**MATH 098: Precalculus (6 ECTS credits)**

**Prerequisite: NONE**

Review of Arithmetic. Basic concepts of algebra. Equations. Systems of linear equations. Inequalities. Review of Elementary Euclidean Geometry. Functions and graphing. Inverse functions. Exponential, trigonometric, and logarithmic functions and equations.

** **

**MATH 101: Calculus I (6 ECTS credits)**

**Prerequisite: MATH 098**

Limits of simple functions (polynomial, rational, roots, trigonometric and superpositions of them). Continuity. The Derivative. Maximum-Minimum problems. Curve sketching. Antiderivatives.

** **

**MATH 102: Calculus II (6 ECTS credits)**

**Prerequisite: MATH 101**

Functions: Limits, Continuity, Derivatives.Taylor’s Expansion Formula. Maxima and Minima. Curve sketching. Second derivative tests. L’Hopitals rule. Mean Value Theorems. Antiderivatives. Techniques of integration. The Definite Integral. Applications of the definite integral. Line integrals. Improper Integrals. Functions of Several Variables. Partial derivatives. Maxima and Minima, Saddle-points. Multiple Integrals. Surface Integrals. Infinite Series. Differential Equations.

** **

**MATH 120: Business Statistics (6 ECTS credits)**

**Prerequisite: MATH 101**

Course builds a foundation on statistics and develops applications to business decision making. Includes descriptive statistics, probabilities, estimation, and regression analysis. Course also provides skills for achieving and maintaining statistical process control methods, problem solving tools. Cases and microcomputer statistical package support the course content.

** **

**MATH 131: Introduction to Linear Algebra (6 ECTS credits)**

**Prerequisite: MATH 098**

Matrices and determinants. Systems of linear equations. Linear transformations. Eigenvalues and eigenvectors.

** **

**MATH 132: Linear Algebra (6 ECTS credits)**

**Prerequisite: MATH 098**

Matrices and determinants. System of linear equations. Eigenvalue (Characteristic value) problem and quadratic form, Bilinear forms. Linear spaces. Euclidean and Unitary spaces. Linear operators.

** **

**MATH 140: Introduction to Discrete Mathematics (6
ECTS credits)**

**Prerequisite: MATH 098**

Finite and infinite sets. Principle of inclusion and exclusion. Permutations and combinations. Relations and functions. Properties of Binary relations. Graphs. Trees. Boolean Algebras.

** **

**MATH 201: Calculus II (6 ECTS credits)**

**Prerequisite: MATH 101**

Definite integrals. Techniques of integration. Infinite series. Functional series. Maxima and minima of functions of several variables.

** **

**MATH 204: Advanced Calculus II (6 ECTS credits)**

**Prerequisite: MATH 104, MATH 132 or consent of the
instructor**

Introduction to set theory. The real number system, including the least upper bound property, completeness. Metric spaces, limits and continuity in metric spaces. Compact sets. Connectedness. Normed spaces. Examples of sequences of real numbers and continuous functions. Hilbert space, orthonormal systems. The Riemann integral, conditions for integrability in term of sets of measure zero.

** **

**MATH 205: Infinite Series (6 ECTS credits)**

**Prerequisite: MATH 101 or consent of the instructor**

Sequences. Convergence, divergence of the sequences. Infinite series. Convergence and divergence of series. Positive series. Absolutely and conditionally convergence. Tests for convergence: Ratio test, Root test. Functional series. Uniform convergence. Power series.

** **

**MATH 206: Complex Analysis (6 ECTS credits)**

**Prerequisite: MATH 102**

Complex numbers, complex valued functions of one complex variable, differentiation. Analytic functions, power series. Integration, Cauchy’s integral Formula. Lourent series, Residues and Poles. Conformal mapping. Analytic continuation. Riemann surfaces.

** **

**MATH 210: Ordinary Differential Equations (6 ECTS credits)**

**Prerequisite: MATH 102**

Linear Differential equations of the first order. Linear Differential equations with constant coefficients. Linear Differential equations with variable coefficients. Existence and uniqueness of solutions of first order equations. Existence and uniqueness of solutions to system and n-th order equations.

