# Theoretical and applied problems of Physics and Mathematics

# Preliminary Group Classification of a Class of Generalized Linear Kolmogorov Equations

The group–theoretic method is a modern research method for studying both linear and nonlinear partial differential equations. By using this method, we construct exact partial classical solutions of equations allowing for non-trivial symmetry groups. In this paper, a class of (2+1)-dimensional generalized linear Kolmogorov equations is considered. Our aim is to investigate symmetry properties of equations from the class and to use them to construct invariant fundamental solutions.

# PRV Conditions of Unbounded of Solution of Stochastic Differential Equation

We consider the behavior of solutions of stochastic differential equation

# The Limit Theorems for Extreme Residuals in Linear Regression Model with Gaussian Stationary Noise

We consider linear regression model with continuous time and strongly dependent stationary Gaussian random noise. The behavior of normalized in some way extreme residuals, that are the maximum differences, or their absolute values, between the observations and the values of the regression function where instead of unknown parameter the least squares estimator is substituted. For linear regression model the conditions of weak convergence of normalized extreme residuals to double exponent curve are obtained which follows from the assumption of normality of random noise.

# Integral Equations with r-Hypergeometric Functions

Some new properties of the r-hypergeometric functions are investigated, in partial, the differential relations for the function

# Periodogram Estimator Properties of the Parameters of the Modulate almost Periodic Signal

The problem of detection of hidden periodicities is considered in the paper. In the capacity of useful signal model the modulated almost periodic signal is taken observed on the background of random noise being the local functional of Gaussian strongly dependent stationary process. For estimation of unknown amplitude and angular frequency of modulated signal periodogram estimators are chosen. Sufficient conditions on consistency and asymptotic normality of the estimators are obtained. The exact form of limiting normal distribution is found.

# Study of Distributive Law in Classical Interval Arithmetic for the General Case

The aim of the article is to study the law of distributivity in classical interval arithmetic. We conduct the research for interval in the center-radius form. A set of intervals is represented as a combination of three subsets defined by values relations of centers and the radii. We prove the lemma about conditions under which the sum of two intervals will belong to the same subset of added intervals. We generalize the distributive law in case of voluntary number of intervals.

# Method for Improving the Convergence of Fourier Series and Interpolating Polynomials Based on Systems of Orthogonal Functions

The aim of this paper is to develop and study the method for improving the convergence of Fourier series based on orthogonal functions systems. This method application allows us obtaining uniformly convergent series for smooth functions. Another goal is to develop and study the method of improving the convergence of interpolating polynomials, based on systems of orthogonal functions, which in many cases allows us to reduce an interpolation error by such polynomials.

# Stochastic Integrals with Respect to a Lévy Process and Stochastic Derivatives on Spaces of Regular Test and Generalized Functions

The extended (Skorohod) stochastic integral with respect to a Lévy process and the corresponding Hida stochastic derivative on the space of square integrable random variables (L2) have many applications in the stochastic analysis, in particular, in the theory of stochastic differential and integral equations. But sometimes (for example, in order to consider so-called normally ordered stochastic equations) it is convenient to introduce and study these operators on certain spaces of test and generalized functions or on spaces of some riggings of (L2).

# On Lyapunov and Ricatti Monotone Differential Matrix Equation

The purpose of the paper is to generalize the Polacik–Terescak theorem for a monotone differential matrix equation of Lyapunov and Ricatti. Our goal is to study the existence of the one-dimensional invariant manifold (corresponding to Lyapunov and Ricatti monotone differential matrix equation). Using the method introduced by Hilbert Birkgoff in the projective contraction fixed point theorem, we determine conditions under which Lyapunov differential matrix equation has a one-dimensional invariant manifold in the cone of positive definite of quadratic form.

# r-Hypergeometric Function and its Application

In this paper with the help of the -generalized confluent hypergeometric function the (τ,β) r-hypergeometric function is considered. The aim of it is to study the main properties of the r hypergeometric function, in particular, to study the relation of Erdelyi’ types, the Mellin transform, the composite relation with integral operator of Erdelyi–Kober’ type. In the study used common methods of the theory of special functions, the theory of integral transforms and operators ofvfractional integration.