Matematik
|

Doktora

Matematik Bölümü Doktora ve  Bütünleşik Doktora Programı:

Matematik Bölümü; Doktora ve Bütünleşik Doktora programlarına Matematik veya diğer disiplinlerde lisans eğitimini tamamlamış olan öğrencileri kabul etmektedir. Matematik, akıl yürütme ve problem çözme sanatı olup, tümdengelim ve tümevarım düşünce yolları ile sayılar ve geometrik şekiller gibi kavramların özelliklerini ve bunların arasındaki bağlantıları inceleyen bir disiplindir. Bilimsel olan her şey bir matematiksel formülasyon gerektirdiğinden Matematik, bilim ve teknolojinin vazgeçilmez aracıdır.

Ankara Yıldırım Beyazıt Matematik Bölümü'nde:

  • Analiz ve Fonksiyonlar Teorisi
  • Cebir ve Sayılar Teorisi 
  • Geometri 
  • Topoloji 
  • Uygulamalı Matematik

alanlarında araştırma faaliyetleri yürütülmektedir.

Doktora programının süresi 4 yıldır. İhtiyaç duyulması halinde 2 yıl ek süre verilebilir. Öğrencinin, doktora programını en fazla 6 yılda tamamlaması gereklidir.

Bütünleşik Doktora programının süresi 5 yıldır. İhtiyaç duyulması halinde 2 yıl ek süre verilebilir. Öğrenci, bütünleşik doktora programını en fazla 7 yılda tamamlamak zorundadır.

Programlara aşağıdaki bölümlerin mezunları başvurabilir:

  • Matematik 
  • Matematik ve Bilgisayar Bilimleri 
  • Matematik Öğretmenliği 
  • Matematik Mühendisliği 
  • İstatistik 
  • Fizik 
  • Bilgisayar Mühendisliği 
  • Yazılım Mühendisliği 
  • Elektrik-Elektronik Mühendisliği

 

Başvuru Şartları:

Resmi başvuru koşulları başvuru döneminden önce Fen Bilimleri Enstitüsü web sayfasında ilan edilmektedir. Aşağıdaki koşullar sadece bilgi amaçlıdır:

  • Doktora programına başvurmak için gerekli yüksek lisans ortalaması en az 2.75/4.00 
  • Bütünleşik doktora programına başvurmak için gerekli lisans ortalaması en az 2.75/4.00
  • Doktora ve bütünleşik doktora programlarına başvurmak için gerekli yabancı dil (YÖKDİL/YDS) puanı en az 60
  • Niyet mektubu
  • Başvurusu kabul edilen adayların, üniversitemiz tarafından yapılan Yabancı Dil Muafiyet Sınavı’na girmeleri gerekmektedir.

 

Programdan Mezun Olmak İçin Koşullar:

Doktora Programı

Öncelikle adaylar ön yeterlilik sınavını başarıyla geçmelidir. Öğrenciler mezun olabilmek için en az 7 ders, seminer dersi ve ek olarak FBE900 Research Methods and Ethics dersini almalıdırlar. Bu öğrenciler ayrıca, derslerini tamamladıktan sonraki dönemde Enstitümüzün belirleyeceği tarihe kadar Doktora Tez Önerisini enstitüye teslim etmelidirler. Aynı zamanda, her dönem Special Studies dersini almalıdırlar. Tez çalışmalarına başlamadan öğrenciler doktora yeterlilik sınavından başarı ile geçmelidir. Tez çalışmalarına başladıklarında da Ph.D. Thesis dersini de mezun oluncaya kadar almak zorundadırlar.

Bütünleşik Doktora Programı

Öncelikle adaylar ön yeterlilik sınavını başarıyla geçmelidirler. Öğrenciler mezun olabilmek için en az 14 ders, seminer dersi ve ek olarak FBE900 Research Methods and Ethics dersini almalıdırlar. Bu öğrenciler ayrıca, derslerini tamamladıktan sonraki dönemde Enstitümüzün belirleyeceği tarihe kadar Doktora Tez Önerisini enstitüye teslim etmelidirler. Aynı zamanda, her dönem Special Studies dersini almalıdırlar. Tez çalışmalarına başlamadan öğrenciler doktora yeterlilik sınavından başarı ile geçmelidir. Tez çalışmalarına başladıklarında da Ph.D. Thesis dersini de mezun oluncaya kadar almak zorundadırlar.

