M.SC. in Computer Engineering (Without Thesis)

MATH 668 | Course Introduction and Application Information

Course Name

Spectral Analysis of Differential Operators

Code	Semester	Theory (hour/week)	Application/Lab (hour/week)	Local Credits	ECTS
MATH 668	Fall/Spring	3	0	3	7.5

Prerequisites	None
Course Language	English
Course Type	Elective
Course Level	Third Cycle
Mode of Delivery	-
Teaching Methods and Techniques of the Course	-
National Occupation Classification	-
Course Coordinator	-
Course Lecturer(s)	Doç. Dr. Baruch SCHNEIDER
Assistant(s)	-

Course Objectives

The objective of this course is to cover the spectral properties of nonselfadjoint StrurmLiouville differential operators.

Learning Outcomes

#	Content	PC Sub	* Contribution Level
#	Content	PC Sub	1	2	3	4	5
1	will be familiar with boundary uniqueness theorems of analytic functions.
2	will be able to asimilate analytic continuation principle.
3	will be able to apply the spectral theory of nonselfadjoint differential equations.
4	will be able to define some concepts such as spectrum, resolvent set, resolvent operator, Jost solution.

Course Description

This course aims to cover an advanced theory and applications of Spectral Analysis.

Course Category	Core Courses
	Major Area Courses
	Supportive Courses
	Media and Management Skills Courses
	Transferable Skill Courses

WEEKLY SUBJECTS AND RELATED PREPARATION STUDIES

Week	Subjects	Related Preparation	Learning Outcome
1	Introduction and Fundamental Concepts.	M. A. Naimark, “Linear Differential Equations, Volume II”., Frederick Ungar Publishing Co.
2	Fourier Transforms; Properties and Applications.	M. A. Naimark, “Linear Differential Equations, Volume II”., Frederick Ungar Publishing Co.
3	NonSelfadjoint Differential Equations.	M. A. Naimark, “Linear Differential Equations, Volume II”., Frederick Ungar Publishing Co.
4	NonSelfadjoint SturmLiouville Differential Operator.	M. A. Naimark, “Linear Differential Equations, Volume II”., Frederick Ungar Publishing Co.
5	Solutions and Their Asymptotic Behaviours.	M. A. Naimark, “Linear Differential Equations, Volume II”., Frederick Ungar Publishing Co.
6	Jost Solution and İts Properties.	M. A. Naimark, “Linear Differential Equations, Volume II”., Frederick Ungar Publishing Co.
7	A Special Integral Representation For Jost Solution.	M. A. Naimark, “Linear Differential Equations, Volume II”., Frederick Ungar Publishing Co.
8	Integral Equations.	M. A. Naimark, “Linear Differential Equations, Volume II”., Frederick Ungar Publishing Co.
9	The Resolvent Operator.	M. A. Naimark, “Linear Differential Equations, Volume II”., Frederick Ungar Publishing Co.
10	Green’s Function and İts Properties.	M. A. Naimark, “Linear Differential Equations, Volume II”., Frederick Ungar Publishing Co.
11	Boundary Uniqueness Theorems of Analytic Functions.	M. A. Naimark, “Linear Differential Equations, Volume II”., Frederick Ungar Publishing Co.
12	Beurling and Pavlov Theorems, and Their Applications.	M. A. Naimark, “Linear Differential Equations, Volume II”., Frederick Ungar Publishing Co.
13	Carleson’s Theorem, and Its Applications.	M. A. Naimark, “Linear Differential Equations, Volume II”., Frederick Ungar Publishing Co.
14	Quantitative Properties of The Spectrum.	M. A. Naimark, “Linear Differential Equations, Volume II”., Frederick Ungar Publishing Co.
15	Spectral Expansion.	M. A. Naimark, “Linear Differential Equations, Volume II”., Frederick Ungar Publishing Co.
16	Review of the Semester

Course Notes/Textbooks	The extracts above and exercises will be given.
Suggested Readings/Materials	B.M. Levitan and I. S. Sargsjan, SturmLiouville and Dirac Operators, Kluwer Academic publishers. Further references and articles related this topic will be delivered in class.

EVALUATION SYSTEM

Semester Activities	Number	Weighting	LO 1	LO 2	LO 3	LO 4
Participation
Laboratory / Application
Field Work
Quizzes / Studio Critiques
Portfolio
Homework / Assignments	5	30
Presentation / Jury
Project
Seminar / Workshop
Oral Exams
Midterm	1	30
Final Exam	1	40
Total

Weighting of Semester Activities on the Final Grade		60
Weighting of End-of-Semester Activities on the Final Grade		40
Total

ECTS / WORKLOAD TABLE

Semester Activities	Number	Duration (Hours)	Workload
Theoretical Course Hours (Including exam week: 16 x total hours)	16	3	48
Laboratory / Application Hours (Including exam week: '.16.' x total hours)	16		0
Study Hours Out of Class	16	5	80
Field Work			0
Quizzes / Studio Critiques			0
Portfolio			0
Homework / Assignments	5	4	20
Presentation / Jury			0
Project			0
Seminar / Workshop			0
Oral Exam			0
Midterms	1	37	37
Final Exam	1	40	40
		Total	225

COURSE LEARNING OUTCOMES AND PROGRAM QUALIFICATIONS RELATIONSHIP

#	PC Sub	Program Competencies/Outcomes	* Contribution Level
#	PC Sub	Program Competencies/Outcomes	1	2	3	4	5
1	Accesses information in breadth and depth by conducting scientific research in Computer Engineering, evaluates, interprets and applies information.		-	-	-	X	-
2	Is well-informed about contemporary techniques and methods used in Computer Engineering and their limitations.		-	-	X	-	-
3	Uses scientific methods to complete and apply information from uncertain, limited or incomplete data, can combine and use information from different disciplines.		-	-	-	X	-
4	Is informed about new and upcoming applications in the field and learns them whenever necessary.		-	-	-	-	X
5	Defines and formulates problems related to Computer Engineering, develops methods to solve them and uses progressive methods in solutions.		-	-	-	-	X
6	Develops novel and/or original methods, designs complex systems or processes and develops progressive/alternative solutions in designs.		-	-	-	X	-
7	Designs and implements studies based on theory, experiments and modelling, analyses and resolves the complex problems that arise in this process.		-	-	-	X	-
8	Can work effectively in interdisciplinary teams as well as teams of the same discipline, can lead such teams and can develop approaches for resolving complex situations, can work independently and takes responsibility.		-	-	-	-	X
9	Engages in written and oral communication at least in Level B2 of the European Language Portfolio Global Scale.		-	-	X	-	-
10	Communicates the process and the results of his/her studies in national and international venues systematically, clearly and in written or oral form.		-	-	X	-	-
11	Is knowledgeable about the social, environmental, health, security and law implications of Computer Engineering applications, knows their project management and business applications, and is aware of their limitations in Computer Engineering applications.		-	-	X	-	-
12	Highly regards scientific and ethical values in data collection, interpretation, communication and in every professional activity.		-	X	-	-	-