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Information about the Department of Mathematics Goals:

The main aim of graduate education of our department is giving an education based on analytic thought. Our department is trying to train strong scientists.

Objectives:

Master:

The M. Sc. program supplies an overall and comprehensive outlook on Mathematics issues to the graduate and gives them an improved knowledge in a selected specific area.

Doctorate:

The main objective of the Ph.D. program is to produce scientists.

Qualification Awarded

The students who successfully complete the program are awarded the degree of Master of Science (M.Sc.) or Doctor of Philosophy (Ph.D.) in Mathematics. Admission Requirements Master:

Bachelor Degree. (4 years minimum)

Academic Personnel and Graduate Education Exam (ALES): Minimum score of 55 at quantitative field.

Foreign Language: Minimum score of 50 from Interuniversity Board Foreign Language Examination (ÜDS), or equivalent score from another valid exam (TOEFL, IELTS, etc.), or being successful at foreign language examination by Balikesir university.

Successful in the scientific interview.

For other requirements please visit http://fbe.balikesir.edu.tr .

Doctorate:

Bachelor Degree (4 years minimum) or Master Degree (Applicants to the Ph.D. program with a bachelor degree must have 85/100 undergraduate grade point average, for applicants with master degree must have 75/100 graduate grade point average)

Academic Personnel and Graduate Education Exam (ALES): Minimum score of 55 at quantitative field (Applicants to the Ph.D. program with a bachelor's degree must have 70 at quantitative field).

Foreign Language: Minimum score of 55 from Interuniversity Board Foreign Language Examination (ÜDS), or equivalent score from other valid exam (TOEFL, IELTS, etc.).

Successful in the scientific interview.

For other requirements please visit http://fbe.balikesir.edu.tr .

Graduation Requirements Master:

A (The?) student is required to complete at least 7 courses, not being less than 42 ECTS credits, a seminar course and a thesis. The midterm, if applicable, and the final exams contribute at specified percentages to the final grade. A student should have a final grade of minimum 65/100 in order to pass a course. Seminar course is evaluated as “satisfactory” or “unsatisfactory”. After completing the courses, a student have to prepare a thesis.

Doctorate:

Students are required to complete at least 7 courses, not being less than 42 ECTS credits, within at least four consecutive semesters, maintain a minimum 75/100. Later a student have to be successful at the Doctoral Qualifying Examination and to prepare a thesis. Assessment and Grading

Examination assessment guidelines are described in presentation form of each course. For detailed information on the related course, please look into the detailed course plan. ECTS Coordinator

Assoc. Prof. Ali GÜVEN

Erasmus Coordinator

Assoc. Prof. Sebahattin İKİKARDEŞ

http://fbe.balikesir.edu.tr/

http://fbe.balikesir.edu.tr/

Program’s Key Learning Outcomes:

1. To be able to understand Mathematical materials in basic and advanced level. 2. To be able to develop research-based solutions for encountered scientific problems. 3. To be able to apply Mathematical principles in real world problems. 4. To be able to use Mathematical knowledge in new technology. 5. To be able to develop new strategic approach and to produce solutions by taking responsibility in unexpected and complicated situations in his/her area. 6. To be able to develop solution methods for problems in his/her field and to solve them. 7. To be able to approach actual mathematical problems in various viewpoints and to develop solution method for them. 8. To be able to use Mathematical thought in the whole area of the life, and to apply his/her knowledge in interdisciplinary studies. 9. To be able to improve the knowledge with scientific methods in his/her field by using limited or missing data. 10. To be able to apply the approach and knowledge of different disciplines in Mathematics. 11. To be able to transfer his/her study and its results to large groups of people in writing or orally. 12. To be able to have a foreign language knowledge in a level for following the developments in mathematics, and to communicate with colleagues. 13. To be able to have knowledge about basic computer programs used in Mathematics. 14. To be able to teach and check the values, which are scientific and social, under the ethic rules in stage of collecting, interpreting and announcing the data in his/her field.

Comparison between Program’s Key Learning Outcomes and National Qualifications

Framework for Higher Education in Turkey (NQF-HETR)

KNOWLEDGE - Theoretical, Factual 1. To understand Mathematical materials in basic and advanced level.

SKILLS - Cognitive, Practical 2. To develop research-based solutions for encountered scientific problems.

3. To apply Mathematical principles in real world problems.

4. To use Mathematical knowledge in new technology.

COMPETENCIES

Ability to work independently and take responsibility 5. To develop new strategic approach and to produce solutions by taking responsibility in unexpected and

complicated situations in his/her area (of practice?). 6. To develop solution methods for problems in his/her field and to solve them.

Learning Competence 7. To approach actual mathematical problems in various viewpoints and to develop solution method for them.

8. To use Mathematical thought in the whole area of the life, and to apply his/her knowledge in interdisciplinary

studies. 9. To improve the knowledge with scientific methods in his/her field by using limited or missing data.

10. To apply the approach and knowledge of different disciplines in Mathematics.

Communication and Social Competence 11. To transfer his/her study and its results to large groups of people in writing or orally.

12. To have a foreign language knowledge in a level for following the developments in mathematics, and to

communicate with colleagues.

Field-based Competence 13. To have knowledge about basic computer programs used in Mathematics.

14. To teach and check the values, which are scientific and social, under the ethic rules in stage of collecting,

interpreting and announcing the data in his/her field.

T.R.

BALIKESIR UNIVERSITY

THE INSTITUTE OF SCIENCE AND TECHNOLOGY

2012-2013 EDUCATION YEAR

MATHEMATICS DIVISION COURSE PLANS

Fall Semester

COURSE CODE

COURSE NAME

HOURS

CREDIT ECTS

CREDIT T A L Total

FMT5101 Topology I 3 3 0 0 3 6

FMT5102 Functional Analysis I 3 3 0 0 3 6

FMT5104 Advanced Group Theory 3 3 0 0 3 6

FMT5106 Module Theory I 3 3 0 0 3 6

FMT5107 Real Analysis I 3 3 0 0 3 6

FMT5108 Quasiconformal Mappings 3 3 0 0 3 6

FMT5109 Advanced Differential Geometry I 3 3 0 0 3 6

FMT5111 N. E. C. Groups 3 3 0 0 3 6

FMT5112 Modular Group and Extended Moduler Group 3 3 0 0 3 6

FMT5114 Approximation Theory I 3 3 0 0 3 6

FMT5115 Riemann Surfaces 3 3 0 0 3 6

FMT5116 Representation Theory On Groups 3 3 0 0 3 6

FMT5119 Riemannian Geometry I 3 3 0 0 3 6

FMT5120 Geometry of Submanifolds I 3 3 0 0 3 6

FMT5125 Advanced Control Theory of Systems I 3 3 0 0 3 6

FMT5126 Convex Functions and Orlicz Spaces I 3 3 0 0 3 6

FMT5128 Contact Manifolds I 3 3 0 0 3 6

FMT5129 Structures on Manifolds I 3 3 0 0 3 6

FMT5130 Commutative Algebra 3 3 0 0 3 6

FMT5131 Introduction to Fractional Calculus 3 3 0 0 3 6

FMT5132 Number Theory I 3 3 0 0 3 6

FMT5133 Function Spaces I 3 3 0 0 3 6

FMT5134 Inversion Theory and Conformal Mappings 3 3 0 0 3 6

FMT5136 Selected Topics in Differential Geometry I 3 3 0 0 3 6

FMT5137 Differentiable Manifolds I 3 3 0 0 3 6

FMT5138 Tensor Geometry I 3 3 0 0 3 6

FMT5139 Seminar 0 0 0 0 0 4

FMT5140 Möbius Transformations I 3 3 0 0 3 6

FMT5141 Averaged Moduli and One Sided Approximation

I

3 3 0 0 3 6

FMT5142 Strong Approximation I 3 3 0 0 3 6

FMT5143 Finite Blascke Products I 3 3 0 0 3 6

FMT5144 Algebra I 3 3 0 0 3 6

FMT5145 Orthogonal Polynomials I 3 3 0 0 3 6

FMT5146 Banach Spaces of Analytic Functions I 3 3 0 0 3 6

FMT5147 Fourier Analysis I 3 3 0 0 3 6

FMT5148 Fourier Series and Approximation I 3 3 0 0 3 6

FMT5149 Applied Mathematics I 3 3 0 0 3 6

FMT5150 Advanced Numerical Analysis I 3 3 0 0 3 6

FMT5151 Differential Geomety of Curves and Surfaces I 3 3 0 0 3 6

FMT5152 Introduction to Fuzzy Topology I 3 3 0 0 3 6

FMT5153 Introduction to Ideal Topological Spaces I 3 3 0 0 3 6

FMT5154 Algebraic Number Theory I 3 3 0 0 3 6

FMT5155 Geometric Theory of Functions I 3 3 0 0 3 6

FMT5156 Numerical Optimization I 3 3 0 0 3 6

FMT5157 Selected Topics in Analysis I 3 3 0 0 3 6

FMT5158 Lorentzian Geometry 3 3 0 0 3 6

FMT5159 Semi-Riemannian Geometry I 3 3 0 0 3 6

FMT5160 Tangent and Cotangent Bundle Theory 3 3 0 0 3 6

FMT6101-6199 Special Topics in Field 5 5 0 0 5 6

2012-2013 EDUCATION YEAR

MATHEMATICS DIVISION LESSON PLANS

Spring Semester

COURSE CODE

COURSE TITLE

HOURS

CREDIT ECTS

CREDIT T A L Total

FMT5202 Functional Analysis II 3 3 0 0 3 6

FMT5205 Module Theory II 3 3 0 0 3 6

FMT5206 Fuchsian Groups 3 3 0 0 3 6

FMT5208 Advanced Differential Geometry II 3 3 0 0 3 6

FMT5210 Hyperbolic Geometry 3 3 0 0 3 6

FMT5212 Dynamic System and Applications 3 3 0 0 3 6

FMT5213 Real Analysis II 3 3 0 0 3 6

FMT5215 Discrete Groups 3 3 0 0 3 6

FMT5216 Approximation Theory II 3 3 0 0 3 6

FMT5221 Riemann Geometry II 3 3 0 0 3 6

FMT5222 Geometry of Submanifolds II 3 3 0 0 3 6

FMT5224 Advanced Control Theory of Systems II 3 3 0 0 3 6

FMT5225 Convex Functions and Orlicz Spaces II 3 3 0 0 3 6

FMT5226 Matrices of Semigroups 3 3 0 0 3 6

FMT5227 Contact Manifolds II 3 3 0 0 3 6

FMT5228 Structures on Manifolds II 3 3 0 0 3 6

FMT5230 Algebraic Geometry 3 3 0 0 3 6

FMT5231 Applications of Fractional Calculus 3 3 0 0 3 6

FMT5232 Number Theory II 3 3 0 0 3 6

FMT5233 Seminar 0 0 0 0 0 4

FMT5234 Bergman Spaces 3 3 0 0 3 6

FMT5235 Differentiable Manifods II 3 3 0 0 3 6

FMT5236 Tensor Geometry II 3 3 0 0 3 6

FMT5237 Möbius Transformations II 3 3 0 0 3 6

FMT5238 Averaged Moduli and One Sided Approximation

II

3 3 0 0 3 6

FMT5239 Strong Approximation II 3 3 0 0 3 6

FMT5240 Finite Blaschke Products II 3 3 0 0 3 6

FMT5241 Algebra II 3 3 0 0 3 6

FMT5243 Function Spaces II 3 3 0 0 3 6

FMT5244 Potential Theory 3 3 0 0 3 6

FMT5245 Banach Spaces of Analytic Functions II 3 3 0 0 3 6

FMT5246 Fourier Analysis II 3 3 0 0 3 6

FMT5247 Fourier Series and Approximation II 3 3 0 0 3 6

FMT5248 Applied Mathematics II 3 3 0 0 3 6

FMT5249 Advanced Numerical Analysis II 3 3 0 0 3 6

FMT5250 Numerical Solutions of Partial Differential

Equations 3 3 0 0 3 6

FMT5251 Differential Geometry of Curves and Surfaces II 3 3 0 0 3 6

FMT5252 Topology II 3 3 0 0 3 6

FMT5253 Introduction to Fuzzy Topology II 3 3 0 0 3 6

FMT5254 Introduction to Ideal Topological Spaces II 3 3 0 0 3 6

FMT5255 Orthogonal Polynomials II 3 3 0 0 3 6

FMT5256 Geometric Theory of Functions II 3 3 0 0 3 6

FMT5257 Algebraic Number Theory II 3 3 0 0 3 6

FMT5258 Numerical Optimization II 3 3 0 0 3 6

FMT5259 Selected Topics in Differential Geometry II 3 3 0 0 3 6

FMT5260 Selected Topics in Analysis II 3 3 0 0 3 6

FMT5261 Semi-Riemannian Geometry II 3 3 0 0 3 6

FMT6201-6299 Special Topics in Field 5 5 0 0 5 6

Fall Semester

The Relationship Table between Courses and Program’s Key Learning Outcomes

Courses PKLO1 PKLO2 PKLO3 PKLO4 PKLO5 PKLO6 PKLO7 PKLO8 PKLO9 PKLO10 PKLO11 PKLO12 PKLO13 PKLO14

Topology I X X X X X X X X X X X X X X

Functional Analysis I X X X X X X X X X X X X X X

Advanced Group Theory X X X X X X X X X X X X X X

Module Theory I X X X X X X X X X X X X X X

Real Analysis I X X X X X X X X X X X X X X

Quasiconformal Mappings X X X X X X X X X X X X X X

Advanced Differential Geometry I X X X X X X X X X X X X X X

N. E. C. Groups X X X X X X X X X X X X X X

Modular Group and Extended

Moduler Group X X X X X X X X X X X X X X

Approximation Theory I X X X X X X X X X X X X X X

Riemann Surfaces X X X X X X X X X X X X X X

Representation Theory On Groups X X X X X X X X X X X X X X

Riemannian Geometry I X X X X X X X X X X X X X X

Geometry of Submanifolds I X X X X X X X X X X X X X X

Advanced Control Theory of

Systems I X X X X X X X X X X X X X X

Convex Functions and Orlicz Spaces

I X X X X X X X X X X X X X X

Contact Manifolds I X X X X X X X X X X X X X X

Structures on Manifolds I X X X X X X X X X X X X X X

Commutative Algebra X X X X X X X X X X X X X X

Introduction to Fractional Calculus X X X X X X X X X X X X X X

Number Theory I X X X X X X X X X X X X X X

Function Spaces I X X X X X X X X X X X X X X

Inversion Theory and Conformal

Mappings X X X X X X X X X X X X X X

Selected Topics in Differential

Geometry I X X X X X X X X X X X X X X

Differentiable Manifolds I X X X X X X X X X X X X X X

Tensor Geometry I X X X X X X X X X X X X X X

Seminar X

Möbius Transformations I X X X X X X X X X X X X X X

Averaged Moduli and One Sided X X X X X X X X X X X X X X

Approximation I

Strong Approximation I X X X X X X X X X X X X X X

Finite Blascke Products I X X X X X X X X X X X X X X

Algebra I X X X X X X X X X X X X X X

Orthogonal Polynomials I X X X X X X X X X X X X X X

Banach Spaces of Analytic Functions

I X X X X X X X X X X X X X X

Fourier Analysis I X X X X X X X X X X X X X X

Fourier Series and Approximation I X X X X X X X X X X X X X X

Applied Mathematics I X X X X X X X X X X X X X X

Advanced Numerical Analysis I X X X X X X X X X X X X X X

Differential Geomety of Curves and

Surfaces I X X X X X X X X X X X X X X

Introduction to Fuzzy Topology I X X X X X X X X X X X X X X

Introduction to Ideal Topological

Spaces I X X X X X X X X X X X X X X

Algebraic Number Theory I X X X X X X X X X X X X X X

Geometric Theory of Functions I X X X X X X X X X X X X X X

Numerical Optimization I X X X X X X X X X X X X X X

Selected Topics in Analysis I X X X X X X X X X X X X X X

Lorentzian Geometry X X X X X X X X X X X X X X

Semi-Riemannian Geometry I X X X X X X X X X X X X X X

Tangent and Cotangent Bundle

Theory X X X X X X X X X X X X X X

Special Topics in Field X X X X X X X X X X X X X X

Spring Semester

The Relationship Table between Courses and Program’s Key Learning Outcomes

Courses PKLO1 PKLO2 PKLO3 PKLO4 PKLO5 PKLO6 PKLO7 PKLO8 PKLO9 PKLO10 PKLO11 PKLO12 PKLO13 PKLO14

Functional Analysis II X X X X X X X X X X X X X X

Module Theory II X X X X X X X X X X X X X X

Fuchsian Groups X X X X X X X X X X X X X X

Advanced Differential Geometry II X X X X X X X X X X X X X X

Hyperbolic Geometry X X X X X X X X X X X X X X

Dynamic System and Applications X X X X X X X X X X X X X X

Real Analysis II X X X X X X X X X X X X X X

Discrete Groups X X X X X X X X X X X X X X

Approximation Theory II X X X X X X X X X X X X X X

Riemann Geometry II X X X X X X X X X X X X X X

Geometry of Submanifolds II X X X X X X X X X X X X X X

Advanced Control Theory of Systems

II X X X X X X X X X X X X X X

Convex Functions and Orlicz Spaces


Matrices of Semigroups X X X X X X X X X X X X X X

Contact Manifolds II X X X X X X X X X X X X X X

Structures on Manifolds II X X X X X X X X X X X X X X

Algebraic Geometry X X X X X X X X X X X X X X

Applications of Fractional Calculus X X X X X X X X X X X X X X

Number Theory II X X X X X X X X X X X X X X

Seminar X

Bergman Spaces X X X X X X X X X X X X X X

Differentiable Manifods II X X X X X X X X X X X X X X

Tensor Geometry II X X X X X X X X X X X X X X

Möbius Transformations II X X X X X X X X X X X X X X

Averaged Moduli and One Sided

Approximation II X X X X X X X X X X X X X X

Strong Approximation II X X X X X X X X X X X X X X

Finite Blaschke Products II X X X X X X X X X X X X X X

Algebra II X X X X X X X X X X X X X X

Function Spaces II X X X X X X X X X X X X X X

Potential Theory X X X X X X X X X X X X X X

Banach Spaces of Analytic Functions


Fourier Analysis II X X X X X X X X X X X X X X

Fourier Series and Approximation II X X X X X X X X X X X X X X

Applied Mathematics II X X X X X X X X X X X X X X

Advanced Numerical Analysis II X X X X X X X X X X X X X X

Numerical Solutions of Partial

Differential Equations X X X X X X X X X X X X X X

Differential Geometry of Curves and

Surfaces II X X X X X X X X X X X X X X

Topology II X X X X X X X X X X X X X X

Introduction to Fuzzy Topology II X X X X X X X X X X X X X X

Introduction to Ideal Topological

Spaces II X X X X X X X X X X X X X X

Orthogonal Polynomials II X X X X X X X X X X X X X X

Geometric Theory of Functions II X X X X X X X X X X X X X X

Algebraic Number Theory II X X X X X X X X X X X X X X

Numerical Optimization II X X X X X X X X X X X X X X

Selected Topics in Differential

Geometry II X X X X X X X X X X X X X X

Selected Topics in Analysis II X X X X X X X X X X X X X X

Semi-Riemannian Geometry II X X X X X X X X X X X X X X

Special Topics in Field X X X X X X X X X X X X X X

GRADUATE COURSE DETAILS

Course Title:

Topology I Code :

FMT5101 Institute: Instute of Science

Field: Mathematics

Education and Teaching Methods Credits

Lecture Application Laboratuary Project/

Field Study

Home

work

Other Total Credit

T+A+L=Credit

ECTS

42 0 0 0 0 198 240 3 6

Semester Fall Language Turkish/English

Course Type

Basic

Scientific Scientific Technical

Elective Social

Elective

Course

Objectives

To teach fundamental concepts of Topology.