Bütünleşik doktora programına kayıt yaptıran öğrenciler, çalışmalarına bağlı olarak aşağıdaki derecelerden birini veya tümünü alabileceklerdir.

Tezli Yüksek Lisans Programı  Doktora Programı Bütünleşik Doktora (Lisans Sonrası Doktora) Programı
7 Ders (en az) Ön Yeterlilik Sınavı Ön Yeterlilik Sınavı

Yüksek Lisans     Seminer Dersi

7 Ders (en az) 14 Ders (en az)
Yüksek Lisans Tezi Doktora Seminer Dersi Doktora Seminer Dersi
  Doktora Yeterlilik Sınavı  Doktora Yeterlilik Sınavı
  Doktora Tezi Doktora Tezi

Öğrenciler, alacakları dersleri danışmanları ile birlikte kararlaştırıp, Akademik takvimde belirtilen süreler içerisinde obs.aybu.edu.tr adresinden giriş yaparak ders kayıtlarını yapmalıdırlar.

GÜZ DÖNEMİ DOKTORA VE BÜTÜNLEŞİK DOKTORA PROGRAMI DERSLERİ

 

CODE

Compulsory/ Elective

Course Name

Credits

ECTS

MATH501

E

Algebra I

3-0-3

8

MATH 531

E

Advanced Linear Algebra

3-0-3

8

MATH 505

E

Real Analysis

3-0-3

8

MATH 507

E

General Topology

3-0-3

8

MATH 509

E

Functional Analysis I

3-0-3

8

MATH 515

E

Differential Equations I

3-0-3

8

MATH 517

E

Coding Theory and  Cryptology 

3-0-3

8

MATH 535

E

Basic Algorithms and Programming

3-0-3

8

MATH 507

E

Introduction to Scientific Computing I

3-0-3

8

MATH 523

E

Partial Differential Equations I

3-0-3

8

MATH 525

E

Methods of Applied Mathematics

3-0-3

8

MATH 527

E

Dynamical Systems

3-0-3

8

MATH 529

E

Numerical Solutions for Ordinary Differential Equations

3-0-3

8

MATH 531

E

Topics in Applied Mathematics I

3-0-3

8

MATH 533

E

Combinatorics

3-0-3

8

MATH 601

E

Boundary Element Method and Its Applications 

3-0-3

8

MATH 603

E

Delay Differential Equations

3-0-3

8

MATH 605

E

Differential Quadrature Method

3-0-3

8

MATH 607

E

Finite Element Methods

3-0-3

8

MATH 609

E

Numerical Analysis

3-0-3

8

MATH 611

E

Mathematics of Fluid Dynamics

3-0-3

8

MATH 613 

E

Introduction to Magnetohydrodynamics

3-0-3

8

MATH 615

E

Algebraic  Number Theory 

3-0-3

8

MATH 617

E

Codes and Algebraic Curves 

3-0-3

8

MATH 619

E

Machine Learning 

3-0-3

8

MATH 621

E

Regression Analysis 

3-0-3

8

MATH 623

E

Introduction to Data Mining 

3-0-3

8

MATH 625                E Applications of Convexity 3-0-3     8
MATH 627                E Digital Communications I 3-0-3     8

MATH 900

E

Special Studies

3-0-3

8

MATH 910

C

PhD Seminar

0-2-0

8

MATH 920

C

PhD Thesis

0-1-0

30

FBE 4000

C

PhD Qualification Exam

0-1-0

30

 BAHAR DÖNEMİ DOKTORA VE BÜTÜNLEŞİK DOKTORA PROGRAMI DERSLERİ

 

