卷 27, 编号 1 (2025)
- 年: 2025
- ##issue.datePublished##: 31.03.2025
- 文章: 6
- URL: https://journals.rcsi.science/2079-6900/issue/view/19733
- DOI: https://doi.org/10.15507/2079-6900.27.202501
完整期次
Mathematics
On the method of solving nonlinear Fredholm integral equation of the second kind with piecewise-smooth kernels
摘要
The work is devoted to the development of iterative methods for solving nonlinear Fredholm integral equations of the second kind with piecewise-smooth kernels. A new approach to constructing their solutions is proposed which combines the method of successive approximations with polynomial interpolations of functions on the segment $[-1,\,1]$. In this case, the original integral equation is reduced to a Volterra-type equation where the unknown function is defined on the segment mentioned. The free term of the equation is chosen as the initial approximation. At each iteration of successive approximations method, the kernel of the integral equation is represented as a partial sum of a series in Chebyshev polynomials orthogonal on the segment $[-1,\,1]$. The coefficients in this expansion are found using the orthogonality of vectors formed by the values of these polynomials at the zeros of the polynomial whose degree is equal to the number of unknown coefficients. An approximation of the solution is made by interpolation of the obtained values of the solution at the Chebyshev nodes at each iteration. The work also constructs a solution to an integral equation whose free term has a discontinuity point of the first kind. The results of the computational experiments are presented, which demonstrate the effectiveness of the proposed approach.
Middle Volga Mathematical Society journal. 2025;27(1):11-24
11-24
Fundamental representations of orthogonal Lie algebra and new simple subalgebras of nonalternating Hamiltonian Lie algebras
摘要
In the paper the action of the orthogonal Lie algebra $\mathfrak{o}(V)$ on the exterior powers of a space $V$ is considered for $n$-dimensional vector space $V$ over a perfect field $K$ of characteristic two with a given nondegenerate orthogonal. The exterior algebra is identified with the algebra of truncated polynomials in $n$ variables. The exterior powers of $V$ taken as modules over $\mathfrak{o}(V)$ are identified with homogeneous subspaces of non-alternating Hamiltonian Lie algebra $P(n)$ with respect to the Poisson bracket corresponding to an orthonormal basis of the space $V$ of variables. It is proved that the exterior powers of the standard representation for Lie algebra $\mathfrak{o}(V)$ are irreducible and pairwise nonequivalent. With respect to subalgebra $so(V)$, $n= 2l+1$ or $n= 2l$, there exist $l$ pairwise nonequivalent fundamental representations in the spaces $\Lambda^{r}V$, $r= 1, \ldots, l$. All of them admit a nondegenerate invariant orthogonal form, being irreducible when $n= 2l+1$. When $n= 2l$ the representations of $so(V)$ in $\Lambda^{r}V$, $r= 1, \ldots, l-1$ are irreducible and the space $\Lambda^{l}V$ possesses the only non-trivial proper invariant subspace $M$, which is a maximal isotropic subspace with respect to an invariant form. Two exceptional simple Lie subalgebras $P_{1}(6)$, $P_{2}(6)$ of $P(n)$, of dimension $2^{5}-1$ and $2^{6}-1$, correspondingly, containing the submodule $M$, and exising only in the case of 6 variables, are found.
Middle Volga Mathematical Society journal. 2025;27(1):25-33
25-33
On some universal criterion for a fixed point
摘要
Fixed point criteria may be applied in various fields of mathematics. The problem of finding sufficient conditions for transformations of a certain class to have a fixed point is well known. In the context of element-wise description for the monoid of all endomorphisms of a groupoid, the following were formulated: a bipolar classification of endomorphisms and related mathematical objects. In particular, the concept of a bipolar type of endomorphism of a groupoid (or simply a bipolar type) was stated. Every endomorphism of an arbitrary groupoid has exactly one bipolar type. In this paper, using bipolar types, a fixed point criterion for an arbitrary transformation of a nonempty set (hereinafter, the universal fixed point criterion) is formulated and proved. This criterion is not easy to apply. Further expansion of the range of problems to which this criterion can be applied depends directly on the success in studying the properties of groupoids’ endomorphisms. The paper formulates such general problems (unsolved for today), that the success in their study will expand the possibilities of using the universal fixed point criterion. The connection between the formulated problems and the obtained criterion is discussed. In particular, necessary and sufficient conditions for the Riemann hypothesis on the zeros of the Riemann zeta function to be satisfied are obtained using the universal fixed point criterion.
