Pub Date : 2023-05-01DOI: 10.35634/2226-3594-2023-61-08
A. B. Khasanov, U. Hoitmetov, Sh.Q. Sobirov
In this paper, we consider the Cauchy problem for the non-stationary modified Korteweg–de Vries equation with an additional term and a self-consistent source in the case of moving eigenvalues. Also, the evolution of the scattering data of the Dirac operator is obtained, the potential of which is the solution of the loaded modified Korteweg–de Vries equation with a self-consistent source in the class of rapidly decreasing functions. Specific examples are given to illustrate the application of the obtained results.
{"title":"Integration of the mKdV Equation with nonstationary coefficients and additional terms in the case of moving eigenvalues","authors":"A. B. Khasanov, U. Hoitmetov, Sh.Q. Sobirov","doi":"10.35634/2226-3594-2023-61-08","DOIUrl":"https://doi.org/10.35634/2226-3594-2023-61-08","url":null,"abstract":"In this paper, we consider the Cauchy problem for the non-stationary modified Korteweg–de Vries equation with an additional term and a self-consistent source in the case of moving eigenvalues. Also, the evolution of the scattering data of the Dirac operator is obtained, the potential of which is the solution of the loaded modified Korteweg–de Vries equation with a self-consistent source in the class of rapidly decreasing functions. Specific examples are given to illustrate the application of the obtained results.","PeriodicalId":42053,"journal":{"name":"Izvestiya Instituta Matematiki i Informatiki-Udmurtskogo Gosudarstvennogo Universiteta","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73710902","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-05-01DOI: 10.35634/2226-3594-2023-61-03
A. R. Danilin, O. O. Kovrizhnykh
In this paper, we investigate a problem of optimal control over a finite time interval for a linear autonomous system with slow and fast variables in the class of piecewise continuous controls with smooth geometric constraints in the form of a ball. We consider a terminal convex quality index that depends on slow and fast variables. We substantiate a limit relation for the vector determining the optimal control as the small parameter tends to zero. We refine the limit relation for the case of an indirect control problem with a terminal quality index, which is the sum of values of two strictly convex continuously differentiable functions, the first of which depends only on slow variables, and the second one depends only on fast variables and has a minimum at zero. In doing so, we show that the first component of the determining vector converges to the determining vector of the limit problem while the second component tends to zero. In the problem of indirect control of a system of material points in a medium with resistance, we obtain the complete asymptotics of the determining vector in powers of a small parameter.
{"title":"Asymptotic expansion of the solution to an optimal control problem for a linear autonomous system with a terminal convex quality index depending on slow and fast variables","authors":"A. R. Danilin, O. O. Kovrizhnykh","doi":"10.35634/2226-3594-2023-61-03","DOIUrl":"https://doi.org/10.35634/2226-3594-2023-61-03","url":null,"abstract":"In this paper, we investigate a problem of optimal control over a finite time interval for a linear autonomous system with slow and fast variables in the class of piecewise continuous controls with smooth geometric constraints in the form of a ball. We consider a terminal convex quality index that depends on slow and fast variables. We substantiate a limit relation for the vector determining the optimal control as the small parameter tends to zero. We refine the limit relation for the case of an indirect control problem with a terminal quality index, which is the sum of values of two strictly convex continuously differentiable functions, the first of which depends only on slow variables, and the second one depends only on fast variables and has a minimum at zero. In doing so, we show that the first component of the determining vector converges to the determining vector of the limit problem while the second component tends to zero. In the problem of indirect control of a system of material points in a medium with resistance, we obtain the complete asymptotics of the determining vector in powers of a small parameter.","PeriodicalId":42053,"journal":{"name":"Izvestiya Instituta Matematiki i Informatiki-Udmurtskogo Gosudarstvennogo Universiteta","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75283811","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-05-01DOI: 10.35634/2226-3594-2023-61-09
A. G. Chentsov
We consider a minimax routing problem related to visiting megacities under precedence conditions and cost functions with task list dependence. It is supposed that some megacity system requiring visiting above all is selected. For solving, an approach with decomposition into a set of two minimax routing problems is proposed. A two-step widely understood dynamic programming procedure realizing an optimal composition solution is constructed. The above-mentioned optimality is established by theoretical methods. Application of the results obtained is possible under investigation of multi-stage processes connected with regular allocation of resources. Another variant of application concerns the particular case of one-element megacities (i.e., cities) and may be related to the issues of aviation logistics under organization of flights using one tool (airplane or helicopter) under system of tasks on the realization of passing cargo transportation with prioritization of visits realized above all.
