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Lisans Ders İçerikleri

////Lisans Ders İçerikleri
Lisans Ders İçerikleri 2017-06-14T11:02:21+00:00
Course Code Course Name Course Content
MATH 123

Zorunlu Ders

Ayrık Matematik Propositional Logic, Predicates and Quantifiers, Techniques of Proofs, Set Theory, Induction and Recursion, The Basics of Countings, The Pigeonhole Principle, Permutations and Combinations, Recurrence Relations, Inclusion-Exclusion and its Applications, Relations: Equivalence Relations, Partial Orders, Total Orders, Zorn’s Lemma, Axiom of Choice and its Equivalences, Graphs
MATH 124

Zorunlu Ders

Analitik Geometri Cartesian Coordinates in the Plane, Review of Trigonometry, Polar Coordinates, Change of Coordinates, Vectors in the Plane, Cartesian Coordinates in 3-Space, Surfaces, Cylindrical and Spherical Coordinates, Conic Sections, Vectors in 3-Space
MATH 151

Zorunlu Ders

Analiz I Real numbers, intervals, absolute value, Inequalities, Graphs of Quadratic Equations, parabolas, Functions, Trigonometric functions, Transcendental functions, Limits and Continuity,  Limits at infinity and infinite limits, Differentiation, chain rule, The Mean Value Theorem, implicit differentiation, Applications of Derivatives (Related rates, Indeterminate Forms and L’Hôpital’s Rule, Curve Sketching, Optimization), Integration, Fundamental Theorem of Calculus, Substitution, Integration by parts
MATH 152

Zorunlu Ders

Analiz II Applications of Integrals (Area, Volume, Arc length, Surface Area), Sequences and Series, Convergence Tests, Power Series, Infinite Products, Vectors, Planes and Lines, Functions of Several Variables, Limits and Continuity, Partial Derivatives, The Chain Rule, Gradients and Directional Derivative, Implicit Differentiation, Extreme Values, Lagrange Multipliers, Double Integrals, Triple Integrals, Line Integrals, Conservative Vector Fields, Divergence and Curl, Green’s Theorem, Stoke’s Theorem.
MATH 223

Zorunlu Ders

Cebir I Groups, Subgroups, Cyclic Groups, Generators of Groups, Groups of Permutations, Alternating Groups, Cosets, Theorem of Lagrange, Direct Products, Finitely Generated Abelian Groups, Homomorphisms, Isomorphisms, Factor Groups, Simple Groups, Group action on a set, Isomorphism Theorems, Sylow Theorems
MATH 231

Zorunlu Ders

Lineer Cebir I Systems of Linear Equations, Row Echelon Form, Matrix Algebra, Elementary Matrices, Determinants, Vector Spaces, Linear Independence, Basis and Dimension, Row Space and Column Space, Null Spaces and Ranges, Linear Transformations, Similarity
MATH 232

Zorunlu Ders

Lineer Cebir II Inner Product Spaces, Orthogonality, Orthonormal Sets, The Gram-Schmidt Orthogonalization Process, Eigenvalues and Eigenvectors, Diagonalization, Hermitian Matrices, Positive Matrices, Normal Matrices, Real Symmetric Matrices, Unitary and Orthogonal Matrices, Bilinear and Quadratic Forms, Canonical Forms, Decompositions.
MATH 245

Zorunlu Ders

Diferensiyel Denklemler Existence-uniqueness theorem of first order initial value problems. First order equations. Higher order linear ordinary differential equations. Constant coefficient equations. Reduction of order method, method of undetermined coefficients, method of variation of parameters. Riccati equations, Cauchy-Euler equations. Power series solutions. The Laplace transforms. Convolution integral. Solution of initial value problems using Laplace transform. Solution of systems of linear differential equations.
MATH 251

Zorunlu Ders

İleri Analiz I The Real line and Euclidean space, The topology of Euclidean space (open sets, Interior of a set, closed sets, accumulation points, closure of a set, boundary of a set, sequences, completeness), Compact and Connected sets (compactness, the Heine-Borel theorem, Nested set property, Path-connected sets, connected sets), Continuous Mappings, Uniform, pointwise convergences, the Weierstrass M-test, Integration and differentiation of series, the space of continuous functions, the Arzela – Ascoli theorem.
MATH 252

Zorunlu Ders

İleri Analiz II Continuous functions, the Arzela – Ascoli theorem, the contraction mapping, the Stone-Weierstrass theorem, Differentiable Mappings, Differentials of transformations, Matrix representation, Differentiable paths,The chain rule, Product rule and gradients, Higher derivatives, Maxima and Minima of the functions defined on R^n, The Inverse and Implicit Function Theorems, Lagrange Multipliers, Integration, Sets of measure zero, Improper Integrals, Fubini’s Theorem and the change of variables formula, Fourier Analysis, Inner Product Spaces, Orthogonal Families of Functions, Fourier Series.
MATH 282

