Balıkesir Üniversitesimatematik.balikesir.edu.tr/userfiles/Bologna FBE Mathematics (engli… · Information about the Department of Mathematics . Goals: The main aim of graduate

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 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 Prof. Dr. Ali GÜVEN

Erasmus Coordinator Assist. Prof. Dr. Derya AVCI

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

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

Program’s Key Learning Outcomes: Master:

1) To be able to understand Mathematical materials in basic and advanced level. 2) Doing analysis with the scientific method for having faced the problem or the subject, to produce suggestions based

on the potential solutions and rese arch 3) Making a research at the national and international level with the independent way in the post-graduate subjects of

the Mathematics, to do papers on common subjects, 4) To have a foregein language at the level being able to follow the improvments in the Mathematics science, and being

able to get into touch with their coleauges, 5) To have sufficient software informations at the level of being able to use basic computer programs about the

Mathematical subjects, 6) To have a personality that is respectful for professional and academic ethics values, 7) To have an ability on making proper solutions, by looking the different aspects, for current Mathematical problems, 8) To use Mathematical thought in the whole area of the life.

Doctorate:

1) Comprehending the basic Mathematics subjects such a good way, to have an education for being able to understand new informations,

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

2017-2018 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

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 Master 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

FMT5161 Introduction to Scientific Computing I 3 3 0 0 3 6

FMT5162 Differential Equation Systems 3 3 0 0 3 6

FMT5163 Faber Series I 3 3 0 0 3 6

FMT5164 Inequalities and extremal problems in approximation theory 3 3 0 0 3 6

FMT5165 Analytic Theory of Polynomials I 3 3 0 0 3 6

FMT5166 *Advanced Linear Algebra I 3 3 0 0 3 6

FMT5167 **Advanced Differential Equations I 3 3 0 0 3 6

FMT5168 Optimal Control Theory 3 3 0 0 3 6

FMT5169 Research Methods for Science and Ethics 3 3 0 0 3 6

FMT8101-8199 Special Topics in Field 8 8 0 0 8 6

2017-2018 Fall Semester Newly Added Courses

2017-2018 Fall Semester Deleted Courses

*FMT5166 Advanced Linear Algebra I is compulsory lecture for master programme in the fall semester. ∆ FMT5169 Research Methods for Science and Ethics is compulsory lecture for master programme in the fall semester. Also This lecture is compulsory for students who do not take this lecture at the master stage and register at the doctorate programme. **FMT5167 Advanced Differential Equations I is compulsory lecture for doctorate programme in the fall semester.

2017-2018 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 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 Doctorate 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 FMT5262 Introduction to Scientific Computing II 3 3 0 0 3 6 FMT5263 Control of Nonlinear Systems 3 3 0 0 3 6 FMT5264 Faber Series II 3 3 0 0 3 6

FMT5265 Analytic Theory of Polynomials II 3 3 0 0 3 6

FMT5266 *Advanced Linear Algebra II 3 3 0 0 3 6

FMT5267 **Advanced Differential Equations II 3 3 0 0 3 6

FMT5268 Fractional Optimal Control Theory 3 3 0 0 3 6

FMT8201-8299 Special Topics in Field 8 8 0 0 8 6

2017-2018 Spring Semester Newly Added Courses

2017-2018 Spring Semester Deleted Courses

*FMT5266 Advanced Linear Algebra II is compulsory lecture for master programme in the spring semester. **FMT5267 Advanced Differential Equations II is compulsory lecture for doctorate programme in the spring semester.

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

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


Convex Functions and Orlicz Spaces I


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


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

Master 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


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


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


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

Introduction to Scientific Computing I


Differential Equation Systems X X X X X X X X X X X X X X

Faber Series I X X X X X X X X X X X X X X

Inequalities and extremal problems in approximation theory


Analytic Theory of Polynomials I X X X X X X X X X X X X X X

*Advanced Linear Algebra I X X X X X X X X X X X X X X

**Advanced Differential Equations I X X X X X X X X X X X X X X

Optimal Control Theory X X X X X X X X X X X X X X

Research Methods for Science and Ethics


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

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


Convex Functions and Orlicz Spaces II


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

Doctorate 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


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 II


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


Differential Geometry of Curves and Surfaces II


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


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 Introduction to Scientific Computing II X X X X X X X X X X X X X X

Control of Nonlinear Systems X X X X X X X X X X X X X X Faber Series II X X X X X X X X X X X X X X Analytic Theory of Polynomials II X X X X X X X X X X X X X X *Advanced Linear Algebra II X X X X X X X X X X X X X X **Advanced Differential Equations II X X X X X X X X X X X X X X

Fractional Optimal Control 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

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 Homework

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 % 100 Other Other (Class Performance)

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 : FMT5102





ECTS

42 0 0 0 0 198 240 3 6


Course Type


Technical Elective

Social Elective

Course Objectives To introduce fundamental concepts and theorems of Functional analysis.


• 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.


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 (%)





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 Prof. Dr. Ali GÜVEN




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

Education and Teaching Methods Credits Lecture Application Laboratuary Project/Field

Study Homework


ECTS

42 0 0 0 0 198 240 3 6


Course Type Basic Scientific Scientific

Technical Elective

Social Elective

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


● 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 Prof. Dr. Fırat ATEŞ




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


Study Homework


ECTS

42 0 0 0 0 198 240 3 6



Technical Elective

Social Elective

Course Objectives To teach the module theory with a comprehensive manner.


● 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.


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 (%)






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 Prof.Dr.Fırat ATEŞ




Course Title: Real Analysis I

Code : FMT5107





ECTS

42 0 0 0 0 198 240 3 6


Course Type


Technical Elective

Social Elective

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


• 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.


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





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




ECTS

42 0 0 0 0 198 240 3 6


Course Type


Technical Elective

Social Elective

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


• 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.


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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Final Exam x 100 Other

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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: N.E.C. Groups

Code : FMT5111





ECTS

42 0 0 0 0 198 240 3 6


Course Type


Technical Elective

Social Elective

Course Objectives

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


• 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.


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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Week Subjects 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





ECTS

42 0 0 0 0 198 240 3 6


Course Type


Technical Elective

Social Elective

Course Objectives

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


• 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.


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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Week Subjects 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





ECTS

42 0 0 0 0 198 240 3 6


Course Type


Technical Elective

Social Elective

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


• 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.


