Ek-1b-yuksek-lisans 6106007001995 M.Sc. Thesis ( Non-credit )

spiritualblurtedAI and Robotics

Nov 24, 2013 (3 years and 6 months ago)



Matematik Anabilim Dalı Yüksek Lisans Dersleri ve İçerikleri


. Thesis (



Program of


leading to M.S. degree arranged between student and a

faculty member. Students register to this course in

all semesters

starting from

the begining of their second semester while the research program or write

of thesis is in progress.


Complex Analysis I

( 3 0 3 )

Conform transformations, Riemann transformation theorem, Analytical
n, monodomy theorem, Picard theorem


Complex Analysis II
( 3 0 3 )

Compactness and Convergence in the Space of Analytic Functions, Space of
Analytic Functions, Spaces of Meromorphic Functions, The Riemann Mapping
Theorem, Weirstrass Factoriz
ation Theorem, Runge’s Theorem, Simple
Connectedness, Analytic Continuation along a Path, Harmonic Functions, The
Dirichlet Problem, Green’s Functions, Entire Functions.


Differential Topology
( 3 0 3 )

Differential Manifolds: Differentiabl
e manifolds, local coordinates, Induced
structures and examples, germs, tangent vectors and differentials Sard’s theorem
and regular values, Local properties of immersions and submersions, Vector fields
and flows tangent bundles, Embedding in Euclidean s
pace, Tubular neighborhoods
and approximations, Classical Lie groups, Fiber bundles induced bundles, Vector
bundles and Whitney sums, Transversality. Cohomology: Multilinear algebra and
tensors, Differential forms, Volume element and Orientation integratio
n on forms,
Stokes theorem, Relationship to singular homology, de Rham’s theorem and
singular co homology, Products and duality: Cross product and the Kunneth
theorem, co homology cross product, Cup and cap product, Orientation, Bundle
Duality on compact
manifolds, Intersection theory: The Euler class, Lefchetz
numbers and vector fields, Gysin map and Stiefel Whitney classes, Cobordism and
bordism: cobordism and orientable cobordism, Thom space and Thom
homomorphism, bordism of a topological space.


Computer Algebra

( 3 0 3 )

The introduction to Gröbner Basis, Symbolic recipes for polynomial equations,
factorisation of polynomials, Groebner Basis and Integer programming, Groebner
Basis for codes, Groebner Basis for decoding, Automatic G
eometry Theorem
Proving, The Inverse Kinematics Problem in Robotic.


Perturbation Methods in Applied Mathematics
( 3 0 3 )

Ordering, Asymptotic Sequences and Expansions, Limit Process, Expansions,
Matching, Regular Perturbation, Singular Problem, Singular Perturbation Problems
with Variable Coefficients, Theorem of Erdelyi, Nonlinear Example for Singular
Perturbation, Singula
r Boundary Problems, Method of Strained, Coordinates for
Periodic Solutions, Two Variable Expansions Procedure, Weakly Nonlinear
Systems, Strongly Nonlinear Oscillations, Limit Process Expansions for Second
Order Partial Differential Equations, Singular Bo
undary Value Problems.


Variational Methods of Approximation
( 3 0 3 )

Existence and Uniqueness of Solutions, Linear Algebraic Equations, Linear
Operator Equations, The Contraction Mapping Theorem, Variational Boundary
Value Problems, Regul
arity of the Solutions The Lax MilgramTheorem, The
Neumann Boundary Value Problems, The Boundary Value Problems with Equality
Constraints, The Ritz Method Description Convergence and Stability, The
Weighted Residual Method, The Bubnov
Galerkin Method, The

Method of Least
Squares, Collocation and Sub domain Methods, Time Dependent Problems.


Dynamical Systems
( 3 0 3 )

Second order differential equations in phase plane, linear systems and exponential
operators, Canonical forms, Stability for
ms, Lyapunov functions, The existence of
periodic solutions, Applications to various fields, Oscillation theory.