Middle Volga Mathematical Society journal. 2025;27(1):34-48
34-48
Group classification of nonlinear time-fractional dual-phase-lag heat equation with a small parameter
摘要
{In this paper, we solve the group classification problem for a nonlinear one-dimensional time-fractional heat conduction equation with full memory and dual-phase-lag, including thermal relaxation and thermal damping. The characteristic times of relaxation processes are assumed to be small enough and therefore a small parameter for fractional differential relaxation terms is introduced. All thermal properties of a medium are considered as functions of temperature. Group classification is performed with respect to groups of approximate point transformations (groups of approximate symmetries) admitted by the equation up to equivalence transformations. We prove that generally the equation admits five-parameter group of approximate transformations, and the cases of its extension to seven- and nine-parameters groups are found. Also, it is shown that the considered nonlinear equation has an infinite approximate symmetry group if the corresponding unperturbed equation is linear. We find that the equation in question always exactly inherits the symmetries of the unperturbed equation. The obtained results make it possible to construct approximately invariant solutions of equation under consideration. In particular, it follows from the classification found that the equation always has a traveling wave solution. The self-similar solutions can be constructed only if the medium thermal properties have power-law dependences on temperature. Ansatzes of these types of solutions are obtained and symmetry reductions of the equation under consideration to the corresponding ordinary fractional differential equations are performed.
Middle Volga Mathematical Society journal. 2025;27(1):49-68
49-68
On the Similarity of Upper Triangular Nilpotent Matrices of the $4$th and the $5$th Orders to a Generalized Jordan Block over the Ring of Integers
摘要
In this paper conditions for similarity of an upper triangular nilpotent matrix and a generalized Jordan block (i.,e. a matrix where only the elements of the first superdiagonal are non-zero) are considered. The problem is solved over the ring of integers. Necessary and sufficient conditions for similarity to a generalized Jordan block are obtained for the following classes of matrices: the fourth-order matrices of rank $3$ with nonzero elements of the first superdiagonal; the fifth-order matrices of rank $4$ and some additional restrictions on the elements of the first superdiagonal. These conditions are formulated in simple terms of divisibility and greatest common divisors of matrix elements. It is proved that if the first and last elements of the first superdiagonal are coprime, and the product of the remaining elements of this superdiagonal is equal to $1,$ then this matrix is similar to a generalized Jordan block. To obtain the similarity criterion, the following statement is used: if two nilpotent upper triangular matrices of order $n$ and rank $n - 1$ are similar over the ring of integers, then among the transforming matrices there is a triangular matrix. This statement reduces the problem of recognizing similarity to solving a system of linear equations in integers. The main tool for obtaining the results in the article is the criterion of consistency of a system of linear equations over the ring of integers.
Middle Volga Mathematical Society journal. 2025;27(1):69-80
69-80
The Krzyz conjecture and convex univalent functions
摘要
We obtained the sharp estimates of the moduli of the initial Taylor coefficients for functions $f$ of the class $B$ of bounded nonvanishing functions in the unit circle. Two types of estimates are obtained: one for ``large'' values of $|f(0)|$ and another one for ``small'' values of $|f(0)|$. The first type of estimates is asymptotic in the sense that it applies to an increasing number of initial coefficients as $|f(0)|$ increases. Similarly, the second type of estimates is asymptotic in the sense that it applies to an increasing number of initial coefficients as $|f(0)|$ decreases. Both types of estimates are deduced using methods of subordinate function theory and the Caratheodory-Toeplitz theorem for the Caratheodory class. This became possible due to the relation we found between the coefficients of convex univalent functions (class $S^0$) and the coefficients of the majorizing functions in the studied subclasses of the class $B$. The bounds for the applicability of the method are provided depending on $|f(0)|$ and on the coefficient number. The obtained results are applied to the theory of Laguerre polynomials. These results are compared with the previously known ones. The methods outlined here can be applied to arbitrary classes of subordinate functions.
Middle Volga Mathematical Society journal. 2025;27(1):81-96
81-96