Infopaket

Advanced course of Analysis and Probability Theory

Вибіркова дисципліна для ОП

First

2021/2022

6 Semester

5

PLO21.3. Understand the fundamental areas of mathematics and computer science, to the extent necessary for learning mathematical disciplines, applied disciplines and using their methods in a chosen profession. PLO22.3. Understand the main areas of mathematical logic, theory of algorithms and computational theory, programming theory, probability theory and mathematical statistics. PLO23.3. Be able to use professional knowledge, skills and abilities in the field of fundamental sections of mathematics and computer science for research of real processes of different nature. PLO24.3. Be able to independently analyze the relevant subject area, be able to develop mathematical and structural algorithmic models.

To successfully learn the discipline Advanced course of Analysis and Probability Theory the student should satisfy the following requirements. They know (a) fundamentals of mathematical methods for construction, verification and investigation of qualitative characteristics of deterministic and stochastic mathematical models; (b) classical methods of Linear Algebra, Probability Theory, Calculus and Functional Analysis. They can (a) investigate qualitative characteristics of available mathematical models; (b) apply classical methods for solving applied problems in deterministic and stochastic models; (c) analуze the nature and goals of construction of mathematical structures and models. They should be able to (a) apply classical methods of Linear Algebra, Probability Theory, Calculus and Functional Analysis; (b) seek information in open sources and properly analyze it.

The discipline is aimed at learning basic results of Renewal Theory for random walks with nonegative steps, the Operator Algebras theory and methods of the proofs and mastering technical tools which are intrinsic to this subject area. The subject matter includes classical theorems of Renewal Theorem like the elementary renewal theorem, Blackwell’s theorem, the key renewal theorem, the strong law of large numbers for the number of renewals; spectral theory, commutative Banach and C*-algebras, realisation of abstract C*-algebras by Hilbert space operators. The present course is a natural continuation of the disciplines “Calculus”, “Probability Theory”, “Linear Algebra” and “Functional Analysis”.

1. Iksanov O.M. Elementy teoriyi vidnovlennya ta yiyi zastosuvannya: Elektronnyy navchalʹnyy posibnyk.-2022.-121 s. http://do.unicyb.kiev.ua/index.php/uk/2011-01-03-10-24-53/11-2011-01-03-10-44-09 2.Iksanov A. Renewal theory for perturbed random walks and similar processes. Cham: Birkhauser, 2016.-250 p. 3. Mitov K.V., Omey E. Renewal processes. Cham: Springer, 2014. -122 p. 4.Gut A. Stopped random walks: Limit theorems and applications. 2nd edition. New York: Springer-Verlag, 2009.—263 p. 5. Wegge-Olsen N.E. K-Theory and C*-algebras: a friendly approach. New-York: Oxford Science Publications. The Clarendon Press, Oxford University Press, 1993. – 370 p. 6. Hille E., Phillips R.S. Functional analysis and semigroups. Providence: AMS, 1974.—808 p. 7. Conway J.W.. A Course in Functional Analysis. Second edition. Graduate Texts in Mathematics, 96. New York: Springer-Verlag,1990. – 399 p.

Lectures, consultations, test works, independent work.

Intermediate assessment: The maximal number of available points is 60. Test work no. 1: 30/18 points. Test work no. 2: 30/18 points. Final assessment (in the form of exam): The maximal number of available points is 40. The form of exam: writing. The types of assignments are 4 writing assignments (2 theoretical and 2 practical).

Ukrainian