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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Week Subjects

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





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.


• 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.


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

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Homeworks X % 40 Term Paper Term Paper (Projects,reports, ….) Oral Examination


Final Exam X % 60 Other

Other (Class Performance)

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 Institute: Instute of Science

Field: Mathematics Education and Teaching Methods Credits

Lecture Application Laboratuary

Project/Field Study

Homework


ECTS

42 0 0 0 0 198 240 3 6



Technical Elective

Social Elective

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


● 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.


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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Other 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 13 General applications 14 General applications

Instructors Prof.Dr.Fırat ATEŞ




Course Title: Riemannian Geometry I Code : FMT 5119 Institute: Instute of Science

Field: Mathematics

Education and Teaching Methods Credits 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


Technical Elective

Social Elective

Course Objectives

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


• 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.


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

Instructors Prof. Dr. Cihan ÖZGÜR




Course Title: Geometry of Submanifolds I Code : FMT 5120 Institute: Instute of Science

Field: Mathematics



T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type


Technical Elective

Social Elective

Course Objectives

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


• 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.


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




ECTS

42 0 0 0 0 198 240 3 6


Course Type


Technical Elective

Social Elective

Course Objectives To teach the concept of Mathematical control theory.


• 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.


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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Week Subjects 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 Institute: Instute of Science


Lecture Application Laboratuary Project/ Field Study

Homework


ECTS

42 0 0 0 0 198 240 3 6


Course Type


Technical Elective

Social Elective

Course Objectives To teach basic structure of Orlicz spaces.


• 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.


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 Institute: Instute of Science

Field: Mathematics

Education and Teaching Methods Credits Lecture Application Laboratory. Project/Field


T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type


Technical Elective

Social Elective

Course Objectives

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


• 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 φ-sectional curvature and Sasakian space form.


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

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Week Subjects

1 Symplectic manifolds 2 Principal S1-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 φ-sectional curvature, Sasakian space form 14 Examples of Sasakian space forms, locally φ-symmetric spaces





Course Title: Structures on Manifolds I Code : FMT5129 Institute: Instute of Science

Field: Mathematics



T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type


Technical 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.


• 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

Education and Teaching Methods Credits Lecture Application Lab. Project/

Field Study Homework Other Total Credit

T+A+L=Credit ECTS

42 0 0 0 100 98 240 3 6


Course Type


Technical 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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(%) Midterm Exams Midterm Exams


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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




ECTS

42 0 0 0 0 198 240 3 6


Course Type


Technical Elective

Social Elective

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


• 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.


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





ECTS

42 0 0 0 0 198 240 3 6


Course Type


Technical Elective

Social Elective

Course Objectives

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


• 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.


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). ASSESSMENT CRITERIA


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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




GRADUATE COURSE DETAILS Course Title: Function Spaces I

Code : FMT5133





ECTS

42 0 0 0 0 198 240 3 6


Course Type


Technical Elective

Social Elective

Course Objectives To teach several function spaces and relations among them.


• 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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Week Subjects

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 Prof.Dr. Ramazan AKGÜN




Course Title: Inversion Theory and Conformal Mappings

Code : FMT5134





ECTS

42 0 0 0 0 198 240 3 6


Course Type


Social Elective

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


• 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.


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




T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type


Technical Elective

Social Elective

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


• 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.


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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Homework 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

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



T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type


Technical Elective

Social Elective

Course Objectives

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


• 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.


Boothby, William M. An introduction to differentiable manifolds and Riemannian geometry. Second edition. Academic Press, Inc., Orlando, FL, 1986.

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Week Subjects

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 Institute: Instute of Science

Field: Mathematics



T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type


Technical Elective

Social Elective

Course Objectives

To teach the fundamental knowledge about tensors.


• 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.


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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Week Subjects 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 : FMT5140





ECTS

42 0 0 0 0 198 240 3 6


Course Type


Social Elective

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


• 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.


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 : FMT5141





ECTS

42 0 0 0 0 198 240 3 6


Course Type


Social Elective

Course Objectives To teach the averaged moduli and their applications.


• 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.


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 : FMT5142





ECTS

42 0 0 0 0 198 240 3 6


Course Type


Social Elective

Course Objectives To teach the fundemantal properties of strong approximation.


• 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


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 : FMT5143





ECTS

42 0 0 0 0 198 240 3 6


Course Type


Social Elective

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


• 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.


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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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




T+A+L=Credit ECTS

42 0 0 0 100 98 240 3 6


Course Type


Technical Elective

Social Elective

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


• 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.


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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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


Technical Elective

Social Elective

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


• 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.


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,

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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 Assoc. Prof. Dr. Burcin OKTAY YÖNET e-mail [email protected]



Course Title: Banach Spaces of Analytic Functions I

Code : FMT5146




Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type


Technical Elective

Social Elective

Course Objectives To introduce fundamental properties of Hp and hp Spaces.


• 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.


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 : FMT5147




Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type


Technical Elective

Social Elective

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


• 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.


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 : FMT5148




Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type


Technical Elective

Social Elective

Course Objectives To introduce Fundamental properties of Trigonometric Fourier series.


• 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.


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



T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6

Semester Spring Language Turkish

Course Type


Technical Elective

Social Elective

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


• 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.


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



T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type


Technical Elective

Social Elective

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


• 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.


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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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




T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type


Technical Elective

Social Elective

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


• 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.


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

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Week Subjects 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





ECTS

42 0 0 0 0 198 240 3 6


Course Type


Technical Elective

Social Elective

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


• 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.


1. Şaziye Yüksel, Genel Topoloji, 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.

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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





ECTS

42 0 0 0 0 198 240 3 6


Course Type


Technical Elective

Social Elective

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


• 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.


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). 4. John L.Kelley, General Topology, Springer-Verlag 1955. 5. K.Kuratowski, Topology, Academic Press 1966.

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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





ECTS

42 0 0 0 0 198 240 3 6


Course Type


Technical Elective

Social Elective

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


• 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.


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 Prof. Dr. Sebahattin İkikardes





Course Title: Geometric Theory of Functions I

Code : FMT5155




Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type


Technical Elective

Social Elective

Course Objectives

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


• 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.


