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Symmetries and reductions of some algebraic equations are considered. Transformations that preserve the form of several algebraic equations, as well as transformations that reduce the degree of these equations, are described. Illustrative examples are provided. The obtained results and solutions can be used as test problems for numerical methods of solving algebraic equations., Comment: 6 pages

The problems of estimating the similarity index of mathematical and other scientific publications containing equations and formulas are discussed for the first time. It is shown that the presence of equations and formulas (as well as figures, drawings, and tables) is a complicating factor that significantly complicates the study of such texts. It is shown that the method for determining the similarity index of publications, based on taking into account individual mathematical symbols and parts of equations and formulas, is ineffective and can lead to erroneous and even completely absurd conclusions. The possibilities of the most popular software system iThenticate, currently used in scientific journals, are investigated for detecting plagiarism and self-plagiarism. The results of processing by the iThenticate system of specific examples and special test problems containing equations (PDEs and ODEs), exact solutions, and some formulas are presented. It has been established that this software system when analyzing inhomogeneous texts, is often unable to distinguish self-plagiarism from pseudo-self-plagiarism (false self-plagiarism). A model complex situation is considered, in which the identification of self-plagiarism requires the involvement of highly qualified specialists of a narrow profile. Various ways to improve the work of software systems for comparing inhomogeneous texts are proposed. This article will be useful to researchers and university teachers in mathematics, physics, and engineering sciences, programmers dealing with problems in image recognition and research topics of digital image processing, as well as a wide range of readers who are interested in issues of plagiarism and self-plagiarism., Comment: 23 pages, 3 figures, 2 photos

For the first time, some hypersingular nonlinear boundary-value problems with a small parameter~$\varepsilon$ at the highest derivative are described. These problems essentially (qualitatively and quantitatively) differ from the usual linear and quasilinear singularly perturbed boundary-value problems and have the following unusual properties: (i) in hypersingular boundary-value problems, super thin boundary layers arise, and the derivative at the boundary layer can have very large values of the order of $e^{1/\varepsilon}$ and more (in standard problems with boundary layers, the derivative at the boundary has the order of $\varepsilon^{-1}$ or less); (ii) in hypersingular boundary-value problems, the position of the boundary layer is determined by the values of the unknown function at the boundaries (in standard problems with boundary layers, the position of the boundary layer is determined by coefficients of the given equation, and the values of the unknown function at the boundaries do not play a role here); (iii) hypersingular boundary-value problems do not admit a direct application of the method of matched asymptotic expansions (without a preliminary nonlinear point transformation of the equation under consideration). Examples of hypersingular nonlinear boundary-value problems with ODEs and PDEs are given and their exact solutions are obtained. It is important to note that the exact solutions presented in this paper can be used to compare the effectiveness of various methods of numerical integration of singularly perturbed problems with boundary layers, and also to develop new numerical and approximate analytical methods., Comment: 9 pages, 2 figures

Several new methods of numerical integration of Cauchy problems with blow-up solutions for nonlinear ordinary differential equations of the first- and second-order are described. Solutions of such problems have singularities whose positions are unknown a priori (the standard numerical methods for solving problems with blow-up solutions can lead to significant errors). The first proposed method is based on the transition to an equivalent system of equations by introducing a new independent variable chosen as the first derivative. The second method is based on introducing a new auxiliary non-local variable with the subsequent transformation to the Cauchy problem for the corresponding system of ODEs. The third method is based on adding to the original equation of a differential constraint, which is an auxiliary ODE connecting the given variables and a new variable. The proposed methods lead to problems whose solutions are represented in parametric form and do not have blowing-up singular points; therefore the transformed problems admit the application of standard fixed-step numerical methods. The efficiency of these methods is illustrated by solving a number of test problems that admit an exact analytical solution. It is shown that: (i) the methods based on non-local transformations of a special kind are more efficient than several other methods, (ii) among the proposed methods, the most general method is the method based on the differential constraints. Some examples of nonclassical blow-up problems are considered. Simple theoretical estimates are derived for the critical value of an independent variable. It is shown that the method based on a non-local transformation of the general form as well as the method based on the differential constraints admit generalizations to the $n$th-order ODEs and systems of coupled ODEs., Comment: 51 pages, 9 figures, 3 tables, 38 references

Two new methods of numerical integration of Cauchy problems for ODEs with blow-up solutions are described. The first method is based on applying a differential transformation, where the first derivative (given in the original equation) is chosen as a new independent variable. The second method is based on introducing a new non-local variable that reduces ODE to a system of coupled ODEs. Both methods lead to problems whose solutions do not have blowing-up singular points, therefore the standard numerical methods can be applied. The efficiency of the proposed methods is illustrated with several test problems., Comment: 8 pages, 2 figures