# Courses

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**MATH541** - Algebra
(3 + 0) 5

Groups: quotient groups, isomorphism theorems, direct products, finitely generated abelian groups, actions, Sylow theorems, nilpotent and solvable groups; rings: ring homomorphisms, ideals, factorization in commutative rings, rings of quotients, polynomial rings; modules: exact sequences, vector spaces, tensor products; fields: field extensions, th

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**MATH587** - Applied Mathematics
(3 + 0) 5

Calculus of variations: Euler-Lagrange equation, the first and second variations, necessary and sufficient conditions for extrema, Hamilton`s principle, and applications to Sturm-Liouville problems and mechanics; integral equations: Fredholm and Volterra integral equations, the Green?s function, Hilbert-Schmidt theory, the Neumann series and Fredho

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**MDES600** - Research Methodology and Communication Skills
(3 + 0) 5

Rigorous, scholarly research, particularly theses or dissertations. Literature review, surveys, meta-analysis, empirical research design, formulating research questions, theory building, qualitative and quantitative data collection and analysis methods, validity, reliability, triangulation, building evidences, writing research proposal

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**MATH500** - Graduation Project
(0 + 0) 40

Introduction, sources of information, new definitions and concepts, theoretical grounding, relevant examples and related problems, presentation of the subject in an informative scientific style using modern text formats (TEX, Word, WordScientific, etc.), submission and presentation of the report.

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**MATH316** - Mathematics of Financial Derivatives
(3 + 0) 6

Introduction to options and markets, European call and put options, arbitrage, put call parity, asset price random walks, Brownian motion, Ito?s Lemma, derivation of Black-Scholes formula for European options, Greeks, options for dividend paying assets, multi-step binomial models, American call and put options, early exercise on calls and puts on a

*General Elective*

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**MATH427** - Introduction to Crytopgraphy
(3 + 0) 6

Basics of cryptography, classical cryptosystems, substitution, review of number theory and algebra, public-key and private-key cryptosystems, RSA cryptosystem, Diffie-Hellman key exchange, El-Gamal cryptosystem, digital signatures, basic cryptographic protocols.

*General Elective*

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**MATH467** - Dynamical Systems and Chaos
(4 + 0) 6

One-dimensional dynamic systems, stability of equilibria, bifurcation, linear systems and their stability, two-dimensional dynamic systems, Liapunov?s direct method and method of linearization, 3-dimensional dynamic systems.

*General Elective*

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**MATH482** - Numerical Methods for Ordinary Differential Equations
(3 + 0) 6

Existence, uniqueness and stability theory; IVP: Euler?s method, Taylor series method, Runge-Kutta methods, explicit and implicit methods; multistep methods based on integration and differentiation; predictor?corrector methods; stability, convergence and error estimates of the methods; boundary value problems: finite difference methods, shooting me

*General Elective*

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**MATH485** - Theory of Difference Equations
(3 + 0) 6

The difference calculus, linear difference equations, linear systems of difference equations, self-adjoint second-order linear equations, the Sturm-Liouville eigenvalue problem, boundary value problems for nonlinear equations.

*General Elective*

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**MATH486** - Mathematical Modeling
(3 + 0) 6

Differetial equations and solutions, models of vertical motion, single-species population models, multiple-species population models, mechanical oscillators, modeling electric circuits, diffusion models.

*General Elective*

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**MATH521** - Numerical Analysis I
(3 + 0) 5

Matrix and vector norms, error analysis, solution of linear systems: Gaussian elimination and LU decomposition, condition number, stability analysis and computational complexity; least square problems: singular value decomposition, QR algorithm, stability analysis; matrix eigenvalue problems; iterative methods for solving linear systems: Jacobi, Ga

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**MATH522** - Numerical Analysis II
(3 + 0) 5

Iterative methods for nonlinear equations and nonlinear systems, interpolation and approximation: polynomial trigonometric, spline interpolation; least squares and minimax approximations; numerical differentiation and integration: Newton-Cotes, Gauss, Romberg methods, extrapolation, error analysis.

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**MATH524** - Finite Difference Methods for PDEs
(3 + 0) 5

Finite difference method, parabolic equations: explicit and implicit methods, Richardson, Dufort-Frankel and Crank-Nicolson schemes; hyperbolic equations: Lax-Wendroff, Crank-Nicolson, box and leap-frog schemes; elliptic equations: consistency, stability and convergence of finite different methods for numerical solutions of partial differential equ

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**MATH543** - Group Theory I
(3 + 0) 5

Review of elementary group theory, groups of matrices, normal closure and core of a group, group actions on sets, the wreath product of permutation groups, decompositions of a group, series and composition series, chain conditions, some simple groups, Sylow`s theorem, the simplicity of the projective special linear groups, solvable groups and nilpo

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**MATH552** - Complex Analysis
(3 + 0) 5

Analytic functions as mappings, conformal mappings, complex integration, harmonic functions, series and product developments, entire functions, analytic continuation, algebraic functions.