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

Instructors Assoc. Prof. Dr. Burcin OKTAY YÖNET




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




ECTS

42 0 0 0 0 198 240 3 6


Course Type


Technical Elective

Social Elective

Course Objectives

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


• 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.


1) Bazaraa M.S., Sherali H.D. and Shetty S.M., Nonlinear programming: Theory and Applications, 3rd edition, John Wiley & Sons, Inc., 2006.

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 : FMT5157





ECTS

42 0 0 0 0 198 240 3 6


Course Type


Social Elective

Course Objectives

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


• 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.


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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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




GRADUATE COURSE DETAILS Course Title: Introduction to Scientific Computation I Code : FMT5161 Institute: Institute of Science



Homework-


ECTS

42 0 0 0 0 198 240 3 6


Course Type


Technical Elective

Social Elective

Course Objectives

To solve the problems that is encountered in the different branches of science by using suitable programs.


• To be able to use MATLAB • To be able to define matrix and compute matrix operations in computer environment • To be able to construct the algorithm of a given problem • To be able to plot two or three dimensional graphics • To be able to define MATLAB functions-functions


6. U. Arifoğlu, MATLAB, Simulink ve Mühendislik Uygulamaları, Alfa Yayınları, Ağustos 2005. 7. E. P. Enander, A. Sjöberg, The Matlab Handbook, Addision-Wesley, 1999. 8. İ. Yüksel, Matlab ile Mühendislik Sistemlerinin Analizi ve Çözümü, Nobel Yayıncılık, 2004. 9. B. Çelik, Mapla ve Maple ile Matematik, Dora Yayıncılık, 2010.

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Final Exam X % 70 Other Other (Class Performance) X % 30

Week Subjects 1 Introduction to Matlab program 2 Variable assignment, simple mathematical functions 3 File procedure in Matlab 4 Numbers, coordinates systems and graphics 5 Definition and operations of matrix 6 Mathematical functions 7 Logical functions 8 Programming in Matlab 9 Matlab function-functions 10 Introduction to Maple 11 Symbolic programming in Matlab and Maple 12 Differential and solutions of differential equations 13 Nonlinear equations systems and their solutions 14 Linear equations system and eigenvalues

Instructors Assist. Prof. Dr. Beyza Billur İSKENDER EROĞLU




Course Title: Differential Equations Systems Code : FMT5162





ECTS

42 0 0 0 0 198 240 3 6


Course Type


Technical Elective

Social Elective

Course Objectives

To teach existence and uniqueness theorems for differential equations systems, methods of solutions and to give stability analysis.


• To be able to identify existence and uniqueness of a differential equations system. • To be able to reduce a differential equations system to a normal form. • To be able to find the solution of a constant coefficient linear differential • To be able to draw phase curves, to draw equilibrium points a differential equations system and to

analysis stability of the system. • To be able to state Bendixson, Poincare-Bendixson theorems.


10. S. L. Ross, Differential Equations, John Wiley and Sons, New York, 1974. 11. M. Çağlıyan, N. Çelik, S. Doğan, Adi Diferansiyel Denklemler, Dora Yayıncılık, Bursa, 2010. 12. E. Hasanov, G. Uzgören, A. Büyükaksoy, Diferansiyel Denklemler Teorisi, Papatya Yayıncılık,

İstanbul, 2002.

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Week Subjects 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Existence and uniqueness theorems for differential equations Existence and uniqueness theorems for differential equations Continuity of solutions Existence and uniqueness theorems for differential equations systems Differential equations systems and normal forms Homogeneous normal linear systems with constant coefficient and eigenvalue and eigenvector methods Nonhomogeneous normal linear systems with constant coefficient and solution methods Exponential matrix, elimination method, Laplace transformation method Phase plane, phase curve and phase portraits Equilibrium points, types of equilibrium points and vector fields Stable solutions in the sense of Lyapunov Phase curves for linear systems Phase curves for nonlinear systems and equilibrium points Limiting loops, Bendixson, Poincare-Bendixson theorems




GRADUATE COURSE DETAILS Course Title : Faber series I

Code : FMT5163 Institute: Instute of Science Field : Mathematics



T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6

Semester Autumn Language Turkish

Course Type Basic Scientific Scientific Technical

Elective Social Elective

Course Objectives To investigations of the main properties of Faber polynomials and Faber series in the complex plane.


· To be able to define the Faber polynomials as the generalization of the power functions · To be able to explane the relation betweeen the Faber polynomials and Riemann conformal mapping · To be able to find the Faber polynomials of some simple domains · To be able to construct the Faber series represenation · To be able to state the simple properties of the Faber polynomials

Textbooks and/or References

1. V. I. Smirnov, N. A. Lebedev. Functions of a complex variable, Massachusetts Institute of Technology, 1968. 2. A. I. Markushevich. Theory of Analytic functions, Nauka, 1968. 3. D. Gaier. Lecturers on Approximation theory, Mir, 1986. 4. P. K. Suetin. Faber Series, Gordon and Breach Science Publishers, Australia… 1998.

ASSESSMENT KRITERIA


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Homework Term Paper Term Paper, Project, Reports, etc. Oral Examination



Other

Week Subjects 1 Best approximations of the real functions 2 Lebesgue constants for Fourier and Taylor series 3 Jeckson sums for approximation 4 Analogs of Weierstrass theorem on approximation 5 Approximate aggregates in the simple connected domains 6 Examples for Faber polynomials 7 The algebraic and assymptotical estimates of Faber polynomials 8 The algebraic and assymptotical estimates of the generalized Faber polynomials 9 The Faber series in the simple connected domains 10 The Faber series of the analytic functions on the continuums 11 The Bernstein-Walsh theorems 12 The Bernstein-Walsh theorems 13 Faber-Laurent partial sums 14 The approximate properties of the Faber-Laurent partial sums

Instructor Prof. Dr. Daniyal M. Israfilzade

E-mail [email protected] Website http://matematik.balikesir.edu.tr/


Course Title: Inequalities and Extremal Problems in Approximation Theory

Code : FMT5164




Credit ECTS

42 0 0 0 0 198 240 3 6

Semester Autumn Language Turkish/English

Course Type


Technical Elective

Social Elective

Course Objectives To teach some inequalities used in and extremal problems approximation theory.


• To be able to express some inequalities related to uniform norm for trigonometric and algebraic polynomials.

