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Course detail

Mathematics - Selected Topics I

Course unit code: FSI-T1K
Academic year: 2016/2017
Type of course unit: compulsory
Level of course unit: Bachelor's (1st cycle)
Year of study: 2
Semester: summer
Number of ECTS credits:
Learning outcomes of the course unit:
Basic knowledge of functional analysis, metric, vector, unitary spaces, Hilbert space, orthogonal systems of functions, orthogonal transforms, Fourier transform and spectral analysis, application of the mentioned subjects in physics.
Mode of delivery:
90 % face-to-face, 10 % distance learning
Prerequisites:
Real and complex analysis
Co-requisites:
Not applicable.
Recommended optional programme components:
Not applicable.
Course contents (annotation):
The course includes selected topics of functional analysis which are necessary for application in physics. It focuses on functional spaces, orthogonal systems and orthogonal transformations.
Recommended or required reading:
Kolmogorov,A.N.,Fomin,S.V.: Základy teorie funkcí a funkcionální analýzy, SNTL Praha 1975 Kolmogorov,A.N.,Fomin,S.V.: Základy teorie funkcí a funkcionální analýzy, SNTL Praha 1975
Kolmogorov,A.N.,Fomin,S.V.: Základy teorie funkcí a funkcionální analýzy, SNTL Praha 1975
Lang, S. Real and Functional Analysis. Third Edition, Springer-Verlag 1993
Planned learning activities and teaching methods:
The course is taught through lectures explaining the basic principles and theory of the discipline. Exercises are focused on practical topics presented in lectures.
Assesment methods and criteria linked to learning outcomes:
Course-unit credit - based on a written test
Exam has a written and an oral part.
Language of instruction:
Czech
Work placements:
Not applicable.
Course curriculum:
Not applicable.
Aims:
The aim of the course is to extend students´ knowledge acquired in the basic mathematical course by the topics necessary for study of physical engineering.
Specification of controlled education, way of implementation and compensation for absences:
Missed lessons can be compensated for via a written test.

Type of course unit:

Lecture: 26 hours, optionally
Teacher / Lecturer: prof. RNDr. Miloslav Druckmüller, CSc.
Syllabus: 1. Introduction
2. Metric space
3. Contraction, fix point Banach's theorem
4. Vector space, base, dimension, Vector spaces of functions
5. Unitary space orthogonal a orthonormal spaces
6. Hilbert space, L2 and l2 space
7. Orthogonal bases, Fourier series
8. Orthogonal transforms, Fourier transform
9. Usage of Fourier transform, convolution theorem
10.2D Fourier transform
11.Filtration in space and frequency domain, applications in physics
12. Operators and functionals
13. Variation methods
seminars: 13 hours, compulsory
Teacher / Lecturer: prof. RNDr. Miloslav Druckmüller, CSc.
Syllabus: 1. Introduction
2. Metric space
3. Fix point Banach's theorem applications
4. Vector space, base, dimension, Vector spaces of functions
5. Unitary space orthogonal a orthonormal spaces
6. Hilbert space, L2 and l2 space
7. Orthogonal bases, Fourier series
8. Orthogonal transforms, Fourier transform
9. Usage of Fourier transform, convolution theorem
10. 2D Fourier transform
11. Filtration in space and frequency domain, applications in physics
12. Operators and functionals
13. Variation methods

The study programmes with the given course