CODE

Compulsory/ Elective

Course Name

Credits

ECTS

MATH 502

               E

Algebra II

3-0-3

8

MATH 504

               E

Complex Analysis

3-0-3

8

MATH 506

               E

Differential Geometry

3-0-3

8

MATH 508

              E

Numerical Linear Algebra

3-0-3

8

MATH 510

               E

Functional Analysis II

3-0-3

8

MATH 516

               E

Differential Equations II

3-0-3

8

MATH 533

               E

Finite Fields and Their Applications

3-0-3

8

MATH 520

               E

Integral Equations

3-0-3

8

MATH 522

               E

Introduction to Scientific Computing II

3-0-3

8

MATH 524

E

Partial Differential Equations II

3-0-3

8

MATH 526

E

Special Functions

3-0-3

8

MATH 528

E

Mathematical Modelling

3-0-3

8

MATH 530

E

Numerical Solutions for Partial Differential Equations

3-0-3

8

MATH 532

E

Topics in Applied Mathematics II

3-0-3

8

MATH 534

E

Probability Theory

3-0-3

8

MATH 536

E

Graph Theory

3-0-3

8

MATH 616

E

Rational Points on Curves

3-0-3

8

MATH 618

E

Introduction to Digital Audio

3-0-3

8

MATH 620

E

Computational Geometry

3-0-3

8

MATH 622 

E

Algebraic Geometry

3-0-3

8

MATH 624

E

Algebraic Topology

3-0-3

8

MATH 626

E

Cryptography and Network Security

3-0-3

8

MATH 628

E

Digital Communications II

3-0-3

8

MATH 900

E

Special Studies

3-0-3

8

MATH 910

C

PhD Seminar

0-2-0

8

MATH 920

C

PhD Thesis

0-1-0

30

FBE 4000

C

PhD Qualification Exam

0-1-0

3

 

GÜZ DÖNEMİ DOKTORA VE BÜTÜNLEŞİK DOKTORA PROGRAMI DERS İÇERİKLERİ

MATH 501 Algebra I (3-0-3)

Groups, cyclic groups, finite groups, alternating groups, quotient groups, isomorphism theorems, direct products of groups, free groups, free abelian groups, finitely generated abelian groups, Group actions on sets, Sylow theorems. Rings, ring homomorphisms, ideals, quotient rings, factorization in commutative rings, principal ideal domains, Euclidean domains, unique factorization domains, polynomial rings, factorization in polynomial rings, power series.

MATH 503 Advanced Linear Algebra (3-0-3)

Vector spaces, linear transformations, isomorphism theorems, modules, linear operators,  eigenvalues and eigenvectors,  real and complex inner product spaces, normal operators, metric vector spaces.

MATH 505 Real Analysis (3-0-3)

Set theory and real numbers, general measure and integration theory. Lebesgue measurable sets, measurable functions, convergence theorems,  Radon-Nikodym theorem, outer measure, Carathe-odory extension theorem, product measures, Riesz representation, Baire Category, Banach Spaces.

MATH 507 General Topology (3-0-3)

Topological spaces and continuous functions,  subspace topology, product  topology, quotient topology,  connectedness and  compactness, countability and separation properties,  Urysohn's lemma, The Tychonoff Theorem, metrization theorems and paracompactness.

MATH 509 Functional Analysis I (3-0-3)

Metric Spaces, Normed spaces, Banach spaces. Linear operators, Spaces of bounded linear operators. The uniform boundedness principle and the open mapping theorem, Bounded linear functionals. Dual spaces. The Hahn-Banach extension theorem. Separation of convex sets.

MATH 515 Differential Equations I (3-0-3)

Existence and Uniqueness for the solution of Initial Value Problems, Continuation of Solutions,Picard Successive Approximation Method, Systems of Differential Equations, Properties of System Solutions, Systems with Constant and Periodic Coefficients; Higher Order Linear Differential Equations, Autonomous Systems , Systems of Nonlinear Equations. 

MATH 517 Coding Theory and  Cryptology (3-0-3)

Basic concepts and codes, linear codes, some good codes, bounds on codes, perfect codes,  cyclic codes,  Goppa codes,  cryptology, stream ciphers, cryptosytems.

MATH 519 Basic Algorithms and Programming (3-0-3)

Fundamentals of Programming, Introduction to MATLAB and programming with MATLAB, Basic Algorithms and problem solving in Linear Algebra and Differential Equations, introducing LATEX, typesetting text using LATEX packages, constructing tables, bibliography and mathematical formulae, graphing.