• To be able to express the inequalities in which uniform norm and Lp norm are estimated. • To be able to use inequalities in the problems in approximation theory. • To be able to understand the properties of extremal polynomials. • To be able to use the extremal polynomials in the problems in approximation theory.


1) G.V. Milovanovic, D.S. Mitrinovic, Th. M. Rassias, Topics in Polynomials : Extremal Problems, Inequalities, Zeros, 1994.

2) Ronald A. Devore, George G. Lorentz, Constructive Approximation, 1993. ASSESSMENT CRITERIA


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Week Subjects 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Inequalities in uniform norm for trigonometric polynomials: Bernstein inequality Inequalities in uniform norm for algebraic polynomials: Markov inequality Inequalities in Lp spaces Inequalities in which different norms are estimated Estimation of the Lp norm with the uniform norm Inequalities of Nikols’skii type Applications of inequalities in approximation theory, Polynomials with minimal uniform norm Polynomials with minimal Lp norm Estimates for coefficients of polynomials On the maximum modulus of polynomials Zeros of polynomials Extremal problems on a circle and on some complex domains Bieberbach polynomials and their applications in approximation theory.

Instructors Assoc. Prof. Burcin OKTAY YÖNET




Course Title: Analytic Theory of Polynomials I

Code : FMT5165





ECTS

42 0 0 0 0 198 240 3 6


Course Type


Social Elective

Course Objectives To introduce the basic knowledge about analytic theory of polynomials.


• To be able to express the Gauss-Lucas theorem and to do applications of it, • To be able to express the extensions of the Gauss-Lucas theorem and to do applications of it, • To be able to express the basic concepts about the real polynomials, • To be able to determine bounds for the zeros of polynomials.


1) Q. I. Rahman and G. Schmeisser, Analytic Theory of Polynomials, Oxford University Press, Oxford, (2002). 2) M. Marden, Geometry of Polynomials, Second edition. Mathematical Surveys, No. 3 American Mathematical

Society, Providence, R.I. (1966). 3) M. Dehmer and A. Moshowitz, Bounds on the moduli of polynomial zeros.

Appl. Math. Comput. 218 (2011), no. 8, 4128-4137.

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Week Subjects 1 The fundamental theorem of algebra, basic concepts 2 The Gauss-Lucas theorem 3 Extensions of the Gauss-Lucas theorem 4 Real polynomials and Jensen’s theorem 5 Applications 6 Circular domains and polar derivative 7 Grace’s theorem and equivalent forms 8 Products and quotients of polynomials 9 Applications

10 The moduli of the zeros 11 The Cauchy bound 12 Bounds for Complex Polynomials 13 Bounds for Special Lacunary Polynomials 14 Applications




http://www.ams.org/mathscinet/search/series.html?cn=Second_edition_Mathematical_Surveys_No_3

http://www.ams.org/mathscinet/search/journaldoc.html?cn=Appl_Math_Comput

http://www.ams.org/mathscinet/search/publications.html?pg1=ISSI&s1=297933




Course Title: Advanced Linear Algebra I

Code : FMT5166

Institute: Instute of Science Field : Mathematics



T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic Scientific Scientific Technical


Course Objectives

This course is intended to introduce concepts of vector spaces and linear maps as a graduate level. These concepts are fundamental to many branches of mathematics, physics and engineering. After a review of basic concepts of vector spaces and linear maps, we will make an in-depth study on relation between linear maps and matrices. Then, to understand linear maps, we will state the theory of polynomials with real and complex coefficients. We will define the determinant function and will give the relation with the alternating linear functions.


Students should be able: to understand the concept of vector space and its properties. to establish the relationship between matrices and linear maps. to understand the concept of dual space. to define the algebra of polynomials over a field. to understand the determinant as an alternating n-linear function of the rows of a square matrix.


1. K. Hoffmann, R. Kunze, Linear Algebra, Prentice-Hall,1971. 2. S. Axler, Linear Algebra Done Right, Springer ,1991. 3.S. Roman, Advanced Linear Algebra, Springer, 2000.

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Week Subjects 1 2 3 4 5 6 7 8 9

10 11 12 13 14

Vector Spaces, Subspaces Bases and Dimension Linear Maps Nullity of Linear Maps The Matrix of a Linear Map Rank and Nullity Theorem Elementary Matrices, Invertibility Linear Functionals Dual Spaces The Algebra of Polynomials Polynomials of Real and Complex Coefficients Determinant Functions Permutations and the Uniqueness of Determinants Properties of Determinants

Instructor/s Asst. Prof. Dr. Pınar Mete e-mail [email protected] Website http://matematik.balikesir.edu.tr


Course Title: Advanced Differential Equations I

Code : FMT5167




T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type

Basic Scientific Scientific Technical


Course Objectives

To construct theoretical concepts of differential equations. To give applications of initial value, boundary value problems. To teach differential equations with power series and special functions. To give qualitative theory of differential equations.


1. To be able to identify existence and uniqueness of initial value problems 2. To be examine dependency of the solutions to the initial conditions and parameters. 3. To be able to analyze stability of differential equations. 4. To be able to find the solutions of differential equations with power series, special functions.


1. R.P. Agarwal, D. O’Regan, An Introduction to Ordinary Differential Equations, Springer, 2008. 2. S.L. Ross, Differential Equations, John Wiley and Sons, New York, 1974. 3. M. Çağlıyan, N. Çelik, S. Doğan, Adi Diferansiyel Denklemler, Dora Yayıncılık, Bursa, 2010. 4. E. Hasanov, G. Uzgören, A. Büyükaksoy, Diferansiyel Denklemler Teorisi, Papatya Yayıncılık, İstanbul, 2002.


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(%) Midterm Exam Midterm Exams


Homework Term Paper Term Paper, Project Reports, etc. X 40 Oral Examination



Other

Week Subjects 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Initial value problems, their classifications and examples. Existence theorems of initial value problems Uniqueness theorems of initial value problems Continuation of solutions, continuous dependence of solutions on initial conditions and parameters. Sturm-Liouville boundary value problems Green function, Hilbert-Schmidt theorem Power series solutions Bessel equation Legendre equation Differential equations systems Phase plane, phase curves, phase flows Equilibrium points and vector fields Stability of solutions Limit cycles, Bendixson-Poincare-Bendixson theorems

Instructor/s Assist. Prof. Dr. Beyza Billur İSKENDER EROĞLU e-mail [email protected] Website http://matematik.balikesir.edu.tr


Course Title: Optimal Control Theory

Code : FMT5168

Institute: Institute of Science Field : Mathematics



T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6

Semester Fall Language Turkish/English Course Type


Technical Elective

Social Elective

Course Objectives

To teach the types of optimal control problems and solution methods; to gain the ability to interpret the solutions and to introduce new problems and solve them.