** **

**MATH 211: Numerical Solution of Ordinary Differential
Equations ****(6 ECTS credits)**

**Prerequisite: MATH 210**

Euler’s method and its modification. Picard’s methods of successive approximation. Equations of the second order and systems of simultaneous equations. The Runge-Cutta method.

** **

**MATH 220: Calculus and Linear Algebra (6 ECTS credits)**

**Prerequisite: MATH 101**

Definite integrals. Infinite series. Functional series. Maxima and minima of multivariate functions. Partial differentiations, optimization, constrained extreme. Linear algebra and its applications.

** **

**MATH 221: Mathematics for Life and Social Sciences (6
ECTS credits)**

**Prerequisite: MATH 098**

Algebra and its applications. Matrices, systems of linear equations and their applications. Techniques and applications of differential calculus. Introduction to integration theory. Linear difference and differential equations. Probability and Statistics.

** **

**MATH** **223: Mathematics for Elementary
Teachers I** **(6
ECTS credits)**

**Prerequisite: NONE**

Mathematical development of arithmetic and geometry; problem solving as those subjects are taught in elementary schools.

**MATH** **224: Mathematics for Elementary
Teachers II** **(6
ECTS credits)**

**Prerequisite: MATH 235**

Development of algebra. Basic ideas of calculus, problem solving, and probability and statistics as those subjects are taught in elementary schools.

** **

**MATH 225: Mathematics for Economics and Business (6
ECTS credits)**

**Prerequisite: MATH 101**

Algebra and its applications. The Mathematics of Finance (Interest, annuities, amortization of loans). Linear Mathematics and its applications. Calculus with applications to economics. Functions of several variables. Partial differentiation, optimization, constrained extrema.

**MATH 231: Applied Linear Algebra (6 ECTS credits)**

**Prerequisite: MATH 132**

Operations on matrices. Inverse matrix. Systems of linear equations. Eigenvalue (Characteristic value) problem. Positive definite matrices. Applications to optimization theory and to linear programming. Linear differential and difference equations.

** **

** **

**MATH 233: Numerical Methods of Linear Algebra (6
ECTS credits)**

**Prerequisite: MATH 132, MATH 204**

Elimination methods. Orthogonalization methods. Iterative and gradient methods. Transformation methods for latent roots and vectors. Corrections to approximate roots and vectors. Numerical evaluation of eigenvalues and eigenvectors.

** **

**MATH 235: Introduction to Mathematical Logic (6
ECTS credits)**

**Prerequisite: MATH 098 or consent of the instructor**

Operations with logical options. Functions of algebra of logic, normal forms. Arithmetical operations of algebra of logic. Negation, conjunction and alternation. Duality. Boolean Schemata.

**MATH 236: Set Theory (6 ECTS credits)**

**Prerequisite: MATH 098 or consent of the instructor**

Algebra of Sets, Relations and functions, numbers. Construction of the real numbers. Axioms of set theory. Cardinality, axiom of choice, transfinite numbers. Continuum hypothesis. Ordinals and order types.

** **

**MATH 237: General Topology (6 ECTS credits)**

**Prerequisite: MATH 236, MATH 101 or consent of the
instructor**

Metric spaces. Topoloqical spaces. Separation axioms. Generalized convergence, continuity, compactness, connectedness. Completion of a metric space. Product topology.

** **

**MATH 238: Number Theory (6 ECTS credits)**

**Prerequisite: MATH 098 or consent of the instructor**

Principle of mathematical induction. Euclid’s division lemma. Factorization and the primes, congruencies. The linear Diophantine equation. The fundamental theorem of arithmetic. Combinatorial number theory. Fermat’s little theorem. Wilson’s theorem. Generating functions. Geometric number theory. Gauss’s circle problem. Dirichlet’s divisor problem.

** **

**MATH 239: Elementary Probability (6 ECTS credits)**

**Prerequisite: MATH 102 or consent of the instructor**

Theory of sets. The sample space. Conditional Probability. Discrete Random Variables. Continuous Random Variables. Univariate Discrete Probability distributions.