{"title":"A bottleneck routing problem with a system of priority tasks","authors":"A. G. Chentsov","doi":"10.35634/2226-3594-2023-61-09","DOIUrl":"https://doi.org/10.35634/2226-3594-2023-61-09","url":null,"abstract":"We consider a minimax routing problem related to visiting megacities under precedence conditions and cost functions with task list dependence. It is supposed that some megacity system requiring visiting above all is selected. For solving, an approach with decomposition into a set of two minimax routing problems is proposed. A two-step widely understood dynamic programming procedure realizing an optimal composition solution is constructed. The above-mentioned optimality is established by theoretical methods. Application of the results obtained is possible under investigation of multi-stage processes connected with regular allocation of resources. Another variant of application concerns the particular case of one-element megacities (i.e., cities) and may be related to the issues of aviation logistics under organization of flights using one tool (airplane or helicopter) under system of tasks on the realization of passing cargo transportation with prioritization of visits realized above all.","PeriodicalId":42053,"journal":{"name":"Izvestiya Instituta Matematiki i Informatiki-Udmurtskogo Gosudarstvennogo Universiteta","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86332206","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-05-01DOI: 10.35634/2226-3594-2023-61-04
L. I. Danilov
Let ${mathcal B}$ be a Banach space and let ${mathcal M}^p({mathbb R};{mathcal B})$, $pgeqslant 1$, be the Marcinkiewicz space with a seminorm $| cdot | _{{mathcal M}^p}$. By $widetilde {mathfrak B}^p_c({mathbb R};{mathcal B})$ we denote the set of functions ${mathcal F}in {mathcal M}^p({mathbb R};{mathcal B})$ that satisfy the following three conditions: (1) $| {mathcal F}(cdot )-{mathcal F}(cdot +tau )| _{{mathcal M}^p}to 0$ as $tau to 0$, (2) for every $varepsilon >0$ the set of ($varepsilon ,| cdot | _{{mathcal M}^p}$)-almost periods of the function ${mathcal F}$ is relatively dense, (3) for every $varepsilon >0$ there exists a set $X(varepsilon )subseteq {mathbb R}$ such that $| chi _{X(varepsilon )}| _{{mathcal M}^1({mathbb R};{mathbb R})}0$ there is a number $delta >0$ such that the estimate $| chi _X{mathcal F}| _{{mathcal M}^p}0$, we prove under the condition $rho (u,F(cdot ))in widetilde {mathcal M}^{p,circ }({mathbb R};{mathbb R})$ the existence of a function ${mathcal F}in widetilde {mathfrak B}^p_c({mathbb R};U)cap widetilde {mathcal M}^{p,circ }({mathbb R};U)$ such that ${mathcal F}(t)in F(t)$ and $rho (u,{mathcal F}(t))
{"title":"On a class of Besicovitch almost periodic type selections of multivalued maps","authors":"L. I. Danilov","doi":"10.35634/2226-3594-2023-61-04","DOIUrl":"https://doi.org/10.35634/2226-3594-2023-61-04","url":null,"abstract":"Let ${mathcal B}$ be a Banach space and let ${mathcal M}^p({mathbb R};{mathcal B})$, $pgeqslant 1$, be the Marcinkiewicz space with a seminorm $| cdot | _{{mathcal M}^p}$. By $widetilde {mathfrak B}^p_c({mathbb R};{mathcal B})$ we denote the set of functions ${mathcal F}in {mathcal M}^p({mathbb R};{mathcal B})$ that satisfy the following three conditions: (1) $| {mathcal F}(cdot )-{mathcal F}(cdot +tau )| _{{mathcal M}^p}to 0$ as $tau to 0$, (2) for every $varepsilon >0$ the set of ($varepsilon ,| cdot | _{{mathcal M}^p}$)-almost periods of the function ${mathcal F}$ is relatively dense, (3) for every $varepsilon >0$ there exists a set $X(varepsilon )subseteq {mathbb R}$ such that $| chi _{X(varepsilon )}| _{{mathcal M}^1({mathbb R};{mathbb R})}<varepsilon $ and the set ${ {mathcal F}(t):tin {mathbb R}, backslash , X(varepsilon )} $ has a finite $varepsilon $-net. Let $widetilde {mathcal M}^{p,circ }({mathbb R};{mathcal B})$ be the set of functions ${mathcal F}in {mathcal M}^p({mathbb R};{mathcal B})$ that satisfy the condition (3) and the following condition: for any $varepsilon >0$ there is a number $delta >0$ such that the estimate $| chi _X{mathcal F}| _{{mathcal M}^p}<varepsilon $ is fulfilled for all sets $Xsubseteq {mathbb R}$ with $| chi _X| _{{mathcal M}^1({mathbb R};{mathbb R})}<delta $. The sets $widetilde {mathfrak B}^p_c({mathbb R};U)$ and $widetilde {mathcal M}^{p,circ }({mathbb R};U)$ for a complete metric space $(U,rho )$ are defined analogously. By ${mathrm {cl}}, U$ denote the metric space of nonempty, closed, and bounded subsets of the