last theorem; selected topics.

Prerequisite: COI

Credit: 3 u.

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Math 220.1

Theory of Functions of a Real Variable I
Lebesque and other integrals; differentiation; measure theory.

Prerequisite: COI

Credit: 3 u.

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Math 220.2

Theory of Functions of a Real Variable II
Continuation of Math 220.1; selected topics

Prerequisite: COI

Credit: 3 u.

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Math 221

Partial Differential Equations
Equations of the firsts and second order; Green's functions; boundary value problems.

Prerequisite: COI

Credit: 3 u.

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Math 222

Approximation Theory
Taylor's theorem, Weierstrass approximation theorem, approximation in Hilbert spaces, Fourier series and Fourier transform, direct and inverse theorems, algebraic and trigonometric interpolation, Whittaker-Shannon sampling theory, wavelet analysis.

Prerequisite: Math 220.1 or equivalent

Credit: 3 u.

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Math 224

Control Theory
Elements of Calculus of variations. Naïve optimal control theory. Functional Analysis. Generalized optimal control Theory. The Pontajagin's maximum principle for chattering controls; research problem.

Prerequisite: Math 126 and Math 142 or COI

Credit: 3 u.

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Math 227

Calculus of Variations
Euler's equation; Legendre conditions; Jacobi's conditions; Isoperimitic problems; Lagrange method; Dirichlet's principle.

Prerequisite: COI

Credit: 3 u.

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Math 228

Theory of Functions of a Complex Variable
Analytic functions; geometric functions theory; analytic continuation; Reimann mapping theorem

Prerequisite: Math 128.2 or equivalent

Credit: 3 u.

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Math 229

Functional Analysis
Linear operators; linear functionals; topological linear spaces; normed spaces; Hilbert spaces; functional equations; Randon measures; distributive and linear partial differential equations and spectral analysis.

Prerequisite: Math 226.1

Credit: 3 u.

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Math 235

Mathematics in Population Biology
Continuous and discrete population models for single species, models for interacting populations, evolutionary models, dynamics of infectious diseases.

Prerequisite: Math 121.1/equivalent or COI

Credit: 3 u.

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Math 236

Mathematics in Biological Processes
Biological oscillators and switches, perturbed and coupled oscillators, reaction diffusion, enzyme kinetics, chemotaxis, circadian systems models, coupled cell networks.

Prerequisite: COI

Credit: 3 u.

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Math 240

Geometric Crystallography
Isometries, frieze groups, crystallographic groups, lattices and invariant sublattices, finite groups of isometries, geometric and arithmetic crystal classes.

Prerequisite: COI

Credit: 3 u.

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Math 241

Hyperbolic Geometry
Moebius transformations, hyperbolic plane and hyperbolic metric, geometry of geodesics, hyperbolic trigonometry, groups of isometries on the hyperbolic plane.

Prerequisite: COI

Credit: 3 u.

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Math 242

General Topology
Topological spaces; metric spaces; theory of convergence; bases; axioms of countability; subspaces; homomorphisms; selected topics.

Prerequisite: COI

Credit: 3 u.

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Math 243

Algebraic Topology
Homotopy, fundamental group, singular homology, simplicial complexes, degree and fixed point theorems.

Prerequisite: Math 242

Credit: 3 u.

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Math 246

Differential Geometry
Classsical Theory of curves and surfaces; mappings of surfaces; differential structures; Lie groups and frame bundles.

Prerequisite: Math 123.2 of COI

Credit: 3 u.

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Math 247

Algebraic Geometry
The general projective space; collineation and correlation in a projective space; algebraic manifolds; plane curves; quadratic transformation of systems of plane curves.

Prerequisite: COI

Credit: 3 u.

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Math 249

Selected Topics in Geometry and Topology
Metric and projective treatment of the hyperbolic and elliptic geometries.

Prerequisite: COI

Credit: 3 u.

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Math 250

Probability Theory
Random variables, laws of large numbers, special probability distributions, central limit theorem, Markov chains, Poisson process, martingales.

Prerequisite: Math 220.1 or COI

Credit: 3 u.

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Math 255

Mathematics of Decision Making
Some applications of Bayesan statistics; use of experiments in decision problems; group decision making and risk sharing.

Prerequisite: Math 155

Credit: 3 u.

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Math 258

Combinatorial Mathematics
Permutations and combinations; generating functions; principle of inclusion and exclusion; recurrence relations; occupancy; matrices of zeros and ones; partitions; orthogonal latin squares; combinatorial designs.

Prerequisite: Math COI

Credit: 3 u.

