# ECTS - - Mathematics

### Compulsory Departmental Courses

####
**CMPE221** - Introduction to Object Oriented Programming
(2 + 2) 6

Object-oriented thinking, review of programming paradigms, abstract data type, scope rules and access controls, classes, constructors and destructors, operator overloading, introduction to object oriented concepts: inheritance, polymorphism. Templates.

####
**MATH112** - Discrete Mathematics and Combinatorics
(3 + 0) 6

Numbers and counting, countable and uncountable sets, continuum, the Pigeonhole Principle and its applications, permutations and combinations, combinatorial formulas, recurrence relations, principle of inclusion and exclusion, binary relations, elementary graph theory.

####
**MATH136** - Mathematical Analysis II
(4 + 2) 8.5

Riemann integral, the fundamental theorem of calculus, integration techniques, applications of integrals: area, volume, arc length, improper integrals, sequences, infinite series, tests for convergence, functional sequences and series, interval of convergence, power series, Taylor series and its applications.

####
**MATH247** - Introduction to Object-Oriented Programming
(2 + 2) 6

Object-oriented thinking, review of programming paradigms, abstract data type, scope rules and access controls, classes, constructors and destructors, operator overloading, introduction to object oriented concepts: inheritance, polymorphism. templates.

####
**MATH251** - Advanced Calculus I
(3 + 2) 8

Vector and matrix algebra, functions of several variables: limit, continuity, partial derivatives, chain rule; implicit functions. inverse functions, directional derivatives, maxima and minima of functions of several variables, extrema for functions with side conditions.

####
**MATH331** - Abstract Algebra
(4 + 0) 7

Groups: subgroups, cyclic groups, permutation groups, Lagrange Theorem, normal subgroups and factor groups, homomorphisms, isomorphism theorems, rings and fields: subrings, integral domains, ideals and factor rings, maximal and prime ideals, homomorphisms of rings,field of quotients, polynomial rings, principal ideal domain (PID), irreducibility of

####
**MATH351** - Introduction to Real Analysis
(4 + 0) 7

A review of sets and functions, real numbers (or system), countable and uncountable sets, sequences of real Numbers (Cauchy sequences), Uniform Convergence of Sequences of functions, Metric Spaces, Compactness and Connectedness, Contraction Mapping Theorem, Arzela-Ascoli Theorem, Extension Theorem fo Tietze, Baire?s Theorem.

####
**MATH374** - Differential Geometry
(3 + 0) 6

Curves in the plane and space, curvature and torsion, global properties of plane curves, surfaces in space, the First Fundamental Form, curvatures of surfaces, Gaussian curvature and the Gauss Map, geodesics, minimal surfaces, Gauss`s Theorema Egregium, the Gauss-Bonnet Theorem.

####
**MATH392** - Probability Theory and Statistics
(4 + 0) 7

Probability spaces, conditional probability and independence, random variables and probability distributions, numerical characteristics of random variables, classical probability distributions, random vectors, descriptive statistics, sampling, point estimation, interval estimation, testing hypotheses.

####
**TURK401** - Turkish Language I
(2 + 0) 2

Languages and their classification; history of Turkish language, its spread over the world and its place among other languages; Turkish language in the republic era; orthography; expressions; foreign words, suffixes and prefixes; punctuation; language and verbalism.

### Elective Courses

####
**MATH313** - Introduction to Mathematical Finance
(3 + 0) 6

Introduction to theory of interest: simple and compound interest, time value of money, rate of interest, rate of discount, nominal rates, effective rates, compound interest functions, generalized cash flow modelling, loans, present value analysis, accumulated profit, and internal rate of return for investment projects, annuities, perpetuities, meas

####
**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

####
**MATH318** - Hıstory of Mathematics I
(3 + 0) 6

Prehistoric mathematics, Ancient Near East mathematics (Mesopotamia-Egypt, 3rd millenium BC?500 BC), Greek and Hellenistic mathematics (c. 600 BC?300 AD), Chinese mathematics (c. 2nd millenium BC?1300 AD), Indian mathematics (c. 800 BC?1600 AD), Islamic mathematics (c. 800?1500).

####
**MATH332** - Finite Fields
(3 + 0) 6

Characterization of finite fields, roots of irreducible polynomials, trace, norm, roots of unity and cyclotomic polynomials, order of polynomials and primitive polynomials, irreducible polynomials, construction of irreducible polynomials, factorization of polynomials

####
**MATH333** - Matrix Analysis
(3 + 0) 6

Preliminaries, eigenvalues, eigenvectors and similarity, unitary equivalence and normal matrices, Canonical forms, Hermitian and symmetric matrices, norms for vectors and matrices, location and perturbation of eigenvalues, positive definite matrices, nonnegative matrices.

