Магистратура 2020/2021

Математический анализ. Продвинутый уровень

Статус: Курс адаптационный (Когнитивные науки и технологии: от нейрона к познанию)

Направление: 37.04.01. Психология

Кто читает: Департамент психологии

Где читается: Институт когнитивных нейронаук

Когда читается: 1-й курс, 1, 2 модуль

Формат изучения: без онлайн-курса

Преподаватели: Захаров Денис Геннадьевич

Прогр. обучения: Когнитивные науки и технологии: от нейрона к познанию

Язык: английский

Кредиты: 3

Контактные часы: 40

Course Syllabus

Abstract

The bridging course “Advanced Calculus” has an aim to train the master students to be ready for the courses devoted to Bayesian Statistics, Qualitative and Quantitative Research Methods in Psychology, Computational Neuroscience, Digital Signal Processing and some others. In the framework of this course the students study the real- and complex-valued functions, theory of derivatives and integrals, differential equations and dynamical systems, as well as Taylor, Fourier and Laplace series.

Learning Objectives

• • Gain understanding of the concept of functions, continuous functions and different kind of discontinuous functions
• • Gain skills in evaluating derivatives and integrals.
• • Gain skills in expansion of a function into a Taylor series and finding its convergence interval.
• • Gain skills in solving of homogeneous and nonhomogeneous linear differential equations.
• • Gain understanding of Fourier series, Fourier transforms and their applications.
• • Gain understanding of a Laplace transform and its applications for solving of differential equations
• • Gain understanding of dynamical systems analysis.

Expected Learning Outcomes

• • Know main operations, rules and properties of sets, real numbers, functions, continued functionsю
• Know man rules and properties of derivatives and integrals
• Know main operations, rules and properties of infinite series, functional series as well as Taylor series
• Know basic facts about Fourier series, Fourier transforms and their applications
• Know basic methods for solving of linear differential equations
• Know basic facts about Laplace transforms and their applications in solving of ordinary differential equations
• Be able to analyze one dimensional and two dimensional flows

Course Contents

• Special elementary functions. Limit function of one variable. Continuous and discontinuous functions.
• Infinite series. Functional series. Taylor Series. Interval of convergence.
• Derivatives and antiderivatives. Definite and improper integrals.
• Fourier Analysis.
• Linear homogenous and non-homogeneous differential equations. Rules of solving.
• Laplace transform. Solving of ordinary differential equations using a Laplace transform.
• Dynamical systems. One dimensional flows: steady states, stability and bifurcations. Two dimensional flows: steady states, limit cycles, stability, bifurcations, phase portraits. Dynamical chaos.

Assessment Elements

• tests
  The tests, which are given at the end of each topics, consists of 5-6 typical tasks which have been considered during the seminars. Test duration is 45 minutes.
• Final exam
  The final exam is written and consists of 2 theoretical questions and 2 tasks for solving. Grading formula for the first and the second re-takings is exactly the same as the grading formula of the basic exam.

Interim Assessment

• Interim assessment (2 module)
  0.6 * Final exam + 0.4 * tests

Bibliography

Recommended Core Bibliography

• Anton, H., Bivens, I. C., & Davis, S. (2016). Calculus (Vol. 11th ed). New York: Wiley. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1639210
• Gorain, G. C. (2014). Introductory Course on Differential Equations. New Delhi: Alpha Science Internation Limited. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1878058
• Strogatz, S. H. (2000). Nonlinear Dynamics and Chaos : With Applications to Physics, Biology, Chemistry, and Engineering (Vol. 1st pbk. print). Cambridge, MA: Westview Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=421098

Recommended Additional Bibliography

• Nekorkin, V. I. (2015). Introduction to Nonlinear Oscillations. Weinheim: Wiley-VCH. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1099772