MATH 521 Introduction to Scientific Computing I (3-0-3)

Introduction, Error Analysis, Numerical Solution of Equations of One Variable, Direct Methods for Solving Linear Systems, Iterative Techniques for Solving Linear Systems, Approximating Eigenvalues, Numerical Solutions of Systems of Nonlinear Equations.

MATH 523 Partial Differential Equations I (3-0-3)

Introduction to Partial Differential Equations Theory, Important Linear Partial Differential Equations: Transport Equation, Laplace Equation, Heat Equation, Wave Equation ;  Nonlinear First Order Partial Differential Equations, Separation of Variables, Transform Methods (Fourier Transform, Laplace Transform)

MATH 525 Methods of Applied Mathematics (3-0-3)

Calculus of Variations, Euler-Lagrange Equations, Various applications including isoperimetric problems and computer vision, Basics of PDEs, Derivation of basic PDEs via vectorial approaches, Derivation of basic PDEs via variational principle, Separation of variables & boundary conditions, Importance of the boundary conditions, Basics of Fourier Series, Sturm-Liouville Problems, Green's Functions, Eigenfunction Expansion, Linear Integral Equations.

MATH 527 Dynamical Systems (3-0-3)

Linear dynamical systems, Solutions of nonlinear dynamical systems, Linearization methods for nonlinear dynamical systems, Lagrangian and Hamiltonian systems, Global theory of nonlinear dynamical systems, Bifurcation theory for nonlinear dynamical systems.

MATH 529 Numerical Solutions for Ordinary Differential Equations (3-0-3)

Introduction to Numerical Methods, The Elementary Theory of Initial Value Problems, Taylor Methods, Runge-Kutta Methods, Error Control and Runge-Kutta-Fehlberg Method, Multi-Step Methods, Numerical Solution of Higher Order Equations and Systems of Equations, Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, Finite Difference Methods for Nonlinear Problems, Stability.

MATH 531 Topics in Applied Mathematics I (3-0-3)

Introduction to ordinary differential equations, linear first order ODEs, integrating factors, integral curves, singular points, series solution, convergence, existence and uniqueness, the view in the complex plane, nonlinear first order ODEs, Picard's existence and uniqueness theorem, second order linear IVPs, reduction of order, variation of parameters, the Laplace transform, convolution, initial value problems with discontinuous and impulsive forcing functions, inverting Laplace transforms with the Mellin inversion formula and the Bromwich contour, higher order linear IVPs, conversion to first order systems, the fundamental matrix, eigenvalues, eigenvectors and generalized eigenvectors, decoupling systems via similarity transformations, phase plane interpretations, nonlinear first order ODEs.

MATH 533 Combinatorics (3-0-3)

Generating functions, recurrence relations, extremal problems for graphs and set systems, probabilistic methods in combinatorics, algebraic methods in combinatorics.

MATH 601 Boundary Element Method and Its Applications  (3-0-3)

Fundamental concepts, Boundary integral equations for the one- and two- dimensional Poisson equation, Approximate solutions, weighted residual techniques, Weak formulations, Boundary and domain solutions, Fundamental integral equations, Boundary element method, constant elements, linear elements, introduction to dual reciprocity boundary element method, applications.

MATH 603 Delay Differential Equations  (3-0-3)

Introduction to delay differential equations, examples, Delay types, Terminology, Delayed negative feedback, Oscillations of solutions, Solutions backward in time, Existence of solutions, the method of steps for discrete delay equations, Positivity of solutions, a more general existence result, Continuation of solutions, stability definitions, Linear systems and delay, Delay logistic equation, Delayed microbial growth model.

MATH 605 Differential Quadrature Method (3-0-3)

Introduction to differential quadrature method, Lagrange interpolated polynomials, Computation of coefficients for the derivatives in DQM, Fourier based DQM, Choice of Grid Points, Error analysis in DQM, Relationship between DQM and FDM, Extension of DQM to two dimensional problems, Implementation of boundary conditions, Solution of Burger’s equation with DQM, Solution of Poisson equation with DQM, Solution of Helmholtz eigenvalue problem with DQM, DQ application to heat transfer problems and to chemical reactor, Solution of square-cavity problems with DQM, DQ analysis of beams.