• To know basic concepts of optimal control theory. • To be able to know the types of optimal control problems, analytical and numerical solution methods. • To be able to comprehend the calculus of variations and to apply to the optimal control problems. • To be able to comprehend the Lagrange multiplier method and Hamiltonian principle by comparatively. • To be able to categorize the works of optimal control theory in the literature and to introduce new problems.


1) D.E. Kirk, Optimal Control Theory, Prentice Hall, 1970. 2) D.S. Naidu, Optimal Control Systems, CRC Press, 2003. 3) D. G. Hull, Optimal Control Theory for Applications, Springer, 2003. 4) S. Anita, V. Arnautu, V. Capasso, An Introduction to Optimal Control Problems in Life

Sciences and Economics, Springer, 2011. ASSESSMENT CRITERIA

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Midterm Exam Midterm Exams





Other

Week Subjects 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Some basic concepts of linear algebra and control theory Introduction to optimal control theory and fundamental definitions Problem classification and main problems Calculus of variations and optimal control applications Euler-Lagrange Equations Lagrange multiplier technique Hamiltonian principle Pontragin’s minimum principle: Minimum time and control effort problems Linear quadratic optimal control systems I: The matrix Riccati equation Linear quadratic optimal control systems I: Sample problem solutions Discrete-time optimal control problems Numerical methods for optimal control problems Literature research I: Recent studies on optimal control of mechanical systems Literature research II: Recent studies on optimal control of physical and biological systems

Instructor/s Assist. Prof. Dr. Derya AVCI e-mail [email protected] Website http://matematik.balikesir.edu.tr


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

GRADUATE COURSE DETAILS Course Title: Research Methods for Science and Ethics

Code : FMT5169



FieldStudy Homework


ECTS

42 0 0 0 0 198 240 3 6

Semester Spring Language Turkish/English

Course Type


Social Elective

Course Objectives

To teach how make use of scientific research methods in the scope of natural sciences and the concept of ethics in scientific research.


• Identifying the fundamental characteristics of scientific information and research • Identifying the fundamental characteristics of different scientific research methods • Designing a scientific research by applying experiment method • Writing and presenting a scientific research • Comprehending the concept of scientific ethic and its importance


1) Michael P. Marder, (2011), Research Methods for Science. Cambridge University Press. 2) Ahmet Aypay, Necati Cemaloğlu, Ruhi Sarpkaya, (2014), Bilimsel Araştırma Yöntemleri, Anı Press

ASSESSMENT CRITERIA

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MidtermExams X %20 MidtermExams

Quizzes MidtermControls

Homeworks TermPaper

TermPaper (Projects,reports, ….) X %20 Oral Examination

LaboratoryWork Final Exam

Final Exam X %60 Other


Week Subjects

1 Science, Scientific Information, Scientific Research, Goals of Scientific Research 2 Reliability and Validity in Scientific Research, Scientific Research Design Process 3 Sampling and Sampling Methods 4 Qualitative Research Methods 5 Quantitative Research Methods, Hypothesis Testing 6 Experiments, Fundamental Experimental Designs 7 Statistical Analyses and Tests Used in Fundamental Experimental Designs 8 Midterm exam 9 Mathematical models (Linear Regression, Matching functions to data)

10 Mathematical models (Fourier Transforms) 11 Preparing the scientific research proposal, searching the litarature, reading the articles 12 Writing Scientific Papers and Presenting Them 13 The concept of Ethics in Scientific Research 14 The ethical problems in research

Instructors Assoc. Prof. Dr. Burçin OKTAY YÖNET




Course Title: Functional Analysis II

Code : FMT5202





ECTS

42 0 0 0 0 198 240 3 6


Course Type


Technical Elective

Social Elective

Course Objectives To teach some advanced topics of functional analysis.


• 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.


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


ECTS

42 0 0 0 0 198 240 3 6



Technical Elective

Social Elective

Course Objectives To teach fundamental concepts of the module theory .


● 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.


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 Prof.Dr.Fırat ATEŞ




Course Title: Fuchsian Groups

Code : FMT5206





ECTS

42 0 0 0 0 198 240 3 6


Course Type


Social Elective

Course Objectives To teach Fuchsian groups and their elementary algebraic properties.


• 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.


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: Hyperbolic Geometry

Code : FMT5210





ECTS

42 0 0 0 0 198 240 3 6


Course Type


Technical Elective

Social Elective

Course Objectives

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


• 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.


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




ECTS

42 0 0 0 0 198 240 3 6


Course Type


Technical Elective

Social Elective

Course Objectives To teach the fundamental concepts of dynamic system theory.


• 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.


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 : FMT5213





ECTS

42 0 0 0 0 198 240 3 6


Course Type


Technical Elective

Social Elective

Course Objectives To teach fundamental theorems of Real analysis.


• 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,


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




GRADUATE COURSE DETAILS Course Title: Discrete Groups

Code : FMT5215

Institute: Instute of Science Field: Mathematics.



Other

Total Credit T+A+L=Credit

ECTS

42 0 0 0 0 198 240 3 6


Course Type


Social Elective

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


• 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.


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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Final Exam X % 60 Other Other (Class Performance)

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




ECTS

42 0 0 0 0 198 240 3 6


Course Type


Technical Elective

Social Elective

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


• 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.


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 Institute: Instute of Science

Field: Mathematics



T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type


Technical Elective

Social Elective

Course Objectives

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


• 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.


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 Institute: Instute of Science

Field: Mathematics



T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type


Technical 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.


• 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.


B. Y. Chen , Geometry of Submanifolds, Pure and applied mathematics (Marcel Dekker, Inc.), New York, 1973

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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




ECTS

42 0 0 0 0 198 240 3 6


Course Type


Technical Elective

Social Elective

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


• 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.


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 Institute: Instute of Science



Homework


ECTS

42 0 0 0 0 198 240 3 6


Course Type


Technical Elective

Social Elective

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


• 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.


