The textbook includes topics of discrete mathematics. Relations: binary relations, pictorial representatives of relations, inverse relation, functional relation. Mathematical logic: propositions and compound statements, basic laws of logical operations.

Using a generalized Dunkl translation, we obtain an analog of theorem 5.2 in Younis’ paper for the Dunkl transform for functions satisfying the (d, g)-Dunkl Lipschitz condition in the space L2(R,|x|2a+1dx). Consideration generalized Dunkl translation.

Arithmetic operations and properties of decimals, proportion and percents. Fundamental concepts of algebra, functions and graphs. The fundamental operations with complex numbers. Limits of function values, one-sided limits, infinitesimal functions.

Further examples of population dynamics. Representation of real numbers in an arbitrary base. A geometrical model for continued fractions. The idea of computational complexity. Elementary applications of congruence. The fundamental theorem of arithmetic.

The quart formula - a tool for describing the composition of functions, the modification of which directly changes properties, which are easily detected by numerical modeling. A complete infinite power-factorial functional series of the exponent.

Properties of Probability Distributions. Conditional Distributions and Expectation. Characteristic Functions, Moments and Cumulants. Parametric Families of Distributions. Distribution Theory for Functions of Random Variables, Approximation of Integrals.

Combinatorial description. Teichmuller curves. Recurrence relation. Asymptotics of numbers of branched coverings of a torus, volumes of moduli spaces of holomorphic differentials. The character of the infinite wedge representation. Cylinder decomposition.

Аналіз функціонально-дискретного методу розв’язування задач з крайовими умовами третього роду, періодичними і антиперіодичними умовами. Розгляд обставин геометричної прогресії. Дослідження асимптотичних розвинень для значень задачі Штурма-Ліувілля.

The theory of the scalar field, directed derivative. The calculation of the line integral. The divergence of vector fields, their properties. Complex numbers and operations with them. The concept of differentiability and analytic function of the complex.

Conditions for nite sequences of positive numbers to be certain parts of spectra of the Dirichlet-Dirichlet, Dirichlet-Neumann, Neumann-Dirichlet and Neumann-Neumann boundary value problems generated by the same Stieltjes string recurrence relations.

Forgotten motives: varieties of scientific experience. Motives and homological algebra, physics. Early history of motives. Determination of the category of (pure) motives. Preservation algebraic-geometric objects, projective varieties over a Wark field.

Monotonic Functions and Unordered Lists. The Pigeonhole Principle. Basic Concepts of Decision Trees. The Principle of Inclusion and Exclusion. Counting Structures with Symmetries. Concepts in Graph Theory. Equivalence Relations and Unlabeled Graphs.

Normed linear spaces. Contraction mapping theorem. Applications to differential and integral equations. Linear transformations. Product spaces and Fubini's theorem. Projection and self-adjoint operators. Gram-Schmidt orthonormalization. Fourier analysis.

Ways of representation of functions of two variables. Partial increment and derivatives of the first order. Application of the total differential to approximate calculations. Description of the largest and smallest values of a function of two variables.

Finding common solutions of linear differential equations of the fourth order, the coefficients of which satisfy the system of three differential equations of the first order. The problem of conformal mapping of polygons, whose sides are arcs of circles.

Several aspects of systems theory. Arithmetic of cardinal numbers. Crises of naive set theory. Constructions of ordinal and cardinal number systems. Growth of the polish school of mathematics. Bellman’s principle of optimality and its generalizations.

Subject and method of statistical science. Elements of probability theory. Random variables and their distribution laws. Fundamental of statistical observation. Grouping, consolidated return and data presentation. Basics of averages statistics method.

The unsupervised methods required to reduce the dimension of the data set and to extract meaningful biological information. This work shows that Independent Component Analysis is a promising approach for the analysis of genome-wide transcriptomic data.

Model of motion, which takes into account the asymmetry of hulls, infiltration processes and surface waves in the form of a system of nonlinear differential equations. Estimation of the error as a function of the magnitude of the initial velocities.

Provides the analysis of the connection existing between constituent components of a binomial expression. The need to study this connection is predetermined by the lack of a clear distinction between binomials and other types of formulaic language.

The natural connection between interassociates and variants of a semigroup. Variants of regular semigroups. Connection interassociates with commutative dimonoids. Study interassociates of the free semigroup with two generators using computer modeling.

Determining the extent of interpolated approximation spaces generated by regular elliptic operators on compact manifolds. Features and through the application of Jackson-Bernstein inequalities for spectral approximation regular elliptic operators.

The study of the theory of probability. Determination of the relative frequency. Moments of random variables. Central limit teorema. Aksioma Richesky Probability. Studies Cumulative Distribution Functions. A communication system for random fluctuations.

Дослідження властивостей L-пiдгруп на скiнченних групах. Характеристика множини L-пiдгруп на циклічних групах деяких порядків з простими числами, квазіциклічній та нециклічній групі. Доведення математичних тверджень та приклади деяких L-пiдгруп.

Complete discrete valuation fields. Extensions of discrete valuation fields. Cyclic extensions of prime degree. The Hasse–Herbrand function. The norm and Ramication groups. The milnor K-groups of a local field. The group of units of local number fields.

Вивчення пар класів неспадних тотальних одномісних арифметичних функцій. Встановлення критеріїв рефлективності, транзитивності. Вивчення ґрат m-звідностей з інформаційними обмеженнями. Дослідження структури рекурсивно перераховних ступенів нерозв’язності.

Measure theory, discrete time martingales and discrete time option pricing. Continuous time martingales. Stochastic integrals, calculus and differential equations. Option pricing in continuous time. Random measures, stochastic calculus (characteristics).

Изучение методов обработки информации при помощи Mathcad, возможных алгоритмов написания программ и способов отображения информации. Построение графиков функций. Расчет количества информации при кодировании равномерным и неравномерным двоичным кодом.

The mathematical principles of the operation of cryptocurrencies. Mathematical formulas for calculating the public key from the private key. Properties of elliptic curves and the principle of their application during the creation of cryptocurrency.

Dynamic Modeling with Difference Equations. Linear Models of Structured Populations. Nonlinear Models of Interactions. Modeling Molecular Evolution. Constructing Phylogenetic Trees. Infectious Disease Modeling. Curve Fitting and Biological Modeling.