** **

**MATH 240: Elementary Statistics (6 ECTS credits)**

**Prerequisite: MATH 102 or consent of the instructor**

Probability spaces, discrete and continuous random variables and their distributions, expected value and Central Limit Theorem. Sampling distributions, estimation of parameters and tests of hypothesis.

** **

**MATH 301: Calculus on Manifolds (6 ECTS credits)**

**Prerequisite: MATH 204, MATH 237**

Multivariable Calculus. Differential mappings of real n-spaces, the inverse and implicit function theorems. Manifolds. Two-dimensional manifolds. Differential forms, tensors and integration on manifolds. The general Stoke’s theorem and its applications.

** **

**MATH 302: Fourier Series and Integrals (6 ECTS credits)**

**Prerequisite: MATH 204, MATH 205**

Trigonometric series and periodic functions. Fourier series. Fourier sine and cosine series. Complex form of Fourier series. Theorem of Riemann. Dirichlet’s formulae, localization principle. Parseval’s equation for the Fourier series. Summability of trigonometric series. Generalized Fourier series. Fourier integral and Fourier transform. Complex form of Fourier integral. Laplace`s integral. Fourier transform of derivatives.

** **

**MATH 304: Integral Transforms (6 ECTS credits)**

**Prerequisite: MATH 204 or consent of the instructor**

Fourier transform. Laplace`s transform. Reverse Laplace`s transform. Hankel and Fourier-Bessel transforms.

** **

**MATH 305: Measure and Integration (6 ECTS credits)**

**Prerequisite: MATH 204**

General topological spaces. The set of continuous functions on a compact metric space. Normal and Hilbert Spaces. Measure and measure space. Lebesgue Integral theory. Lebesgue integrable functions spaces, their completeness. Functions of bounded variation and absolutely continuous functions. Radon - Nikodym theorem.

** **

**MATH 310: Applied Differential Equations (6 ECTS credits)**

**Prerequisite: MATH 102 or consent of the instructor**

Differential equations and special functions. Series solutions. Differential equations of mathematical physics. Methods of separation of variables. Green’s functions. Asymptotics, perturbation methods. Linear systems of differential equations. Phase portrait. Stability theory. Application of differential equations for the solution of economics problems. Transportation. Efficiency of advertisement. Supply and demand. Differential models in ecology. Application of differential equations for the solution of military problems.

** **

**MATH 311: Partial Differential Equations (6 ECTS credits)**

**Prerequisite: MATH 210**

Linear partial differential equations. Second order differential equations. Wave equation, heat equation and Laplace equation. The Method of Separation of variables. Initial-boundary value problems with two or more space - variables.

** **

**MATH 312: Differential Equations of Mathematical
Physics (6 ECTS credits)**

**Prerequisite: MATH 311**

Wave equation. Equation of crossing vibrations. Equation of Helmholtz. Diffusion equation. The equation of conduction of heat. Poisson`s equation. Laplace equation. Stationary equations. The solution of Cauchy`s problem for the equation of crossing vibrations using Dalamber`s method. Method of Separation of variables.

** **

**MATH 320: Numerical Analysis (6 ECTS credits)**

**Prerequisite: MATH 210**

Numerical modelling. Iterative methods for non-linear equations. Simple iteration by algebraic transformation. The bisection method. The Newton-Raphson method. Interpolation. Finite differences and difference operators. Interpolation formulae involving forward and backward differences. Divided differences and Newton’s divided difference formula. Approximation of the integrals. The Trapezium rule. Simpson’s rule.

** **

**MATH 321: Numerical Solution of Differential Equations
(6 ECTS credits)**

**Prerequisite: MATH 312**

Difference equations. Solution of difference equations by iteration. Solution of difference equations by relaxation. Galerkin and Rayleigh-Ritz procedures. Perturbation methods.