Zorunlu Ders

Sayısal Analiz Roundoff errors, algorithms and convergence, bisection method, fixed point iteration, Newton’s method, error analysis, accelerating convergence. Interpolation and Lagrange polynomial, divided differences. Cubic splines. Numerical differentiation, Richardson’s extrapolation. Numerical integration, trapezoid, Simpson’s and Boole’s rules. Romberg integration, adaptive quadrature. Gaussian quadrature. Multiple integrals.
MATH 311

Seçmeli Ders

Varyasyon Analizi Lagrange multipliers, Maxima and minima, the first variation. The Euler-Lagrange equations. Constraints. Natural and boundary conditions.Transversality conditions. The Hamiltonian formulation.
MATH 315

Zorunlu Ders

Kısmi Türevli Denklemler First order equations; linear, quasilinear, and nonlinear equations; classification of second order linear partial differential equations; canonical forms; the Cauchy problem for the wave equation; Laplace and heat equations (Sturm-Liouville Problems)
MATH 322

Seçmeli Ders

Belirtisiz Kümeler The notion of membership, The concept of fuzzy subsets, Simple operations on fuzzy sets. Properties of the set of fuzzy subsets. Product and algebraic sum of two fuzzy subsets. Fuzzy graphs. Fuzzy relations. Properties of fuzzy binary relations.
MATH 323

Zorunlu Ders

Cebir II Rings, Fields, Integral Domains,Fermat’s and Euler’s Theorems, The Field of quotients of an integral domain, Rings of Polynomials, Factorization of Polynomials over a Field, Homomorphisms,  Factor Rings, Prime Ideals, Maximal Ideals, Unique Factorization Domains, Euclidean Domains, Gaussian Integers, Field Extensions, Algebraic Extensions, Finite Fields.
MATH 324

Zorunlu Ders

Olasılık ve İstatistiğe Giriş Statistical Inference, Sampling Procedures, Statistical Modeling, Graphical Methods, Data Description, Sample Spaces, Events, Algebra of Events, Probability of Events, Conditional Probability, Bayes’ Rule, Random Variables, Joint Random Variables, Mathematical Expectation, Variance, Covariance, Discrete Random Variables: Binomial, Hypergeometric, Negative Binomial, Geometric and Poison Distribution, Continuous Random Variables: Normal, Gamma and Exponential Distribution, Random Sampling, Sampling Distributions, Central Limit Theorem, t-Distribution, F-Distribution
MATH 342

Zorunlu Ders

Uygulamalı Matematik Classification of Linear Integral Equations, Solution of an Integral Equation, Converting Volterra Equation to an ODE, Converting IVP to Volterra Equation, Converting BVP to Fredholm Equation, Fredholm Integral Equations, Volterra Integral Equations, Fredholm Integro-Differential Equations, Volterra Integro-Differential Equations, Real World Applications
MATH 352

Zorunlu Ders

Kompleks Analiz Complex  numbers. Complex functions and linear mappings of regions. Limits and continuity. Branches of functions. Differentiable and analytic functions. Harmonic, Elementary functions. Contours and contour integrals. The Cauchy-Goursat theorem. Cauchy integral formula. Taylor and Laurent series representations. Singularities, zeros, and poles. The residue theorem and its applications to evaluation of trigonometric and improper integrals. The argument principle and Rouché’s theorem.
MATH 366

Seçmeli Ders

Elementer Sayı Teorisi Divisibility. The linear Diophantine equation. Primes. Congruences. Euler, Fermat, Wilson, Lagrange and Chinese Remainder Theorems. Arithmetical functions.
MATH 371

Seçmeli Ders

Kesirli Diferensiyel Denklemlere Giriş Mittlag-Leffler Functions; Riemann-Liouville fractional integrals and derivatives; Caputo fractional derivatives; Grünwald-Letnikov fractional derivative; Riesz fractional integro-differentiation ordinary differential equations; fractional Laplace transform; Cauchy type problems.
MATH 373

Seçmeli Ders

 Matematik Tarihi  The following periods of history of mathematics will be studied: Mathematical Periods, Egyptian and Babylonian Period (2000 B.C.- 500 B.C.) Greek Mathematics Period, (500 B.C- A.D.500) Hindu, Islamic and Period of Transmission (A.D.500-A.D.1700), Classic Period (A.D. 1700-A.D.1900) Modern Period (A.D.1900- present).
MATH 381

