# Distribution of the Antiferromagnetism Vector for an Isolated Antidot and a System of Remote Antidots in Antiferromagnets

In the paper, an antiferromagnetism vector distribution in an antiferromagnetic film composed of an uniaxial or isotropic two-sublattice antiferromagnet with a set system of circular antidots is investigated. For such a system, the Landau-Lifshitz equation is written and its solution is obtained.

# Symmetry Analysis of a Class of (2+1)-Dimensional Linear Ultra-Parabolic Equations

In this paper, a class of (2+1)-dimensional linear ultra-parabolic equations of the second order is investigated by using the methods of group analysis of differential equations. The class under study generalizes a number of the classical equations of mathematical physics such as the free Kramers equation, the linear Kolmogorov equation etc. The classification of the symmetry properties of equations from the class is carried out by using the well-known Lie–Ovsiannikov algorithm.

# Minimal Systems of Generators and Relations and Properties of Wreath Products of Perfect Groups

Generators and defining relations for wreath products of perfect group which is two generating and alternating groups (m 2 times) are given. System of generators of metaperfect groups are found. Generators and defining relations for wreath products of 2-generating perfect groups were found, including alternating groups, i.e. (m 2 time). Systems of generators for metaperfect groups were investigated. A constructive proof of the minimality found system of generators was presented. It is shown that metaperfect group is not locally finite group.

# Complex-Valued Functions with Nondegenerate Groups of Regular Points

Complex-valued functions with nondegenerate groups of regular points are studied in the paper. A class of functions , which takes value on the complex plain and for which the limit exists, and is nonzero and finite for some from subset of positive real numbers is considered. It was received that this subset is multiplicative group, and it is called the group of regular points. Functions with nondegenerate groups of regular points generalize the class of RV functions. The corresponding limit functions are defined for complex-valued functions with nondegenerate groups of regular points.

# Consistency of Least Squares Estimator of Linear Regression Parameter in Case of Discrete Tme and Long-Range or Weak Dependent Regressors

Linear regression model with discrete time, long-range/weak dependent random noise and time dependent regressors, which are observed with long L range/weak dependent errors, is considered. Parameter estimation of such models is one of the important problems of statistics of random processes. Least squares estimator is chosen for the estimation. The aim of the work is to prove consistency of least squares estimator of such regression model.

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

In this paper non-linear regression model with Gaussian stationary random noise and continuous time is considered. The behavior of normalized in some way maximum residuals and maximum of residuals absolute values in which its the least squares estimator is substituted instead of unknown parameter of regression function. The convergence of distribution of these normalized maximum to double exponent law is proved which follows from the assumption of random noise normality.

# Moments Asymptotic Expansion of the Least Squares Estimator of the Vector-Parameter of Nonlinear Regression with Correlated Observations

A nonlinear regression model with continuous time and mean square continuous separable measurable Gaussian stationary random noise with zero mean and integrable covariance function is considered. Parameter estimation in the models of such kind is an important problem of statistics of random processes. In this paper, the first terms of asymptotic expansions of the bias vector and covariance matrix of the least square estimator of nonlinear regression function vector parameter are obtained.

# Asymptotic Uniqueness of the Non-Linear Regression Model Parameter with Least Squares Estimator

In the paper the nonlinear regression model with continuous time and random noise, which is a local functional of strongly dependent stationary Gaussian random process, is considered. Sufficient conditions of asymptotic uniqueness of the least squares estimator of regression function parameters are obtained. This result is applied to the least squares estimator of amplitude and angular frequencies of harmonic oscillations sum observed on the background of given random noise.

# Study of Distributive Law in Extended Interval Space

A study of the law of distributivity in the extended interval space is suggested. The research for interval in the center–radius form was conducted. The classification of the intervals is proposed. A set of intervals is represented as a union of three subsets which have defined by the relations of values the centers and the radii. We proved the lemma about the conditions under which the sum of the two intervals will own to same subset of the intervals you want to add. The conditions in which the sum of the two intervals belongs to the same subset as intervals, which are added.