** **

**MATH 330: Differential Geometry. Theory of Curves and
Surfaces (6 ECTS credits)**

**Prerequisite: MATH 132, MATH 104**

Curves and surfaces in Euclidean space. The problem of curve’s theory, connected with curvature and twist notions. The basic notions for surfaces connected with contact notion. First and second fundamental forms. The main equations of surface’s theory. Inside geometry of surfaces.

** **

**MATH 331: Algebra: Polynomials and Fields (6 ECTS credits)**

**Prerequisite: MATH 238 or consent of the instructor**

Ring of integers, fields, polynomial domains, unique factorization, and field extension. Galois theory. Solvability by radicals.

** **

**MATH 332: Linear Programming (6 ECTS credits)**

**Prerequisite: MATH 102, MATH 131**

Principles of linear programming, duality theorem, simplex method. Applications to industrial and business problems. Sensitivity analysis, integer programming, distribution and transportation algorithms.

** **

**MATH 333: Mathematical Programming (6 ECTS credits)**

**Prerequisite: MATH 332 or consent of the instructor**

Convex sets. Liner programming . The Dual problem. Optimization of non-linear function. Replacement decisions. Game theory. Statistical quality control. Dynamic programming. Decision analysis.

**MATH 334: Introduction to Operations Research (6
ECTS credits)**

**Prerequisite: MATH 132, MATH 102**

Linear programming: Simplex method, Duality. The transportation problem. Forward and Backward computations. Deterministic models. Non-linear Programming. Kuhn-Tucker conditions. The constraint qualification. Concave programming. Game theory.

** **

**MATH 335: Introduction to Mathematical Economics (6
ECTS credits)**

**Prerequisite: MATH 222**

The notion of mathematical modelling. Examples of mathematical models. Elements of linear programming. Leontiev Input-Output models. Problem of working regime of energy system. Ration problem. Transportation problem. Differential models in economics.

** **

**MATH 340: Applied Statistical Analysis (6 ECTS credits)**

**Prerequisite: MATH 102**

Sample space, events, probability. Conditional probability, Bayes Theorem. Independence, independent trials. Combinatorial probability. Densities and distributions. Joint densities and independence. Expectation. Analysis of variance. Estimation. Covariance and correlation. Linear regression.

** **

**MATH 341: Theory of Probability (6 ECTS credits)**

**Prerequisite: MATH 102**

Combinatorial Analysis. Axioms of Probability. Conditional Probability and Independence Random Variables (Discrete and continuous). Jointly Distributed Random Variables. Expectation. Limits Theorems. Elements of stochastic processes, Markov chains.

** **

**MATH 342: Mathematical Statistics (6 ECTS credits)**

**Prerequisite: MATH 102**

Samples, Histograms and Ogives, Exploratory Data Analysis. Graphical Comparisons of Data Sets, Time Sequences and Digitots, Least Squares, and Correlation. Random Variables of the Discrete type. The Mean, Variance, and Standard Deviation. Random Variables of Continuous type. The Central Limit Theorem. Distributions. Properties of Estimators, Confidence Intervals for Means, Variances and Proportions. The Correlation Coefficient, Conditional Distributions.

** **

**MATH 349: Applied Complex Analysis (6 ECTS credits)**

**Prerequisite: MATH 102**

Introduction to basic formulas and calculation procedures of complex analysis of one variable relevant to applications. Topics Cauchy - Riemann equations. Cauchy’s Integral Formula. Power Series Expansion. Residue Calculus. Application to real analysis and the theory of harmonic functions. Applications to boundary-value problems.

** **

**MATH 351: Introduction to Functional Analysis (6
ECTS credits)**

**Prerequisite: MATH 237, MATH 305**

Normed spaces. Completeness. Examples of normed spaces, including the space of all continuous functions on a compact space. The spaces of Lebesgue integrable functions. Linear Bounded operators. Topological vector spaces. The Banach-Steinhaus theorem (the uniform boundedness principle). The open mapping theorem. The closed graph theorem. The Hahn-Banach theorems. Weak convergence. Reflexive Banach spaces. Hilbert Space. Bilinear forms and operators in Hilbert space. Compact operators. Banach algebras.

