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.

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.

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.

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.

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.

Writing interactive math tests in the LaTeX system. Using the system for writing interactive tests on the theory of probabilities in distance learning. Creation of tasks of the type "True/False", "Short answer", "Essay" ("Clarifying the definition").

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.

Characteristics of laminar and turbulent fluid flows. Analysis of the relationship between the Reynolds number and turbulence. Solution of the Navier-Stokes hydrodynamic equations. Investigation of the structure and inhomogeneities of a cellular flame.

Consideration of the process of evolution and prospects of mathematical methods and models in economic research. Characterization of advantages and identification of disadvantages of mathematical formalization in the conditions of a new economic paradigm.

Fundamentals of the method of least squares for the case of vector and matrix of observations. Advantages of MNCs to predict performance in the telemedia business. MNCs use algorithm for the matrix of observations with the ability to scale the data.

The methods developed by Hamilton 1989 and Chib 1996 to identied multiple-equation models. It details Bayesian estimation and inference for a class of models with different degrees of time variation and discuss analytical and computational difculties.

Teaching students of mathematical fields of training to inverse problems. Implementation of the scientific, educational potential of teaching university students inverse problems for differential equations. Computational methods of mathematical physics.

Examination of unlimited closed convex subsets of Banach space X, having the same recessive cone, and metric spaces, which they form with the Hausdorff metric. Receiving an analog of the theorem of approximation of convex compacts by normal polyhedrons.

The discussion about boolean algebras and their application to switching circuits. The rotation groups of the regular solids are investigated. New material on order of an element and cyclic groups, more details about the lattice of divisors of an integer.

Studies objects of geometric nature by means of some algebraic invariants defined over the category of these objects. Algebraic K-theory spectrum of DG-category. Definding the different versions of cyclic homology are via the mixed complex functor.

The self regulation of the parameters of the algorithm is a major step towards the establishment of the method as a general tool of nonlinear data analysis. Algorithms for the general task of extracting nonlinear principal manifolds from high dimensional.