Courses

Vectors, introduction to matrix algebra, determinants, systems of linear equations, Cramer's theorem, inverse matrices, elementary matrix transformations, solution of linear equation systems using inverse matrix, eigenvalues and eigenvectors.

Service

Introduction to basic mathematics, coordinates and vectors, functions, limit, continuity, derivative, tangent lines, mean value theorem, graphs, critical points, maximum and minimum problems, linearization and differentials, integral, Riemann sums and definite integrals, fundamental theorem of mathematics, natural logarithms, exponential functions, inverse trigonometric functions, L'Hospital rule, integral methods, applications of integral.

Service

Sequences and series, Taylor and Maclaurin series, lengths of planar curves, polar coordinates and complex numbers, lines in space, planes and quadratic surfaces, multivariable functions, limit and continuity, segmented derivatives, chain rule, directional derivatives, critical points, multiple integrals integrals in polar, cylindrical and spherical coordinates, line integrals and surface integrals.

Service

Real numbers, algebraic operations and basic identities, drawing of lines, linear and quadratic equations, functions, limit, continuity, derivative, derivation techniques, marginal analysis, higher order derivatives, extremum points, exponential functions and logarithmic functions and derivatives, finance mathematics , simple and compound interest calculations, matrices and determinants, inverse matrices, systems of linear equations, Cramer's rule, introduction to linear programming problems.

Service

Review of exponential and natural logarithm functions, definite and indefinite integrals, field calculus, basic theorem of mathematics, applications to business and economics, functions of two variables, partial derivatives, chain rule, local maximum and minimums, introduction to probability, Lagrange multiplier, double integrals and applications, volume calculation.

Service

Symbolic logic. Set theory. Cartesian product. Relations and functions. Equipotential sets. Countability of sets. Equivalence relations, equivalence classes and decompositions. Section sets. Sort relations. Mathematical induction and iterative function definitions.

Compulsory

Repetitive and non-repeated permutations, combinations. Inclusion-exclusion principle. Generator function. Basic concepts of graph theory. Planar graphs. Hamilton paths and graph coloring. Trees. Optimization and matching. Transport networks.

Compulsory

Functions of one variable, limit and derivative. Basic theorems of differential calculus: intermediate value theorem, extreme value theorem and mean value theorem. Applications: graphing and maximum-minimum problems.

Compulsory

Riemann integral. Mean Value Theorem of Integral Calculus. Fundamental Theorem of Analysis. Techniques for integral calculus. Various geometric and physical applications. Series. Generalized integrals. Infinite series. Power series, Taylor series and applications.

Compulsory

Axiomatic structures in geometry; finite geometries, Euclidean and non-Euclidean geometries. Polygons, similar shapes. Properties of the circle. Illustrations. Solid objects. Angle measurement in degrees and radians, trigonometric functions and applications, sine and cosine theorems, graphs of trigonometric functions, trigonometric identities. Polar coordinates. Vectors in space and plane. Lines in plane, lines and planes in space. Basic information about conics and second order surfaces.

Compulsory

Matrices, line equivalence, inverse matrix, systems of linear equations, determinants, Cramer's rule, vector spaces, linear dependence and independence, bases, inner product spaces, Gramm-Schmidt method, orthogonal projections, Fourier series, eigenvalues, eigenvectors, exponential matrices , diagonalization and its applications, linear transformations and matrices.

Service

Basic concepts of differential equations, first order differential equations, solution of linear differential equations, constant coefficient differential equations, Cauch-Euler equations, systems of linear differential equations, Laplace transformations, applications of linear equations to linear systems, solutions of linear equations with power series, introduction to linear differential equations Allocation to variables.

Functions of several variables: limit and continuity. Partial derivatives, directional derivatives. Tangent plane. Mean Value Theorem, closed and inverse function theorems. Maximum-minimum values. Introduction to differential calculus for vectors: gradient, divergence and rotation. Double integrals, polar coordinates. Change of variables in multiple integrals. Triple integrals: Cylindrical and spherical coordinates. Linear integrals. Green's theorem. Independence from the road, full differentials. Surface integrals. Divergence and Stokes theorems.

Compulsory

Complex numbers algebra. Polar notation. Analyticity, Cauchy-Riemann equations. Power series. Elementary functions. Transformation with elementary functions. Linear fractional transformations. Linear integrals, Cauchy theorem, Cauchy integral formula. Taylor series, Laurent series, residues, residual theorem. Generalized integrals.

Compulsory

Differential equations and their solutions. Existence and uniqueness theorems. First order equations and various applications. Higher order linear differential equations. Solutions with power series: ordinary and uniform singular points. Laplace transform: solution of initial value problems. Systems of linear differential equations: solutions by operator method, solutions by Laplace transform.