** **

**MATH 352: Linear Operators (6 ECTS credits)**

**Prerequisite: MATH 351**

Banach and Hilbert Spaces. Review of fundamental theorems of functional analysis and their consequences. Compact operators. Linear operators in Hilbert space. Basic properties of spectra. Fredholm theory. Introduction to the theory of unbounded operators.

** **

**MATH 420: Boundary Value Problems (6 ECTS credits)**

**Prerequisite: MATH 451, MATH 311**

Basic boundary value problems for second-order linear differential equations. Cauchy`s problem for parabolic and hyperbolic partial differential equations. Boundary - value problems for elliptic equations. The first - Dirichlet`s boundary value problem. The second Neumann`s boundary value problem. The third boundary value problem. Initial - boundary value problem for hyperbolic and parabolic partial differential equations. Function Spaces. Classic and generalized solutions.

** **

**MATH 430: Algebra: Theory of Groups, Rings and Fields
(6 ECTS credits)**

**Prerequisite: MATH 132**

Vector spaces and Linear transformations. Groups. Theory of finite groups. Rings, ideals, and modules. Structure of Rings. Fields, field extension, Galois theory.

** **

**MATH 431: Tensor Products. Algebraic Theory (6
ECTS credits)**

**Prerequisite: MATH 231**

Polylinear algebra. Tensor products. Tensor algebra of linear space. Exterior algebra of linear space. Tensor product of operators and matrices. Spectrum, tensor products and operator equations.

** **

**MATH 432: The Theory of Finite-dimensional Groups (6
ECTS credits)**

**Prerequisite: MATH 132 and consent of the instructor**

Symmetry, geometrical symmetry. Geometry and the measurement of symmetry. Group operations. Simple finite groups. Classification of Simple finite groups. Group transformations. Representations of groups.

** **

**MATH 438: Game Theory (6 ECTS credits)**

**Prerequisite: MATH 332**

Mathematical models and determining of optimal solution by conflict conditions. Formal definition of game. Set of strategies, set of situations, classification of games. General position game. Differential game. Matrix game.

** **

**MATH 440: Estimation Theory (6 ECTS credits)**

**Prerequisite: MATH 342**

Some properties of Estimators. Some Common Unbiased Point Estimators. Evaluating the Goodness of a Point Estimate. Confidence Intervals. A large-Confidence Interval. Selecting the Sample Size. Methods of estimations and properties of point estimators.

** **

**MATH 441: Theory of Regression and Analysis of
Variance (6 ECTS credits)**

**Prerequisite: MATH 342**

Analysis of Variance. An Analysis of Variance Table for a Completely Randomized Design. Estimation for the Completely Randomized Design. The Analysis of Variance for a Randomized Block Design. Selecting the Sample Size. Simple Linear Regressions. Non-linear regression. Multiple regression and correlation analysis.

** **

**MATH 442: Hypothesis Testing (6 ECTS credits)**

**Prerequisite: MATH 342**

Elements of a statistical test. Common Large-Sample Tests. Calculating Type II Error Probabilities and finding the Sample Size for the Z test, Testing Hypotheses concerning Variances. Power of tests. Likelihood Ratio Tests. Minima and Bayes tests.

** **

**MATH 451: Function Spaces (6 ECTS credits)**

**Prerequisite: MATH 351**

Continuous functions on a compact and locally compact spaces. Stone-Weierstrass theorem. Banach spaces of Lebesgue integrable functions. Locally integrable functions. Introduction to the theory of distributions. Fourier transforms. Sobolev`s spaces. Applications to differential equations.

** **

**MATH 452: Introduction to Spectral Theory of Operators
(6 ECTS credits)**

**Prerequisite: MATH 352, MATH 206**

Linear bounded operators in Banach and Hilbert spaces. Banach algebras. Vector-valued holomorphic functions. The spectrum, the resolvent set and the spectral radius of operators. Holomorphic (Symbolic) calculus of operators. Spectral theory of compact operators. Fredholm theory. Normal, self-adjoint and symmetric operators.