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**MATH557** - Functional Analysis
(3 + 0) 5

Sets and mappings, countable sets, metric spaces, complete metric spaces, Baire category theorem, compactness, connectednes, normed spaces, linear topological invariants, Hilbert spaces, Cauchy-Schwartz inequality, linear operators, bounded operators, unbounded operators, inverse operators, Hahn-Banach extension theorems, open mapping and closed gr

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**MATH562** - Theory of Differential Equations
(3 + 0) 5

IVP: existence and uniqueness, continuation and continuous dependence of solutions; linear systems: linear (non)homogeneous systems with constant and variable coefficients; structure of solutions of systems with periodic coefficients; higher order linear differential equations; Sturmian theory, stability: Lyapunov (in)stability, Lyapunov functions

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**MATH564** - Impulsive Differential Equations
(3 + 0) 5

General description of IDE, systems with impulses at fixed times, systems with impulses at variable times, discontinuous dynamical systems, general properties of solutions, stability of solutions, adjoint systems, Perron theorem, linear Hamiltonian systems of IDE, direct Lyapunov method, periodic and almost periodic systems of IDE, almost periodic

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**MATH571** - Topology
(3 + 0) 5

Topological spaces, homeomorphisms and homotopy, product and quotient topologies, separation axioms, compactness, connectedness, metric spaces and metrizability, covering spaces, fundamental groups, the Euler characteristic, classification of surfaces, homology of surfaces, simple applications to geometry and analysis.

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**MATH575** - Calculus on Manifolds
(3 + 0) 5

Euclidean spaces, manifolds, the tangent spaces, vector fields, differential forms, integration on manifolds, Stokes? theorem.

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**MATH577** - An Introduction to Low Dimensional Topology
(3 + 0) 5

Knots, links and their invariants, Seifert surfaces, braids, mapping class groups, Heegaard decompositions, lens spaces and surface homeomorphisms, surgery of 3-manifolds, branched coverings.

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**MATH545** - Introduction to Authenticated Encryption
(3 + 0) 5

Fundamentals of cryptography, block ciphers, DES, AES competition, authentication, mode of operations, cryptographic hash functions, collision resistance, birthday paradox, Merkle Damgard construction, MD5, SHA-1, SHA-3 competition, Keccak, authenticated encryption, CAESAR competition, success probability of cryptanalytic attacks, LLR method, hypot

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**MATH542** - Algebraic Number Theory
(3 + 0) 5

Integers, norm, trace, discriminant, algebraic integers, quadratic integers, Dedekind domains, valuations, ramification in an extension of Dedekind domains, different, ramification in Galois extensions, ramification and arithmetic in quadratic fields, the quadratic reciprocity law, ramification and integers in cyclotomic fields, Kronecker-Weber the

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**MATH546** - Galois Theory
(3 + 0) 5

Characteristic of a field, the Frobenius morphism, field extensions, algebraic extensions, primitive elements, Galois extensions, automorphisms, normal extensions, separable and inseparable extensions, the fundamental theorem of Galois theory, finite fields, cyclotomic extensions, norms and traces, cyclic extensions, discriminants, polynomials of d

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**MATH565** - Dynamic Systems on Time Scales
(3 + 0) 5

Differentiation on time scales, integration on time scales, the first-order linear differential equations on time scales, initial value problem, the exponential function on time scales, the second-order linear differential equations on time scales, boundary value problem, Green?s function, the Sturm-Liouville eigenvalue problem.

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**MATH572** - Differentiable Manifolds
(3 + 0) 5

Topological manifolds, differentiable manifolds, tangent and cotangent bundles, differential of a map, vector fields, submanifolds, tensors, differential forms, orientations on manifolds, integration on manifolds, Stokes` theorem.

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**MATH574** - Riemannian Geometry
(3 + 0) 5

Review of differentiable manifolds and tensor fields, Riemannian metrics, the Levi-Civita connections, geodesics and exponential map, curvature tensor, sectional curvature, Ricci tensor, scalar curvature, Riemannian submanifolds, the Gauss and Codazzi equations.

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**MATH582** - Approximation Theory
(3 + 0) 5

Uniform convergence, uniform approximation, Weierstrass approximation theorems, best approximation, Chebyshev polynomials, modulus of continuity, rate of approximation, Jackson?s theorems, positive linear operators, Korovkin?s theorem, Müntz theorems.