Algebra I
( 3 0 3 )

Groups: Generalities groups acting on a set, Sylow Theorems, free group, direct
product and sums. Rings
: Generalities, commutative ring, principal ideal domains,
unique factorization domains. Euclidean domains, Noetherian rings, Hilbert’s
Euclidean domains. Noetherian rings Hilbert’s Theorem, Field of fractions,


Algebra II
( 3

0 3 )

Galois Theory: Categories and functors, Generalities, Additive abelian categories,
Yoneda’s lemma. Module categories: Definitions, Projective, injective modules,
simple rings, Modules over Noetherian rings and principal ideal domains,
Morita t
heory. Homological methods: The functors Ext, Tot, (co
) homology,
Derived categories, Derived functors, Stable categories, Applications to co
homology of groups, Schemes.


Topology I
( 3 0 3 )

Euler’s Theorem, Topological equivalence, surfa
ces, Abstract spaces, The
Classification Theorem, Topological Invariants continuity, open and closed sets,
cont. Functions, Peano curve, The Tietze ext. Theorem, Compactness and
Connectedness, Heine
Borel Theorem, Product spaces , Path connectedness,
ification spaces, the torus, the cone construction, glueing lemma, Projective
spaces attaching maps.


Topology II
( 3 0 3 )

The Fundamental group, homotopic maps, Construction of the fundamental group.
Calculations. Homotopy type, The
Brouwer fixed
point Theorem, The simplical
complex surfaces. Classification, Orientation, Euler characteristic, Surgery
momology theory, Cyeles and boundaries, The calculation of momology groups,
degree and Litsehetz number, Euler Roinear’e formula, Bor
Ulam theorem,
The Lifschets fixed

point theorem, Dimension, knots and covering spaces,
Examples of knots, Group covering spaces, Alexander polynomials


Topological Groups
( 3 0 3 )

Topological Groups and continuous homogeneous spaces,

Lie groups and
Differentiable manifolds, Tangent spaces, Adjoint representations, Lie algebra,
Root decompositions, Weyl Groups and Dynkin diagrams


Algebraic Topology

( 3 0 3 )

General Topology
, Group Theory, Modules, Euclidean spaces, Categories,
Functors, Chain complexes, Chain homotopy, Singular homology, Exactness,
Vietuoris sequences, Some applications of homology, Axiomatic
characterization of homology, Homology with coefficients
, Universal coefficient
theorem for homology, The Künnetth formula, Cohomology cup end cop products,
Universal coefficient theorem for co homology of fiber bundle, The co homology
algebra and the Steenrod squaring operations, Hurewietz homomorphism, CW

complex spectral sequences.


Partial Differential Equations and Boundary Value Problems
( 3 0 3 )

Nonhomogeneous Problems, Heat Flow with Sources Method of Eigen function,
Expansion with homogeneous Boundary Conditions, Method of Ei
gen function,
Expansion Using Green’s Formula, Poisson’s Equation, Green’sFunctions for time

independent problems, Fredholm Alternative and Modified Green’s Functions,
Green’s Functions for Poisson’s Equations, Perturbed Eigenvalue Problem, Infinite
Domain Problems, Complex Form of Fourier Series, Fourier Transform and the
Heat Equation, Fourier Sine and Cosine Transforms, Green’s Functions for Time
Dependent Problems, Green’s Functions for the Wave Equation, Green’s Functions
for the Heat Equation,

The Method of Characteristics for Linear and Quasi Linear
Wave Equations, Characteristics for First Order Wave Equations ,The method of
Characteristics for Quasi
Linear Partial Differential Equations, The Laplace
Transform Solution of Partial Differential

Equations, Elementary Properties of the
Laplace Transform, Green’s Functions, Initial Value Problems for Ordinary
Differential Equations, Inversion of Laplace Transforms Using Contour Integrals.