Learning

Outcomes

and

Competences

To be able to construct Topological structures by using Topological Construction Methods,

To be able to define the concepts of Normality and Expansion of Functions,

To be able to express the Characterizations related to connectedness,

To be able to express the relations between Connectedness and Derived Spaces,

To be able to express the relations among Components, Local Connectedness, Connectedness and T2-

Spaces.

Textbook

and /or

References

1. Şaziye Yüksel, Genel Topoloji (in Turkish), Eğitim Kitapevi, (2011).

2. John L.Kelley, General Topology, Springer-Verlag 1955.

3. K.Kuratowski, Topology, Academic Press 1966.

4. Michael C.Gemignani, Elementary Topology, Dover publications 1990.

5. Nicolas Bourbaki, General Topology, Springer-Verlag 1998.

ASSESSMENT CRITERIA

Theoretical Courses Project Course and Graduation Study

If any,

mark as

(X)

Percent

(%)

If any,

mark as

(X)

Percent

(%)

Midterm Exams Midterm Exams

Quizzes Midterm Controls

Homeworks Term Paper

Term Paper

(Projects,reports, ….) Oral Examination

Laboratory Work Final Exam

Final Exam X % 80 Other

Other (Class

Performance) X % 20

Week Subjects

1 Topology Concepts

2 Topology Construction Methods

3 Base, Subbase

4 Open neighborhoods System

5 First and Second Countable Spaces

6 Subspaces

7 Continuity, Homeomorfizm

8 Part spaces, product spaces

9 T1-spaces, regular spaces and normal spaces

10 Normality and Expansion of Functions

11 The Concept of Connectedness 12 Characterizations related to connectedness

13 Connectedness and Derived Spaces

14 Components, Local Connectedness, Connectedness and T2-Spaces

Instructors Assoc.Prof. Dr. Ahu Açıkgöz

e-mail [email protected]

Website http://matematik.balikesir.edu.tr/

mailto:[email protected]


Course Title:

Functional Analysis I Code :


Field: Mathematics



Field Study

Home

work

Other Total Credit

T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective Social

Elective

Course

Objectives To introduce fundamental concepts and theorems of Functional analysis.

Learning

Outcomes

and

Competences

To be able to define the concepts of Banach space and Hilbert space,

To be able to define the concepts of orthogonal set and orthonormal base,

To be able to define the concept of bounded linear operator,

To be able to state the uniform boundedness principle, open mapping theorem and closed graph

theorem,

To be able to state the Hahn-Banach theorem,

To be able to define the concept of quotient space.

Textbook

and /or

References

1. Barbara D. MacCluer, Elementary Functional Analysis, Springer (2009).

2. J. B. Conway, A Course in Functional Analysis, Springer (1985).

3. W. Rudin, Functional Analysis, McGraw Hill (1991).

ASSESSMENT CRITERIA


If any,

mark as

(X)

Percent

(%)

If any,

mark as

(X)

Percent

(%)




Term Paper



Final Exam X 100 Other

Other

Week Subjects

1

2

3

4

5

6

7

8

9

10

11

12

13

14

Hilbert Spaces

Normed Spaces

Orthogonality

The Geometry of Hilbert Spaces

Linear Functionals

Orthonormal Bases

Bounded Linear Transformations

Adjoints of Operators on Hilbert Spaces

Dual Spaces

Adjoints of Operators on Banach Spaces

The Hahn-Banach Theorem

Uniform Boundedness Principle

Open Mapping and Closed Graph Theorems

Quotent Spaces

Instructors Assoc. Prof. Dr. Ali GÜVEN




Course Title : Advanced Group Theory Code : FMT5104 Institute: Instute of Science

Field: Mathematics


Lecture Application Laboratuary Project/Field

Study

Homework

Other Total Credit

T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type Basic


Elective Social

Elective

Course

Objectives To teach the structure and properties of free groups and some graphs which is very important in group theory.

Learning

Outcomes and

Competences

● to be able to define the free groups,

● to be able to create the presentations of groups,

● to be able to compare the properties of free groups by graphs,

● to be able to express the 1-complexes and their Fundamentals properties,

● to be able to define the Cayley graphs.

Textbook and/or

References

1) D. L. Johnson , Presentatıons of groups, lms student texts 15, Cambrıdge Unıversıty Press, (1997).

2) R. C. Lyndon, P. E. Schupp, Combınatorıal Group Theory, Sprınger-Verlag, (1977).

3) G. M. S. Gomes, P. V. Sılva, J. E. Pın, Semıgroups, Algorıthms, automata and languages, World Scıentıfıc, (2002). 4) W. Magnus, A. Karrass, D. Solıtar, Combınatorıal group theory:Presentatıons of groups ın terms of generators and

relatıons, Dover Publıcatıons, (1975).

5) R. V. Book, F. Otto, Strıng rewrıtıng systems, Sprınger-Verlag, (1993).

ASSESSMENT CRITERIA


If any,

mark as

(X)

Percent

(%)

If any,

mark as

(X)

Percent

(%)

Midterm Exams - - Midterm Exams - -

Quizzes - - Midterm Controls - -

Homeworks - - Term Paper - -

Term Paper

(Projects,reports, ….) - - Oral Examination - -

Laboratory Work - - Final Exam - -


Other

Week Subjects

1 Free groups and theır propertıes

2 Presentatıons of groups

3 Graphs and mappıng of graphs

4 Fundamental group of graph ıs free

5 Applıcatıons of nıelsen-screıer theorem

6 To construct the graph groups

7 Propertıes of free groups by graphs

8 1-complexes and theır Fundamentals properties

9 Homomorphısms over 1-complexes

10 General applıcatıons

11 2-complexes

12 Cayley graphs

13 The fundamental propertıes of cayley graphs

14 General applıcatıons

Instructors Assoc.Prof.Dr.Fırat ATEŞ




Course Title : Module Theory I Code : FMT5106 Institute: Instute of Science

Field: Mathematics



Study

Homework

Other Total Credit

T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type Basic


Elective Social

Elective

Course

Objectives To teach the module theory with a comprehensive manner.

Learning

Outcomes and

Competences

● To be able to express the concepts of abelian groups and their properties,

● To be able to define the concepts of commutator subgroups and their properties,

● To be able to create the exact sequences on abelian groups,

● To be able to define the concepts of module, submodule and to do their applications,

● To be able to define the concepts of Artin and Noether modules.

Textbook and/or

References

1) Harmancı, Cebir II, Hacettepe yayınları, (1987).

2) V. P. Snaıth, Groups, rıngs and galoıs theory, World scıentıfıc, (2003).

3) J. J. Rotman, An ıntroductıon to the theory of groups, Sprınger- Verlag, (1995).

ASSESSMENT CRITERIA


If any,

mark as

(X)

Percent

(%)

If any,

mark as

(X)

Percent

(%)




Term Paper




Other

Week Subjects

1 Remind the fundamental algebraic structures 2 Finitely generated Abelian groups and properties 3 Series of groups and their types (compozıtıon series etc. vs.) 4 Commutator subgroups 5 Nilpotent and solvable groups 6 General applications 7 Exact sequences on f.g. Abelian groups 8 Basics of module, submodule and applications 9 Factor modules and homomorphisms

10 Direct sum and direct product 11 Free module and its properties 12 Injective and projective modules 13 Artin and noether modules 14 General applications

İnstructors Assoc.Prof.Dr.Fırat ATEŞ




Course Title:

Real Analysis I Code :


Field: Mathematics



Field Study

Home

work

Other Total Credit

T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective Social

Elective

Course

Objectives To teach fundamental concepts of Measure and integration theory in advanced level.

Learning

Outcomes

and

Competences

To be able to express the concepts of σ- Algebra and measure,

To be able to define the concepts of outer measure and measurable set,

To be able to define the concept of Lebesgue measure,

To be able to express the concept of measurable function,

To be able to the express the Lebesgue integral and its some properties,

To be able to define the product measures.

Textbook

and /or

References

1. C. D. Aliprantis, O. Burkinshaw, Principles of Real Analysis, Academic Press, (1998).

2. W. Rudin, Real and Complex Analysis, McGraw Hill, (1987).

3. G. B. Folland, Real Analysis, John Wiley & Sons, Inc., (1999).

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Week Subjects

1

2

3

4

5

6

7

8

9

10

11

12

13

14

σ- Algebras

Measures

Outer measures and measurable sets

Lebesgue measure

Measurable functions

Simple functions

Integration of simple functions

Integration of nonnegative functions

Fatou Lemma and Monotone convergence theorem

İntegrable functions

Lebesgue dominated convergence theorem

Integration of Complex functions

Product measures

Double integrals and Fubini’s theorem

Instructors Assoc. Prof. Ali GÜVEN




Course Title: Quasiconformal Mappings Code : FMT5108 Institute: Instute of Science

Field: Mathematics



Field Study

Home

work

Other Total Credit

T+A+L=Credit

ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective Social

Elective

Course

Objectives To teach some selected topics of Complex Analysis and Quasiconformal mapping theory.

Learning

Outcomes

and

Competences

To be able to define the concept of Conformal mapping,

To be able to state the concept of normal family and Montel’s theorem,

To be able to state The Riemann conformal mapping theorem,

To be able to define the concept of Quasiconformal mappings,

To be able to explain the relation between conformal and quasiconformal mappings.

Textbook

and /or

References

1. V. V. Andrievskii, V. I. Beyli, V. K. Dzyadyk, Conformal invariants in constructive theory of functions of

complex variable, World Scientific, (2000).

2. L. Ahlfors, Lectures on Quasiconformal mappings, Mir, Moscow, (1969).

3. O. Lehto, K. I. Virtonen, Quasiconformal mappings in the plane, Springer-Verlag, (1987).

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Week Subjects

1

2

3

4

5

6

7

8

9

10

11

12

13

14

Conformal mappings

Some simple conformal mappings

Conformal automorphisms and izomorphisms

The normal families

The Montel compactness criterion

The Riemann conformal mapping theorem

Conformal mappings on the boundaries of the domains

Quasiconformal mappings

Different definitions of the quasiconformal mappings

Relation between conformal and quasiconformal mappings

The conformity modulus

Properties of the modulus

The quasiinvariantness of the modulus

Applications of the quasiinvariants in the Approximation theory

Instructors Prof. Dr. Daniyal Israfilzade




Course Title:

Advanced Differential Geometry I

Code : FMT 5109 Institute: Instute of Science

Field: Mathematics


Lecture Application. Laboratory. Project/Field

Study

Homework Other Total Credit

T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective Social

Elective

Course

Objectives

To teach the general properties of curves and surfaces in three dimensional Euclidean space and

manifolds.

Learning

Outcomes

and

Competences

To be able to express the general properties of curves in 3-dimensional Euclidean space,

To be able to express the general properties of 1-forms and differential forms,

To be able to express the fundamental conceptes about surfaces and manifolds,

To be able to define the concepts of regular surface and oriented surface,

To be able to define the mappings of surfaces.

Textbook

and /or

References

1) B. O’Neill, Elementary Differential Geometry, Academic Pres, Inc., 1966.

2) H. H. Hacısalihoğlu, Yüksek Diferensiyel Geometri’ ye Giriş, Fırat Ünv. Fen Fak. 1980.

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Other

Week Subjects

1 Curves in 3-dimensional Euclidean space, examples of some curves 2 1-forms 3 Differential forms 4 Frame fields, connection forms 5 The structural equations 6 Isometries 7 Orientation 8 Surfaces in 3-dimensional Euclidena space 9 Regular surfaces 10 Oriented surfaces 11 Mappings of surfaces 12 Topological properties of surfaces 13 Manifolds I 14 Manifolds II

Instructors Prof. Dr. Cihan ÖZGÜR




Course Title:

N.E.C. Groups

Code :

FMT5111

Institute: Instute of Science

Field: Mathematics



Field Study

Home

work

Other Total Credit

T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective Social

Elective

Course

Objectives To teach some fundamental definitions and theorems related with N.E.C. groups.

Learning

Outcomes

and

Competences

To be able to define the concepts of NEC group and Fuchsian group,

To be able to define the concepts of discrete group and fundamental region,

To be able to find the presentation and the signature of NEC groups,

To be able to define the fundamental concepts of Hyperbolic geometry,

To be able to explain the relationships between Fuchsian groups and NEC groups.

Textbook

and /or

References

1) T. Başkan, Discrete Groups (in Turkish), Hacettepe Üniversitesi Fen Fakültesi Yayınları, (1980).

2) E. Bujalance, J. J. Etayo, J. M. Gamboa, G. Gromadzki , Automorphisms Groups of Compact

Bordered Klein Surfaces. A Combinatorial Approach, Lecture Notes in Mathematics, Springer-

Verlag, (1990).

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1 Topological transformation groups 2 NEC groups 3 The properties of the NEC groups 4 Fuchsian groups 5 The elementary properties of the Fuchsian groups 6 The relationships between Fuchsian groups and NEC groups 7 Linear transformations with real coefficients 8 The elementary properties of the linear transformations with real coefficients 9 Discrete groups 10 The properties of discrete groups 11 Hyperbolic geometry 12 Fundamental regions 13 Surface signatures 14 The presentation of NEC groups

Instructors Prof. Dr. Recep Şahin





Course Title: Modular Group and Extended

Modular Group

Code :

FMT5112


Field: Mathematics



Field Study

Home

work

Other Total Credit

T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective Social

Elective

Course

Objectives

To give some fundamental definitions and theorems related with modular group and extended

modular group.

Learning

Outcomes

and

Competences

To be able to define the fundamental properties of the Modular group,

To be able to define the concepts of Power subgroup, commutator subgroup and congruence

subgroup of the modular group,

To be able to obtain the generators and presentations of these subgroups,

To be able to express the relationships among these subgroups,

To be able to express the fundamental properties of the extended modular group and its subgroups.

Textbook

and /or

References

1. M. Newman, Integral Matrices, Academic Press, (1972).

2. H.S.M. Coxeter and W.O.J. Moser, Generators and Relations for Discrete Groups, Springer,

(1972).

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1 Modular group and its properties 2 Generators and abstract presentation of the modular group 3 Fundamental region of the modular group 4 Power subgroups of the modular group 5 Commutator subgroups of the modular group 6 The relationships between the commutator subgroups and power subgroups of the modular

group 7 Congruence subgroups of the modular group 8 Principal congruence subgroups of the modular group 9 Extended modular group 10 Generators and abstract presentation of the extended modular group 11 Power subgroups and commutator subgroups of the extended modular group 12 The relationships between the commutator subgroups and power subgroups of the extended

modular group 13 Fundamental region of the extended modular group 14 The properties of the extended modular group






Course Title: Theory of Approximation I

Code : FMT5114


Field: Mathematics



Field Study

Home

work

Other Total Credit

T+A+L=Credit

ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective Social

Elective

Course

Objectives To teach fundamental concepts and theorems of approximation theory in the real axis.

Learning

Outcomes

and

Competences

To be able to express the fundamental concepts of approximation theory,

To be able to express Weierstrass’s theorems for approximation by algebraic and trigonometric

polynomials,

To be able to express the direct and converse of approximation theory,

To be able to express the concepts of modulus of continuity,

To be able to define the local and global estimations of approximation theory.

Textbook

and /or

References

1.V. K. Dzyadyk, Introduction to the theory of uniform approximation of functions by polynomials (Russian),

Moscow, (1977).

2. R. A. De Vore and G. G. Lorentz, Constructive Approximation, Springer, (1993).

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1

2

3

4

5

6

7

8

9

10

11

12

13

14

Function Spaces

Fundamental problems of Approximation Theory

Approximation by algebraic polynomials and Weierstrass theorems

Approximation by trigonometric polynomials and Weierstrass theorems

The modulus of continuity and its properties

The direct theorems of polynomial approximation on the real line, Jackson’s theorems

The inverse theorems of polynomial approximation on the real line, Bernstein’s theorems

Local and global estimations of Approximation Theory

Lebesgue spaces

Modulus of smoothness in Lebesgue spaces

Approximation in the Lebesgue spaces

Direct theorems

Inverse theorems

Comparsion of the results





Course Title:

Riemann Surfaces

Code :

FMT5115


Field: Mathematics



Field Study

Home

work Other Total Credit

T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic

Scientific Scientific Technical Elective Social

Elective

Course

Objectives To introduce the basic knowledge about Riemann surfaces.

Learning

Outcomes

and

Competences

To be able to express the concepts of analytic and meromorphic continuation,

To be able to define the concepts of Riemann surface and abstract Riemann surface,

To be able to express the Monodromy theorem,

To be able to define the concepts of analytic, meromorphic and holomorphic functions on Riemann

surfaces,

To be able to define the Riemann surface of an algebraic function.

Textbook

and /or

References

G. A. Jones and D. Singerman, Complex Functions, Cambridge University Press (1987).

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Week Subjects

1 Meromorphic and analytic continuation

2 Analytic continuation using power series

3 Regular and singular points

4 Meromorphic continuation along a path

5 The Monodromy theorem

6 The Fundamental group

7 Riemann surfaces of the functions Log(z) and z1/q

8 Abstract Riemann surfaces

9 Analytic, meromorphic and holomorphic functions on Riemann surfaces

10 The Riemann surface of an algebraic function

11 Oriantable and non-oriantable surfaces

12 The genus of a compact Riemann surface

13 Conformal equivalence and automorphisms of Riemann surfaces

14 Covering surfaces of Riemann surfaces

Instructors Prof. Dr. Nihal YILMAZ ÖZGÜR




Course Title : Representation Theory on

Groups Code : FMT5116


Field: Mathematics


Lecture Application Laboratuar

y

Project/Field

Study

Homework

Other Total Credit

T+A+L=Credi

t

ECTS

42 0 0 0 0 198 240 3 6


Course Type Basic

Scientific Scientific

Technical

Elective

Social

Elective

Course

Objectives To teach the definitions and theorems of advanced group theory in a comprehensive manner.

Learning

Outcomes and

Competences

● To be able to define the Jacabson radicals of an algebra,

● To be able to express the exact factorization modules,

● To be able to express the Burnside theorem ,

● To be able to construct the characters over different algebras,

● To be able to define semi simple and simple algebras.

Textbook

and/or

References

1) J. L. Alperin, R. B. Bell, Groups and representations, Springer, (1995).

2) J. J. Rotman, An introduction to the theory of groups, Brown Publ., (1988).

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Week Subjects

1 Remind the fundamental algebraic structures

2 Finitely generated Abelian groups and applications

3 C-algebras

4 Modules and homeomorphisms

5 Jacabson radicals of an algebra

6 General applications

7 Exact factorization modules

8 Sem simple and simple algebras

9 The characters over different algebras

10 Algebraic integers

11 Burnside theorem on p^a q^b

12 Applications of this theorem



Instructors Assoc.Prof.Dr.Fırat ATEŞ




Course Title:

Riemannian Geometry I Code : FMT 5119


Field: Mathematics



Study


T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective Social

Elective

Course

Objectives

To teach the general properties of differentiable manifolds, tensors, immersion and imbeddings,

connections and geodesics.

Learning

Outcomes

and

Competences

To be able to define the notion of a differentiable manifold and to give examples,

To be able to define the general properties of tensors,

To be able to define the notions of affine connections and Riemannian connections,

To be able to define the notions of curvature tensor and sectional curvature,

To be able to define the notion of tensor on Manifolds.

