# ECTS - - Ph.D. on B.S. in Mathematics

### Compulsory Departmental Courses

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**MATH611** - Mathematical Analysis
(3 + 0) 5

Sets and mappings, countable and uncountable sets, real number system, completeness; metric spaces, complete metric spaces; Banach fixed point theorem; sequences and series of functions, sigma algebras, measures, integral with respect to measure, convegence theorems (monotone and dominated), modes of convergence.

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**MATH622** - Advanced Linear Algebra
(3 + 0) 5

Basic linear algebra, linear transformations, the structure of a linear operator, eigenvalues and eigenvectors, real and complex inner product spaces, structure theory for normal operators, metric vector spaces: the theory of bilinear forms, Hilbert spaces, tensor products, operator factorizations: QR and singular values.

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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

### Elective Courses

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**IE519** - Forecasting
(3 + 0) 5

Forecasting methodology and techniques; dynamic Bayesian modelling; methodological forecasting and analysis; polynomial, seasonal, harmonic and regression systems; superpositioning; variance learning; forecast monitoring and applications; time series analysis and forecasting; moving averages; estimation and forecasting for arma models; arma models;

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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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**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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**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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**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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**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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**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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**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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**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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**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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**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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**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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**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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**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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**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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**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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**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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**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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**MATH640** - Time Series Analysis
(3 + 0) 5

Graphical representation of economic time series, ergodicity and stationarity, stochastic difference equation models, autoregressive processes, moving average processes, mixed processes, forecasting, the relation between econometric models and ARMA processes, Granger causality, causality tests, vector autoregressive processes, nonstationary processes, cointegration in single equation models, cointegration in vector autoregressive models.

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**MATH651** - Measure Theory and Functional Analysis
(3 + 0) 5

Topological and metric spaces, measure spaces and integration, Banach spaces and the four structural Banach space theorems, an analysis of C(X) and the Stone - Weierstrass theorem and its applications, Hilbert spaces and Hilbert bundles, the continuous and Borel functional calculus and the spectral theorem.

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**MATH658** - Operator Theory
(3 + 0) 5

This is an introductory course in Operator Theory and its applications. We will start with a short review of Banach and Hilbert Spaces and their linear bounded operators, then we turn to the Spectral Theory of linear bounded operators. Classes of Hilbert Space operators that appear frequently in applications, e.g. self adjoint, positive operators, are considered in detail.

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**MATH659** - Spectral Representations and Unbounded Operator Theory
(3 + 0) 5

Projection operators and their properties on Hilbert Spaces, Spectral family, Spectral representation of self adjoint operators, Unbounded operators on a Hilbert space, Spectral representation of unitary operators and not necessarily bounded self adjoint operators, applications of unbounded operators in Quantum Mechanics.

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**MATH670** - Mathematical Models in Biology
(3 + 0) 5

Linear and nonlinear biological models via difference equations; linear and nonlinear biological models via differential equations; special topics in mathematical biology including predator-prey models, SI,SIS,SIR epidemic models, competition models of two and three species, Van Der Pol equation, Hodgkin-Huxley and FitzHugh-Nagumo models.

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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.

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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

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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

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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.

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**ME683** - Boundary Element Method
(3 + 0) 5

Introduction; preliminary concepts: vector and tensor algebra, indicial notation; divergence theorem, Dirac delta function; singular integrals; Cauchy principal value integrals in 1 and 2D; boundary element formulation for Laplace equation; Laplace equation: discretization; boundary element formulation for elastostatics; elastostatics: discretization.

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**MFGE505** - Practical Finite Elements (Linear Finite Element)
(3 + 0) 5

Background and application of FE, direct approach, strong and weak forms, weight functions and Gauss quadrature, FE formulation for 1D problems, plane strain/stress and axisymmetric problems, displacement based FE formulation, isoparametric elements, performance of displacement based elements and volumetric locking; reduced selective integration.

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**MFGE508** - Boundary Element Method
(3 + 0) 5

Introduction, preliminary concepts, vector and tensor algebra, indicial notation, divergence theorem, Dirac delta function; singular integrals, Cauchy principal value integrals in 1 and 2D, boundary element formulation for Laplace equation, Laplace equation; discretization, boundary element formulation for elastostatics, elastostatics, discretizati