####
**MATH337** - Bernstein Polynamials
(3 + 0) 6

Weierstrass Approximation Theorem, Definition of Bernstein Polynomials, Derivatives of Bernstein Polynomials, Approximation of the Derivatives, The Degree of Approximation by Bernstein Polynomials, Theorems of Popoviciu and Voronovskaya, Shape-Preserving Properties of Bernstein Polynomials, Generalizations of Bernstein Polynomials, Kantorovich, Durrmeyer and Chlodowskii Polynomials, Bernstein Polynomials in Complex Domain. ●Prerequisite: MATH 136

####
**MATH357** - Functional Analysis
(3 + 0) 6

Vector spaces, Hamel basis, linear operators, equations in operators, ordered vector spaces, extension of positive linear functionals, convex functions, Hahn-Banach Theorem, The Minkowski functional, Separation Theorem, metric spaces, continuity and uniform continuity, completeness, Baire Theorem, normed spaces, Banach spaces, the algebra of bounde

####
**MATH360** - Theory of Ordinary Differential Equations
(3 + 0) 6

First-order ordinary differential equations, the Existence and Uniqueness Theorem, systems and higher-order ordinary differential equations, linear differential equations, boundary value problems and eigenvalue problems, oscillation and comparison theorems.

####
**MATH372** - Topology
(3 + 0) 6

Fundamental concepts, functions, relations, sets and Axiom of Choice, well-ordered sets, topological spaces, basis, the Order Topology, the Subspace Topology, closed sets and limit points, continuous functions, the Product Topology, Metric Topology, the Quotient Topology, connectedness and compactness, Countability and Separation Axioms, the fundam

####
**MATH378** - Partial Differential Equations
(3 + 0) 6

Basic concepts; first-order partial differential equations; types and normal forms of second-order linear partial differential equations; separation of variables; Fourier series; hyperbolic, parabolic and elliptic equations; solution of the Wave Equation.

####
**MATH381** - Numerical Analysis
(3 + 0) 7

Computational and mathematical preliminaries, numerical solution of nonlinear equations and systems of nonlinear equations, numerical solution of systems of linear equations, direct and iterative methods, the Algebraic Eigenvalue Problem, interpolation and approximation, numerical differentiation and integration, numerical solution of ODE`s.

####
**MATH417** - Computational Methods of Mathematical Finance
(2 + 0) 6

Introduction to MATLAB, finite difference formulae, the explicit and implicit finite difference methods, The Crank-Nicolson method, European option pricing by the heat equation, pricing by the Black-Scholes equation, pricing by an explicit, an implicit and Crank-Nicolson method, pricing American options, projected SOR and tree methods, pseudo-rando

####
**MATH419** - History of Mathematics II
(3 + 0) 6

Early Middle Ages European mathematics (c. 500?1100), mathematics of the Renaissance: rebirth of mathematics in Europe (1100?1400), early modern European mathematics (c. 1400?1600): solution of the cubic equation and consequences, invention of logarithms, time of Fermat and Descartes, development of the limit concept, Newton and Leibniz, the age of

####
**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.

####
**MATH437** - Statistical Methods and Financial Applications
(3 + 0) 6

Central tendency/dispersion measures, moments, maximum likelihood estimation, correlation and simple linear regression, multi-regression model, autocorrelation and multi-collinearity on regression models, portfolio theory, CAPM and ARMA approaches.

####
**MATH441** - Group Theory
(3 + 0) 0

Review of Elementary Group Theory, Group Actions on Sets, Finite p-groups and Sylow's Theorem, Groups of Small Orders, Compositions Series and Jordan-Hölder's Theorem, Soluble groups and Nilpotent groups, The Frattini Subgroups and Burnside's Basis Theorem, Direct Products, Direct Sums and the Structure of Finitely Generated Abelian Groups, Free Groups and Presentations.

####
**MATH443** - Algebraic Number Theory
(3 + 0) 0

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, The Kronecker-Weber Theorem on Abelian Extensions, The Dirichlet's Theorem on the Finiteness of the Class Group, The Dirichlet's Theorem on Units, The Hermite-Minkowski Theorem, The Fermat's Last Theorem.

####
**MATH447** - Computer Algebra
(3 + 0) 6

Introduction to Maple, complexity of arithmetic operations, computing with algebraic numbers, bit operations, computer arithmetic with big integers; Euclidean and extended Euclidian algorithms, polynomial interpolation, factorization and applications, primality testing, arithmetic on finite fields, operations in linear algebra, computing in lattice

####
**MATH473** - Algebraic Topology
(3 + 0) 6

Homotopic Mappings, Homotopy Equivalence Versus Homeomorphism, the Fundamental Group, Covering Spaces, Higher Homotopy Groups, Singular Complexes and Singular Homology, Relationship Between the Fundamental Groupand the First Homology Group, Homotopy Invariance of Homology Groups, Relative Homology, Exacteness, Excision and Mayer-Vietories Sequence, Applications to Spheres, Applications to Euclidean Spaces, Finite Cell Complex, Betti Numbers and Euler Charateristic, Outline of Singular Cohomology, Poincare Duality for Topological Manifolds and Applications of Cohomology Algebras.

####
**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

####
**MATH483** - Special Functions of Applied Mathematics
(3 + 0) 6

Gamma and Beta functions; Pochhammer`s symbol; hypergeometric series; hypergeometric differential equation; generalized hypergeometric functions; Bessel function; the functional relationships, Bessel`s differential equation; orthogonality of Bessel functions.