ECTS - Mathematical Analysis
Mathematical Analysis (MATH611) Course Detail
| Course Name | Course Code | Season | Lecture Hours | Application Hours | Lab Hours | Credit | ECTS |
|---|---|---|---|---|---|---|---|
| Mathematical Analysis | MATH611 | 1. Semester | 3 | 0 | 0 | 3 | 5 |
| Pre-requisite Course(s) |
|---|
| N/A |
| Course Language | English |
|---|---|
| Course Type | Compulsory Departmental Courses |
| Course Level | Ph.D. |
| Mode of Delivery | Face To Face |
| Learning and Teaching Strategies | Lecture, Discussion, Question and Answer, Problem Solving. |
| Course Lecturer(s) |
|
| Course Objectives | Mathematical analysis is a foundation for mathematical disciplines such as functional analysis, complex analysis, differential equations, numerical analysis, and the others. In addition, methods of mathematical analysis are used in probability and mathematical statistics, approximation theory, number theory, optimization, and many others. This course starts with the fundamentals and principles of mathematical analysis with a focus on the main notions and theorems. |
| Course Learning Outcomes |
The students who succeeded in this course;
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| Course Content | Sets and mappings, countable and uncountable sets, real number system, completeness; metric spaces, complete metric spaces; Banach fixed point theorem; sequences and series of functions, sigma algebras, measures, integral with respect to measure, convegence theorems (monotone and dominated), modes of convergence. |
Weekly Subjects and Releated Preparation Studies
| Week | Subjects | Preparation |
|---|---|---|
| 1 | Sets and mappings. Countable sets. Completeness of real numbers. Bolzano-Weierstrass theorem. | [2], 1.1-1.21, [1], 3.7 |
| 2 | Numerical sequences and series. Upper and lower limits. Cauchy sequences. Some special sequences. | [2], 3.11-3.10 |
| 3 | Continuity and uniform continuity. The uniform continuity theorem. | [2], 4.1-4.11, 4.18-4.20 |
| 4 | Sequences and series of functions. Uniform convergence, theorem on uniform convergence and continuity, differentiation, integration. The Weierstrass approximation theorem. | [2], 7.1 – 7.16, 7.26 |
| 5 | Metric spaces: limit of a sequence, closed and open sets, closure, continuous functions. Examples of classical metric spaces. Important inequalities (Hölder, Minkowski). | [1], Sec. 5 |
| 6 | Complete metric spaces. The completion of metric spaces. Theorems on complete metric spaces (the theorem on nested spheres, Bolzano-Weierstarss theorem). | [1], Sec. 7 |
| 7 | The contraction mapping principle (the Banach fixed point theorem). Linear spaces. Subspaces. Normed linear spaces, Banach spaces. Classical sequence and function spaces. | [1], Sec. 8.1-8.3, Sec.15 |
| 8 | Set functions, sigma-algebras of sets. Measurable functions, sequences of measurable functions, and their properties. | [3], Ch. 2 |
| 9 | Measures, examples of measures and measure spaces. | [3], Ch. 3 |
| 10 | Integral with respect to measure, Monotone Convergence Theorem. | [3], Ch. 4, 4.1-4.7 |
| 11 | Fatou’s Lemma and its applications. | [3], Ch. 4, 4 – 4.12 |
| 12 | Integrable functions. Dominated Convergence Theorem. | [3], Ch. 5 |
| 13 | The Lebesgue spaces Lp . The completeness theorem. | [3], Ch. 6 |
| 14 | Modes of convergence. | [3], Ch.7 |
| 15 | Relations between different types of convergence. | [3], Ch.7 |
| 16 | Review and Final Exam |
Sources
| Course Book | 1. A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Funtional Analysis, Dover, New York, 1999. |
|---|---|
| 2. W. Rudin, Principles of mathematical analysis, McGrawHill. | |
| 3. R. G. Bartle, The elements of integration and Lebesgue measure. Wiley, New York, 1995. |
Evaluation System
| Requirements | Number | Percentage of Grade |
|---|---|---|
| Attendance/Participation | - | - |
| Laboratory | - | - |
| Application | - | - |
| Field Work | - | - |
| Special Course Internship | - | - |
| Quizzes/Studio Critics | - | - |
| Homework Assignments | 2 | 30 |
| Presentation | - | - |
| Project | - | - |
| Report | - | - |
| Seminar | - | - |
| Midterms Exams/Midterms Jury | 1 | 30 |
| Final Exam/Final Jury | 1 | 40 |
| Toplam | 4 | 100 |
| Percentage of Semester Work | |
|---|---|
| Percentage of Final Work | 100 |
| Total | 100 |
Course Category
| Core Courses | X |
|---|---|
| Major Area Courses | |
| Supportive Courses | |
| Media and Managment Skills Courses | |
| Transferable Skill Courses |
The Relation Between Course Learning Competencies and Program Qualifications
| # | Program Qualifications / Competencies | Level of Contribution | ||||
|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | ||
| 1 | Is independently able to build a problem in the area of study, solve the problem by developing solution techniques and assess the solutions. | X | ||||
| 2 | Is capable of creating a groundwork in the fundamental branches of mathematics as well as in his/her research area | X | ||||
| 3 | follows the latest national and international literature in Mathematics and in his/her area of research; and uses them in his/her related studies | X | ||||
| 4 | observes and adopts the scientific ethical values in his/her professional and social life | X | ||||
| 5 | presents in Turkish and English in academic/scientific events the results of his/her research or the latest studies and findings on a special topic and participates in discussions | X | ||||
| 6 | Develops skills to work independently or as a member of a team | X | ||||
| 7 | Develops competences in the areas of creative and critical thinking, problem solving and producing original studies. Follows recent scientific studies, is capable of making an analysis, synthesis and assessment of the knowledge acquired | X | ||||
| 8 | Is open to lifelong improvement of his/her acquired knowledge, skills and competences. | X | ||||
| 9 | Is able to apply the acquired knowledge and problem-solving skills to interdisciplinary studies, proposes different solution methods to problems in terms of mathematical models and from a mathematical point of view | X | ||||
| 10 | Uses the mathematical based softwares, informatics and communication technologies for scientific purposes | X | ||||
ECTS/Workload Table
| Activities | Number | Duration (Hours) | Total Workload |
|---|---|---|---|
| Course Hours (Including Exam Week: 16 x Total Hours) | 16 | 3 | 48 |
| Laboratory | |||
| Application | |||
| Special Course Internship | |||
| Field Work | |||
| Study Hours Out of Class | 14 | 3 | 42 |
| Presentation/Seminar Prepration | |||
| Project | |||
| Report | |||
| Homework Assignments | 2 | 5 | 10 |
| Quizzes/Studio Critics | |||
| Prepration of Midterm Exams/Midterm Jury | 1 | 10 | 10 |
| Prepration of Final Exams/Final Jury | 1 | 15 | 15 |
| Total Workload | 125 | ||