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


ECTS

42 0 0 0 0 198 240 3 6



Technical Elective

Social Elective

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


● 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


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 Prof. Dr. Fırat ATEŞ




Course Title: Contact Manifolds II Code : FMT 5227 Institute: Instute of Science

Field: Mathematics



T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type


Technical 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.


• 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.


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(ξ), 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 Institute: Instute of Science

Field: Mathematics



T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type


Technical 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.


• 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.


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




T+A+L=Credit ECTS

42 0 0 0 100 98 240 3 6


Course Type


Technical Elective

Social Elective

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


• 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.


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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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




ECTS

42 0 0 0 0 198 240 3 6


Course Type


Technical Elective

Social Elective

Course Objectives

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


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

• To be able to make comparison between fractional PI Dλ µ 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.


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 Dλ µ 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





ECTS

42 0 0 0 0 198 240 3 6


Course Type


Technical Elective

Social Elective

Course Objectives

To teach the concepts of quadratic and cubic residue.


• 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].


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





ECTS

42 0 0 0 0 198 240 3 6


Course Type


Social Elective

Course Objectives To teach the structure of Bergman spaces.


• 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.


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

Code : FMT 5235 Institute: Instute of Science Field: Mathematics



T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type


Technical 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.


• 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.


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 Institute: Instute of Science

Field: Mathematics



T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type


Technical Elective

Social Elective

Course Objectives To teach fundamental knowledge about tensors.


• 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.


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 : FMT5237





ECTS

42 0 0 0 0 198 240 3 6


Course Type


Social Elective

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


• 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.


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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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 Applications 7 Rotations of the sphere I 8 Rotations of the sphere II 9 Applications 10 Finite groups of Möbius transformations I 11 Finite groups of Möbius transformations II 12 Applications 13 The most general transformations of plane regions which transform circles into circles 14 Applications





Course Title: Averaged moduli and one sided approximation II

Code : FMT5238





ECTS

42 0 0 0 0 198 240 3 6


Course Type


Social Elective

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


• 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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Week Subjects

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





ECTS

42 0 0 0 0 198 240 3 6


Course Type


Social Elective

Course Objectives To teach the strong approximation and the embedding theorems.


• 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


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 : FMT5240





ECTS

42 0 0 0 0 198 240 3 6


Course Type


Social Elective

Course Objectives

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


• 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.


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




T+A+L=Credit ECTS

42 0 0 0 100 98 240 3 6


Course Type


Technical Elective

Social Elective

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


• 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.


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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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





ECTS

42 0 0 0 0 198 240 3 6


Course Type


Social Elective

Course Objectives To teach several function spaces and relations among them.


• 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/%7Epharjule/varsob/pdf/book.pdf

http://www.helsinki.fi/%7Epharjule/varsob/pdf/book.pdf


Course Title: Potential Theory

Code : FMT5244




Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type


Technical Elective

Social Elective

Course Objectives To teach the concepts and tecniques in potential theory.


• 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.


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 : FMT5245




Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type


Technical Elective

Social Elective

Course Objectives To teach fundamental properties of Smirnov and Bergman spaces.


• 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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Week Subjects

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 : FMT5246




Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type


Technical Elective

Social Elective

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


• 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.


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 : FMT5247




Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type


Technical Elective

Social Elective

Course Objectives To teach the fundamental theorems of trigonometric approximation theory.


• 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.


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) ASSESSMENT CRITERIA


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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



T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type


Technical Elective

Social Elective

Course Objectives To teach the concepts of feedback linearization of nonlinear systems, Lyapunov stablity.


• 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.


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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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



T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type


Technical Elective

Social Elective

Course Objectives To teach numerical solution methods for ordinary differential equartions.


• 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.


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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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 Institute: Instute of Science

Field: Mathematics



T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type


Technical Elective

Social Elective

Course Objectives To teach Numerical Methods for Solutions of Partial Differential Equations.


• 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.


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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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




T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type


Technical Elective

Social Elective

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


• 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.


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

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Week Subjects

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





ECTS

42 0 0 0 0 198 240 3 6


Course Type


Technical Elective

Social Elective

Course Objectives To teach the concepts of general topology in advanced level.


• 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.


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

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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 Instructors Assoc. Prof. Dr. Ahu Açıkgöz




GRADUATE COURSE DETAILS Course Title: Introduction to Fuzzy Topology II

Code : FMT5253





ECTS

42 0 0 0 0 198 240 3 6


Course Type


Technical Elective

Social Elective

Course Objectives

To teach the corresponding concepts of general topology in fuzzy topological spaces.


• 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.


1. Şaziye Yüksel, Genel Topoloji, 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.

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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





GRADUATE COURSE DETAILS Course Title: Introduction to Ideal Topological Spaces II

Code : FMT5254





ECTS

42 0 0 0 0 198 240 3 6


Course Type


Technical Elective

Social Elective

Course Objectives To teach the concept of delta-I-continuous function and to compare with the other types of functions.


• 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.


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). 4. John L.Kelley, General Topology, Springer-Verlag 1955. 5. K.Kuratowski, Topology, Academic Press 1966.

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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 : FMT5255




Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type


Technical Elective

Social Elective

Course Objectives To teach the approximation properties of orthogonal polynomials in the complex plane.


• 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.


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.

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Final Exam X 100 Other 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 : FMT5256




Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type


Technical Elective

Social Elective

Course Objectives To introduce the problems of convergence in the geometric theory of functions.


• 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.


G. M. Goluzin, Geometric Theory of Functions of a complex variable, 1969.

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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





ECTS

42 0 0 0 0 198 240 3 6


Course Type


Technical Elective

Social Elective

Course Objectives

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


• 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.


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 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 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




ECTS

42 0 0 0 0 198 240 3 6


Course Type


Technical Elective

Social Elective

Course Objectives

To teach optimality conditions of unconstrained and constrained nonlinear optimization problems with fundamental solution methods.


• 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.


1) Bazaraa M.S., Sherali H.D. and Shetty S.M., Nonlinear programming: Theory and Applications, 3rd edition, John Wiley & Sons, Inc., 2006.

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.

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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 and 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




T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type


Technical Elective

Social Elective

Course Objectives To teach fundamental concepts of Riemannian Geometry and the concept of submanifold of finite type.