Textbook

and /or

References

1) Manfredo Perdigao do Carmo , Riemannian Geometry , Birkhauser, 1992.

2) W. M. Boothby, An introduction to Differentiable manifolds and Riemannian Geometry, Elsevier,

2003.

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Visa examination Midterm Exams

Quiz Midterm Controls

Homework Term Paper

Term project (project,

report, etc) Oral Examination

Laboratory Final Exam

Final examination X 100 Other

Other

Week Subjects

1 Differentiable manifolds

2 Tangent spaces

3 Immersions and Imbeddings and some examples

4 Orientations

5 Vector fields, Lie brackets

6 Topology of Manifolds

7 Riemann metrics

8 Affine connections and Riemann connections

9 Geodesics

10 Convex neighborhoods

11 Curvature tensor and sectional curvature

12 Ricci curvature and scalar curvature

13 Tensors on Manifolds I

14 Tensors on Manifolds II





Course Title:

Geometry of Submanifolds I Code : FMT 5120


Field: Mathematics



Study


T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective Social

Elective

Course

Objectives

To teach the general properties of differentiable manifolds, tensors, Riemannian and semi-

Riemannian manifolds and their submanifolds.

Learning

Outcomes

and

Competences

To be able to define the notions of Riemannian and semi-Riemannian manifolds and to

give some examples of them,

To be able to express general properties of tensors,

Tobe able to define general properties of submanifolds

Tobe able to define the notion of second fundamental form and to do its applications,

Tobe able to define the notion of submanifolds with flat normal connection.

Textbook

and /or

References

B. Y. Chen , Geometry of Submanifolds, Pure and applied mathematics (Marcel Dekker, Inc.), New York,

1973

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Week Subjects

1 Differentiable manifolds 2 Tensors 3 Riemannian manifolds 4 Semi-Riemannian manifolds 5 Exponential map and normal coordinates 6 Weyl conformal curvature tensor 7 Kaehler manifolds 8 Submersions and Projective spaces 9 Submanifolds 10 Induced connections 11 Second fundamental form and its properties I 12 Second fundamental form and its properties II 13 Curvature tensor of submanifolds 14 Submanifolds with flat normal connection





Course Title: Advanced Systems Theory I Code : FMT5125 Institute: Instute of Science

Field: Mathematics



Field Study

Home

work

Other Total Credit

T+A+L=Credit

ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective Social

Elective

Course

Objectives To teach the concept of Mathematical control theory.

Learning

Outcomes

and

Competences

To be able to express continuous and discrete time state space systems,

To be able to express the concepts of Laplace and Z transformations,

To be able to define the concept of stability analysis,

To be able to define the concept of Lyapunov stability,

To be able to define the concepts of controllability and observabilty.

Textbook

and /or

References

1. C. T. Chen, Linear System Theory and Design, Oxford University Press, 1999.

2. E. D. Sontag, Mathematical Control Theory, Springer-Verlag, 1990.

3. S. Barnett, R. G. Cameron, Introduction to Mathematical Control Theory, Oxford University Press, 1985.

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1

2

3

4

5

6

7

8

9

10

11

12

13

14

Matrix Algebra

Continuous and discrete time state space systems.

Laplace transform, transfer function.

z transform.

General solutions using with similarity transformations.

Stability Theory and phase portraits.

Stability theory for linear systems

Lyapunov stability method.

Lyapunov stability method for linear systems.

Controllability.

Controllability Canonic Form.

Stabilizability.

Pole placement.

Observability, observers.

Instructors Assoc. Prof. Dr. Necati ÖZDEMİR





Course Title:

Convex functions and Orlicz spaces I Code : FMT5126


Field: Mathematics



Field Study

Home

work

Other Total Credit

T+A+L=Credit

ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective Social

Elective

Course

Objectives To teach basic structure of Orlicz spaces.

Learning

Outcomes

and

Competences

To be able to define the fundamental properties of the convex functions,

To be able to define the notions of N function and complementary N function,

To be able to define the Notion of Orlicz space,

To be able to express the relation between Orlicz spaces and Lebesgue spaces,

To be able to define the quivalent norms on the Orlicz spaces.

Textbook

and /or

References

1. M. A. Krasnosel’ski and Ya. B. Rutickii, Convex funktions and Orlicz Spaces, Noordhoff, (1961).

2. C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press, (1988).

3. M. M. Rao, Z. D. Ren, Applications of Orlicz Spaces, Marcel Dekker, (2002).

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Week Subjects

1

2

3

4

5

6

7

8

9

10

11

12

13

14

Convex functions and continuous functions

Properties of the convex functions

N function and its properties

Complementary N function and its properties

Young inequality

Some inequalities for the N functions and complementary N functions

Comparsion of the N functions

The fundamental part of the N function

2 and ’ conditions

2 and ’ conditions for the complementary N functions

Orlicz classes

Relation with Orlicz classes and Lebesgue spaces

Orlicz spaces

Equivalent norms on the Orlicz spaces





Course Title:

Contact Manifolds I Code : FMT 5128


Field: Mathematics


Lecture Application Laboratory. Project/Field

Study


T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective Social

Elective

Course

Objectives

To teach the general properties of contact structures and contact manifolds.

Learning

Outcomes

and

Competences

To be able to define the notions of a contact structure and complex structure and to give some

examples of these kinds of structures,

To be able to define the notions of an integral submanifold and a contact transformation,

To be able to define the notions of Legendre curve and CR-submanifold and to give some

applications of them,

To be able to define the curvature of a contact metric manifold,

To be able to define the notions of -sectional curvature and Sasakian space form.

Textbook

and /or

References

D. Blair , Riemannian Geometry of Contact and Symplectic Manifolds, Birkhauser, 2002.

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Week Subjects

1 Symplectic manifolds 2 Principal S

1-bundles

3 Contact manifolds, examples 4 Almost complex and contact structures, examples of contact manifolds 5 Almost contact metric manifolds, examples 6 Integral submanifolds and contact transformations 7 Examples of contact integral submanifolds 8 Legendre curves and Withney spheres 9 Sasakian and cosymplectic manifolds 10 CR-manifolds 11 Product of almost contact manifolds 12 Curvature of contact metric manifolds 13 -sectional curvature, Sasakian space form 14 Examples of Sasakian space forms, locally -symmetric spaces





Course Title:

Structures on Manifolds I Code : FMT5129


Field: Mathematics



Study


T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective Social

Elective

Course

Objectives

To teach the general properties of Riemannian manifolds, tensors, almost complex and complex manifolds,

Hermitian manifolds, Kaehler Manifolds, Nearly Kaehlerian manifolds and Quaternion Kaehlerian

manifolds.

Learning

Outcomes

and

Competences

To be able to define the notion of a Riemannian manifold,

To be able to define the notions of tensor, Riemannian curvature tensor, Ricci tensor, sectional

curvature, scalar curvature and to give examples.

To be able to express the Gauss, Codazzi and Ricci equations,

To be able to define the notions of almost complex and complex manifolds,

To be able to define the notions of Hermitian manifold, Kaehler Manifold, Nearly Kaehlerian

manifold and Quaternion Kaehlerian manifolds.

Textbook

and /or

References

Kentaro Yano and Mashiro Kon , Structures On Manifolds, World Sci. 1984.

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Week Subjects

1 Riemannian manifolds 2 Tensors 3 Connections and covariant derivatives 4 Riemannian curvature tensor, Ricci tensor, sectional curvature, scalar curvature 5 Fibre bundles and covering spaces 6 Induced connection and second fundamental form 7 Gauss, Codazzi and Ricci equations 8 The Laplacian of the second fundamental form, submanifolds of space forms 9 Minimal submanifolds 10 Almost complex and complex manifolds 11 Hermitian manifolds 12 Kaehlerian Manifolds 13 Nearly Kaehlerian manifolds 14 Quaternion Kaehlerian manifolds





Course Title:

Commutative Algebra Code :

FMT5130

Institute: Graduate School of Natural and Applied

Sciences

Field : Mathematics


Lecture Application Lab. Project/

Field Study


T+A+L=Credit

ECTS

42 0 0 0 100 98 240 3 6


Course Type

Basic


Elective Social

Elective

Course

Objectives To teach the commutative rings including algebraic geometry, number theory and invariant theory.

Learning

Outcomes

and

Competencies

To be able to define the concepts of ring, ideal and module,

To be able to express the Hilbert basis theorem,

To be able to define the integral extensions,

To be able to define the concept of an irreducible variete,

To be able to define the concept of Artinian ring.

Textbooks

and /or

References

1. D. Eisenbud , Commutative Algebra with a View Toward Algebraic Geometry, Springer 1995.

2. M.F Atiyah and I.G. MacDonald, Introduction to Commutative Algebra, Perseus Books 1994.

3. E. Kunz , Introduction to Algebra and Algebraic Geometry, Birkhäuser Boston 1984.

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Homework X 60 Term Paper

Term Paper, Project

Reports, etc. Oral Examination



Other

Week Subjects

1

2

3

4

5

6

7

8

9

10

11

12

13

14

Rings and Ideals

Radicals

Modules

The determinant trick

Noetherian rings

The Hilbert Basis Theorem

Integral Extensions

Noether Normalization

The Nullstellensatz

Irreducible Varieties

Ring of Fractions and Localization

Primary Decomposition

Artinian Rings

Discrete Valuation Rings

Instructor/s Asst.Prof.Dr. Pınar Mete


Website http://matematik.balikesir.edu.tr


Course Title: Introduction to Fractional Calculus Code : FMT5131 Institute: Instute of Science

Field: Mathematics



Field Study

Home

work

Other Total Credit

T+A+L=Credit

ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective Social

Elective

Course

Objectives To teach the concept of fractional derivative and fractional integral.

Learning

Outcomes

and

Competences

To be able to define special functions of fractional analysis,

To be able to express the concepts of Riemann-Liouville fractional integral and derivative,

To be able to express Grünwald-Letnikov fractional derivative and its properties,

To be able to express Caputo fractional derivative and its properties,

To be able to calculate the Laplace transforms of fractional derivatives,

To be able to express solution methods of fractional-order differential equations.

Textbook

and /or

References

1. I. Podlubny, Fractional Differential Equations, Academic Pres, 1999.

2. K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, 1974.

3. K. S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations,

John Wiley & Sons, Inc., 1993.

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Week Subjects

1

2

3

4

5

6

7

8

9

10

11

12

13

14

The origin of the fractional calculus.

Special functions of the fractional calculus.

Riemann-Liouville fractional integral and derivative.

Grünwald-Letnikov fractional derivative and its properties.

Caputo fractional derivative and its properties

Comparison of fractional derivative approaches.

Laplace transforms of fractional derivatives

Fractional-order differential equations.

Fractional Green functions.

Solution methods of fractional-order differential equations.

Numerical evaluation of fractional derivatives.

Comparison the analytical and numerical solutions of fractional-order differential equations.

Physical problems defined by fractional-order differential equations

MATLAB applications of problem solutions.






Course Title:

Number Theory I Code :

FMT5132


Field: Mathematics



Field Study

Home

work

Other Total Credit

T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective Social

Elective

Course

Objectives To give some fundamental definitions and theorems related with the number theory.

Learning

Outcomes

and

Competences

To be able to solve the linear Diophantine equations,

To be able to express Euler’s and Fermat’s Theorems,

To be able to solve systems of linear equations and congruence systems,

To be able to define the fundamental notions related to Fermat and Mersenne primes, Gauss and

Jacobi sums,

To be able to apply division and Euclid’s algorithms.

Textbook

and /or

References

1. K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Springer, (1990).

2. İ.N.Cangül, B. Çelik, Sayılar Teorisi Problemleri, Nobel Yayınları, (2004).

3.G. A.Jones and J.M. Jones, Elementary Number Theory, Springer, (2004).

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Week Subjects

1 Divisibility and Euclid’s Algorithm 2 Linear Diophantine Equations 3 Euler’s Function 4 Congruences and The Chinese Remainder Theorem 5 Euler’s Theorem and Fermat’s Theorem 6 Congruences Systems 7 Fermat prime and Mersenne prime 8 The ring Z[i] and Z[w] 9 Primitive Roots 10 The Group Structure of Un 11 Sums of Squares 12 Gauss Sums 13 Jacobi Sums 14 Divisibility and Euclid’s Algorithm

Instructors Assist. Prof. Dr. Dilek Namlı





Course Title:

Function Spaces I Code :

FMT5133


Field: Mathematics



Field Study

Home

work

Other Total Credit

T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective Social

Elective

Course

Objectives To teach several function spaces and relations among them.

Learning

Outcomes

and

Competences

To be able to define the notion of Lebesgue space,

To be able to define the notion of Orlicz space,

To be able to Express the relation between Orlicz and Lebesgue spaces,

To be able to define the concept of Rearrangement invariant Banach function space,

To be able to Express the relation between Orlicz and Rearrangement invariant Banach function

spaces.

Textbook

and /or

References

1) C. Bennet and R. Sharpley, Interpolation of operators, Academic Pres, 1987.

2) M. A. Krasnosel’ski and Ya. B. Rutickii, Convex funktions and Orlicz Spaces, Noordhoff, (1961).

3) L. Grafakos, Classical Fourier Analysis, Springer, 2008.

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1 Lebesgue spaces 2 Lebesgue spaces 3 Lebesgue spaces 4 Inequalities in Lebesgue spaces 5 Inequalities in Lebesgue spaces 6 Orlicz spaces 7 Orlicz spaces 8 Structure properties of Orlicz spaces 9 Rearrangement invariant Banach function spaces 10 Rearrangement invariant Banach function spaces 11 Main inequalities in Rearrangement invariant Banach function spaces 12 Main inequalities in Rearrangement invariant Banach function spaces 13 Particular cases of Rearrangement invariant Banach function spaces 14 Particular cases Rearrangement invariant Banach function spaces

Instructors Assoc.Prof.Dr. Ramazan AKGÜN




Course Title:

Inversion Theory and Conformal Mappings

Code :

FMT5134


Field: Mathematics



Field Study

Home


T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective

Course

Objectives To introduce the basic knowledge about inversion theory and conformal mapping.

Learning

Outcomes

and

Competences

To be able to define and to apply the concept of cross ratio,

To be able to express the definition and fundamnetal properties of fractional linear transformations

and to apply them,

To be able to define the concept of conformal mapping and to apply it,

To be able to define the Poincaré model of Hyperbolic geometry,

To be able to define the concept of inversion.

Textbook

and /or

References

1) D. E. Blair, Inversion Theory and Conformal Mapping, AMS, Providence, RI, (2000).

2) G. A. Jones and D. Singerman, Complex Functions, Cambridge University Press, (1987).

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Week Subjects

1 Classical inversion theory in the plane

2 Cross ratio

3 Applications: Miquel’s Theorem

4 Applications: Feuerbach’s Theorem

5 The extended complex plane and stereographic projection

6 Linear fractional transformations

7 Some special linear fractional transformations

8 Extended Möbius transformations

9 The Poincaré models of hyperbolic geometry

10 Conformal maps in the plane

11 Inversion in spheres, conformal maps in Euclidean space

12 Sphere preserving transformations

13 Surface theory, the classical proof of Liouville’s theorem

14 Curve theory and convexity





Course Title: Selected Topics in Differential

Geometry I

Code : FMT5136


Field: Mathematics



Field Study


T+A+L=Credit

ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective Social

Elective

Course

Objectives To teach fundamental concepts of Riemannian geometry and finite-type submanifolds.

Learning

Outcomes

and

Competencies

To be able to define the concept of differentiable manifold and to give examples,

To be able to define the concept of tangent space,

To be able to define the topology of manifolds,

To be able to define the concepts of Riemannian metric, affine and Riemannian connection and to

give examples,

To be able to define the concept of geodesic.

Textbooks

and /or

References

1) M.P. do Carmo, Riemannian Geometry, Birkhauser Boston 1992.

2) Bang-yen Chen, Total Mean Curvature and Submanifolds of Finite Type, World Scientific 1984.

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Other

Week Subjects

1

2

3

4

5

6

7

8

9

10

11

12

13

14

Differentiable manifolds,

Differentiable manifolds,

Tangent space

Tangent space

Immersions and Embeddings

Immersions and Embeddings

Orientation

Vector fields,

Topology of Manifolds

Topology of Manifolds

Riemannian metrics, affine and Riemannian connections

Riemannian metrics, affine and Riemannian connections

Geodesics

Geodesics

Instructor/s Assoc.Prof.Dr.BENGÜ BAYRAM




Course Title:

Differentiable Manifolds I

Code : FMT5137 Institute: Instute of Science

Field: Mathematics



Study


T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective Social

Elective

Course

Objectives To teach the general properties of differentiable manifolds, vector fields and Lie groups.

Learning

Outcomes

and

Competences

To be able to define the concept of a differentiable manifold and to give some examples,

To be able to define the concept of submanifold,

To be able to express the fundamental geometrical structures of Lie groups,

To be able to define the concept of vector field on manifolds,

To be able to define one parameter subgroups of Lie groups.

Textbook

and /or

References

Boothby, William M. An introduction to differentiable manifolds and Riemannian geometry. Second

edition. Academic Press, Inc., Orlando, FL, 1986.

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1 An introduction to manifolds 2 Multi variables functions and mappings 3 Vector fields and inverse function theorem 4 The rank of a mapping 5 Differentiable manifolds and examples 6 Differentiable functions and mappings 7 Applications 8 Submanifolds 9 Lie groups 10 Applications 11 Vector fields on manifolds 12 One parameter subgroups of Lie groups 13 Frobenius Theorem 14 Applications





Course Title:

Tensor Geometry I Code : FMT5138


Field: Mathematics



Study


T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective Social

Elective

Course

Objectives To teach the fundamental knowledge about tensors.

Learning

Outcomes

and

Competences

To be able to define the notions of tensors, covariant and contravariant tensors and to

give their examples,

To be able to use tensors on Riemannian manifolds,

To be able to define and calculate the derivative of a tensor,

To be able to define the Christoffel symbols,

To be able to define the notions of Riemannian curvature tensor and sectional curvature.

Textbook

and /or

References

1) H. Hilmi Hacısalihoğlu , Tensör Geometri, Ankara Ünv. Fen-Fakültesi, 2003.

2) D. C. Kay, , Schaum’s outline of theory and problems, McGraw-Hill, 1988.

3) C. T. J. Dodson, T. Poston, Tensor geometry, Springer-Verlag, Berlin, 1991.

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1 Tensors, covariant and contravariant tensors 2 Applications 3 Tensor products of two tensors 4 Applications 5 Metric tensor 6 Applications 7 The derivative of a tensor 8 Applications 9 Tensors on Riemannian manifolds 10 Applications 11 Christoffel symbols 12 Applications 13 Riemannian curvature tensor, sectional curvature 14 Applications





Course Title:

Möbius Transformations I

Code :


Field: Mathematics



Field Study

Home


T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective

Course

Objectives To introduce the basic knowledge about Möbius transformations and their elementary properties.

Learning

Outcomes

and

Competences

To be able to define and apply basic properties of Möbius transformations on the extended complex

plane,

To be able to explain the relations between Möbius transformations and circles,

To be able to explain fundamental properties of the inversion in a circle,

To be able to define types of transformations and to give examples,

To be able to define the notion of isometric circle.

Textbook

and /or

References

1) L. R. Ford, Automorphic Functions, Chelsea Pub. Co., 1951.

2) G. A. Jones and D. Singerman,Complex Functions, Cambridge University Press, 1987.