Applied Functional Analysis and Variati
onal Methods in
( 3 0 3 )

General Concepts and Formulas, A Review of the Field Equations of Engineering
Kinematics, Kinetics and Mechanical Balance Laws, Thermodynamics Principles,
Constitutive Laws, Concepts from Functional Analysis, Vector Sp
aces, Linear
Transformations and Functional Theory of Normed Spaces, Theory of Inner
Product Spaces, Variational Formulations of Boundary Value Problems, Linear
Functionals and Operators on Hilbert Spaces, Soboley Spaces and Concept of
Generalized soluti
on, The Minimum of a Quadratic Functional Problems with
Equality Constraints, Existence and Uniqueness of Solutions, Linear Algebraic
Equations, Linear Operator Equations, Variational Boundary Value Problems,
Boundary Value Problems with Equality Constrain
ts, Eigenvalue Problems,
Variational Methods of Approximation, The Ritz Method, The Weighted Residual
Method, Time Dependent Problems.


Rings and Modules
( 3 0 3 )

Rings: Ideals, Factorization in Commutative Rings, Rings of Quotients and
Localization. Modules: Modules, Homomorphisms and Exact Sequences, Free
Modules and Vector Spaces ,Projective and Injective Modules, Commutative Rings
and Modules, Chain Conditions,
Prime and Primary Ideals, Noetherian Rings and
Modules, Dedekind Domains, The Structure of Rings, Simple and Primitive Rings,
The Jacobson Radical, Semi
Simple Rings, The Prime Radical, Prime and Semi
Prime Rings.


mples of Group I
(3 0


The Ascending Central Series and Nilpotent Groups
The Derived Series and
Solvable Groups
Free Groups and Presantations

6106005661995 Examples of Groups II
(3 0 3)

Finite Groups

Infinite abelian groups
Infinite Nonabelian Groups
The group

6106005681995 Generating Functions
(3 0 3)

1. Classical Ortogonal Polynomials

2. Generating Functions ( Linear, Bilinear, Bilateral )

3. Obtaining Generating Functions

3.1. Series Rearrangement Technique

3.2. Decomposition


3.3. Operational Technique

3.4. Lie Algebraic Technique

4. Lagrange’s Expansion and Gould’s Identity


Combinatorial Commutat
ive Algebra and Toric Varieties

(3 0 3)

Convex Bodies. Combinatorial Theory of Polytopes and polyhedral sets.
Polyhedral spheres.

Minkowski sum and mixed volume.

Lattice Polytopes and fans.
Toric Varieties.

Enumeration, sampling and integer programming
Grobner Bases,
Betti numbers and localiz
ations of toric ideals.


ntroduction to Operator Theory
(3 0 3)


Spectral Properties of Bounded Linear Operators.


Use of Complex Analysis in Spectral Theory.


Compact Linear Operators on Normed Spaces.


Spectral Properties of Compact
Linear Operators on Normed Spaces.


Spectral Prtoperties of Bounded Self
Adjoints Linear Operatorts.


Positive Operators.


Projection Operators


Unbounded Linear Operators and their Hilbert

Adjoint Operators.


Adjoint Operators , Symmetric and Self
Adjoint Linear Operators.


Closed Linear Operators and Closeres.


Spectral Properties of Self
Adjoint Linear Operators.

Representation of Unitary Operators


ical Methods in Linear Algebra
(3 0 3)


Linear Systems o
f Equations.


Matrix Algebra with Mathlab.


LU, LQ and QR Decomposition.


Orthogonal Vectors and Matrices.


Norms, Vector Norms and Matris Norms.


Bases and Dimension.


Linear Systems Revisited.


The Solution of Linear Systems Ax=B with Mathlab.


Find the
Eigenvalues and Eigenvectors with Mathlab.


Eigenvalues and Eigenvectors of symmetric and non
symmetric matrices.


Orthogonal Diagonalization.


The Singular Value Decomposition.


Real Analysis
( 3 0 3 )

Introduction to measure theory,
Abstract Integral, Positive Borel Measures, Riesz
representation theorem, Lebesque measure.

6106005741995 Harmonic Analysis
(3 0 3)

Orthogonal systems, Orthogonal series, Fourier series, Some summability
techniques, fundamental properties of harmonic f
unctions, Convolution, Poisson’s
integral, properties of the kernels of Abel
Poisson, Poisson, Weierstrass and Fejer
integrals, Fejer type integral operator. Integral operators with positive kernel,
Approximation properties of family of integral operator,


rier Analysis and Approximation
(3 0 3)

Fourier series, Fourier transformations in L1 space, Sequences of integral
operators, Kernel function and their properties, Characteristic points of functions in
L1 and Lp, Weierstrass theorem an
d approximation properties, Approximation
properties of family of integral operator, Modulus of contiuous of functions in L1
and thier properties, Modulus of contiuous of functions in Lp and their properties,
Convergence of integral operators on characteri
stic points, Convergence rate of
integral operators on characteristic points.