MATH 607 Finite Element Methods  (3-0-3)

Sobolev spaces, Weak solution, Finite element spaces, Construction of finite elements spaces, Finite elements on rectangular and brick meshes, Interpolation and discretization error, Local and global interpolation error, Parabolic problems, Time discretization, in fluid mechanics, Conservation of mass and momentum, Weak formulation of the Stokes problem, Navier–Stokes problem with mixed boundary conditions, Time discretization and linearization of the Navier–Stokes problem Implementation of the finite element method.

MATH 609 Numerical Analysis  (3-0-3)

Linear Spaces, Linear Operators on Normed Spaces, Approximation Theory, Best Approximation, Finite Difference Method, Lax equivalence theorem, Sobolev Spaces, Weak derivatives, Sobolev spaces, Periodic Sobolev spaces, Weak Formulations of Elliptic Boundary Value Problems, The Lax-Milgram Lemma, The Galerkin Method and Its Variants, Finite Element Analysis, Basics of the finite element method, Error estimates of finite element interpolations Local interpolation error estimates, Global interpolation error estimates), Convergence and error estimates.

MATH 611 Mathematics of Fluid Dynamics  (3-0-3)

Euler’s equation of motion. Viscosity. Navier-Stoke’s equations for viscous fluids. Vorticity transport equation. Nondimensionalization. Dimensionless parameters. Dynamical similarity. Cartesian tensors. Stress-strain relations. Flows with high-low Reynold’s number. Turbulent flow. Boundary layer theory. Reynold’s equations of turbulent motion.Water waves. Fluid drifts. Stability and equilibrium problems.

MATH 613  Introduction to Magnetohydrodynamics (3-0-3)

Equations of electrodynamics. Equations of fluid dynamics. Ohm’s law. Equations of magnetohydrodynamics. Motion of an incompressible fluid. Motion of a viscous electrically conducting fluid with linear current flow. Steady state motion along a magnetic field. Wave motion of an ideal fluid. MHD wavesDamping and excitation of MHD waves. Characteristics lines and surfaces. Simple waves and shock waves in Magnetohydrodynamics.

MATH 615 Algebraic  Number Theory  (3-0-3)

Algebraic numbers, Ring of integers of an algebraic number field, Integral bases, Norms and traces, The discriminant, Factorization into irreducibles, Euclidean domains, Dedekind domains, Prime factorization of ideals, Principal ideal rings, Lattices, Minkowski’s Theorem, Geometric Representation of Algebraic Numbers, Class-group and class number,  Computational Methods, Fermat’s Last Theorem, Dirichlet’s Units Theorem, Quadratic Residues.

MATH 617 Codes and Algebraic Curves  (3-0-3)

Curves and codes, Algebraic curves, Functions on algebraic curves, Some important properties of algebraic curves, Geometric Goppa codes, Basic error processing, Full error processing, Fields of algebraic functions, Function fields and places, Valuations, Divisors, Repartitions and differentials, Extensions of function fields, Curves and function fields.

MATH 619 Machine Learning  (3-0-3)

Supervised learning, Decision theory, Parametric Methods, Nonparametric Methods, Dimension reduction algorithms, Clustering and classification, Decision trees, Linear discrimination, Multilayer perceptron, Radial basis function, Support vector machine, Graphical Modelling, Markov models, Combining multiple learners.

MATH 621 Regression Analysis  (3-0-3)

Regression and Model Building, Simple Linear Regression, Multiple Linear Regression, Model Adequacy and Regression Diagnostics, Transformations and Weighting to Correct Model Inadequacies, Polynomial Regression Models, Indicator Variables, Variable Selection and Model Building, Variable and Model Building, Validation of Regression Models.

MATH 623  Introduction to Data Mining  (3-0-3)

Supervised learning, Decision theory, Parametric Methods, Nonparametric Methods, Dimension reduction algorithms, Clustering and classification, Decision trees, Linear discrimination, Multilayer perceptron, Radial basis function, Support vector machine, Graphical Modelling, Markov models, Combining multiple learners.

MATH 625  Applications of Converxity (3-0-3)

The Euclidean space R^n, Convex sets, Convex polytopes, Linear programming, Convex functions, Mixed volumes and extremum problems.

MATH 627  Digital Communications I (3-0-3)

Introduction, Deterministic and Random Signal Analysis, Digital Modulation Schemes, Optimum Receivers for AWGN Channels, Carrier and Symbol Synchronization, An Introduction to Information Theory, Linear Block Codes, Trellis and Graph Based Codes.