3) A. F. Beardon, Algebra and geometry, Cambridge University Press, Cambridge, 2005.

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Week Subjects

1 The Riemann sphere and behaviour of functions at infinity

2 The definition and basic properties of Möbius transformations (linear fractional transformations)

3 The connection between Möbius transformations and matrices, and the group PGL(2,C)

4 Fixed points of the Möbius transformations

5 Transitivity and cross-ratios

6 Möbius transformations and circles

7 Inversion in a circle

8 The Multiplier, K

9 Hyperbolic transformations

10 Elliptic transformations

11 Loxodromic transformations

12 Parabolic transformations

13 The isometric circle

14 The unit circle





Course Title:

Averaged moduli and one sided approximation I Code :


Field: Mathematics



Field Study

Home


T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective

Course

Objectives To teach the averaged moduli and their applications.

Learning

Outcomes

and

Competences

To be able to define the notions of integral moduli and averaged moduli,

To be able to express Whitney type inequalities,

To be able to express interpolation theorems,

To be able to express the quadrature formulas for periodic functions,

To be able to define the notions of Bernstein and Szasz-Mirakian operators.

Textbook

and /or

References

Bl. Sendov and V. A. Popov, The avaraged moduli of smoothness, 1988.

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Week Subjects

1 Preliminaries 2 Integral moduli and averaged moduli 3 Interrelations of two moduli 4 Whitney type inequalities 5 Intermediate approximation 6 Intermediate approximation 7 Interpolation theorems 8 Quadrature formulas for periodic functions 9 Quadrature formulas for periodic functions 10 Bernstein operators, Szasz-Mirakian operators 11 Bernstein operators, Szasz-Mirakian operators 12 Korovkin theorems in Lp 13 Interpolation splines 14 Interpolation splines





Course Title:

Strong Approximation I Code :


Field: Mathematics



Field Study

Home


T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective

Course

Objectives To teach the fundemantal properties of strong approximation.

Learning

Outcomes

and

Competences

To be able to define the order of strong approximation in Lipschitz class,

To be able to define the order of strong approximation in WrHw class,

To be able to express the basic theorems of strong approximation by (C,alpha) means of negative

order,

To be able to define the strong approximation by matrix means,

To be able to apply these concepts

Textbook

and /or

References

Laszlo Leindler, Strong approximation by Fourier series, Akademiai Kiado., 1985.

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Week Subjects

1 Preliminaries 2 Order of strong approximation in Lipschitz class 3 Order of strong approximation in Lipschitz class 4 Order of strong approximation in Lipschitz class 5 Order of strong approximation in WrHw class 6 Order of strong approximation in WrHw class 7 Order of strong approximation in WrHw class 8 Order of strong approximation in WrHw class 9 Strong approximation by (C,alpha) means of negative order 10 Strong approximation by (C,alpha) means of negative order 11 Strong approximation by (C,alpha) means of negative order 12 Strong approximation by (C,alpha) means of negative order 13 Some applications 14 Some applications





Course Title:

Finite Blaschke Products I

Code :


Field: Mathematics



Field Study

Home


T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective

Course

Objectives To introduce the basic knowledge about Finite Blaschke Products and their elementary properties.

Learning

Outcomes

and

Competences

To be able to define the concepts of Möbius transformation and finite Blaschke product,

To be able to prove the basic theorems about finite Blaschke products,

To be able to define and apply geometric properties of finite Blaschke products,

To be able to express the uniqueness theorem for monic Blaschke products,

To be able to express the relations between ellipses and finite Blaschke products.

Textbook

and /or

References

1) L. R. Ford, Automorphic Functions, Chelsea Pub. Co., 1951.

2) R. L. Craighead and F. W. Carroll, A decomposition of finite Blaschke products. Complex Variables

Theory Appl. 26 (1995), no. 4, 333-341.

3) A. L. Horwitz and A. L. Rubel, A uniqueness theorem for monic Blaschke products. Proc. Amer.

Math. Soc. 96 (1986), no. 1, 180-182.

4) J. Mashreghi, Expanding a finite Blaschke product. Complex Var. Theory Appl. 47 (2002), no. 3,

255-258.

5) U. Daepp, P. Gorkin and R. Mortini, Ellipses and finite Blaschke products. Amer. Math. Monthly

109 (2002), no.9, 785-795.

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Performance) X % 20

Week Subjects

1 Möbius transformations

2 Basic properties of Möbius transformations

3 The Multiplier, K

4 The isometric circle

5 The unit circle

6 The definition and basic properties of finite Blaschke products

7 A decomposition of finite Blaschke products I

8 A decomposition of finite Blaschke products II

9 A uniqueness theorem for monic Blaschke products

10 Expanding a finite Blaschke product I

11 Expanding a finite Blaschke product II

12 Basic geometric properties of finite Blaschke products

13 Ellipses and finite Blaschke products I

14 Ellipses and finite Blaschke products II





Course Title:

Algebra I

Code :

FMT5144


Sciences

Field : Mathematics



Field Study


T+A+L=Credit

ECTS

42 0 0 0 100 98 240 3 6


Course Type

Basic


Elective Social

Elective

Course

Objectives To teach the basic concepts of algebra in graduate level.

Learning

Outcomes

and

Competencies

To be able to state and prove some of the classical theorems of finite group theory,

To be able to determine whether or not there can be a simple group of a given order,

To be able to present the facts in the theory of rings,

To be able to construct a factor ring from an ideal in a ring,

To be able to define the ideal structure of Euclidean domains.

Textbooks

and /or

References

1. T. W. Hungerford, Algebra, Springer 1996.

2. D.S. Dummit and R.M.Foote, Abstract Algebra, Wiley 2nd edition ,1999.

3. N. Jacobson, Basic Algebra I-II, Dover Publications, 2009.

4. H.İ. Karakaş, Cebir Dersleri, TUBA 2008.

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Other

Week Subjects

1

2

3

4

5

6

7

8

9

10

11

12

13

14

Groups: Review basic group theory

Isomorphisms theorems

Symmetric, Alternating and Dihedral Groups

Direct Products and Direct Sums

Free groups, Free Abelian groups, Group actions

The Sylow Theorems

Classification of Finite Groups

Nilpotent and Solvable Groups

Normal and Subnormal Series

Introduction to Rings: Homomorphisms, Ideals

Factorization in Commutative Rings

Rings of Quotients and Localization

Ring of Polynomials and Formal Power Series

Factorization in Polynomial Rings

Instructor/s Asst. Prof.Dr. Pınar Mete




Course Title:

Orthogonal Polynomials I Code :

FMT5145 Institute: Institute of Science

Field: Mathematics



Field Study

Hw. Other Total T+A+L=

Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective Social

Elective

Course

Objectives To introduce properties of orthogonal polynomials and expansions on complex plane.

Learning

Outcomes

and

Competences

To be able to express the fundamental properties of orthogonal polynomials,

To be able to define the properties of orthogonal polynomials on an interval,

To be able to define the properties of orthogonal polynomials over a region,

To be able to express the general properties of the polynomials which are expressed with the help of

orthogonal polynomials,

To be able to define the approximation properties of the polynomials which are expressed with the

help of orthogonal polynomials.

Textbook

and /or

References

1) P. K. Suetin, Fundamental Properties of Polynomials Orthogonal on a Contour, Russ. Math. Surv., 1966.

2) P. K .Suetin, Polynomials Orthogonal over a region and Bieberbach Polynomials, Proceedings of the

Steklov Institute of Mathematics, AMS, 1974.

3) D.Gaier, Lectures on Complex Approximation,1985.

4) V.V. Andrievskii, H. P. Blatt, Discrepancy of Signed Measures and Polynomial Approximation,

Springer, 2001.

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Week Subjects

1

2

3

4

5

6

7

8

9

10

11

12

13

14

The fundamental properties of orthogonal polynomials

The construction of orthogonal polynomials by Gram-Schmidt method

The construction of orthogonal polynomials by moments

Orthogonal polynomials on an interval

Orthogonal polynomials over a region

Orthogonal polynomials on the boundary of a region

Estimation of the leading coefficient

The polynomials which are expressed orthogonal polynomials: Bieberbach polynomials

Approximation of Bieberbach polynomials

The zeros of orthogonal polynomials

Estimations the rate of approximation of zeros

Erdös-Turan type theorems

Asymptotic behavior of zeros of Bieberbach polynomials

Relations with potential theory

Instructors Assist. Prof. Dr. Burcin OKTAY




Course Title:

Banach Spaces of Analytic Functions I Code :


Field: Mathematics



Field Study


Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective Social

Elective

Course

Objectives To introduce fundamental properties of Hp and hp Spaces.

Learning

Outcomes

and

Competences

To be able to express some properties of harmonic functions,

To be able to define the Poisson integral of a function,

To be able to express the fundamental properties of hp Spaces,

To be able to define the Blaschke products,

To be able to express the fundamental properties of Hp Spaces,

To be able to define the concepts iner and outer functions.

Textbook

and /or

References

1) P. Koosis, Introduction to Hp Spaces, Cambridge University Press (1998).

2) P. L. Duren, Teory of Hp spaces, Academic Press (1970).

3) J. B. Garnett, Bounded Analytic Functions, Academic Press (1981).

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Week Subjects

1

2

3

4

5

6

7

8

9

10

11

12

13

14

Harmonic functions in the unit disk

Poisson kernel and the Poisson integral

Boundary behaviour of harmonic functions

Subharmonic functions

The spaces hp and Hp

The Nevanlinna class N

Boundary behaviour of analytic functions

Blaschke products

Inner and outer functions

Mean convergence to boundary values

The class N+

Harmonik majorants

The space H1 and Cauchy integral

Description of boundary values





Course Title:

Fourier Analysis I Code :


Field: Mathematics



Field Study


Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective Social

Elective

Course

Objectives To introduce fundamental concepts and theorems related to Fourier analysis.

Learning

Outcomes

and

Competences

To be able to define the concept of distribution function,

To be able to express the approximate identities,

To be able to express the Marcinkiewicz interpolation theorem,

To be able to express the Riesz-Thorin interpolation theorem,

To be able to define the Hardy-Littlewood maximal function,

To be able to define the Fourier and inverse Fourier transforms.

Textbook

and /or

References

1) L. Grafakos, Classical Fourier Analysis, Springer (2008).

2) J. Duoandikoetxea, Fourier Analysis, American Math. Soc. (2001).

3) E.M.Stein, G.Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press (1971).

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Week Subjects

1

2

3

4

5

6

7

8

9

10

11

12

13

14

Lp and weak Lp spaces

The distribution function

Topological groups

Convolution

Approximate identities

Marcinkiewicz interpolation theorem

Riesz-Thorin interpolation theorem

Decreasing rearrangements

Lorentz spaces

Duals of Lorentz spaces

The Hardy-Littlewood maximal function

The class of Schwartz functions

Fourier transforms of Schwartz functions

The Inverse Fourier transform





Course Title:

Fourier Series and Approximation I Code :


Field: Mathematics



Field Study


Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective Social

Elective

Course

Objectives To introduce Fundamental properties of Trigonometric Fourier series.

Learning

Outcomes

and

Competences

To able to define Fourier series,

To able to define the notions of Dirichlet, Fejer and Poisson kernels,

To able to express summability of Fourier series by Cesaro method,

To able to express summability of Fourier series by Abel’s method,

To able to define the concept of conjugate function and M. Riesz’s theorem,

To able to define the norm convergence of Fourier series.

Textbook

and /or

References

1. A. Zygmund, Trigonometric Series, Cambridge Univ. Press (1959).

2. Y. Katznelson, An Introduction to Harmonic Analysis, Cambridge Univ. Press (2004)

3. R.A. DeVore, G.G.Lorentz, Constructive Approximation, Springer-Verlag (1993).

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Week Subjects

1

2

3

4

5

6

7

8

9

10

11

12

13

14

The spaces C and Lp

Best approximation

Weierstrass approximation theorems

Trigonometric series and conjugate series

Fourier series

Partial sums and the Dirichlet kernel

Fejer kernel and ve Fejer means

Convergence of the Fejer mean, Fejer’s theorem

Pointwise convergence of Fourier series

Almost everywhere convergence of Fourier series, the Carleson-Hunt theorem

Poisson kernel and Abel-Poisson means

Conjugate functions and theorem of M. Riesz

Convergence of Fourier series in the norm

Marcinkiewicz multiplier theorem and Littlewood-Paley theorem





Course Title: Applied Mathematics I Code : FMT5149 Institute: Instute of Science

Field: Mathematics



Field Study


T+A+L=Credit

ECTS

42 0 0 0 0 198 240 3 6

Semester Spring Language Turkish

Course Type

Basic


Elective Social

Elective

Course

Objectives To teach the methods which are usually used in applied mathematics and give their Maple applications.

Learning

Outcomes

and

Competencies

To be able to express the class of first order ordinary differential equation,

To be able to solve first order linear differential equation and do MAPLE applications,

To be able to express high order ordinary differential equations and do MAPLE

applications,

To be able to apply Laplace, inverse Lapalce and Fourier transformation in MAPLE,

To be able to express the concept of Legendre equations and polynomials.

Textbooks

and /or

References

1. E. Hasanov, G. Uzgören, A. Büyükaksoy, Diferansiyel Denklemler Teorisi, Papatya, 2002.

2. B. Karaoğlu, Fizikte ve Mühendislikte Matematik Yöntemler, Seyir, 2004.

3. C. T. J. Dodson, E. A. Gonzalez, Experiments in Mathematics Using Maple, Springer, 1991.

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Week Subjects

1

2

3

4

5

6

7

8

9

10

11

12

13

14

Classes of first order ordinary differential equations.

Classes of first order ordinary differential equations., Bernoulli, Riccati etc.

Higher order differential equations.

Laplace transformations.

Inverse Laplace transformations.

Solving differential equations with Laplace Transformations.

Fourier transformations.

Legendre equations and polynomials.

Introduction to maple.

Plotting with Maple.

Solving first order differential equations with Maple.

Solving higher order differential equations with Maple.

Laplace applications with Maple.

Fourier applications with Maple.






Course Title: Advanced Numerical Analysis I Code : FMT5150 Institute: Instute of Science

Field: Mathematics



Field Study


T+A+L=Credit

ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective Social

Elective

Course

Objectives To teach advanced techniques of methods which are used while make numerical calculation.

Learning

Outcomes

and

Competencies

To be able to solve nonlinear equations by applying numerical analysis methods,

To be able to do approximation by using polynomials,

To be able to apply numerical derivation and integration operations,

To be able to solve the problems of eigenvalues and eigenvectors,

To be able to find inverse with Sequential Iteration Methods.

Textbooks

and /or

References

1) G. Amirali, H. Duru, Nümerik Analiz, Pegem A Yayınları, 2002,

2) A. Ralston, A First Course in Numerical Analysis, McGraw-Hill,1978,

3) S.C. Chapra, R.P. Canale, Numerical Methods for Engineers, McGraw-Hill, 1990.

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Other

Week Subjects

1

2

3

4

5

6

7

8

9

10

11

12

13

14

Nonlinear Equations, Existence Theorems

Newton and semi-Newton Methods, Optimization,

Local and Maximum Notions, Methods of Foundation of True,

The Method of Foundation of Maximum Variable, Conjugate Gradient Method,

Minimization of Quadratic Function,

Conjugate Direction Methods,

Lagrange Multipliers, Kuhn-Tucker Conditions,

Approximation Method of Polynomials,

Orthogonal Polynomials,

Approximation in Maximum Norm,

Numerical Differentiable, Richardson Extrapolation,

Numerical Integration, Gaussian Integration Formulas, Calculation of Generalized Integrals,

Eigenvalues and Eigenvectors Problem,

Foundation of Inverse with Sequential Iteration Methods

Instructors Assist. Prof. Dr. Figen KİRAZ




Course Title: Differential Geometry of Curves

and Surfaces I

Code : FMT5151


Field: Mathematics



Field Study


T+A+L=Credit

ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective Social

Elective

Course

Objectives To teach the differential geometry of curves and surfaces both in local and global aspects.

Learning

Outcomes

and

Competencies

To able to define the concepts of parametrized curves and regular curves,

To able to express the local Canonical form,

To able to express the global properties of plane curves,

To able to express the notions of the tangent plane, the differential of a map, the first fundamental form,

To able to characterize the compact orientable surfaces.

Textbooks

and /or

References

Manfredo P. do Carmo, Differential Geometry of Curves and Surfaces, 1976.

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1

2

3

4

5

6

7

8

9

10

11

12

13

14

Parametrized curves, Regular curves,

Parametrized curves, Regular curves,

The vector product in R^3, The local theory of curves parametrized by arc length,

The vector product in R^3, The local theory of curves parametrized by arc length,

The local Canonical form, Global properties of plane curves.

The local Canonical form, Global properties of plane curves.

Regular surfaces, Inverse images of regular values ,

Regular surfaces, Inverse images of regular values ,

Change of parameters, Differential functions on surfaces ,

Change of parameters, Differential functions on surfaces

The tangent plane, The differential of a map, The first fundamental form ,

The tangent plane, The differential of a map, The first fundamental form

Orientation of surfaces, A characterization of compact orientable surfaces,

Orientation of surfaces, A characterization of compact orientable surfaces,

Instructor/s Assoc. Prof. Dr. Bengü Bayram




Course Title:

Introduction to Fuzzy Topology I

Code :

FMT5152


Field: Mathematics



Field Study

Home

work

Other Total Credit

T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective Social

Elective

Course

Objectives To teach the fundamental concepts and theorems of Fuzzy topological spaces.

Learning

Outcomes

and

Competences

To be able to define the basic concepts about Fuzzy sets and to state theorems,

To be able to do algebraic operations on Fuzzy sets,

To be able to define the concept of convexity in Fuzzy sets,

To be able to do Cartesian Product of Fuzzy sets,

To be able to find the image and reverse image of Fuzzy Sets under a function.

Textbook

and /or

References

1. Şaziye Yüksel, Genel Topoloji, Eğitim Kitapevi, 2011.





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Week Subjects

1 Fuzzy Sets 2 Fuzzy Set Concept 3 Fuzzy sets Transactions 4 Algebraic Operations on Fuzzy sets 5 Problem solving 6 Convexity of fuzzy sets 7 The Concept of Fuzzy Relation, 8 Cartesian Product of Fuzzy sets 9 Family of Fuzzy Sets 10 The image of Fuzzy Sets under a function 11 The reverse image of Fuzzy Sets Under a function 12 Problem solving 13 The concept of fuzzy point. 14 General review of the issues.

Instructors Assoc. Prof. Dr. Ahu Açıkgöz





Course Title:

Introduction to Ideal Topological Spaces I

Code :

FMT5153


Field: Mathematics



Field Study

Home

work

Other Total Credit

T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective Social

Elective

Course

Objectives To teach properties and several examples of Ideal topological spaces.

Learning

Outcomes

and

Competences

To be able to define the basic concepts and the seperation properties of Ideal topological spaces,

To be able to construct topologies by using maximal and minimal Ideals,

To be able to express several Ideal examples and their properties,

To be able to define the seperation axioms in Ideal topological spaces,

To be able to define the concept of compactness in Ideal topological spaces.

Textbook

and /or

References

1. Şaziye Yüksel, Genel Topoloji, Eğitim Kitapevi, (2011).