Compulsory

First order equations; linear, linear and nonlinear equations. Classification of second order linear partial differential equations, canonical forms. Cauchy problem for wave equation. Dirichlet and Neumann problems for Laplace equation, maximum principle. Heat equation on ribbon.

Compulsory

Matrices and systems of linear equations. Vector spaces, subspaces, sum and direct sums of subspaces. Linear dependence, independence, bases, dimension, partition spaces. Linear transformations, kernel, image, isomorphism. Linear transformations space, Hom (V, W), V *, V **, transpose. Representation of linear transformations with matrices, similarity. Determinants.

Compulsory

Characteristic and minimal polynomials of an operator, eigenvalues, diagonalization. Canonical forms: Smith normal form, Jordan and rational forms of matrices. Inner product spaces, norm and orthogonality, projections. Linner operators on inner product spaces, adjoint of an operator, normal, self adjoint, unitary and positive operators. Double linear and quadratic forms.

Compulsory

Language and axioms of set theory. Sequential pairs, relations and functions. Sort relation and well ordered sets. Ordinal numbers, transfinite induction, arithmetic of ordinal numbers. Arithmetic of quantity and quantitative numbers. Axiom of choice, generalized continent hypothesis.

Departmental Elective

Mathematics in Egypt and Mesopotamia, Ionian and Pythagoreans, Zeno paradoxes. Plato, Aristotle, Euclid of Alexandria, Archimedes, Appolonyus and Diophantus. Mathematics in China and India. Renaissance mathematics, contribution of Muslims. Fermat and Descartes period. Development of the concept of limit, Newton and Leibniz's work, Gauss and Cauchy's contributions. Non-Euclidean geometries. Arithmetization of the analysis. The emergence of abstract algebra. Various aspects of the twentieth century.

Departmental Elective

Definition of risk, risk management, risk and risk mitigation. Modeling of risk distribution, basic premium rate and prudential separation techniques. Reinsurance, risk theory for individual and community, reliability theory, wholesale destruction theory.

Departmental Elective

Multiple shortening lifespan models, actuarial functions, design and financing of retirement plans.

Departmental Elective

Simple and compound interest. Initial and future value of investment, money flow. Discrete probability, conditional probability, discrete random variables. Distribution function and expectations. Continuous random variables, distribution function and expectations. Variance and Standard deviation. Normal random variables. Central limit theorem and longnormal variables. Linear programming. Dual Problems. Fundamental theorem and applications of finance mathematics. Random gait theory and Brownian motion. Poisson and Laplace equations. Stochastic process and market applications. Wiener and Ito processes. Market options, pricing. Black-Scholes differential equation. Solutions of the Black-Scholes equation.

Departmental Elective

Real numbers system. Metric spaces. Completion of metric space. Continuous functions in metric spaces. Compactness and connectedness. Continuity and compactness. Contraction mapping theorem and its applications. Arzela-Ascoli Theorem.

Compulsory

Elements of the set theory. Definition of Riemann integral with step functions. Outer measurement at R ^ n; measurable sets and Lebesgue measurement. Properties of measurable sets; immeasurable clusters. Cantor set, Sigma-Algebra, Borel sets and Borel measurements. Measurable functions. Lusin and Egoroff theorems. Lebesgue integral. Fundamental theorem of Lebesque integral: Lebesgue, Fatou and Levi theorems. General measurement in measurable spaces and Lebesque integral. L_p-spaces. Applications: L_p and Sobolev Spaces.

Compulsory

Fourier series. Fourier transform, inverse Fourier transform. Laplace transform. Inverse integral for Laplace transform (complex perimeter integration). Applications of Laplace transform to ordinary, partial differential and integral equations. Z-transform. Inversion integral for Z-transform. Applications of Z-transform to difference equations and linear networks.

Departmental Elective

Gamma and beta functions. Pochhammer icon. Hypergeometric series. Hypergeometric differential equations; ordinary and confluent hypergeometric functions. Generalized hypergeometric functions. Bessel functions; functional relations, Bessel differential equation. Orthogonality of Bessel functions. Legendre functions, Hermite polynomials.

Departmental Elective

Groups, subgroups, normal subgroups and division groups. Isomorphism theorems. Direct products. Groups working on clusters. Class equation. Sylow theorems and basic theorem of finite commutative groups. Rings, isomorphism theorems. Prime and maximal ideals. Completion regions, fraction bodies. Euclidean regions, TÜİB, TÇB, polynomials, very uncertain polynomials. Field extensions. Impossibility of some geometric drawings. Finite fields.

