**Dynamic systems.**

For 3rd year students of specialties

MATHEMATICS, APPLIED MATHEMATICS

Semester 5

**Lecturer** : Senior Lecturer *Colbasin Stanislav Aleksandrovich*

**Course structure** : 2 years. ( **Lectures** ) + 2 years ( **practice** )

**Basic knowledge: the** basics of mathematical analysis and ordinary differential equations

**Reporting form** : credit

**Tentative content:**

The theory of dynamical systems examines the qualitative behavior of complex objects over time. The main example of such objects for us will be differential equations, for which it is difficult to find and / or use exact solutions in the general case. Based on the properties of the differential equation as a dynamic system, we can predict the behavior of solutions, without having to look them explicitly. The theory of dynamical systems generalizes such a qualitative approach for more abstract situations.

At the beginning of the course, we will analyze the methods of constructing a dynamic system and its differences from other models of complex objects. Consider numerous examples of systems with discrete and continuous time. On the basis of examples we introduce the basic concepts of the theory, such as: phase space, evolution operator, trajectories, invariant and boundary sets.

The main part of the course is devoted to the properties that the system demonstrates at large values of time. The main such property that adequately characterizes the system is the “steady” behavior; For example, the presence of a “convenient” boundary set, to which in one way or another the trajectories of the system converge. Formally, this can be described by the notion of dissipation and asymptotic compactness, and even more fully – through the existence of a global attractor. We will get acquainted with these concepts and establish the basic theorems concerning them and their mutual relations.

At the end of the course, we will study some more special issues, such as: fractal dimensionality, structure of the global attractor, behavior of the family of dynamical systems, depending on the parameter.

**Fourier series and the Fourier integral.**

For 3rd year students of specialties

MATHEMATICS, APPLIED MATHEMATICS

Training semester 5

**Lecturer** : Doctor of sciences, professor Dubovy Volodymyr Kirillovych

**Course structure** : 2 years. ( **Lectures** ) + 2 hours ( **lectures / practice** )

**Basic knowledge:** mathematical analysis, linear algebra, Ordinary differential equations

**Reporting form** : credit

**Tentative content:**

- Functions of limited variation.
- The Styletes Integral.
- Total Fourier series. Bessel’s inequality, Parseval’s equality.
- Fourier series of trigonometric series. Complex form of the Fourier series.
- Convergence of Fourier series in averaged quadratic.
- Convergence of the Fourier series at the point. Sign of Dirichlet – Jordana.
- Image of functions in Fourier series.
- Feyer’s theorem.
- Complexity of the trigonometric system.
- Properties of Fourier series.
- Fourier method for solving problems of mathematical physics.
- Fourier integral. Formula of inversion.
- Convergence of the Fourier integral.
- Fourier transform and its properties.
- Application of the Fourier transform to the solution of differential equations.

**Object-Oriented Programming (C ++ language)**

For 3rd year students of a specialty

MATH

Semester 5 or 6

**Lecturer** : Ph.D. – M.Sc., Associate Professor *Anoshchenko Olga Alekseevna*

**Course structure** : 2 years. ( **Lectures** ) + 2 hours ( **practice** )

**Basic knowledge: the** basics of structural programming

**Reporting form** : credit

**Tentative content:**

- The main paradigms of object-oriented programming
- The concept of class. The protocol of the class, its structure. Members of the class.
- Inheritance, its types, peculiarities of application.
- Polymorphism. Virtual functions Pure virtual features. Abstract classes.
- Overloading operations.
- Multiple inheritance. Virtual base classes.
- Function templates. Template classes.
- Exceptional situations. Processing of exceptional situations.

**Classical geometry problems**

For 3rd year students of specialties

MATHEMATICS, APPLIED MATHEMATICS

Semester 5

**Lecturer** : Doctor of sciences, associate professor *Gorkiev Vasiliy Alekseevich*

**Course structure** : 2 years. (Lectures) + 2 years (lectures / practice)

**Basic knowledge:** linear algebra, mathematical analysis, ordinary differential equations

**Reporting form** : credit

**Tentative content:**

- Definition of curves for given curvilinear. The Nikolaev theorem.
- Integral inequalities for curves
*.*Fennel’s inequality. Inequality of Feri Milnor. - Restoration of closed curves over spherical indicatrices. Vygodsky’s theorem.
- Analogues of curvature and difficulty for plane and spatial polygons.
- Theorem on 4 oval vertices
- Isoperimetric inequality. Isoperimetric property of a circle.
- Jordan’s theorem on closed curves.
- Cauchy’s theorem on unambiguous certainty of convex polyhedra.
- Flexori The hypothesis of blacksmith’s bags. Sabitov’s theorem.
- Liebman’s theorem on the sphere.
- Theorem on unambiguous certainty of ovaloid.
- Hilbert’s theorem on the impossibility of isometric immersion in the “whole” plane of Lobachevsky. Efimov’s theorem.
- Pseudospherical surfaces and their transformation of Biancai-Bucklund.