2. Osman Mucuk, Topoloji, Nobel Kitapevi, (2009).

3. Mahmut Koçak, Genel Topoloji I ve II, Gülen Ofset Yayınevi, (2006).



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Week Subjects

1 The concept of Ideally 2 Maximal ideal 3 Minimal ideal 4 Comparisons 5 Local function 6 *- topology, and generalized open sets 7 The ideal characteristics and a variety of the ideal samples 8 Problem solving 9 Ideal topological spaces and separation axioms 10 *- topological features 11 Compactness in ideal topological spaces 12 Various sets in ideal topological spaces. 13 Some properties of the sets 14 General review of the issues






Course Title: Algebraic number theory I Code :

FMT5154


Field: Mathematics



Field Study

Home

work

Other Total Credit

T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective Social

Elective

Course

Objectives To give fundamental concepts and theorems related with the algebraic number theory.

Learning

Outcomes

and

Competences

To be able to define the concepts of ring, field and algebraic field extensions,

To be able to define the Dedekind domains,

To be able to define the norms of ideals,

To be able to define the prime factors in a number field,

To be able to find units in quadratic fields.

Textbook

and /or

References

1) E. Weiss, Algebraic Number Theory, Dover publications, 1998.

2) I. Stewart, D. Tall, Algebraic Number Theory and Fermat’s Last Theorem, A K Peters Ltd., 2002.

3) M.R. Murty, J. Esmonde, Problems in Algebraic Number Theory, Springer,2005.

4) Ş. Alaca, K. S. Williams, Introductory Algebraic Number Theory, Cambridge Univ. Press, 2004.

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Week Subjects

1 Rings 2 Fields 3 Algebraic Extensions of a Field 4 Algebraic Extensions of a Field 5 Algebraic Number Fields 6 Algebraic Number Fields 7 Conjugates 8 Dedekind Domains 9 Dedekind Domains 10 Norms of Ideals 11 Norms of Ideals 12 Prime factoring in a number field 13 Units in Real Quadratic Fields 14 Units in Real Quadratic Fields

Instructors Assoc. Prof. Dr. Sebahattin İkikardes


Website http://w3.balikesir.edu.tr/~skardes/



Course Title:

Geometric Theory of Functions I Code :


Field: Mathematics



Field Study


Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective Social

Elective

Course

Objectives

To teach the one-to-one correspondence between analytic properties of the functions and geometric properties

of the domains.

Learning

Outcomes

and

Competences

To be able to define the concepts of curve, domain, simply connected domain and multiply connected

domain,

To be able to express the fundamental properties of conformal mappings,

To be able to define the boundary behavior of derivatives,

To be able to define the modulus of continuity and its properties,

To be able to express the fundamental properties of Smirnov Lavrentiev domains.

Textbook

and /or

References

1. Ch. Pommerenke, Boundary Behaviour of Conformal Maps,1992

2. Zeev Nehari, Conformal Mapping, 1952.

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Week Subjects

1

2

3

4

5

6

7

8

9

10

11

12

13

14

Curve, Domain, Simply connected domain, Multiply connected domain

Conformal mappings

Analytic curves

Smooth Jordan curves

Domains by bounded boundary rotation

The analytic characterization of smoothness

The boundary behavior of derivatives

Modulus of continuity

Quasidisks

John Domains

Quasiconformal extension

Rectifiable curves

Smirnov Domains

Lavrentiev domains





Course Title: Numerical Optimization I Code : FMT 5156 Institute: Instute of Science

Field: Mathematics



Field Study

Home

work

Other Total Credit

T+A+L=Credit

ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective Social

Elective

Course

Objectives

To teach the fundamental concepts of linear programming and unconstrained optimization problems with

solution methods.

Learning

Outcomes

and

Competences

To be able to express the fundamental concept of optimization problems,

To be able to define linear programming problems,

To be able to solve LP problems by Simplex method,

To be able to express optimality conditions for unconstrained optimization problems,

To be able to express line search method,

To be able to apply basic descent, conjugate gradient and quasi newton methods.

Textbook

and /or

References

1) Bazaraa M.S., Sherali H.D. and Shetty S.M., Nonlinear programming: Theory and Applications, 3rd edition,


2) Chong E.K. and Zak S.H., An introduction to optimization, 2nd edition, John Wiley & Sons, Inc., 2001.

3) Griva I., Nash S.G. and Sofer A., Linear and nonlinear optimization, 2nd edition, SIAM, 2008.

4) Luenberger D.G. and Ye Y., Linear and nonlinear programming, 3rd edition, Springer, 2008.

5) Nocedal J. and Wright S.J., Numerical optimization, Springer, 1999.

6) Sun W. and Yuan Y-X, Optimization Theory and Method: Nonlinear Programming, Springer, 2006.

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Week Subjects

1

2

3

4

5

6

7

8

9

10

11

12

13

14

Mathematical review and background

Fundamentals of optimization

Basic properties of linear programming

The simplex method

The simplex method and analysis

Duality

İnterior-point method

Unconstrained optimization

Optimality conditions and basic properties

Line search methods

Basic descent methods

Conjugate direction method

Quasi-newton method

Trust-region method

Instructors Assist. Prof. Dr. Fırat EVİRGEN





Course Title:

Selected Topics in Analysis I

Code :


Field: Mathematics



Field Study

Home


T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective

Course

Objectives

To introduce the basic knowledge about Fibonacci, Lucas and generalized Fibonacci polynomials and their

elementary properties.

Learning

Outcomes

and

Competences

To be able to define the concepts of Fibonacci, Lucas and generalized Fibonacci polynomials and

their basic properties,

To be able to use and apply these basic properties in some analysis problems,

To be able to find the generating functions,

To be able to find the zeros of Fibonacci and Lucas polynomials,

To be able to define the Jacobsthal polynomials.

Textbook

and /or

References

1) T. Koshy, Fibonacci and Lucas numbers with applications, Wiley, 2001.

2) V. E. Hoggatt and M. Bicknell, Generalized Fibonacci polynomials, Fibonacci Quart. 11(5), 457-465,

1973.

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Performance) X % 20

Week Subjects

1 Fibonacci and Lucas numbers

2 Generalized Fibonacci numbers

3 Generating functions

4 Fibonacci and Lucas series I

5 Fibonacci and Lucas series II

6 Fibonacci polynomials

7 Byrd’s Fibonacci polynomials

8 Applications

9 Lucas polynomials

10 Jacobsthal polynomials

11 Applications

12 Zeros of Fibonacci and Lucas polynomials I

13 Zeros of Fibonacci and Lucas polynomials II

14 Applications





Course Title:

Lorentzian Geometry Code : FMT5158

Institute: Institute of Science

Field: Mathematics



Field Study


T+A+L=Credit

ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective Social

Elective

Course

Objectives To teach general properties of Lorentzian manifolds.

Learning

Outcomes

and

Competencies

To be able to define the concepts of Lorentzian metric and Lorentzian space,

To be able to express the fundamental properties of Lorentzian manifolds,

To be able to define the concepts of Minkowski space time and Robertson-Walker space time,

To be able to express the fundamental properties of the Schwarzschild and Kerr space time,

To be able to define bi-linear Lorentzian metrics on Lie groups.

Textbooks

and /or

References

J. K. Beem, P. E. Ehrlich and K. L. Easley, Global Lorentzian Geometry, Second Edition, Pure and Applied

Mathematics, Marcel Dekker, Inc., 1996.

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Other

Week Subjects

1

2

3

4

5

6

7

8

9

10

11

12

13

14

Lorentzian metric and Lorentzian space

Lorentzian manifolds

Convex normal neighborhoods

Curves and topology on curves

Two dimensional space times

The second fundamental form

Warped products

Homothetic maps

Minkowski space time

Schwarzschild-Kerr space times

Spaces of constant curvature

Robertson-Walker space times

Bi-linear Lorentzian metrics on Lie groups

Lorentzian sectional curvature

Instructor/s Assist. Prof. Dr. Sibel SULAR





Course Title:

Semi-Riemannian Geometry I Code : FMT5159


Field: Mathematics



Field Study


T+A+L=Credit

ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective Social

Elective

Course

Objectives To teach general properties of Semi-Riemannian manifolds.

Learning

Outcomes

and

Competencies

To be able to express the fundamental properties of Semi-Riemannian manifolds,

To be able to define the concepts of type changing and metric contraction,

To be able to define the geometrical structure of warped product manifolds,

To be able to express the fundamental properties of Lightlike manifolds,

To be able to define Non-Degenerate and Null curves in Semi-Riemannian manifolds.

Textbooks

and /or

References

1) B. O’Neill, Semi-Riemannian Geometry with Applications to Relativity, Academic Press, Inc., 1983.

2) K. L. Duggal D. H. Jin, Null Curves and Hypersurfaces of Semi-Riemannian Manifolds, World Sci.,

2007.

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Other

Week Subjects

1

2

3

4

5

6

7

8

9

10

11

12

13

14

Symmetric bilinear forms

Isometries

The Levi-Civita connection

Parallel translation

Geodesics

Curvatures

Semi-Riemannian surfaces

Type changing and metric contraction

Frame fields

Some differential operators

Semi-Riemannian manifolds

Warped product manifolds and curvatures of warped product manifolds

Lightlike manifolds

Non-Degenerate and Null curves in Semi-Riemannian manifolds






Course Title:

Tangent and Cotangent Bundle Theory Code : FMT5160


Field: Mathematics



Field Study


T+A+L=Credit

ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective Social

Elective

Course

Objectives To teach fundamnetal properties of tangent and cotangent bundles.

Learning

Outcomes

and

Competencies

To be able to express the general properties of tangent bundles,

To be able to define tangent bundles of Riemannian manifolds,

To be able to define Non-linear connections of tangent bundles

To be able to express the general properties of cotangent bundles,

To be able to express the fundamental properties of tangent and cotangent bundles of order 2.

Textbooks

and /or

References

K. Yano and S. Ishihara, Tangent and Cotangent Bundles, Marcel Dekker, Inc., New York, 1973.

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Other

Week Subjects

1

2

3

4

5

6

7

8

9

10

11

12

13

14

Tangent bundles

Vertical and complete lifts from a manifold to its tangent bundle

Metrics on tangent bundle

Complete lifts of vector fields to the tangent bundle

Complete lifts of affine connections to the tangent bundle

Horizontal lifts from a manifold to its tangent bundle

Tangent bundles of Riemannian manifolds

Non-linear connections of tangent bundles

Cotangent bundles

Vertical and complete lifts from a manifold to its cotangent bundle

Horizontal and complete lifts from a manifold to its cotangent bundle

Complete lifts of affine connections to the cotangent bundle

Tangent bundles of order 2

Cotangent bundles of order 2





Course Title:

Functional Analysis II Code :


Field: Mathematics



Field Study

Home


T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6

Semester Spring Language Turkish/English

Course Type

Basic


Elective Social

Elective

Course

Objectives To teach some advanced topics of functional analysis.

Learning

Outcomes

and

Competences

To be able to define the concept of compact operator,

To be able to define the concept of Banach algebra,

To able to define the spectrum of an operator,

To be able to define the concept of C* Algebra,

To be able to define the concept of weak topology

To be able to define the concept of Fredholm operator.

Textbook

and /or

References

1. Barbara D. MacCluer, Elementary Functional Analysis, Springer, (2009).

2. J. B. Conway, A Course in Functional Analysis, Springer, (1985).

3. W. Rudin, Functional Analysis, McGraw Hill, (1991).

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Week Subjects

1

2

3

4

5

6

7

8

9

10

11

12

13

14

Finite Dimensional Spaces

Compact Operators

The Invariant Subspace Problem

Banach Algebras

Spectrum

Analytic Functions in Banach Spaces

Ideals and Homomorphisms

Commutative Banach Algebras

C* Algebras

Weak Topologies

Fredholm Operators

Lp Spaces

Stone-Weierstrass Theorem

Positive Linear Functionals on C(X)





Course Title : Module Theory II Code : FMT5205 Institute: Instute of Science

Field: Mathematics



Study

Homework

Other Total Credit

T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type Basic


Elective Social

Elective

Course

Objectives To teach fundamental concepts of the module theory .

Learning

Outcomes and

Competences

● to be able to define the Noetherian and Artinian modules,

● to be able to express the semi simple modules,

● to be able to express the Goldie theorem for rings,

● to be able to define the modules on Goldie rings,

● to be able to express the bimodules and Noetherian bimodules.

Textbook and/or

References

1. A. Harmancı, Cebir II, Hacettepe yayınları, (1987).

2. V. P. Snaith, Groups, rings and Galois theory, World Scientıfıc, (2003).

3. J. J. Rotman, An introductıon to the theory of groups, Springer- Verlag, (1995).

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Week Subjects

1 Remind some material over abelıan groups 2 Remind some material over module theory ı 3 The classical ring definition and applications 4 Noetherian and artinian modules 5 Semı simple modules 6 General applications 7 Injective hull 8 The Goldie theorem for rıngs 9 Modules defined on goldie rıngs

10 Bimodüles, noetherian bimodüles 11 Modules of factors 12 Submodules of factors 13 General applications 14 General applications

İnstructors Assoc.Prof.Dr.Fırat ATEŞ




Course Title:

Fuchsian Groups

Code :

FMT5206


Field: Mathematics



Field Study

Home


T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective

Course

Objectives To teach Fuchsian groups and their elementary algebraic properties.

Learning

Outcomes

and

Competences

To be able to state and apply the basic properties of the group PGL(2,C),

To be able to express the definition and basic properties of Möbius transformations on the extended

complex plane,

To be able to express the definition and basic properties of the group PSL(2,R) and its

transformations,

To be able to define the concepts of Elliptic function and topological group,

To be able to express the automorphisms of compact Riemann surfaces.

Textbook

and /or

References

1. G. A. Jones and D. Singerman,Complex Functions, Cambridge University Press, (1987).

2. A. F. Beardon, The Geometry of Discrete Groups, Springer-Verlag, New York, (1983).

3. B. Iversen, Hyperbolic Geometry, , Cambridge University Press, (1992).

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Week Subjects

1 The Riemann sphere

2 Möbius transformations

3 Generators for PGL(2,C)

4 Transitivity and cross-ratios

5 Conjugacy classes in PGL(2,C)

6 Geometric classification of Möbius transformations

7 The area of a spherical triangle

8 Elliptic functions, topological groups

9 Lattices and fundamental regions

10 PSL(2,R) and its discrete subgroups

11 The hyperbolic metric

12 Hyperbolic area and the Gauss-Bonnet formula

13 Fuchsian groups and elementary algebraic properties of Fuchsian groups

14 Automorphisms of compact Riemann surfaces





Course Title:

Advanced Differential Geometry II


Field: Mathematics



Study


T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective Social

Elective

Course

Objectives To teach fundamantel concepts of Differential Geometry.

Learning

Outcomes

and

Competences

To be able to find the shape operator, Gaussian curvature and the mean curvature of a

surface,

To be able to define the orientatiability of a surface,

To be able to calculate the Euler-Poincare charactersitic of a surface,

To be able to state and prove the Gauss-Bonnet theorem,

To be able to define the congruence of surfaces.

Textbook

and /or

References

1) B. O’Neill, Elementary Differential Geometry, Academic Pres, Inc., 1966.

2) H. H. Hacısalihoğlu, Yüksek Diferensiyel Geometri’ ye Giriş, Fırat Ünv. Fen Fak. 1980.

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Week Subjects

1 Shape operator 2 Normal curvature, Gaussian curvature 3 Gauss map, minimal surfaces 4 Computational techniques 5 Special curves on a surface 6 Surfaces of revolution 7 Form computations 8 Isometries and local isometries 9 Integration and Orientation 10 Congruence of surfaces 11 Geodesics 12 Mappings that preserve inner products 13 Euler-Poincare characteristic of a surface 14 Gauss-Bonnet Theorem





Course Title: Hyperbolic Geometry

Code :

FMT5210


Field: Mathematics



Field Study

Home

work

Other Total Credit

T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective Social

Elective

Course

Objectives To teach the fundamental definitions and theorems related with Hyperbolic geometry.

Learning

Outcomes

and

Competences

To be able to define define the concepts of hyperbolic metric and hyperbolic area,

To be able to state the fundamental teorems related with hyperbolic geometry,

To be able to state the Gauss-Bonnet thorem,

To be able to define the fundamental concepts of Hyperbolic trigonometry,

To be able express the relations in a Hyperbolic triangle.

Textbook

and /or

References

1) G. A. Jones and D. Singerman,Complex functions, Cambridge University Press, (1987).

2) A.F. Beardon, The geometry of Discrete Groups, Springer, (1983).

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Week Subjects

1 Hyperbolic geometry 2 The isometry of the hyperbolic plane 3 Hyperbolic metric 4 The properties of the hyperbolic metric 5 Hyperbolic metric in the upper half plane 6 Hyperbolic metric in the unit disk 7 Topology induced by hyperbolic metric 8 Hyperbolic disk and its presentation 9 Hyperbolic area 10 The theorem of Gauss-Bonnet 11 Hyperbolic polygons 12 Hyperbolic trygonometry 13 The relations on hyperbolic triangle 14 Some theorems of hyperbolic trigonometry






Course Title: Dynamic Systems and Applications Code : FMT5212 Institute: Instute of Science

Field: Mathematics



Field Study

Home

work

Other Total Credit

T+A+L=Credit

ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective Social

Elective

Course

Objectives To teach the fundamental concepts of dynamic system theory.

Learning

Outcomes

and

Competences

To be able to define Laplace and invere Laplace transformations,

To be able to express the concept of state space and transfer function,

To be able to express the fundamental concepts of stability theory,

To be able to define Routh-Hurwitz stability criteria and to do MATLAB application,

To be able to define Nyquist criteria and to do MATLAB application.

Textbook

and /or

References

1. R. S. Burns, Advanced Control Engineering, Butterworth Heinemann, 2001.

2. B. C. Kuo, Otomatik Kontrol Sistemleri, Literatür Yayınları,2002.

3. J.Wilkie, M. Johnson, R. Katebi, Control Engineering Introductory Course, Palgrave Macmillan,2002.

4. E.P. Erander, A. Sjöberg, The Matlab Handbook 5, Addison-Wesleys,1999.

5. İ. Yüksel, Matlab ile Mühendislik sistemlerin Analizi, Vipaş A.Ş.,2000.

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Week Subjects

1

2

3

4

5

6

7

8

9

10

11

12

13

14

Fundamental Matrix Theory.

S-plane and Laplace transformations

Inverse Laplace transformations.

State space and Transfer functions.

Time domain input functions and time domain. Response of systems.

Step response and Performance identification.

Stability analysis.

Routh-Hurwitz Stability criterion.

Routh-Hurwitz criterion and MATLAB application.

Root Locus methods.

Root Locus methods MATLAB application.

Nyquist criterion.

Nyquist criterion MATLAB application.

Bode diagram and its MATLAB application.






Course Title:

Real Analysis II Code :


Field: Mathematics



Field Study

Home


T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective Social

Elective

Course

Objectives To teach fundamental theorems of Real analysis.

Learning

Outcomes

and

Competences

To be able to define Lp Spaces and state their fundamental properties,

To be able to express the duals of Lp Spaces,

To be able to state the Radon-Nikodym Theorem,

To be able to state the Riesz Representation Theorem,

To be able to define the concepts of function of bounded variation and absolutely continuous

function,

Textbook

and /or

References

1. C.D. Aliprantis, O. Burkinshaw, Principles of Real Analysis, Academic Pres (1998).

2. W. Rudin, Real and Complex Analysis, McGraw Hill (1987).

3. G. B. Folland, Real Analysis, John Wiley & Sons, Inc. (1999).

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Week Subjects

1

2

3

4

5

6

7

8

9

10

11

12

13

14

Normed Linear Spaces and Banach Spaces

Bounded Linear Transformations

Linear Functionals and Dual Spaces

Lp Spaces (1 ≤p<∞)

The space L∞

Linear Functionals on Lp Spaces

Signed Measures

Comparison of Measures

Decomposition of Measures

Radon-Nikodym Theorem

Riesz Representation Theorem

Functions of Bounded Variation

Absolutely Continuous Functions

Lebesgue differentiation theorem





Course Title:

Discrete Groups

Code :

FMT5215


Field: Mathematics.