** **

**MATH 453: Spectral Theory of Self-adjoint Operators (6
ECTS credits)**

**Prerequisite: MATH 452, MATH 305**

Bilinear form, linear operators and numerical range of operators in Hilbert space. Self-adjoint, unitary and normal operators. Spectra and eigenvalues. Vector-valued measures and resolutions of the identity (spectral measures). The spectral theorem and the functional (symbolic) calculus for self-adjoint (normal) operators. Representation of one-parameter semigroups.

** **

**MATH 454: Introduction to Harmonic Analysis (6
ECTS credits)**

**Prerequisite: Consent of the instructor**

The Basic Cosine-Sine Fourier series. Uniform convergence of Fourier series. Orthogonal sets of functions. Generalized Fourier Series. Sturm-Liouville problems. Bessel Functions and Legendre Polynomials. Groups and introduction to the representation theory. Introduction to general harmonic analysis.

** **

**MATH 455: Banach Algebras (6 ECTS credits)**

**Prerequisite: MATH 351**

Axioms, algebras of continuous functions and bounded linear operators. Complex homomorphisms. Spectrum, resolvent set, spectral radius. Holomorphic functional (symbolic) calculus. Commutative Banach algebras, ideals and homomorphisms. Gelfand transforms. Gelfand-Naimark theorem. Positive functionals, a characterization of B* - algebras. Introduction to operator algebras.

** **

**MATH 462: Advanced Linear Algebra (6 ECTS credits)**

**Prerequisite: MATH 131 or consent of the instructor**

Linear equations, and polynomials. Vector spaces. Linear transformations. Inner Product Spaces. Operators on inner product Spaces. Bilinear forms. Jordan canonical forms. Multilinear algebra and tensor products.

** **

**MATH 469: History and Development of Mathematics (6
ECTS credits)**

**Prerequisite: Consent of the instructor**

The origin of the first mathematical notions and methods. Mathematics in Ancient Egypt and Babylon. Greek schools of Mathematics. Euclid’s Elements. Archimedes - the leading mathematician of Antiquity. Arabic and Hindu mathematics. Mathematics in the near East and the Middle Asia. Mathematics in Middle Ages and European Renaissance. Non-Euclidean geometries. Lobachevski and mathematical mentality. A modern view of development of mathematics.

** **

**MATH 470: Philosophy of Mathematics (6 ECTS credits)**

**Prerequisite: Consent of the instructor**

Mathematics and antique philosophy. Antique philosophy and Euclid’s elements. The Pythagorean mathematical and philosophical school. “Numerical mysticism”. Philosophy and mathematics in the Near East. Philosophical directions of mathematics in India and China. Philosophical directions of mathematics in European Renaissance. Modern situation in mathematical philosophy.

** **

**MATH 550: Linear Differential Operators (6 ECTS credits)**

**Prerequisite: MATH 210, MATH 452**

Definition and examples of boundary value problems. Separation of variables. Regular Sturm Liouville Problems. Linear operators in Hilbert space. The Spectrum, eigenvalues and eigenfunctions. The Basic Fourier series and Fourier series of orthogohal functions. The eigenfunction expansion theorem for self-adjoint regular ordinary differential operators. Singular differential operators. Self-adjoint extensions. On the discreteness of spectra. The expansion problems.

** **

**MATH 551: Inverse Spectral Problems (6 ECTS credits)**

**Prerequisite: MATH 550**

Reviews of spectra of second-order linear ordinary differential operators. Expansion theorem and Parseval`s equation. The determination of potential - function from the spectral measure function. The construction of potential from the S- matrix. Inverse scattering theory for periodic potentials.

** **

**MATH 552: Non-self Adjoint Operators (6 ECTS credits)**

**Prerequisite: MATH 452 or consent of the instructor**

Compact linear operators. The spectrum. Holomorphic calculus. Riesz theory of compact operators Fredholm theory. Introduction to perturbation theory. Norm ideals of compact operators (Von Neumann - Schatten algebras), Hilbert-Schmidt operators, nuclear operators. Completeness and eigenfunction expansions associated with non- self-adjoint operators. The spectral theory of certain general and differential non- self-adjoint operators.