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Math 260

Actuarial Theory and Practice
Multiple life theory, multiple decrement theory, applications of multiple decrement theory, risk theory, ruin theory and introduction to credibility theory.

Prerequisite: Math 164 or COI

Credit: 3 u.

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Math 261

Survival and Loss Models.
Hazard rate function, analysis of various survival and loss models, credibility theory.

Prerequisite: Math 164 or COI

Credit: 3 u.

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Math 262.1

Actuarial Science I
Gross premiums and asset shares, non forfeiture values, expense analysis, distribution of surplus, valuation of liabilities, product development process, introduction to life insurance accounting.

Prerequisite: Math 261 or COI

Credit: 3 u.

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Math 262.2

Actuarial Science II
Selection of risks, reinsurance, introduction to investments analysis and finance management, insurance code, actuarial principles in special lines of insurance.

Prerequisite: Math 262.1 or COI

Credit: 3 u.

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Math 265

Stochastic Calculus
Conditional expectations, martingales, Brownian motion, Ito integral, Ito formula, stochastic differential equation, Girsanov Theorem, applications to mathematical finance.

Prerequisite: Math 150.1 or COI

Credit: 3 u.

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Math 266

Mathematical Finance
Binomial asset pricing model, vanilla options, exotic options, American options, arbitrage probabilities, profit and loss, stochastic interest rates.

Prerequisite: Math 265 or COI

Credit: 3 u.

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Math 271.1

Numerical Analysis I
Floating point representation, condition numbers, iterative methods for solving systems of linear and non-linear equations, numerical integration, numerical linear algebra.

Prerequisite: COI

Credit: 3 u.

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Math 271.2

Numerical Analysis II
Numerical methods for ordinary differential equations, finite-difference methods for partial differential equations, numerical methods for conservation laws, multi-grid methods.

Prerequisite: Math 271.1 or COI

Credit: 3 u.

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Math 272

Automata Theory
Finite state automata; Regular expression; Decomposition of Finite Automata and their realization, Turning Machines, Introduction to formal Languages.

Prerequisite: Math 271.1 or COI

Credit: 3 u.

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Math 276

Introduction to Computer Simulation
Introduction to computer simulation of theoretical system and real-time processes; examples of simulation for the solution of both theoretical and practical problems in various fields of applications.

Prerequisite: COI

Credit: 3 u.

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Math 280

Linear Programming
Simplex method; duality; geometry of linear programs; parametric programming; decomposition and upper bound variables.

Prerequisite: Math 271.1 or COI

Credit: 3 u.

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Math 281

Non-linear Programming
Properties of convex sets and functions; unconstrained optimization; Kuhn - Tucker theorem; lagrange multipliers. Saddle point theorems; algorithms.

Prerequisite: Math 280 or equivalent

Credit: 3 u.

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Math 282

Integer Programming and Combinatorial Optimization
Applications of integer programming; converging dual and primal cutting plane algorithms; Branch-bound methods; total unimodularity and the transportation problem; application of graph theory to mathematical programming.

Prerequisite: Math 280 or equivalent

Credit: 3 u.

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Math 283

Applied Dynamic Programming
Deterministic decision problems; analytical and computational methods; applications to problems of equipment replacement; resource allocation; scheduling; search and routing.

Prerequisite: Graduate standing and COI

Credit: 3 u.

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Math 285

Introduction to Stochastic Optimization
Probability theory and application to discrete and continuous – time Markov chains; classifications of states; algebraic methods; birth and death processes, renewal theory, limit theorems.

Prerequisite: Math 114 and Math 150.1

Credit: 3 u.

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Math 286

Finite graphs and Networks
Basic graph theory and applications to optimal path problems; flows in network; combinatorial problems.

Prerequisite: Math 285 or COI

Credit: 3 u.

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Math 288

Numerical Optimization.
Deterministic descent type methods, stochastic optimization methods, numerical implementation.

Prerequisite: Math 271.1 or COI

Credit: 3 u.

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Math 290

Prerequisite: Math 271.1 or COI

Prerequisite: COI

Credit: 3 u.

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Math 294

Independent Study

Credit: 3 u. May be credited once in the M.S. Mathematics/ Applied Mathematics programs and twice in the Ph.D. Mathematics program.

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Math 295

Special Project

Prerequisite: COI

Credit: 3 u.

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Math 296

Graduate Seminar

Credit: 1 u.

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Math 297

Special Topics

Credit: 3 u. ; topic to be specified for record purposes.

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Math 300

Master's Thesis

Credit: 6 u.

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Math 400

Ph.D. Dissertation

Credit: 12 u.

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