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**MATH547** - Algebraic Geometry
(3 + 0) 5

Affine spaces, the Hilbert`s basissatz, the Hilbert`s nullstellensatz, the Zariski`s topology, irreducible sets, algebraic varieties, curves, surfaces, sheafs, ringed spaces, preschemes, affine schemes, the equivalence between affine schemes and commutative rings, projective varieties, dimension, singular points, divisors, differentials.

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**MATH556** - Applied Functional Analysis
(3 + 0) 5

Review on basics in functional analysis, spectral theory of self-adjoint operators in Hilbert spaces, semigroups of operators and their applications to evolution equations, optimal control in Banach spaces.

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**MATH563** - Difference Equations
(3 + 0) 5

The difference calculus, first-order linear difference equations, second-order linear difference equations, the discrete Sturmian theory, Green?s functions, disconjugacy, the discrete Riccati equation, oscillation, the discrete Sturm-Liouville eigenvalue problem, linear difference equations of higher order, systems of difference equations.

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**MATH573** - Algebraic Topology
(3 + 0) 5

The fundamental group, covering spaces, the singular homology, the cellular homology, the simplicial homology, cohomology, universal coefficient theorems for homology, cup product and cross product and cohomology, the Künneth formula.

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**MATH591** - Analytical Probability Theory
(3 + 0) 5

Definition and properties of probability, conditional probability and independence, random variables, probability distributions, their types, classical distributions, moments, random vectors, independent random variables, moment-generating and characteristic function, sums of independent random variables, limit theorems.

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**MATH584** - Nonlinear Problems in Applied Mathematics
(3 + 0) 5

Boundary-value problems for nonlinear second order ordinary differential equations on finite intervals, reducing nonlinear boundary value problems to a fixed point problem, application of the Banach and Schauder fixed point theorems, boundary value problems for nonlinear difference equations, application of the Brouwer fixed point theorem, positive

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**MATH576** - Differential Topology
(3 + 0) 5

Manifolds and differentiable structures, tangent space, vector bundles, immersions, submersions, embeddings, transversality, the Sard?s theorem, the Whitney?s embedding theorem, the exponential map and tubular neighborhoods, manifolds with boundary, the Thom?s transversality theorem.

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**MATH585** - Mathematical Modeling
(3 + 0) 5

Modeling with first-order differential equations: radioactivity, rate of growth and decay; single-species population models, a heat flow model, modeling RL and RC electric circuits; modeling with second order DEs: the motion of a mass on an elastic spring, modeling RLC electric circuits, diffusion models; modeling with systems of DEs: multiple-spe

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**MDES610** - Mathematical Modeling via Differential and Difference Equations
(3 + 0) 5

Differential equations and solutions, models of vertical motion, single-species population models, multiple-species population models, mechanical oscillators, modeling electric circuits, diffusion models, modeling by means of difference equations.

*General Elective*

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**MDES615** - Analytical Probability Theory
(3 + 0) 5

Sigma-algebra of sets, measure, integral with respect to measure; probability space; independent events and independent experiments; random variables and probability distributions; moments and numerical characteristics; random vectors and independent random variables; convergence of random variables; transform methods; sums of independent random v

*General Elective*

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**MDES618** - Probabilistic Methods in Engineering
(3 + 0) 5

Basic notions of probability theory, reliability theory, notion of a stochastic process, Poisson processes, Markov chains, statistical inference.

*General Elective*

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**MDES620** - Numerical Solution of Differential Equations
(3 + 0) 5

Numerical solution of initial value problems; Euler, multistep and Runge-Kutta methods; numerical solution of boundary value problems; shooting and finite difference methods; stability, convergence and accuracy; numerical solution of partial differential equations; finite difference methods for parabolic, hyperbolic and elliptic equations; explic

*General Elective*

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**MDES621** - Numerical Linear Algebra
(3 + 0) 5

Floating point computations, vector and matrix norms, direct methods for the solution of linear systems, least squares problems, eigenvalue problems, singular value decomposition, iterative methods for linear systems.

*General Elective*

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**MDES655** - Linear Optimization
(3 + 0) 5

Sets of linear equations, linear feasibility and optimization, local and global optima, the Simplex method and its variants, theory of duality and the dual-Simplex method, network-Simplex algorithms, computational complexity issues and interior-point algorithms.

*General Elective*

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**MATH555** - Bernstein Polynomials
(3 + 0) 5

Uniform continuity, uniform convergence, Bernstein polynomials, Weierstrass approximation theorem, positive linear operators, Popoviciu theorem, Voronovskaya theorem, simultaneous approximation, shape-preserving properties, De Casteljau algorithm, complex Bernstein polynomials, Kantorovich polynomials.

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**ECON552** - Advance Data Modeling
(3 + 0) 5

Statistical inference, regression, generalized least squares, instrumental variables, simultaneous equations models, and evaluation of policies and programs.

*General Elective*