• 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.


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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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 : FMT5260





ECTS

42 0 0 0 0 198 240 3 6


Course Type


Social Elective

Course Objectives

To teach the basic knowledge about r-bonacci polynomials and generalized complex Fibonacci and Lucas functions.


• 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.


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.

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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 9 Complex factorizations of the Lucas numbers 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




GRADUATE COURSE DETAILS Course Title: Introduction to Scientific Computation II Code : FMT5262 Institute: Institute of Science



Homework-


ECTS

42 0 0 0 0 198 240 3 6


Course Type


Technical Elective

Social Elective

Course Objectives

To evaluate numerical computation and analyze control systems by MATLAB.


• To be able programming numerical computation in MATLAB. • To be able o write transfer function in MATLAB • To be able to analyze state-space. • To be able to design a system by using Simulink • To be able to simulate simple control design.


13. U. Arifoğlu, MATLAB, Simulink ve Mühendislik Uygulamaları, Alfa Yayınları, Ağustos 2005. 14. İ. Yüksel, Matlab ile Mühendislik Sistemlerinin Analizi ve Çözümü, Nobel Yayıncılık, 2004. 15. Z. Bingül, MATLAB ve SIMULINK’le Modelleme ve Kontrol I-II, Birsen Yayınevi, 2005. 16. C. T. Chen, Linear System Theory and Design, Oxford University Press, 1999.

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Final Exam X % 70 Other Other (Class Performance) X % 30

Week Subjects 1 Curve fitting, interpolation 2 Curve Fitting toolbox 3 Numerical integration and derivative 4 State-space analysis in continuous 5 Transfer function 6 Discrete time systems 7 Steady state response 8 Simple control design 9 Optimization toolbox 10 Simulink toolbox 11 Modeling by Simulink 12 Embedding function into a Simulink model 13 Simulink and control design 14 Applications





Course Title: Control of Nonlinear Systems Code : FMT5263





ECTS

42 0 0 0 0 198 240 3 6


Course Type


Technical Elective

Social Elective

Course Objectives To design suitable control for nonlinear systems.


• Tobe able to identify existence and uniqueness of nonlinear systems • To be able to interpret solution curves and equilibrium • To be able to analyze stability of nonlinear • To be able to realize the linearization of a nonlinear system. • To be able to know control design methods for nonlinear systems and apply these methods.


17. H. K. Khalil, Nonlinear Systems, Prentice Hall, 1996 18. A. Isidori, Nonlinear Control Systems, Springer, 1995 19. S. Sastry, Nonlinear System Analysis, Stability and Control, Springer, 1999 20. J.J.E. Slotine, Applied Nonlinear Control, Prentice Hall, 1991.

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Other X 20

Week Subjects 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Examples of nonlinear systems Second order systems and solution curves Existence and uniqueness theorems of nonlinear systems Dependence of initial conditions Lyapunov stability Input-output stability Periodical solutions and stability Feedback linearization Lyapunov based control design Backsteeping control Sliding mode control PID control design Comparison of control methods Application problems





Course Title : Faber series II

Code : FMT5264 Institute: Institute of Science Field : Mathematics



T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6

Semester spring Language Turkish

Course Type Basic Field Course

Field Course

Technic Elective Social

Elective

Course Objective To study the investigation and application of the Faber polynomials and Faber operators in the approximation theory and Univalent function theory


· To be able to investigate the convergeness conditions of the Faber series into a domain · To be able to investigate the uniqueness of the Faber series representations · To be able to definite the Faber operats · To be able to apply the Faber operators in the approximation theory · To be able to apply the Faber polynomials in the univalent function theory

Textbooks and/or References

1. V. I. Smirnov, N. A. Lebedev. Functions of a complex variable, Massachusetts Institute of Technology, 1968. 2. A. I. Markushevich. Theory of Analytic functions, Nauka, 1968. 3. D. Gaier. Lecturers on Approximation theory, Mir, 1986. 4. P. K. Suetin. Faber Series, Gordon and Breach Science Publishers, Australia… 1998

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Midterm Examination Midterm Examination

Quiz Midterm controls

Homework Midterm delivering

Project, Report Oral examination

Laboratory Final Examination

Final Examination x 100 Other

Other

Week Subjects 1 The convergeness conditions for hhe Faber series into the domain 2 The unicueness of the Faber series presentation 3 The Faber operators and their’s properties 4 The boundary properties of the Faber series 5 The estimation of the Faber operators norm 6 The inverse Faber operator 7 Applications of the Faber operators in the approximation theory 8 The applications of the Faber operators in the Univalent function theory 9 The method of areas

10 The investigations on the estimation of the Faber polinomials on the closed domains 11 The estimation of the Faber polynomials on the quasidisc 12 The estimation of the Faber polynomials on the closed domains with smooth boundary 13 The estimation of the Faber polynomials on the continua 14 The estimation of the Faber polynomials in the Bergman spaces

Instructor Prof. Dr. Daniyal M. Israfilzade

E-mail [email protected]

Web site http://matematik.balikesir.edu.tr/


Course Title: Analytic theory of Polynomials II

Code : FMT5265





ECTS

42 0 0 0 0 198 240 3 6


Course Type


Social Elective

Course Objectives

To introduce the basic knowledge about the bounds for the zeros of complex and real valued polynomials.


• To be able to determine bounds for the zeros of complex polynomials with special conditions for the coefficients, • To be able to determine the annular regions for the zeros of polynomials, • To be able to find roots of Fibonacci polynomials, • To be able to determine the regions which are contain the zeros of generalized Fibonacci polynomials.


4) Q. I. Rahman and G. Schmeisser, Analytic Theory of Polynomials, Oxford University Press, Oxford, (2002). 5) M. Dehmer, On the location of zeros of complex polynomials. JIPAM. J. Inequal. Pure Appl. Math. 7 (2006), no. 1, Article 26, 13

pp. 6) M. Dehmer and Y. R. Tsoy, The quality of zero bounds for complex polynomials. PLoS ONE 7(7): e39537.

doi:10.1371/journal.pone.0039537. 7) V. E. Hoggatt and M. Bicknell, Roots of Fibonacci polynomials. Fibonacci Quart. 11 (1973), no. 3, 271-274. 8) M. X. He, P. E. Ricci and D.Simon, Numerical results on the zeros of generalized Fibonacci polynomials.