Compulsory

Divisibility, congruences, Euler's theorem, China's theorem and Wilson's theorem. Arithmetic functions. Primitive roots. Quadratic residues and quadratic reciprocity. Diophantine equations.

Departmental Elective

Curves in R3, Frenet formulas. Regular surfaces. Inverse images of regular values. Derivative functions on surfaces. Tangent plane; a transformation differential, vector fields, first basic form. Gauss transform, second basic form, normal curvature, principal curvature, principal directions and asymptotic directions. Gauss transformation in local coordinates. Covariant derivative, geodesics.

Compulsory

Convergence, stability, error analysis and conditioning. Solution of systems of linear equations: LU and Cholosky factors, pivoting, Gauss elimination error analysis. Matrix eigenvalue problems, force method, orthogonal factorization and least squares problems. Solutions of nonlinear equations. Newton's intersecting and fixed point iteration methods.

Departmental Elective

Introductory information about reductions. Structure of programs, prefix operators. Subroutines. A computer algebra system. How to use a computer algebra system. Representation of polynomials, real functions, algebraic functions, matrices and series. Advanced algorithms. The largest common divisor with different variables. Other applications of modular methods. P-adic methods. Formal integration and differential equations.

Departmental Elective

Topological spaces; base, subbase, subspaces. Closed sets, limit points. Hausdorff spaces. Continuous functions, homeomorphisms. Product topology. Connected spaces, components, path connectedness, path components. Compactness, sequential compactness, compactness in metric spaces. Definition of regular and normal spaces. Urysohn auxiliary theorem, Tietsze expansion theorem.

Departmental Elective

Importance of optimization, basic definition and first information in convex analysis. Linear and convex programming theory, simplex method and its applications, nonlinear programming, search methods, basic ideas of classical variational calculus, optimal control theory. Pontraygin's maximum principle and dynamic programming, linear theory for optimal control.

Compulsory

Unconditional optimization: conditions of optimality, convexity and geometric programming, Newton's method, quazi-Newton and conjugate gradient methods. Conditional optimization: Karucsh-Kuhn-Tucker theory, second order conditions, conditions given by equality and inequalities. Linear programming: optimality and quality, basics of simplex method and inset methods.

Departmental Elective

Advanced statistical methods in insurance, pricing, loss return methods, iterative insurance models, affordability of insurance, simulation models in insurance, insurance companies as financial institutions.

Departmental Elective

Differential. Inverse and implicit function theorems. Integration in subsets of Euclidean space. Tensors. Differential forms. Integral in manifolds. Stokes theorem.

Departmental Elective

Review of Riemann integral. Sets with Lebesgue measurement zero in Rn, characterization of Riemann integrable functions. Lebesgue integrable functions and Lebesgue integral in Rn. Convergence theorems, Lusin and Egorov theorems. Fubini's theorem. Selected applications.

Departmental Elective

Normed linear spaces, Banach spaces. Hahn-Banach theorem and its results. Baire category theorem. Uniform boundary principle. Open transform and closed graph theorems. Hilbert spaces.

Departmental Elective

Groups, division groups, isomorphism theorems, alternating and dihedral groups, direct products, free groups, generators and relations, free abelian groups, finite generated abelian groups. Sylow theorems, nilpotent and solvable groups, normal and semi-normal series. Rings, ring homomorphisms, ideals, factorization in commutative rings, division rings, localization, basic ideal regions, Euclidean regions, region that can be divided into single prime factors, polynomials and power series, separation in polynomial rings.

Departmental Elective

Riemannian metric, Riemannian space, arc length of a curve, angle between two vectors. Absolute derivative, parallel shift of a vector field along a curve, geodesics, geodesic coordinates, Riemann coordinates. Subspaces, hypersurfaces, curvature tensor, Ricci tensor, Bianchi identity. Riemannian curvature, Schur's theorem. Einstein spaces. Mainardi-Codazzi equations for hypersurfaces, Gauss equation.

Departmental Elective

Differences account. Linear difference equations: first order equations, higher order equations. Systems of difference equations. Basic theory. Linear periodic systems. Stability theory. Linear approach. Lyapunov's second method. Z-transform.

Departmental Elective

Mathematical modeling of boundary value problems in partial differential equations. Expression of Dirichlet and Neumann problems. Green functions. Asymptotic analysis of solutions. Perturbation techniques. Introduction to integral equations, Volterra and Fredholm equations, solutions with Neumann series and connections with eigenvalue problems.

Departmental Elective

The aim of this course is to discuss recent developments in pure and / or applied mathematics.

Departmental Elective

The aim of this course is to discuss recent developments in pure and / or applied mathematics.

Departmental Elective

This project is expected to enable students to study a topic independently and prepare a graduation thesis on this subject.

Compulsory