Advanced Differential Geometry
(3 0 3)

Differentiable manifolds, Multilinear algebra, Exterior differential calculus,
Connections, Riemannian geometry, Lie grou
ps and moving frames, Complex
manifolds, The geometry of the gauss map, The intricsic geometry of surfaces,
Global differential geometry.

6106005771995 Homological Algebra I
(3 0 3)

Modules, Categories and Functors, Hom functor, Free Modules, Tensor Product,
Adjoint Isomorphisms,

Projective Modules, Injective Modules, Flat Modules,
Purity, Specific Rings, Localization, Pullback and Pushout Systems, Direct and
Inverse Limit.

781995 Homological Algebra II
(3 0 3)

Abelian Categories and Complexes, homology functors, Projective

resolutions and nth homology, Long exact sequence and Snake Lemma and
homotopy, Comparision Theorem and Left Derived Functors, Horseshoe Le
and covariant

contravariant right derived functors, Tor , Ext, Dimension, Covers
and Envelopes.


Process Dynamics and Control I
(3 0 3)

Laws and languages of process control, time
domain dynamics and control,
conventional control sys
tems, advanced control systems, Laplace
domain dynamics
and control, frequency
domain dynamics and control, process identification.


Functional Analysis
( 3 0 3 )

Normed spaces, Linear and multilinear transformations, The product of normed
spaces, Series concept in normed spaces, Hilbert spaces.


Introduction to Digital Signal Processing (3 0 3)

Time Signals and Systems; The z
Transform; Frequency Analysis of
Signals and Systems; Discrete Fourier Transform;
Filter Structures; Filter Design
Techniques; Applications of Digital Signal Processing


Statistical Signal Processing
(3 0 3)

Introduction to Random Processes; Correlation Function and Power Spectral
Density of Stationary Processes; Noise

Mechanisms, the Gaussian and Poisson
Processes; Introduction to Signal Detection Theory; Linear Mean Square Filtering;
Wiener and Kalman Filtering; Least Square Filtering; Signal Modeling and
Parametric Spectral Estimation.


Mathematical M
ethods for Signal Processing (3 0 3)

Linear operators and Matrix Inverses; Eigenvalues and Eigenvectors; Singular
Value Decomposition; Some Special Matrices and Their Applications; Detection
Theory Algorithms; Estimation Theory Algorithms; Optimal Filter
ing Algorithms;
Methods of Iterative Algorithms; Least Mean Square Adaptive Filtering; Neural
Networks; Clustering; Methods of Optimization.


Process Dynamics and Control II
(3 0 3)

Multivariable process, analysis of multivariable
systems, design of controllers for
multivariable processes, sampled
data control systems, stability analysis of
data systems, applications.


Differential Equations
( 3 0 3 )

Existence and Uniqueness theorems, linear equation with co
nstant coefficients,
nonlinear equations, classification of points solving equations using


Partial Differential Equations
( 3 0 3 )

Diffusion equation, Wave equation, classification of second order Particular
equations, Cauchy problems solution using Fourier series, separation of


Algebraic Geometry
( 3 0 3 )

Introduction, Plane Curves and Conics, Bezout’s Theorem, Affine and Projective
Varieties, Noetherian Rings, Hilbert Basis Theorem
, Nullstellenzats, Zariski
Topology, Noether Normalization, Projective Space and Projective Varieties,
Projective Closure of Affine Varieties, Rational Maps, Rational Map and
Birational Isomorphism, Degree of a Rational Map, Blow
ups, Dimension of
ties, Hilbert Function and Dimension of a Variety, Elementary Properties of
Dimension, Dimension and Algebraic Independence, Dimension and



Special Studies