MATH 900 Special Studies (3-0-3)

MATH 910 PhD Seminar (0-2-0)

It is compalsory that the student will prepare a seminar on a topic to be determined together with his / her supervisor, present it in a suitable way within the pre-defined period and submit the seminar report to his / her supervisor.

MATH 920 PhD Thesis (0-1-0)

FBE 4000  PhD Qualification Exam  (0-1-0)

Students who have successfully completed their credit courses and seminar course must take the doctoral proficiency exam which includes the general knowledge of the mathematics.   

 

BAHAR DÖNEMİ DOKTORA VE BÜTÜNLEŞİK DOKTORA PROGRAMI DERS İÇERİKLERİ

MATH 502 Algebra II (3-0-3)

Modules, homomorphisms, exact sequences, projective and injective modules, free modules, vector spaces, tensor products, modules over a PID.  Fields, field extensions, algebraic extensions, Galois theory, splitting fields, algebraic closure. Finite fields, structure of finite fields.

MATH 504 Complex Analysis (3-0-3)

Complex numbers, differentiation, integraion, Cauchy’s theorem, harmonic fuctions, Taylor and Laurent series, isolated singularities and the residue theorem, applications of the residue theorem.

MATH 506 Differential Geometry (3-0-3)

Curves and surfaces,  plane curves,   geometry  of  hypersurfaces,  lengths and distances, curvature,  Riemannian connection, geodesics, normal coordinates,  conjugate points, isometric immersions, Metric and geodesic completeness, variations of the energy functional.

MATH 508 Numerical Linear Algebra (3-0-3)

Introduction. Summary/recap of basic concepts from linear algebra and numerical analysis: matrices, operation counts. Matrix factorizations. (Cholesky factorization. QR factorization, LU factorization and Gaussian elimination; partial pivoting), Linear systems, Sparse and banded linear systems and iterative methods, Linear least squares problem, Singular value decomposition (SVD), Eigenvalue problem.

MATH 510 Functional Analysis II (3-0-3)

Spaces of continuous functions, Ascoli’s theorem, Stone-Weierstrass’ theorem, Spaces of Holder continuous functions and of k-times differentiable functions, Hilbert spaces, Compact operators on a Hilbert space. Fredholm’s alternative. Spectrum and eigenfunctions of a compact, self-adjoint operatör, Weak derivatives, Sobolev spaces, Embedding theorems.

MATH 516 Differential Equations II (3-0-3)

Nonlinear Periodic Systems, Bifurcation, Boundary Value Problems, Linear Differential Operators, Boundary Conditions, Existence of Solutions of Boundary Value Problems, Eigenvalues and Eigenfunctions for Linear Differential Operators, Green’s Function of a Linear Differential Operator.

MATH 518 Finite Fields and Their Applications (3-0-3)

Structure of finite fields, polynomials over finite fields, factorization of polynomials, exponential sums, lineer recurring sequences, applications of finite fields.

MATH 520 Integral Equations (3-0-3)

Basics of Integral equations, types of integral equations, Volterra and Fredholm integral equations, degenerate kernels, Green's functions, Fredholm Alternative.

MATH 522 Introduction to Scientific Computing II (3-0-3)

Finite Differences, Interpolation ve Polynomial Approximation, Numerical Differentiation and Integration,  Approximation Theory.

MATH 524 Partial Differential Equations II (3-0-3)

Sobolev Spaces, Weak Derivatives, Second Order Elliptic and Parabolic Equations and their Weak Solutions, Regularity of Solutions, Energy Estimates, Maximum Principles, Second Order Hyperbolic Equations.

MATH 526 Special Functions (3-0-3)

Preliminaries, The Gamma Function, The Beta function, Bessel functions, Legendre polynomials, Hermite and Laguere polynomials, Chebyshev and Jacobi polynomials, Hypergeometric functions, Applications of Special functions.

MATH 528 Mathematical Modelling (3-0-3)

The Modelling Process, Discrete Models, Difference Equations, Continuous Models, Ordinary Differential Equations Models, Partial Differential Equations Models.