** **

**MATH 553: Tensor Products. Functional Theory (6
ECTS credits)**

**Prerequisite: MATH 431, MATH 452 or consent of the
instructor**

Tensor product of Hilbert Spaces. Tensor product of normed spaces with cross-norms. The two important topologies on tensor products. Tensor products of linear operators. Spectrum and tensor products. Tensor algebra over Hilbert space. Tensor determinants. Tensor products and operator equations. Topological tensor products. Koszul complex, joint spectrum.

** **

**MATH 554: Multiparameter Spectral Theory (6 ECTS credits)**

**Prerequisite: MATH 453, MATH 553**

Tensor products of Hilbert spaces and linear operators. General method of separation of variables. Multiparameter system of linear operators and joint spectra. Tensor-determinant operators and the multiparameter definiteness. Expansion theorem for self-adjoint multiparameter problems. Multiparameter spectrum and Taylor’s joint spectrum. Fredholm family of operators. Multiparameter differential operators.

** **

**MATH 555: Numerical Range of Operators (6 ECTS credits)**

**Prerequisite: MATH 452, MATH 455 is recommended**

Bilinear Forms and Linear Operators in Hilbert space. Numerical range, convexity. Spectrum, spectral radius, numerical range, numerical radius. The description of numerical range of two-dimensional matrices. Homotopic description of finite-dimensional matrices. Numerical range and classification of operators in Hilbert space. Numerical range and non-self-adjoint (non-normal) operators. Numerical ranges of linear operators in Banach spaces, elements of Banach algebras, operator-functions and their applications.

** **

**MATH 556: Boundary Behavior of Solutions of Partial Differential
Equations (6 ECTS credits)**

**Prerequisite: MATH 420**

Principles of radiation. The principle of limit absorption. The principle of limit amplitude. Partial conditions of radiation by Zommerfield.

** **

**MATH 557: Method of Separation of Variables (6
ECTS credits)**

**Prerequisite: MATH 302, MATH 453 or consent of the
instructor**

Fourier Series and Fourier integrals. Partial Differential Equations and Separation of variables. Examples of Wave, Heat and Laplace equations. Boundary value Problems. Sturm- Liouville problems and applications. Tensor product of Hilbert space and linear operators. Abstract method of separation of variables. Two-parameter spectral theory. Joint spectra of commuting operators. Expansion theorem for self-adjoint two-parameter spectral problem.

** **

**MATH 558: A History and Development of
Functional Analysis (6 ECTS credits)**

The origin of Functional Analysis. The evolution of Functional Analysis thought. The leading mathematicians who have made outstanding contributions to the Functional analysis. The modern advances in Functional analysis.

** **

**MATH 590: Special Topics in Mathematics (6 ECTS credits)**

Courses not listed in the catalogue are given as Special Topics courses. Contents vary from year to year according to interest of student and instructor in charge. Courses include various mathematics topics.

** **

**MATH
600: MS Thesis (****12****
ECTS credits)**

Scientific – research work. Individual investigation of the theme to be arranged with supervisor. The results of the work should be presented in the form of thesis.

** **

**MATH 700:** **PhD Dissertation (24 ECTS credits)**

Program of research leading to PhD degree, arranged between a student and the faculty member. Student register to this course in all semesters starting from the beginning of their second semester but not later then the beginning of the third semester while the research program or write-up of the dissertation is in progress.

** **

**MATH 810: Probability and Theory of Statistics (6
ECTS credits)**

**Prerequisite: BSA 250**

This course presents a general outline of the mathematical theory of probability and statistics. Topics include random variables, discrete and continuous, probability distributions, sampling distribution theory, estimation, and multivariate distributions. It covers advanced topics in the theory of probability and statistics including Bayesian estimation, chi-square tests, quadratic forms, analysis of variance, non-parametric statistics, sufficiency, and the multivariate normal distribution.