Field Study

Home

work Othe

r

Total Credit

T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective

Course

Objectives To teach the discrete group theory at the basic level.

Learning

Outcomes

and

Competences

To be able to express the definition and basic properties of Möbius transformations on Rn,

To be able to express the definition and basic properties of some discontinuous groups of Möbius

transformations,

To be able to express the Discrete groups of isometries,

To be able to define the function groups,

To be able to define the concept of Schottky groups.

Textbook

and /or

References

1) A. F. Beardon, The Geometry of Discrete Groups, Springer-Verlag, New York, (1983).

2) B. Maskit, Kleinian Groups, Springer-Verlag, Berlin, (1988).

3) B. Fine and G. Rosenberger, Algebraic Generalizations of Discrete Groups, Marcel Dekker, (1999).

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Week Subjects

1 Basic Properties of Möbius transformations on Rn

2 Complex Möbius transformations

3 Discontinuous groups

4 Jorgensen’s inequality

5 Fundamental Domains

6 The Dirichlet Polygon

7 Covering spaces

8 Groups of isometries

9 Discrete groups of isometries

10 The geometric basic groups

11 Geometrically finite groups

12 Function groups

13 Signatures

14 Schottky groups





Course Title: Theory of Approximation II Code : FMT5216 Institute: Instute of Science

Field: Mathematics



Field Study

Home

work

Other Total Credit

T+A+L=Credit

ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective Social

Elective

Course

Objectives To teach the fundamental principles of approximation theory in the complex plane.

Learning

Outcomes

and

Competences

To be able to define function spaces in the complex plane,

To be able to construct the approximating polynomials in the complex plane,

To be able to state the Walsh, Keldysh, Lavrentiev and Mergelyan theorems,

To be able to express the asymptotic properties of Faber polynomials,

To be able to state the theorems of rational approximation on the curves.

Textbook

and /or

References

1. V. K. Dzyadyk, Introduction to the theory of uniform approximation of functions by polynomials

(Russian). Moscow, (1977).

2. J. L. Walsh. Approximation and interpolation of the domains of the complex plane

3. V. V. Andrievskii, V. I. Beyli, V. K. Dzyadyk, Conformal invariants in constructive theory of functions of

complex variable, Atalanta, (1995).

4. P. S. Suetin, Series of Faber Polynomials, Moscow, (1984).

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Week Subjects

1

2

3

4

5

6

7

8

9

10

11

12

13

14

Function spaces in the complex plane

Modulus of smoothness on the complex plane

Polynomials of the best approximation on the complex plane

Construction of the approximation polynomials

Theorems of Walsh, Keldysh, Lavrentiev and Mergelyan

Faber polynomials and their’s properties

Generalized Faber polynomials

The asymptotical properties of Faber polynomials

Approximation by Faber polynomials

Approximation by rational functions on the curves

Approximation on the domains

Direct theorems

Inverse theorems

Comparsion of the results

Instructors Prof. Dr. Daniyal İsrafilzade




Course Title:

Riemannian Geometry II Code : FMT 5221


Field: Mathematics



Study


T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective Social

Elective

Course

Objectives

To teach the general properties of Einstein manifolds, submanifolds, surfaces, hypersurfaces and space

forms.

Learning

Outcomes

and

Competences

To be able to define the notions of Einstein manifold and submanifold and to give examples,

To be able to express the general properties of total geodesic , totally umbilical and pseudo-

umbilical submanifolds,

To be able to define and apply the notion of space form,

To be able to state and prove Cartan’s theorem and its corollaries,

To be able to Express the isometries of Hyperbolical space and Liouville’s theorem.

Textbook

and /or

References

1) Manfredo Perdigao do Carmo , Riemannian Geometry , Birkhauser, 1992.

2) W. M. Boothby, An introduction to Differentiable manifolds and Riemannian Geometry, Elsevier,

2003.

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Visa examination Midterm Exams

Quiz Midterm Controls

Homework Term Paper

Term project (project,

report, etc) Oral Examination

Laboratory Final Exam

Final examination X 100 Other

Other

Week Subjects

1 Ricci curvature tensor, definition and geometric meaning of Ricci curvature tensor

2 Some theorems about Ricci curvature tensor

3 Einstein manifolds

4 Submanifolds, definition and basic notions

5 Isometric Immersions

6 Fundamental forms

7 Totally geodesic , totally umbilic and pseudo umbilic submanifolds

8 Curvature of submanifolds

9 Surfaces

10 Hypersurfaces

11 Space forms

12 Cartan Theorem and its results

13 Hyperbolical space

14 Isometries of Hyperbolical space, Liouville Theorem





Course Title:

Geometry of Submanifolds II Code : FMT 5222


Field: Mathematics



Study


T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective Social

Elective

Course

Objectives

To teach the notions of totally umbilical submanifolds, minimal submanifolds, invariant and totally real

submanifolds , quaternionic submanifolds, submanifolds of Kahler manifolds, surfaces in a real space form.

Learning

Outcomes

and

Competences

To be able to define the concepts of totally umbilical submanifold and minimal submanifold, and

to give examples,

To be able to express the concepts of invariant and totally real submanifold,

To be able to define the concepts of quaternionic submanifold and submanifold of a Kahler

manifold,

To be able to define the concept of surfaces in a real space form and to give examples,

To be able to prove the Gauss-Bonnet theorem.

Textbook

and /or

References

B. Y. Chen , Geometry of Submanifolds, Pure and applied mathematics (Marcel Dekker, Inc.), New York,

1973

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Other

Week Subjects

1 Totally umbilical submanifolds

2 Minimal submanifolds

3 The first Standard imbeddings of Projective Spaces I

4 The first Standard imbeddings of Projective Spaces II

5 Invariant and totally real submanifolds I

6 Invariant and totally real submanifolds II

7 Quaternionic submanifolds

8 Riemann submersions

9 Submanifolds of Kahler manifolds, basic definitions and notions I

10 Submanifolds of Kahler manifolds, basic definitions and notions II

11 Surfaces in 3-dimensional Eucliden space and related results

12 Surfaces in a Real space form I

13 Surfaces in a Real space form II

14 Gauss-Bonnet Theorem





Course Title: Advanced Control Systems II Code : FMT5224 Institute: Instute of Science

Field: Mathematics



Field Study

Home

work

Other Total Credit

T+A+L=Credit

ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective Social

Elective

Course

Objectives To teach controllability of nonlinear systems and optimal control theory in advanced level.

Learning

Outcomes

and

Competences

To be able to express controllability of nonlinear systems,

To be able to define unconstrained optimization problems,

To be able to define problems of optimal control theory,

To be able to state Pontryagin maximum principle,

To be able to express sufficient conditions for optimal control.

Textbook

and /or

References

1. E. R. Pinch, Optimal Control And The Calculus Of Variations, Oxford University Press, 1995.

2. J. Macki, A. Strauss, Introduction to Optimal Control Theory, Springer-Verlag, 1982.

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Week Subjects

1

2

3

4

5

6

7

8

9

10

11

12

13

14

Controllability for nonlinear systems.

Controllability for nonlinear systems.

Optimization: functions of one variable, critical points, end points, discontinuity points.

Optimization with constraint, geometrical interpretation.

Calculus of variation, fixed end points problems, minimization curves.

Isometric problems, sufficient problems, extreme fields.

Optimal control theory problems.

Pontryagin maximum principle.

Optimal control to objective curve.

Time optimal control problems of linear systems.

Linear systems and quadratic costs.

Steady State Riccati equations.

Convex sets in n

Sufficient conditions for optimal control.






Course Title:

Convex Functions and Orlicz Spaces II Code : FMT5225


Field: Mathematics



Field Study

Home

work

Other Total Credit

T+A+L=Credit

ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective Social

Elective

Course

Objectives To teach the completeness and separability concepts and compactness criteria in Orlicz spaces.

Learning

Outcomes

and

Competences

To be able to define the concept of completeness in Orlicz spaces,

To be able to Express the Notion of absolute continuity of the norm in Orlicz spaces,

To be able to Express the Kolmogorov compactness criter in Orlicz spaces,

To be able to Express the approximation theorems in Orlicz spaces,

To be able to define the notion of weighted Orlicz space.

Textbook

and /or

References

1) M. A. Krasnosel’ski and Ya. B. Rutickii, Convex funktions and Orlicz Spaces, Noordhoff, 1961.

2) C. Bennett and R. Sharpley, Interpolation of Operators, Academic Pres, 1988.

3) M. M. Rao, Z. D. Ren, Applications of Orlicz Spaces, New York, 2002.

4) R. A. De Vore and G. G. Lorentz, Constructive Approximation, Springer, 1993.

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Week Subjects

1

2

3

4

5

6

7

8

9

10

11

12

13

14

Completeness in the Orlics spaces

Norm of the characteristic functions, Hölder’s inequality

Mean convergence

Separability in the Orlicz spaces, necessary conditions

The absolute continuity of the norm

Compactness criteria

Kolmogorov’s compactness criterion for the Orlics spaces

Riesz’s compactness criterion for the Orlics spaces

Basis in the Orlisz spaces

Comparsion of spaces

An inequality for norms

Approximation in the Orlicz spaces

Direct and inverse theorems

Weighted Orlicz spaces

Instructors Prof. Dr. Daniyal İsrafilzade




Course Title : Matrices of Semigroups Code : FMT5226 Institute: Instute of Science

Field: Mathematics



Study

Homework

Other Total Credit

T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type Basic


Elective Social

Elective

Course

Objectives To introduce semigroups of matrices and to teach the rewriting system.

Learning

Outcomes and

Competences

● to be able to express the definitions of semıgroup and monoıd,

● to be able to understand the construction of lineer semigroup,

● to be able to create the monoids with lie type,

● to be able to express the non-factorization semigroups,

● to be able to create the rewriting systems

Textbook and/or

References

1) J. Okninski, Semigroups of matrices, World Scientific, (1988).

2) C. Kart, Matris metodları ve lineer dönüşümler, Ank. Üniv. , (1985).

3) J. Almedia, Fınıte semigroups and universal algebra, World Scientific, (1994).

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Week Subjects

1 Remind the basics on fundamental algebraic structures 2 Definitions of semigroup and monoid, and applications 3 To extend the usegace of definitions 4 General tecnics 5 Exact linear monoid 6 General applications 7 Construction of linear semigroup 8 Non factorization semigroups 9 Identities of semigroups

10 Monoids with lie type 11 Rewriting systems 12 Rewriting systems-cont. 13 General applications 14 General applications

Instructors Assoc. Prof. Dr. Fırat ATEŞ




Course Title:

Contact Manifolds II Code : FMT 5227


Field: Mathematics



Study


T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective Social

Elective

Course

Objectives

To teach submanifolds Kaehler and Sasakian manifolds, Invariant and anti-invariant

submanifolds, Lagrangian and integral submanifolds and general properties of tangent sphere

bundles.

Learning

Outcomes

and

Competences

To be able to understand the notions of Kaehler and Sasakian manifolds and to give some

examples of them,

To be able to understand the notions of invaryant ve anti-invariant submanifolds,

Lagrangian and integral submanifolds and to do their applications,

To be able to express some general properties of Complex contact manifolds and 3-Sasakian

manifolds,

To be able to express the geometry of tangent sphere bundles and vector bundles,

To be able to define integral submanifolds of 3-Sasakian manifolds.

Textbook

and /or

References

D. Blair , Riemannian Geometry of Contact and Symplectic Manifolds, Birkhauser, 2002.

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Week Subjects

1 Submanifolds of Kaehler and Sasakian manifolds 2 Invariant and anti-invariant submanifolds 3 Lagrangian and integral submanifolds 4 Legendre curves 5 Tangent bundles 6 Tangent sphere bundles, geometry of vector bundles 7 The *-scalar curvature 8 The integral of Ric(), the Webster scalar curvature 9 Complex contact manifolds and associated metrics 10 Examples of complex contact manifolds 11 Normality of complex contact manifolds 12 Holomorphic Legendre curves 13 3-Sasakian manifolds 14 Integral submanifolds of 3-Sasakian manifolds





Course Title:

Structures on Manifolds II Code : FMT5228


Field: Mathematics



Study


T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective Social

Elective

Course

Objectives

To teach the general properties of submanifolds of Kaehlerian manifolds, Almost contact manifolds, contact

manifolds, contact manifolds, locally product manifolds, submanifolds of product manifolds, submersions

and submanifolds.

Learning

Outcomes

and

Competences

To be able define the submanifolds of Kaehlerian manifolds,

To be able to define the almost contact manifolds and contact manifolds, and to give examples of them,

To be able to define the locally product manifolds and submanifolds of product manifolds,

To be able to define the concept of submersions and to give examples,

To be able to define the concept of CR-submanifod and to give examples.

Textbook

and /or

References

Kentaro Yano and Mashiro Kon , Structures On Manifolds, World Sci. 1984.

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Week Subjects

1 Submanifolds of Kaehlerian manifolds 2 Anti-invariant submanifolds of Kaehlerian manifolds 3 CR submanifolds of Kaehlerian manifolds 4 Almost contact manifolds, contact manifolds 5 Sasakian manifolds 6 Invariant submanifolds of Sasakian manifolds 7 Anti-invariant submanifolds of Sasakian manifolds 8 Contact CR-submanifolds 9 Locally product manifolds 10 Submanifolds of product manifolds 11 Submanifolds of Kaehlerian product manifolds 12 Fundamental equations of Submersions 13 Almost Hermitian submersions 14 Submersions and submanifolds





Course Title:

Algebraic Geometry Code :

FMT5230


Sciences

Field : Mathematics



Field Study


T+A+L=Credit

ECTS

42 0 0 0 100 98 240 3 6


Course Type

Basic


Elective Social

Elective

Course

Objectives To teach the algebraic varieties which are the zero sets of polynomials in several variables.

Learning

Outcomes

and

Competencies

To be able to define the concept of Affine Algebraic Variete,

To be able to state Hilbert basis theorem,

To be able to define the concept of projective variete,

To be able to express the Veronese Maps and Product of Varieties,

To be able to define the concept of Hilbert function.

Textbooks

and /or

References

1. Huishi Li - F. Van Oystaeyen, A Primer of Algebraic Geometry, Marcel Dekker 2000.

2. Kenji Ueno, An Introduction to Algebraic Geometry, American Mathematical Society 1997.

3. Karen E. Smith et al, An Invitation to Algebraic Geometry, Springer 2000.

4. J. Harris , Algebraic Geometry, Springer 1992.

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Other

Week Subjects

1

2

3

4

5

6

7

8

9

10

11

12

13

14

Plane curves, conics and cubics

Affine Algebraic Varieties

Hilbert Basis Theorem

The Zariski Topology

Hilbert Nullstellensatz

The Coordinate Ring

Morphisms of Affine Varieties

Projective Varieties

Quasi-Projective Varieties

Veronese Maps and Product of Varieties

Grassmannians, The Hilbert Function

Smoothness, Bertini’s Theorem

Resolution of Singularities

Blowing Up

Instructor/s Assist. Prof.Dr. Pınar Mete




Course Title: Applications of Fractional Calculus Code : FMT5231 Institute: Instute of Science

Field: Mathematics



Field Study

Home

work

Other Total Credit

T+A+L=Credit

ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective Social

Elective

Course

Objectives

To teach fractional-order systems and controllers, fractional optimal control problems and applications of

fractional.

Learning

Outcomes

and

Competences

To be able to define the concept of the fractional order controllers,

To be able to make comparison between fractional PI D and classic PID controllers,

To be able to define Hamiltonian and Euler-Lagrange Equations,

To be able to construct mathematical modeling of fractional diffusion-wave equations,

To be able to construct Fractional mathematical modeling of viscoelastic materials.

Textbook

and /or

References

1. S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional Integrals and Derivatives-Theory and Applications,

CRC Press, 1993.

2. R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, 2000.

3. A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations,

Elsevier Science, 2006.

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Week Subjects

1

2

3

4

5

6

7

8

9

10

11

12

13

14

Fractional-order systems.

Fractional-order controllers.

Fractional-order transfer functions.

Comparison of classic PID and fractional PI D controllers.

Responses of open-loop and closed-loop fractional-order systems.

Stochastic analysis of fractional dynamic systems

Hamiltonian and Euler-Lagrange Equations.

Definition and examples of optimal control problems.

Fractional optimal control problems.

Mathematical modeling of fractional diffusion-wave equations.

Fractional mathematical modeling of viscoelastic materials.

Other applications of fractional calculus in physics.

Applications of fractional calculus in chemistry.

Applications of fractional calculus in biology.






Course Title:

Number Theory II

Code :

FMT 5232


Field: Mathematics



Field Study

Home

work

Other Total Credit

T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective Social

Elective

Course

Objectives To teach the concepts of quadratic and cubic residue.

Learning

Outcomes

and

Competences

To be able to define the reduction rule of second degree and to apply it,

To be able to apply the quadratic residues,

To be able to define the concept of cubic residue,

To be able to solve the cubic equations,

To be able to express the primes in Z[w].

Textbook

and /or

References

1. K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Springer, (1990).

2. D. Namlı, Kübik Rezidüler, Doktora Tezi, Balıkesir, (2001).

3. G. A.Jones and J.M. Jones, Elementary Number Theory, Springer, (2004).

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Week Subjects

1 The ring of congruence class

2 Quadratic Residues and The Legendre Symbol

3 The group of quadratic residues

4 Quadratic Reciprocity

5 Algebraic Numbers

6 The quadratic character of 2

7 Quadratic Gauss Sums

8 An application to quadratic residues

9 Cubic Residue Character

10 The cubic character of 2

11 Primes of Z[w]

12 Index Rules

13 Cubic Equations

14 Cubic Residues

Instructors Assist. Prof. Dr. Dilek Namlı





Course Title:

Bergman Spaces

Code :

FMT5234


Field: Mathematics



Field Study

Home


T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective

Course

Objectives To teach the structure of Bergman spaces.

Learning

Outcomes

and

Competences

To be able to define the Bergman space,

To be able to express the relations between Bergman spaces and other function spaces,

To be able to interpret the density of polynomials,

To be able to express the Hilbert space structure of the Bergman space A2,

To be able to state the appraximation theorems in the Bergman space A2.

Textbook

and /or

References

1) P. L. Duren and Schuster, Bergman Spaces.

2) P. L. Duren, Introduction to Hp spaces, Academic Press, 1970.

3) D. Gaier, Lectures on complex approximation, Birkhauser, 1987.

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Week Subjects

1 Bergman Kernel function

2 Orthonormal bases, conformal invariants

3 Hardy spaces, strict and uniform convexity

4 Bergman projection, Harmonic conjugate

5 Linear isometries, Function multipliers

6 Growth properties of functions

7 Coefficients multipliers

8 Approximation in Bergman space A2

9 Bergman space A2 as a Hilbert space

10 Orthonormal systems

11 Density of polynomials

12 Domains with PA property

13 Domains with PA property

14 Expansions with respect to ON systems





Course Title:

Differentiable Manifolds II


Field: Mathematics



Study


T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective Social

Elective

Course

Objectives

To teach the notions of a tensor on a manifold, integration on a manifold and the general properties of

Riemannian manifolds.

