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Course, academic year 2019/2020
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Advanced Differentiation and Integration 1 - NMMA437
Title in English: Derivace a integrál pro pokročilé 1
Guaranteed by: Department of Mathematical Analysis (32-KMA)
Faculty: Faculty of Mathematics and Physics
Actual: from 2019 to 2019
Semester: winter
E-Credits: 4
Hours per week, examination: winter s.:2/0 Ex [hours/week]
Capacity: unlimited
Min. number of students: unlimited
State of the course: taught
Language: Czech, English
Teaching methods: full-time
Guarantor: prof. RNDr. Jan Malý, DrSc.
Class: M Mgr. MA
M Mgr. MA > Povinně volitelné
Classification: Mathematics > Real and Complex Analysis
Annotation -
Last update: T_KMA (02.05.2013)
Real-analytic properties of Sobolev functions. Change of variables in integral for Lipschitz transformations – area and coarea formula. Differentiation of convex functions. Recommended for master students of mathematical analysis.
Course completion requirements - Czech
Last update: prof. RNDr. Jan Malý, DrSc. (10.10.2017)

Zkouška je ústní. Požadavky ke zkoušce odpovídají sylabu předmětu v rozsahu, jak byl presentován na přednášce.

Literature - Czech
Last update: T_KMA (02.05.2013)

L. C. Evans, R. F. Gariepy: Measure theory and fine properties of functions. CRC Press, Boca Raton, FL, 1992

W. P. Ziemer: Weakly differentiable functions. Sobolev spaces and functions of bounded variation. Springer-Verlag, New York, 1989

Syllabus -
Last update: doc. Mgr. Petr Kaplický, Ph.D. (09.06.2015)

1. Real-analytic properties of Sobolev functions

Riesz potential estimate

Beppo Levi's characterization

Embedding theorems

Approximate differentiability, a.e. differentiability

Examples concerning discontinuity and non-differentiability

2. Change of variables in integral for Lipschitz transforms

Area formula

Coarea formula

Sard type theorems

Luzin's (N) condition

3. Differentiation of convex functions

Lipschitz estimates

Zajíček's theorem (informatively)

Alexandrov's theorem

Entry requirements -
Last update: prof. RNDr. Jan Malý, DrSc. (02.05.2018)

Measures, Radon-Nikodym theorem, Lebesgue integral, Radon measures, convolution, smoothing by convolution, strong, weak and weak* convergence in Banach spaces, elements of theory of distributions, Lipschitz functions and mappings, Hahn-Banach theorem, Hausdorff measure, L^p spaces and spaces of continuous functions

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