Calcolo 34 (1997), no. 1-4, 25-40 (1998).

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Homeworks X % 40 Term Paper

Term Paper (Projects,reports, ….) Oral Examination




Week Subjects

1 Bounds for the zeros of complex polynomials with special conditions for the coefficients I 2 Bounds for the zeros of complex polynomials with special conditions for the coefficients II 3 Examples 4 Annular regions for the zeros of polynomials I 5 Annular regions for the zeros of polynomials II 6 Annular regions for the zeros of polynomials III 7 Annular regions for the zeros of polynomials IV 8 Examples 9 The quality of zero bounds for complex polynomials I

10 The quality of zero bounds for complex polynomials II 11 Examples 12 Roots of Fibonacci polynomials 13 Zeros of generalized Fibonacci polynomials 14 Examples




http://www.ams.org/mathscinet/search/author.html?mrauthid=786101

http://www.ams.org/mathscinet/search/journaldoc.html?cn=JIPAM_J_Inequal_Pure_Appl_Math






http://www.ams.org/mathscinet/search/journaldoc.html?cn=Fibonacci_Quart




http://www.ams.org/mathscinet/search/journaldoc.html?cn=Calcolo





Course Title: Advanced Linear Algebra II

Code : FMT5266




T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type


Technical Elective

Social Elective

Course Objectives

As a continuation of a graduate Advanced Linear Algebra I course, firstly, we will study the concept of eigenvectors which is the result of the idea of a linear operator by restricting it to subspaces. With the result of ``On complex vector spaces, eigenvalues always exist``, we will prove that each linear operator on a complex vector space has an upper-triangular matrix with respect to some basis. We will introduce the inner product spaces and their basic properties. We will state the Spectral Theorem, which characterizes the linear operators. Finally, we will give the definition of minimal polynomial, characteristic polynomial and generalized eigenvectors.


Students should be able: to understand what the invariant subspace is. to comprehend the idea of eigenvalues and eigenvectors. to define the inner product space. to state the Spectral Theorems. to establish the relation between linear operators and forms.


1. K. Hoffmann, R. Kunze, Linear Algebra, Prentice-Hall,1971. 2. S. Axler, Linear Algebra Done Right, Springer ,1991. 3 .S. Roman, Advanced Linear Algebra, Springer, 2000.

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Week Subjects

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Invariant Subspaces Eigenvalues and Eigenvectors Diagonalization of Matrices Inner Product Spaces Norms, Orthonormal Bases Linear Functionals and Adjoints Hermitian (Self-Adjoint) Operators Normal Operators Positive Operators Isometries The Spectral Theorem Generalized Eigenvectors Characteristic and Minimal Polynomial Canonical Forms

Instructor/s Asst. Prof.Dr. Pınar Mete



GRADUATE COURSE DETAILS Course Title: Advanced Differential Equations II

Code : FMT5267




T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6


Course Type


Technical Elective

Social Elective

Course Objectives To teach the solution methods of the basic partial differential equations by integral transforms.


• To be able to define special functions. • To be able to express basic properties of integral transforms. • To be able to do applications of integral transforms.


1. İ.B.Yaşar, İntegral Dönüşümleri ve Uygulamaları, Siyasal Kitabevi, 2003. 2. Y. Pala, Modern Uygulamalı Diferensiyel Denklemler, Nobel Yayın Dağıtım, 2006. 3. N.H. Asmar, Partial Differential Equations with Fourier Series and Boundary Value Problems, prentice

Hall, 2005. 4. A.D. Paulakiras, The Transforms and Applications Handbook, CRC Press LLC, 2000.


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Other

Week Subjects 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Special functions (Gamma,Beta, Hypergeometric functions) Orthoganal functions (Bessel, Legendre, Hermitian, Chebyshev, Laguerre) Separation of variables Fourier transforms Applications of Fourier transforms Laplace transforms Applications of Laplace transforms Henkel transforms Applications of Henkel transforms Mellin transoforms and its applications Laplace equation (separation of variables, integral transforms) Poisson equation (separation of variables, integral transforms) Heat equation (separation of variables, integral transforms) Wave equation (separation of variables, integral transforms)





Course Title: Fractional Optimal Control Theory

Code : FMT5268

Institute: Institute of Science Field : Mathematics



T+A+L=Credit ECTS

42 0 0 0 0 198 240 3 6

Semester Spring Language Turkish/English Course Type


Technical Elective

Social Elective

Course Objectives To teach the basic concepts and problems of fractional optimal control theory.


• To know the basic functions and definitions of fractional calculus. • To be able to associate the optimal and fractional optimal control problems. • To be able to determine the optimality conditions for different types of fractional optimal

control problems. • To know the basic numerical solution methods of fractional optimal control problems.


1) A.C.J. Luo, J.Q. Sun, Complex Systems-Fractionality, Time-delay and Synchronization, Springer, 2011. 2) D. Baleanu, K. Diethelm, E. Scalas, J.J. Trujillo, Fractional Calculus-Models and Numerical Methods,

World Scientific Publishing, 2012. 3) A.B. Malinowska, D.F.M. Torres, Introduction to The Fractional Calculus of Variations, World Scientific

Publishing, 2012. 4) V.V. Uchaikin, Fractional Derivatives for Physicists and Engineers-Volume I Background and Theory,

Springer, 2013. 5) T.M. Atanackovic, S. Pilipovic, B. Stankovic, D. Zorica, Fractional Calculus with Applications in

Mechanics-Wave Propagation, Impact and Variational Principles, John Wiley & Sons, Inc., 2014.


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Other

Week Subjects 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Fundamental functions of fractional calculus Basic definitions and transforms of fractional calculus and their applications The relation between optimal and fractional optimal control theories Classification of fractional optimal control problems Fractional calculus of variations via Riemann-Liouville operators Fractional calculus of variations via Caputo operators Foundation of optimality conditions, fractional Euler-Lagrange equations Calculations of fractional Euler-Lagrange equations for some selected problems Generalized Hamiltonian Principle Fractional Noether’s Theorem Basic numerical solution methods for fractional optimal control problems Comparison of numerical solutions: Advantageous and disadvantageous Analysis of literature related by fractional optimal control problems Applications of fractional optimal control problems on physical and biological systems




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