Learning

Outcomes

and

Competences

To be able to define the notion of a tensor on a manifold and to give some examples,

To be able to define the notion of a Riemannian manifold and to give some examples,

To be able to define the concept of orientiability of manifolds, To be able to express the concept of integration on manifods,

To be able to define the concept of Manifold of constant curvature nad to give examples.

Textbook

and /or

References

Boothby, William M. An introduction to differentiable manifolds and Riemannian geometry. Second

edition. Pure and Applied Mathematics, 120. Academic Press, Inc., Orlando, FL, 1986.

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Week Subjects

1 Tensors on manifolds 2 2-lineer forms, Riemann metrics 3 Riemannian manifolds a metric spaces 4 Tensor fields on manifolds 5 Tensor products 6 Orientation on manifolds 7 Exterior differentiation 8 Applications 9 Integration on Manifolds 10 Differential forms 11 Differentiation on Riemannian manifolds 12 Geodesics on Riemannian manifolds 13 Manifolds of constant curvature 14 Applications




http://www.ams.org/mathscinet/search/publications.html?pg1=IID&s1=39455

http://www.ams.org/mathscinet/search/series.html?cn=Pure_and_Applied_Mathematics


Course Title:

Tensor Geometry II Code : FMT5236


Field: Mathematics



Study


T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective Social

Elective

Course

Objectives To teach fundamental knowledge about tensors.

Learning

Outcomes

and

Competences

To be able to define and to apply the notions of Ricci tensor, scalar curvature,

To be able to apply the concept of tensor in classical mechanics,

To be able to apply the concept of tensor in special relativity,

To be able to define the concept of Einstein Manifold and to give examples,

To be able to define the concept of Quasi-Einstein Manifold and to give examples.

Textbook

and /or

References

1. H. Hilmi Hacısalihoğlu , Tensör Geometri, Ankara Ünv. Fen-Fakültesi, 2003.

2. D. C. Kay, Tensor Calculus, McGraw-Hill, 1988.

3. C. T. J. Dodson, T. Poston, Tensor geometry, Graduate Texts in Mathematics, 130. Springer-

Verlag, Berlin, 1991.

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Week Subjects

1 Ricci tensor, scalar curvature 2 Applications 3 Spaces of constant curvature 4 Applications 5 Einstein manifolds 6 Applications 7 Quasi-Einstein manifolds 8 Applications 9 Tensors in classical mechanics I 10 Tensors in classical mechanics II 11 Applications 12 Tensors in special relativity I 13 Tensors in special relativity II 14 Applications




http://www.ams.org/mathscinet/search/series.html?cn=Graduate_Texts_in_Mathematics


Course Title:

Möbius Transformations II

Code :


Field: Mathematics



Field Study

Home


T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective

Course

Objectives To teach the the fundamental algebraic and geometric properties of Möbius transformations.

Learning

Outcomes

and

Competences

To be able to define and to apply the algebraic properties of Möbius transformations on the extended

complex plane,

To be able to define and to apply the geometric properties of Möbius transformations on the

extended complex plane,

To be able to express the finite groups of Möbius tranfromation,

To be able to define the group of rotations of the shpere,

To be able to express a geormetric definition of the infinity.

Textbook

and /or

References

1) A. F. Beardon, Algebra and geometry, Cambridge University Press, Cambridge, 2005.

2) T. Needham, Visual complex analysis, The Calerendon Press, Oxford University Press, New York,

1997.

3) C. Caratheodory, The most general transformations of plane regions which transform circles into

circles. Bull. Amer. Math. Soc. 43 (1937), no. 8, 573-579.

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Performance) X % 20

Week Subjects

1 The stabilisers of a circle and a disc

2 Conformality

3 Complex lines

4 Fixed points and eigenvectors

5 A geometric view of infinity

6 Rotations of the sphere I

7 Rotations of the sphere II

8 Finite groups of Möbius transformations I

9 Finite groups of Möbius transformations II

10 The most general transformations of plane regions which transform circles into circles

11 The most general planar transformations that map hyperbolas to hypaerbolas I

12 The most general planar transformations that map hyperbolas to hypaerbolas II

13 The most general planar transformations that map parabolas into parabolas I

14 The most general planar transformations that map parabolas into parabolas II





Course Title:

Averaged moduli and one sided approximation II

Code :

FMT5238


Field: Mathematics



Field Study

Home


T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective

Course

Objectives To teach the theorems of one sided approximation in the space Lp, 0<p<infinity.

Learning

Outcomes

and

Competences

To be able to state the direct theorem of one sided approximation in the space Lp, p>1,

To be able to state the converse theorem of one sided approximation in the space Lp, p>1,

To be able to state the direct theorem of one sided approximation in the space Lp, p<1,

To be able to state the converse theorem of one sided approximation in the space Lp, p<1,

To be able to explain the concepts of modulus of smoothness with real order and one sided

approximation.

Textbook

and /or

References

Bl. Sendov and V. A. Popov, The avaraged moduli of smoothness, 1988.

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1 Preliminaries 2 In short, the main trigonometric approximation theorems 3 The direct theorem of one sided approximation in Lp, p>1 4 The direct theorem of one sided approximation in Lp, p>1 5 The inverse theorem of one sided approximation in Lp, p>1 6 The inverse theorem of one sided approximation in Lp, p>1 7 The direct theorem of one sided approximation in Lp, p<1 8 The direct theorem of one sided approximation in Lp, p<1 9 The inverse theorem of one sided approximation in Lp, p<1 10 The inverse theorem of one sided approximation in Lp, p<1 11 Fractional order moduli of smoothness an done sided approximation 12 Fractional order moduli of smoothness an done sided approximation 13 Some exact inequalities 14 Some applications





Course Title:

Strong Approximation II

Code :

FMT5239


Field: Mathematics



Field Study

Home


T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective

Course

Objectives To teach the strong approximation and the embedding theorems.

Learning

Outcomes

and

Competences

To be able to explain the relation between strong approximation and structural properties,

To be able to define the concept of generalized strong de la Vallee Poussin means,

To be able to explain the relation between the order of strong approximation and structural properties,

To be able to the concept of generalized strong approximation,

To be able to state the embedding theorems

Textbook

and /or

References

Laszlo Leindler, Strong approximation by Fourier series, Akademiai Kiado, 1985.

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Week Subjects

1 Preliminaries 2 Generalized strong de la Vallee Poussin means 3 Generalized strong de la Vallee Poussin means 4 Generalized strong de la Vallee Poussin means 5 Order of strong approximation and structural properties 6 Order of strong approximation and structural properties 7 Order of strong approximation and structural properties 8 structural properties function derivatives 9 structural properties function derivatives 10 Generalized strong approximation 11 Generalized strong approximation 12 Imbedding theorems 13 WrH1 class 14 WrH1 class





Course Title:

Finite Blaschke Products II

Code :


Field: Mathematics



Field Study

Home


T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective

Course

Objectives

To teach fundamental definitions and theorems about the notions of centralizers of finite Blaschke products and

commuting finite Blaschke products.

Learning

Outcomes

and

Competences

To be able to define the concept of centralizer of a finite Blaschke product,

To be able to express the theorems about the concept of centralizer of a finite Blaschke product,

To be able define the concept of commuting finite Blaschke products,

To be able to express the theorems about the concept of commuting finite Blaschke products,

To be able to give examples about these topics.

Textbook

and /or

References

1. C. Artega, Centralizers of finite Blaschke products. Bol. Soc. Brasil Mat. (N.S.) 31 (2000), no. 2,

163-173.

2. C. Artega, Commuting finite Blaschke products. Ergodic Theory Dynam. Systems 19 (1999), no. 3,

549-552.

3. I. Chalender and R. Mortini, When do finite Blaschke products commute? Bull. Austral. Math. Soc.

64 (2001), no. 2, 189-200.

4. C. Artega, On a theorem of Ritt for commuting finite Blaschke products. Complex Var. Theory Appl.

48 (2003), no.8, 671-679.

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Week Subjects

1 Centralizers of finite Blaschke products I

2 Centralizers of finite Blaschke products II

3 Centralizers of finite Blaschke products III

4 Examples

5 Commuting finite Blaschke products

6 Commuting finite Blaschke products with a fixed point in the unit disc I

7 Commuting finite Blaschke products with a fixed point in the unit disc II

8 Counterexamples to C. C. Cowen’s Conjectures

9 Commuting finite Blaschke products with no fixed point in the unit disc I

10 Commuting finite Blaschke products with no fixed point in the unit disc II

11 Examples

12 Commuting finite Blaschke products with no fixed point in the unit disc III

13 Commuting finite Blaschke products with no fixed point in the unit disc IV

14 Applications





Course Title:

Algebra II

Code :

FMT5241


Sciences

Field : Mathematics



Field Study


T+A+L=Credit

ECTS

42 0 0 0 100 98 240 3 6


Course Type

Basic


Elective Social

Elective

Course

Objectives To teach the fundamental properties of module and field theories.

Learning

Outcomes

and

Competencies

To be able to classify free modules over a ring and finitely generated module over PID,

To be able to demonstrate various constructions involving modules,

To be able to express the fundamental facts about field extensions,

To be able to state the main theorems,

To be able to classify finite fields.

Textbooks

and /or

References

1. T. W. Hungerford, Algebra, Springer 1996.

2. D.S. Dummit and R. M. Foote, Abstract Algebra, Wiley 2nd edition ,1999.

3. N. Jacobson, Basic Algebra I-II, Dover Publications, 2009.

4. H.İ. Karakaş, Cebir Dersleri, TUBA 2008.

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Other

Week Subjects

1

2

3

4

5

6

7

8

9

10

11

12

13

14

Modules, Homomorphisms and Exact Sequences

Projective and Injective Modules

Free Modules, Vector Spaces

Hom and Duality

Tensor Products

Modules over a Principal Ideal Domain

Basic properties of Fields

Algebraic and transcendental extensions of fields

Fundamental theorem of Galois theory

Splitting fields and Normal extensions

The Galois Group of a Polynomial

Finite Fields

Separability

Cyclic Extensions

Instructor/s Assist. Prof.Dr. Pınar Mete




Course Title:

Function Spaces II

Code :

FMT5243


Field: Mathematics



Field Study

Home


T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective

Course

Objectives To teach several function spaces and relations among them.

Learning

Outcomes

and

Competences

To able to define the concept of Modular space,

To able to define the concept of Musielak Orlicz space,

To be able express the relations between modular spaces and Musielak Orlicz spaces,

To be able to define the Lebesgue spaces with variable exponent,

To be able to express the relation between Musielak Orlicz space and Lebesgue space with variable

exponent.

Textbook

and /or

References

1) J. Musielak, Orlicz spaces and Modular Spaces, Springer, 1982.

2) L. Diening, P. Harjulehto, P. Hästö, M. Růžička Lebesgue and Sobolev spaces with variable

exponents , Springer, 2011.

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Week Subjects

1 Modular space 2 Modular space 3 Modular space 4 Modular space 5 Musielak Orlicz space 6 Musielak Orlicz space 7 Musielak Orlicz space 8 Musielak Orlicz space 9 Musielak Orlicz space 10 Musielak Orlicz space 11 Variable exponent Lebesgue space 12 Variable exponent Lebesgue space 13 Inequalities in Variable exponent Lebesgue space 14 Inequalities Variable exponent Lebesgue space




http://www.helsinki.fi/~pharjule/varsob/pdf/book.pdf

http://www.helsinki.fi/~pharjule/varsob/pdf/book.pdf


Course Title:

Potential Theory Code :


Field: Mathematics



Field Study


Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective Social

Elective

Course

Objectives To teach the concepts and tecniques in potential theory.

Learning

Outcomes

and

Competences

To be able to define the concept of subharmonic function,

To be able to state the maximum principle for potantials,

To be able to define the concepts of potantial equilibrium measure and capacity,

To be able to apply the techniques of potantial theory in analysis of orthogonal polynomials,

To be able to define the concept of Green function.

Textbook

and /or

References

1. E. B. Saff, Orthogonal Polynomials From a Complex Perspective, Kluwer Academic Publisher, 1990.

2. E. B. Saff, V. Totik, Logaritmic Potentials with External Fields, Springer, 1997.

3. H. Stahl, V. Totik, General Orthogonal Polynomials, Cambridge University Press, 1992.

4. T. Ransford, Potential Theory in the Complex Plane, London Math. Soc.Student Texts. Cambridge

Press. 1995.

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Week Subjects

1

2

3

4

5

6

7

8

9

10

11

12

13

14

Harmonic functions

Dirichlet problem

Subharmonic functions

Potentials

Maximum principle,for potentials

Equilibrium measure

Logarithmic capacity

Energy

Relations with orthogonal polynomials

Relations with potential theory

Geometric convergence

Fejer theorem

Green functions

Relations with approximation theory





Course Title:

Banach Spaces of Analytic Functions II Code :


Field: Mathematics



Field Study


Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective Social

Elective

Course

Objectives To teach fundamental properties of Smirnov and Bergman spaces.

Learning

Outcomes

and

Competences

To be able to express the linear space structure of Hp spaces,

To be able to define the dual spaces of Hp spaces,

To be able to express the fundamental properties of Smirnov spaces,

To be able to express the fundamental properties of Bergman spaces,

To be able to express the domains with the PA property and the domains does not have the PA

property.

Textbook

and /or

References

1) P. Koosis, Introduction to Hp Spaces, Cambridge University Press (1998).

2) P. L. Duren, Teory of Hp spaces, Academic Press (1970).

3) D. Gaier, Lectures on Complex Approximation, Birkhauser (1987).

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1

2

3

4

5

6

7

8

9

10

11

12

13

14

Conjugate functions

Theorems of Riesz and Kolmogorov

Zygmund’s theorem

Hp as a linear space

Duals of Hp spaces

Hp spaces over general domains

The Smirnov spaces Ep (G)

The space E1 (G) and Cauchy integral

Smirnov domains

The Bergman space A2(G)

A2(G) as a Hilbert space

Orthonormal systems in A2(G)

Polynomials in A2(G)

Domains with the PA property and domains not having the PA property



Website http://w3.balikesir.edu.tr/~aguven/


Course Title:

Fourier Analysis II Code :


Field: Mathematics



Field Study


Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective Social

Elective

Course

Objectives To teach convergence properties and summability methods of multiple Fourier series.

Learning

Outcomes

and

Competences

To be able to define the square and circular Dirichlet and Fejer kernels,

To be able to state the poisson summation Formula,

To be able to express the convergence propeties of Fejer means,

To be able to express the convergence and divergence of multiple Fourier series,

To be able to express the Bochner-Riesz summability method.

Textbook

and /or

References

1) L. Grafakos, Classical Fourier Analysis, Springer (2008).

2) J. Duoandikoetxea, Fourier Analysis, American Math. Soc. (2001).

3) E.M.Stein, G.Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press (1971).

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Week Subjects

1

2

3

4

5

6

7

8

9

10

11

12

13

14

The n-torus Tn

Multiple Fourier series

The square and circular Dirichlet and Fejer kernels

The Poisson summation formula

Decay of Fourier coefficients

Pointwise convergence of the Fejer means

Almost everywhere convergence of the Fejer means

Pointwise divergence of multiple Fourier series

Pointwise convergence of multiple Fourier series

Bochner-Riesz summability

Divergence of Bochner-Riesz means of Integrable functions

Boundedness of the conjugate function in Lp spaces

Convergence of multiple Fourier series in the norm

Almost everywhere convergence of multiple Fourier series





Course Title:

Fourier Series and Approximation II Code :


Field: Mathematics



Field Study


Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective Social

Elective

Course

Objectives To teach the fundamental theorems of trigonometric approximation theory.

Learning

Outcomes

and

Competences

To be able to define the concepts of modulus of smoothness and modulus of continuity,

To be able to state the direct theorems of trigonometric approximation in the spaces C and Lp,

To be able to state the converse theorems of trigonometric approximation in the spaces C and Lp,

To be able to define the Muckenhoupt (Ap) weights,

To be able to state the fundamental theorems of trigonometric approximation in weighted Lp spaces.

Textbook

and /or

References

1. R.A. DeVore, G.G.Lorentz, Constructive Approximation, Springer-Verlag (1993).

2. G. Mastroianni, G.V.Milovanovic, Interpolation Processes, Springer (2008).

3. J. Garcia Cuerva, J. L. Rubio De Francia, Weighted Norm Inequalities and Related Topics, North

Holland (1985)

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Week Subjects

1

2

3

4

5

6

7

8

9

10

11

12

13

14

Modulus of continuity and modulus of smoothness

Lipschitz and generalized Lipschitz classes

Direct theorems of trigonometric approximation in the spaces C and Lp

Bernstein inequality and inverse theorems of trigonometric approximation

Characterization of Lipschitz and gemneralized Lipschitz classes in terms of best approximation

Improvement of direct and inverse theorems

The Hardy-Littlewood maximal function

The Hilbert transform

Weighted Lp spaces and Ap weights

Weighted norm inequalities for the Hilbert transform and conjugate function

Convergence of Fourier series in weighted Lp spaces

Modulus of smoothness and K-functionals in weighted Lp spaces

Trigonometric approximation in weighted Lp spaces

Analogues of Marcinkiewicz multiplier and Littlewood-Paley theorems in weighted Lp spaces





Course Title: Applied Mathematics II Code : FMT5248 Institute: Instute of Science

Field: Mathematics



Field Study


T+A+L=Credit

ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective Social

Elective

Course

Objectives To teach the concepts of feedback linearization of nonlinear systems, Lyapunov stablity.

Learning

Outcomes

and

Competencies

To be able to state existence and uniqueness theorems of nonlinear systems,

To be able to express and apply Lyapunov stability theorem,

To be able to express the concept of Input-Output stability,

To be able to express the concept of Stability with linearization

To be able to express Input-output Linearization.

Textbooks

and /or

References

1- H. K. Khalil, Nonlineer Systems, Prenice-Hall,1996.

2- F. Verhulst, Nonlineer Differential Equations and Dynamics Systems, Springer-Verlag, 1989.

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Other

Week Subjects

1

2

3

4

5

6

7

8

9

10

11

12

13

14

Introduction to nonlinear systems. (Existence and uniqueness theorems).

Autonomous systems , Phase space, orbits,

Class of Critical points,

Periodic of solutions,

Stability Theory,

Lyapunov Stability Method,

Input-Output stability,

Stability with linearization,

Feedback systems,

Feedback control,

Feedback linearizable systems,

Feedback linearization,

Input-output Linearization,

State feedback control.

Instructors Assoc Prof. Dr. Necati ÖZDEMİR





Course Title: Advanced Numerical Analysis II Code : FMT5249 Institute: Instute of Science

Field: Mathematics



Field Study


T+A+L=Credit

ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective Social

Elective

Course

Objectives To teach numerical solution methods for ordinary differential equartions.

Learning

Outcomes

and

Competencies

To be able to solve first order differential equations with sequential iterative method,

To be able to get numerical solutions of initial value problems for ordinary differential equations,

To be able to express Euler and Runge-Kutta one Step methods for first order ordinary differential

equations,

To be able to use Nystom method for high order ordinary differential equations,

To be able to express stability of numerical methods.

Textbooks

and /or

References

1) G. Amirali, H. Duru, Nümerik Analiz, Pegem A Yayınları, 2002,

2) A. Ralston, A First Course in Numerical Analysis, McGraw-Hill,1978,

3) S.C. Chapra, R.P. Canale, Numerical Methods for Engineers, McGraw-Hill, 1990.