MATH 530 Numerical Solutions for Partial Differential Equations (3-0-3)

Introduction to the Elementary Theory of Partial Differential Equations, Well Posedness for Partial Differential Equations, Finite Difference Method for Elliptic and Parabolic Equations, Iterative Methods , Finite Difference Discretization of Hyperbolic Equations, Systems of Partial Differential Equations and Their Numerical Solutions.

MATH 532 Topics in Applied Mathematics II (3-0-3)

Linear stability analysis, Poincare-Bendixson theorem, linear equations with analytic coefficients, series solutions near ordinary points, solution behavior near singular points, regular singular points, Euler equations, solutions near regular singular points by the method of Frobenius, asymptotics and WKBJ solutions, separation of variables for partial differential equations, boundary value problems, Gibb's phenomenon, Fourier transforms, Bessel functions and other special functions, Green's functions, eigenvalue problems, eigenfunction expansions, Sturm-Liouville theory.

MATH 534 Probability Theory (3-0-3)

Probability distributions, random variables, expectation, convergence of distribution, connection between probability theory and real analysis, weak and strong laws of large numbers, central limit theorem.

MATH 536 Graph Theory (3-0-3)

Types of graphs, matching,  connectivity, planar graphs, coloring of graphs, Ramsey theory.

MATH 616  Rational Points on Curves  (3-0-3)

Background on  Function  Fields, Class Field Theory, Explicit  Function  Fields, Function  Fields with Many Rational Places, Asymptotic Results, Applications to Algebraic Coding Theory, Applications to Cryptography, Applications to Low-Discrepancy Sequences, Curves and Their Function Fields.

MATH 618  Introduction to Digital Audio  (3-0-3)

Introducing digital audio, Conversion, Some essential principles, Digital coding principles, Digital audio interfaces, Digital audio tape recorders, Magnetic disk drives, Digital audio editing, Optical disks in digital audio.

MATH 620 Computational Geometry  (3-0-3)

Introduction to Computational Geometry, Line Segment Intersection, Polygon Triangulation, Linear Programming, Orthogonal Range Searching, Convex Hulls.

MATH 622  Algebraic Geometry  (3-0-3)

Guiding problems, Division algorithm and Gröbner bases, Affine varieties, Elimination, Resultants, Irreducible varieties, Nullstellensatz, Primary decomposition, Projective geometry, Projective elimination theory, Parametrizing linear subspaces, Hilbert polynomials and the Bezout Theorem.

MATH 624  Algebraic Topology  (3-0-3)

Homology Groups  of a Simplicial Complex, Topological lnvariance of the Homology Groups, Relative Homology and the Eilenberg-Steenrod Axioms, Singular  Homology Theory, Cohomology, Homology with  Coefficients, Homological Algebra, Duality  in Manifolds.

MATH 626  Cryptography and Network Security  (3-0-3)

Introduction, Classical Encryption Techniques, Block Ciphers and the Data Encryption Standard, Advanced Encryption Standard, More on Symmetric Ciphers, Confidentiality Using Symmetric Encryption, Public-Key Cryptography and RSA, Key Management; Other Public-Key Cryptosystems, Message Authentication and Hash Functions, Hash and MAC Algorithms, Digital Signatures and Authentication Protocols, Authentication Applications, Electronic Mail Security, IP Security, Web Security, Intruders, Malicious Software, Firewalls.

MATH 628  Digital Communications II (3-0-3)

Digital Communication Through Band-Limited Channels, Adaptive Equalization, Multichannel and Multicarrier Systems, Spread Spectrum Signals for Digital Communications, Fading Channels I: Characterization and Signaling, Fading Channels II: Capacity and Coding, Multiple-Antenna Systems, Multiuser Communications.

MATH 900 Special Studies (3-0-3)

MATH 910 PhD Seminar (0-2-0)

It is compalsory that the student will prepare a seminar on a topic to be determined together with his / her supervisor, present it in a suitable way within the pre-defined period and submit the seminar report to his / her supervisor.

MATH 920 PhD Thesis (0-1-0)

FBE 4000  PhD Qualification Exam  (0-1-0)    

Students who have successfully completed their credit courses and seminar course must take the doctoral proficiency exam which includes the general knowledge of the mathematics.