Zorunlu Ders

Bilimsel Hesaplama Ordinary differential equations and initial value problems. Euler’s method. Higher order Taylor methods. Runge-Kutta methods. Error control, systems of ordinary differential equations and higher order equations. Linear systems of equations. Operations of linear algebra. Gaussian elimination. Pivoting strategies, LU factorization. Eigenvalues. Iterative methods of Gauss-Seidel and Jacobi. Applications with MATLAB.
MATH 386

Seçmeli Ders

Matematiksel Modellemeye Giriş Discrete dynamical systems. Optimization models and Linear Programming. Correlation and regression. Discrete and continuous probabilistic models. Predator-prey models, optimal harvesting, traffic flow. Verification and validation of models.
MATH 402

Seçmeli Ders

 Dinamik Sistemler ve Kaos  One dimensional maps. Fixed points and stability. Periodic points. Sensitive dependence on initial conditions. Chaos. Lyapunov exponents. Chaotic orbits. Fractals. Cantor set. Deterministic systems. Fractal dimensions.
MATH 410

Seçmeli Ders

Özel Fonksiyonlar Gamma function. Beta function. Bessel’s and generalized Bessel’s function. Orthogonal polynomials. Chebyshev, Legendre, Hermite, Laguere, Jacobi  polynomials, Hypergeometric functions.
MATH 414

Zorunlu Ders

 Reel Analiz  Sets of Measure zero, Borel sets, Measurable functions, Lebesgue integrals, Properties, Lebesgue integration for bounded measurable functions,Lebesgue integration for unbounded functions, the monotone and dominated convergence theorems, The L^p Spaces: Completeness and Approximation, the construction of the Lebesgue measure on R^d.
MATH 417

Seçmeli Ders

Fark Denklemlerine Giriş The difference calculus, first order equations, linear equations, equations with constant coefficients, equations with variable coefficients, undetermined coefficients method, variation of parameters method, the Z-transform, linear systems, stability theory.
MATH 418

Seçmeli Ders

Zaman Skalalarında Analiz Basic Definitions, Differentiation, Integration, First Order Linear Equations, Second Order Linear Equations, Laplace Transform, Self-Adjoint Equations, Boundary Value Problems and Green’s Function, Eigenvalue Problems.
MATH 427

Seçmeli Ders

Kriptografiye Giriş History and overview of cryptography, The Basic Principles of Modern Cryptography, Private-Key Cryptography; One time pad and stream ciphers, Block ciphers, PRPs and PRFs, Attacks on block ciphers. Message Integrity; Collision resistant hashing, Authenticated encryption: security against active attacks. Public-Key Cryptography; Cryptography using arithmetic modulo primes, Public key encryption, Arithmetic modulo composites. Digital Signatures.
MATH 428

Seçmeli Ders

 Temsil Teorisine Giriş GroupRepresentations, FG-modules,GroupAlgebras, Maschke’sTheorem, Schur’sLemma, irreduciblemodulesandgroupalgebras, characters, charactertables.
MATH 451

Zorunlu Ders

Topoloji Topological. Spaces; definitions, bases and subbases; continuous closed and open functions; homeomorphisms; countability axioms; separation axioms; compactness; product and quotient topologies; connectedness; metric and normed spaces; function spaces, Ascoli’s theorem, compact open topology.
MATH 452

Seçmeli Ders

Fonksiyonel Analiz Metric Spaces, Completion of Metric Spaces, Normed Spaces, Finite Dimensional Normed Spaces, Bounded Linear operators, Linear Functional, Dual Space, Inner Product Spaces and Hilbert Spaces, Riesz  Representation Theorem, Hilbert Adjoint Operator; Self-Adjoint, Unitary and Normal Operators;  Hahn-Banach Theorem, Uniform Boundedness  Principle, Strong and Weak Convergence.
MATH 461

Zorunlu Ders

Matematikçiler için Modern Bilgisayar Bilimlerine Giriş The course covers a number of modern computer software technologies which have taken significant input from mathematicsThese technologies or applications range from explicit mathematical software such as Matlab, to the likes of web search, e-commerce and social networking.
MATH 476

Zorunlu Ders

Diferensiyel Geometri Curves in R^3, the local theory of curves parametrized by arc length, Frenet-Serret formulas,curvature and torsion. Regular  surfaces, the tangent plane, the differential of a map, diffeomorphism, the first fundamental form, Gauss map,  the second fundamental form,  normal curvature, principal curvature, Gauss map in local coordinates.
MATH 491

Zorunlu Ders

Mezuniyet Projesi I A practical / theoretical training in mathematics / modelling / computer / simulation. A report and presentation are required for the completion of the course.
MATH 492

Zorunlu Ders

Mezuniyet Projesi II A practical / theoretical training in mathematics / modelling / computer / simulation. A report and presentation are required for the completion of the course.