**Elements of the theory of stability and differential equations with stratification** **(memory)**

For 3rd year students of specialties

MATHEMATICS, APPLIED MATHEMATICS

Semester 6

**Lecturer** : Ph.D., Associate Professor **Olexandr V. Rezunenko**

**Course structure** : 2 years. ( **Lectures** ) + 2 hours ( **practice)** per week

**Basic knowledge:** *it is desirable to* have initial skills in mathematical analysis and differential equations, *it is useful to* have an idea of the basic facts of the course ” *Dynamic Systems* “.

**Reporting form** : credit

**Tentative content:**

The existence, uniqueness and properties of solutions of **differential** equations with tuning (memory) are studied. Such equations naturally arise in all applied problems, which takes into account the finiteness of the rate of propagation of signals and, as a consequence, the stratification (delay) of the reaction in biological, chemical and mechanical systems.

Different types of rationalization are considered. Starting from the simplest case – a steady, coercive trick (the simplest example is dx (t) / dt = Ax (t) + Bx (tr), r> 0) and continuing to study the stable distributed and rationing, which depends on the state of the system. The emphasis is on the study of qualitative properties, in particular the stability of solutions. The study of the theoretical material is accompanied by examples and exercises. In particular, such modern biological models as population (predator-victim, cooperative), immunology are considered. In biological tasks, rationalization can be measured from a fraction of a second (motion of the eye on a moving object) to several years (reaching the adult age by members of a particular population). Thus, taking into account the effects of mating is an important component of the formulation of mathematical models and significantly influences the research methods. (Lecturer: O.V.Rezunenko ).

Firing systems are classical and, at the same time, modern examples of *infinite-dimensional dynamic systems* . Another example of *infinite-dimensional dynamical systems* is differential equations in partial derivatives. In this sense, this course naturally embeds in the package of courses in *Mathematical Physics* . The course is constructed in such a way that the basic essential facts from mathematical analysis and differential equations will be recalled when applied.

If you have questions related to the course – *feel* free to write to rezounenko@karazin.ua . Everyone interested in this modern direction of mathematics is invited!

**The theory of operators**

For 3rd year students of specialties

MATHEMATICS, APPLIED MATHEMATICS

Training semester 6

**Lecturer** : Doctor of Philology, Professor *Yegorova Irina Evgenievna*

**Course structure** : 2 years. ( **Lectures** ) + 2 hours ( **practice or lectures** )

**Basic knowledge:** linear algebra, complex analysis, theory of measure and integral.

**Reporting form** : credit

**Tentative content:**

- Geometry of the geometric space. The existence of orthonormal bases. The dimension of the gill space.
- Limited linear operator. Hiding operator.
- Compact operator. Alternative to Fredholm and its application to integral equations.
- Projection operators. Symmetric operators.
- Spectrum and resolvent of self-directed operator.
- Spectral theorem for a compact operator.
- Spectral theorem for a symmetric bounded operator.
- Spectral analysis of an infinite Jacobi matrix.

**Differential geometry.**

(Continuation of the course)

For 3rd year students of a specialty

APPLIED MATHEMATICS

(Together with students of the specialty “mathematics”)

Semester 6

**Lecturer** : Doctor of sciences, associate professor *Yampolsky Alexander Leonidovich*

**Course structure** : 2 years. ( **Lectures** ) + 2 hours ( **practice)** per week

**Basic knowledge:** differential geometry (theory of curves), ordinary differential equations, linear algebra.

**Reporting form** : exam / credit (optional)

**Tentative content:**

- The second fundamental form of the surface.
- Main curvatures and main directions. Gaussian curvature of the surface.
- Lines of curvature and asymptotic lines.
- Gaussian and Weingarten derivational formulas. Gauss and Kodatz equation.
- Absolute (covariant) differential of a vector field.
- Metrics of stable curvature.
- Geodetic as extremal functional length.
- Minimum surfaces.
- Gauss-Bone formula. Gaussian integral formula.
- Elements of tensor analysis and its application in geometry.