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Other

Week Subjects

1

2

3

4

5

6

7

8

9

10

11

12

13

14

Difference Equations,

Solution of First Order Differential Equations with Sequential Iterative Method,

Numerical Solutions of Initial Value Problems for Ordinary Differential Equations,

One Step Methods for Ordinary Equations: Euler and Runge-Kutta,

Multi Step Methods,

Trial and Correction Formulas,

Runge-Kutta Method for Systems of First Order Equations,

Hamming Method,

Solutions of Higher Order Equations, Nystöm Method,

Numerical Solution of Ordinary Differential Equations for Boundary Value Problems,

Ignition Method,

Finite Difference Method,

Variational Difference Methods,

Stability of Numerical Methods.

Instructors Assist Prof. Dr..Figen KİRAZ




Course Title: Numerical Solution of Partial

Differential Equations Code : FMT5250


Field: Mathematics



Field Study


T+A+L=Credit

ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective Social

Elective

Course

Objectives To teach Numerical Methods for Solutions of Partial Differential Equations.

Learning

Outcomes

and

Competencies

To be able to express convergence and stability of Parabolic Equations,

To be able to apply Crank-Nicolson Closed Method,

To be able to apply Finite-Difference Methods,

To be able to solve Hyperbolic equations,

To be able to solve Eliptic Equations.

Textbooks

and /or

References

1. K. W. Morton, D.F. Mayers, Numerical solution of partial differential equations, Cambridge University

Press, 1994

2. G.D. Smith, Numerical solution of partial differential equations, Oxford University Press, 1985.

3. J.Strickwerda, Finite difference schemes and partial differential equations, Wadsworth&Brooks/Cole,

1989.

4. E. Godlewski, P-a. Raviart, Numerical approximation of hyperbolic systems of conservation laws,

Springer, 1996.

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Other

Week Subjects

1

2

3

4

5

6

7

8

9

10

11

12

13

14

Introduction and Finite-Difference Formula,

Parabolic Equations: Finite Difference Methods, Convergence and Stability,

Explicit Method,

Crank-Nicolson Implicit Method,

Fourier Analysis of Eror,

Descriptive Treatment, Convergence, Stability

Gerschgorin’s theorems, Neumann’s Methods, Lax’s equivalence Theorem,

Hyperbolic equations and Characteristics: Analytical Solution of First Order Quasi-Linear

equations,

Numerical Integration Along a Characteristic,

Finite-Difference Methods, Lax-Wendroff Explicit Metod,

The Counrant –Friedrichs-Lewy Condition,

Wendroff’s Implicit Appoximation,

Elliptic Equations and Systematic Iterative Methods,

Systematic Iterative Methods for Large Linear Systems.

Instructors Assist Prof. Dr. Figen KİRAZ




Course Title: Differential Geometry of Curves

and Surfaces II

Code : FMT5251


Field: Mathematics



Field Study


T+A+L=Credit

ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective Social

Elective

Course

Objectives To teach the differential geometry of curves and surfaces both in local and global aspects.

Learning

Outcomes

and

Competencies

To be able to define the Gauss map,

To be able to state the Gauss theorem,

To be able to define the concept of parallel transport,

To be able to express the properties of geodesics,

To be able to define the geodesic polar coordinates.

Textbooks

and /or

References

Manfredo P. do Carmo, Differential Geometry of Curves and Surfaces, 1976.

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1

2

3

4

5

6

7

8

9

10

11

12

13

14

A geometric definition of area.

A geometric definition of area.

The definition of Gauss map and its fundamental properties,

The definition of Gauss map and its fundamental properties,

The Gauss map in local coordinates, Vector fields.

The Gauss map in local coordinates, Vector fields.

Isometries , conformal maps ,

Isometries , conformal maps ,

The Gauss theorem, Parallel transport ,

The Gauss theorem, Parallel transport ,

The exponential map, Geodesic polar coordinates,

The exponential map, Geodesic polar coordinates,

Further properties of geodesics, Convex neighborhoods

Further properties of geodesics, Convex neighborhoods.





Course Title:

Topology II Code :

FMT5252


Field: Mathematics



Field Study

Home

work

Other Total Credit

T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective Social

Elective

Course

Objectives To teach the concepts of general topology in advanced level.

Learning

Outcomes

and

Competences

To be able to construct topological structures by convergence of nets and filters,

To be able to express the countability properties,

To be able to define the concepts of compactness and local compactness,

To be able to express the metrizability properties of topological spaces,

To be able to define the concepts of Cauchy sequence, complete metric space, Baire category

theorem, paracompactness, totally regularity.

Textbook

and /or

References


2. Osman Mucuk, Topoloji , Nobel Kitapevi, 2009.

3. Mahmut Koçak, Genel Topoloji I ve II, Gülen Ofset Yayınevi, 2006.


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Week Subjects

1 Convergence 2 Networks, Convergence of networks 3 Limit Point 4 Continuity and Convergence 5 Countability Features 6 Compactness, Derived Spaces and Compactness

7 Compactness in Rn Compactness, local compactness

8 Kompaktifikasyon, Sequential Compactness and Countable Compactness

9 Metric Space Concept

10 Neighborhoods, Open Sets, Closed Sets

11 Convergence of Sequences

12 Continuity

13 Metrizability

14 Cauchy Sequences, Complete Metric Spaces, Baire Category Theorem, paracompactness,

totally Regularity






Course Title:

Introduction to Fuzzy Topology II

Code :

FMT5253


Field: Mathematics



Field Study

Home

work

Other Total Credit

T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective Social

Elective

Course

Objectives

To teach the corresponding concepts of general topology in fuzzy topological spaces.

Learning

Outcomes

and

Competences

To be able to give examples of interior, closure and boundary of a set in fuzz topological spaces,

To be able to define the concepts of fuzzy regular open set and fuzzy regular closed set,

To be able to define the concepts of fuzzy topology base and subbase,

To be able to define the fuzzy product spaces,

To be able to express the fuzzy separation axioms.

Textbook

and /or

References






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Week Subjects

1 The Concept of Fuzzy Topology 2 Fuzzy Topological Spaces 3 Fuzzy Neighborhoods Family 4 Within the cluster is a fuzzy 5 Closing and Limitation of a fuzzy cluster 6 On Fuzzy Regular Regular Closed Sets and Fuzzy 7 Accumulation Points of a fuzzy cluster 8 Fuzzy Topology Base and Subbase 9 Fuzzy First Countable Space 10 Fuzzy Second Countable Space 11 Fuzzy Subspaces 12 Fuzzy Product Spaces 13 Fuzzy Continuity 14 Fuzzy Separation Axioms






Course Title:

Introduction to Ideal Topological Spaces II

Code :

FMT5254


Field: Mathematics



Field Study

Home

work

Other Total Credit

T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective Social

Elective

Course

Objectives To teach the concept of delta-I-continuous function and to compare with the other types of functions.

Learning

Outcomes

and

Competences

To be able to define a type of continuous function in Ideal topological spaces and to prove related

theorems,

To be able to express the properties of Delta-I-closure point,

To be able to prove the characterization of Delta-I-continuous function,

To be able to compare the functions,

To be able to express the properties of functions in SI-R and AI-R spaces.

Textbook

and /or

References

1. Şaziye Yüksel, Genel Topoloji, Eğitim Kitapevi, (2011).

2. Osman Mucuk, Topoloji, Nobel Kitapevi, (2009).

3. Mahmut Koçak, Genel Topoloji I ve II, Gülen Ofset Yayınevi, (2006).



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Week Subjects

1 Delta-I-sets 2 Delta-I-Cluster Point 3 Properties of Delta-I-Cluster Point 4 R-I-open set 5 Comparison of the Sets 6 Delta-I-continuous function 7 Characterization of Delta-I-continuous function 8 Strongly theta-I-continuous function 9 Almost-I-continuous function 10 Comparison Functions 11 All the reverse examples studies 12 SI-R space 13 AI-R space 14 Investigation of the functions in these spaces






Course Title:

Orthogonal Polynomials II Code :


Field: Mathematics



Field Study


Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective Social

Elective

Course

Objectives To teach the approximation properties of orthogonal polynomials in the complex plane.

Learning

Outcomes

and

Competences

To be able to express the asymptotic representations of orthogonal polynomials,

To be able to express the Bernstein-Walsh maximal convergence theorem,

To be able to express the asymptotic properties of orthogonal polynomials,

To be able to express the approximation properties of Fourier series of orthogonal polynomials on

closed domains,

To be able to define the distribution of zeros of kernel functions.

Textbook

and /or

References

1) V.I.Smirnov and N. A. Lebedev, Functions on a Complex Variable, MIT pres, 1968.

2) P. K. Suetin, Fundamental Properties of Polynomials Orthogonal on a Contour, Russ. Math. Surv.,

1966.

3) P. K .Suetin, Polynomials Orthogonal over a region and Bieberbach Polynomials, Proceedings of the

Steklov Institute of Mathematics, AMS, 1974.

4) D.Gaier, Lectures on Complex Approximation,Birkhauser, 1987.

ASSESSMENT CRITERIA


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mark as

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Other

Week Subjects

1

2

3

4

5

6

7

8

9

10

11

12

13

14

The representation of asymptotic s of othogonal polynomials , Carleman Theorem

The rate of approximation of analytic functions on closure of the domain

Bernstein-Walsh Lemma

The convergence of Fourier Series of orthogonal polynomials on closed domains

In the case of weight function, the convergence of Fourier Series of orthogonal polynomials

Orthogonal polynomials on unit circle

The convergence of Fourier Series of orthogonal polynomials on closed domains on the boundary of the

domain

Ortogonal polynomials from potential theory perspective

Asymptotics of ortogonal polynomials over domains bounded with analytic Jordan curves,

Zeros of ortogonal polynomials over domains bounded with analytic Jordan curves

Asymptotics of Bergman polynomials

Zero distribution of Bergman polynomials

Asymptotics of Kernel polynomials,

Zero distribution of Kernel polynomials





Course Title:

Geometric Theory of Functions II Code :


Field: Mathematics



Field Study


Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective Social

Elective

Course

Objectives To introduce the problems of convergence in the geometric theory of functions.

Learning

Outcomes

and

Competences

To be able to define the convergence of the sequences of analytic and harmonic functions,

To be able to expressthe boundary value problems for analytic functions defined on a disk,

To be able to express the boundary value problems for functions analytic inside a rectifiable contour,

To be able to define the conformal mappings of multiply connected domains,

To be able to make representations of harmonic functions by aim of Poisson integral.

Textbook

and /or

References

G. M. Goluzin, Geometric Theory of Functions of a complex variable, 1969.

ASSESSMENT CRITERIA


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Term Paper




Other

Week Subjects

1

2

3

4

5

6

7

8

9

10

11

12

13

14

Fundamental properties of analytic functions

Fundamental properties of harmonic functions

The convergence of sequence of analytic functions

The convergence of sequence of harmonic functions

Conformal mappings of simply connected domains

Riemann conformal theorem

Conformal mappings of multiply connected domains

Dirichlet problem; Green function

Limiting values of Poisson’s integral

The representation of harmonic functions by means of Poisson integral

Boundary properties of analyic functions in Hardy class

The limiting values of Cauchy integrals

Applications of conformal mappings

Applications of conformal mapping





Course Title: Algebraic Number Theory II Code :

FMT5257


Field: Mathematics



Field Study

Home

work

Other Total Credit

T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective Social

Elective

Course

Objectives

To teach fundamental concepts and theorems related with the algebraic number theory.

Learning

Outcomes

and

Competences

To be able to define the ideal class group,

To be able to apply the algorithms for the ideal class group,

To be able to state the Dirichlet’s unit theorem,

To be able to determine the fundamental units of cubic fields,

To be able to apply the diophantine equations.

Textbook

and /or

References

1) E. Weiss, Algebraic Number Theory, Dover publications, 1998.

2) I. Stewart, D. Tall, Algebraic Number Theory and Fermat’s Last Theorem, A K Peters Ltd., 2002.

3) M.R. Murty, J. Esmonde, Problems in Algebraic Number Theory, Springer,2005.

4) Ş. Alaca, K. S. Williams, Introductory Algebraic Number Theory, Cambridge Univ. Press, 2004.

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Other

Week Subjects

1 The Fundamental Unit 2 Calculating the Fundamental Unit 3 The Ideal Class Group 4 The Ideal Class Group 5 Algorithm to Determine the Ideal Class Group 6 Applications to Binary Quadratic Forms 7 Dirichlet’s Unit Theorem 8 Valuations of an Element of a Number Field 9 Valuations of an Element of a Number Field 10 Fundamental System of Units 11 Fundamental Units in Cubic Fields 12 Fundamental Units in Cubic Fields 13 Applications to Diophantine Equations 14 Applications to Diophantine Equations

Instructors Assoc. Prof. Dr. Sebahattin İkikardes


Website http://w3.balikesir.edu.tr/~skardes/



Course Title: Numerical Optimization II Code : FMT 5258 Institute: Instute of Science

Field: Mathematics



Field Study

Home

work

Other Total Credit

T+A+L=Credit

ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective Social

Elective

Course

Objectives

To teach optimality conditions of unconstrained and constrained nonlinear optimization problems with

fundamental solution methods.

Learning

Outcomes

and

Competences

To be able to express optimality conditions for unconstrained and constrained optimization problems,

To be able to express the concept of Lagrange function and multiplier,

To be able to define Karush-Kuhn-Tucker conditions,

To be able to express optimality conditions for quadratic programming,

To be able to apply penalty, barrier and feasible direction methods.

Textbook

and /or

References

1) Bazaraa M.S., Sherali H.D. and Shetty S.M., Nonlinear programming: Theory and Applications, 3rd edition,


2) Chong E.K. and Zak S.H., An introduction to optimization, 2nd edition, John Wiley & Sons, Inc., 2001.

3) Griva I., Nash S.G. and Sofer A., Linear and nonlinear optimization, 2nd edition, SIAM, 2008.

4) Luenberger D.G. and Ye Y., Linear and nonlinear programming, 3rd edition, Springer, 2008.

5) Sun W. and Yuan Y-X, Optimization Theory and Method: Nonlinear Programming, Springer, 2006.

ASSESSMENT CRITERIA


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mark as

(X)

Percent

(%)

If any,

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(X)

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Term Paper




Other

Week Subjects

1

2

3

4

5

6

7

8

9

10

11

12

13

14

Nonlinear programming and problem formulations

Optimality conditions for equality constraints

Optimality conditions for inequality constraints

Constraint qualifications

The Lagrange multipliers and the Lagrangian functions

Karush-Kuhn-Tucker conditions

Optimality for Quadratic Programming

Methods for quadratic Programming

Penalty an Barrier Methods

Feasible Direction Methods

Sequential Quadratic Programming

Nonsmooth optimization and problems

Generalized gradients

The sub-gradient method

Instructors Assist. Prof. Dr. Fırat EVİRGEN





Course Title: Selected Topics in Differential

Geometry II

Code : FMT5259


Field: Mathematics



Field Study


T+A+L=Credit

ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective Social

Elective

Course

Objectives To teach fundamental concepts of Riemannian Geometry and the concept of submanifold of finite type.

Learning

Outcomes

and

Competencies

To be able to define the concepts of Sectional , Ricci and scalar curvature,

To be able to define the concept of tensor in Riemann manifolds,

To be able to define the concept of submanifold of finite type and to give examples,

To be able to define closed curves of finite type and to give examples,

To be able to define the concept of isometric immersion.

Textbooks

and /or

References

1) M.P. do Carmo, Riemannian Geometry, Birkhauser Boston 1992.

2) Bang-yen Chen, Total Mean Curvature and Submanifolds of Finite Type, World Scientific 1984.

ASSESSMENT CRITERIA


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mark as (X)

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mark as (X)

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Homework Term Paper

Term Paper, Project




Other

Week Subjects

1

2

3

4

5

6

7

8

9

10

11

12

13

14

Curvature; Sectional , Ricci and scalar curvature

Curvature; Sectional , Ricci and scalar curvature

Tensors on Riemannian manifolds

Tensors on Riemannian manifolds

Jacobi fields

Isometric immersions

Submanifolds

Submanifolds

Submanifolds of finite type

Submanifolds of finite type

Characterizations of 2-type submanifolds

Characterizations of 2-type submanifolds

Closed curves of finite type

Closed curves of finite type





Course Title:

Selected Topics in Analysis II

Code :


Field: Mathematics



Field Study

Home


T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective

Course

Objectives

To teach the basic knowledge about r-bonacci polynomials and generalized complex Fibonacci and Lucas

functions.

Learning

Outcomes

and

Competences

To be able to define and apply basic properties of tribonacci, quadranacci polynomials,

To be able to define and apply basic properties of r-bonacci polynomials,

To be able to define and apply basic properties of generalized complex Fibonacci functions,

To be able to define and apply basic properties of Lucas functions,

To be able to express the continuous functions for the Fibonacci and Lucas p-numbers.

Textbook

and /or

References

1) N. D. Cahill, J. R. D’Ericco and J. P. Spence, Complex factorizations of the Fibonacci and Lucas

numbers, Fibonacci Quart., 41(1), 13-19, 2003.

2) A. Stakhov and B. Rozin, Theory of Binet formulas for Fibonacci and Lucas p-numbers, Chaos,

Solitons Fractals, 27(5), 1162-1177, 2006.

3) A. Stakhov and B. Rozin, The continuous functions for the Fibonacci and Lucas p-numbers, Chaos,

Solitons Fractals, 28(4), 1014-1025, 2006.

ASSESSMENT CRITERIA


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mark as

(X)

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(%)

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mark as

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Performance) X % 20

Week Subjects

1 Tribonacci numbers

2 Tribonacci polynomials

3 Factoring Fibonacci and Lucas polynomials I

4 Factoring Fibonacci and Lucas polynomials II

5 Applications

6 Quadranacci and r-bonacci polynomials I

7 Quadranacci and r-bonacci polynomials II

8 Complex factorizations of the Fibonacci numbers I

9 Complex factorizations of the Fibonacci numbers II

10 Applications

11 Generalized complex Fibonacci and Lucas functions

12 Fibonacci and Lucas p-numbers

13 The continuous functions for the Fibonacci and Lucas p-numbers

14 Applications





Course Title:

Semi-Riemannian Geometry II Code : FMT5261


Field: Mathematics



Field Study


T+A+L=Credit

ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic


Elective Social

Elective

Course

Objectives To teach the general properties of hypersurfaces and submanifolds of Semi-Riemannian manifolds.

Learning

Outcomes

and

Competencies

To be able to express the general properties of submanifolds of Semi-Riemannian manifolds,

To be able to define the Non-Degenerate hypersurfaces of Semi-Riemannian manifolds,

To be able to define the Lightlike hypersurfaces of Semi-Riemannian manifolds,

To be able to define the concept of totally umbilical hypersurface,

To be able to define the concept of normal connection.

Textbooks

and /or

References

3) B. O’Neill, Semi-Riemannian Geometry with Applications to Relativity, Academic Press, Inc., 1983.

4) K. L. Duggal D. H. Jin, Null Curves and Hypersurfaces of Semi-Riemannian Manifolds, World Sci.,

2007.

ASSESSMENT CRITERIA


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mark as (X)

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Homework Term Paper

Term Paper, Project




Other

Week Subjects

1

2

3

4

5

6

7

8

9

10

11

12

13

14

Tangents and normal

Induced connections

Geodesic submanifolds

Non-Degenerate hypersurfaces of a Semi-Riemannian manifold

Lightlike hypersurfaces of a Semi-Riemannian manifold

Lightlike submanifolds

Lightlike surfaces in R14

Hyperquadrics

Codazzi equation

Totally umbilical hypersurfaces

The normal connection

A Congruence Theorem

Isometric immersions

Two-parameter maps





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