space $(U,rho )$ with Hausdorff metrics. In the paper, in particular, for any $Fin widetilde {mathfrak B}^p_c({mathbb R};{mathrm {cl}}, U)$, $pgeqslant 1$, and $uin U$, $varepsilon >0$, we prove under the condition $rho (u,F(cdot ))in widetilde {mathcal M}^{p,circ }({mathbb R};{mathbb R})$ the existence of a function ${mathcal F}in widetilde {mathfrak B}^p_c({mathbb R};U)cap widetilde {mathcal M}^{p,circ }({mathbb R};U)$ such that ${mathcal F}(t)in F(t)$ and $rho (u,{mathcal F}(t))<varepsilon +rho (u,F(t))$ for almost every $tin {mathbb R}$.","PeriodicalId":42053,"journal":{"name":"Izvestiya Instituta Matematiki i Informatiki-Udmurtskogo Gosudarstvennogo Universiteta","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90229948","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-05-01DOI: 10.35634/2226-3594-2023-61-05
P. Lebedev, O. Kuvshinov
The problem of covering a compact planar set $M$ with a set of congruent disks is considered. It is assumed that the centers of the circles belong to some lattice. The criterion of optimality in one case is the minimum of the number of elements of the covering, and in the other case — the minimum of the Hausdorff deviation of the union of elements of the covering from the set $M$. To solve the problems, transformations of parallel transfer and rotation with the center at the origin can be applied to the lattice. Statements concerning sufficient conditions for sets of circles that provide solutions to the problems are proved. Numerical algorithms based on minimizing the Hausdorff deviation between two flat compacts are proposed. Solutions of a number of examples are given for various figures of $M$.
{"title":"Algorithms for constructing suboptimal coverings of plane figures with disks in the class of regular lattices","authors":"P. Lebedev, O. Kuvshinov","doi":"10.35634/2226-3594-2023-61-05","DOIUrl":"https://doi.org/10.35634/2226-3594-2023-61-05","url":null,"abstract":"The problem of covering a compact planar set $M$ with a set of congruent disks is considered. It is assumed that the centers of the circles belong to some lattice. The criterion of optimality in one case is the minimum of the number of elements of the covering, and in the other case — the minimum of the Hausdorff deviation of the union of elements of the covering from the set $M$. To solve the problems, transformations of parallel transfer and rotation with the center at the origin can be applied to the lattice. Statements concerning sufficient conditions for sets of circles that provide solutions to the problems are proved. Numerical algorithms based on minimizing the Hausdorff deviation between two flat compacts are proposed. Solutions of a number of examples are given for various figures of $M$.","PeriodicalId":42053,"journal":{"name":"Izvestiya Instituta Matematiki i Informatiki-Udmurtskogo Gosudarstvennogo Universiteta","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83840804","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-05-01DOI: 10.35634/2226-3594-2023-61-01
A. I. Blagodatskikh
The problem of pursuit of a group of $m$ evaders $(mgeqslant 1)$ in conflict-controlled processes with equal opportunities is considered. It is said that in the problem of chasing one evader $(m = 1)$, multiple capture occurs if a given number of pursuers catch him, and the moments of capture may not coincide. In the problem of simultaneous multiple capture of one evader, it is required that the moments of capture coincide. Simultaneous multiple capture of the whole group of evaders $(m geqslant 2)$ occurs if, as a result of pursuit, each evader is repeatedly caught simultaneously, and at the same time. In terms of the initial positions of the participants, necessary and sufficient conditions for the simultaneous multiple capture of the whole group of evaders are obtained.
{"title":"Synchronous implementation of simultaneous multiple captures of evaders","authors":"A. I. Blagodatskikh","doi":"10.35634/2226-3594-2023-61-01","DOIUrl":"https://doi.org/10.35634/2226-3594-2023-61-01","url":null,"abstract":"The problem of pursuit of a group of $m$ evaders $(mgeqslant 1)$ in conflict-controlled processes with equal opportunities is considered. It is said that in the problem of chasing one evader $(m = 1)$, multiple capture occurs if a given number of pursuers catch him, and the moments of capture may not coincide. In the problem of simultaneous multiple capture of one evader, it is required that the moments of capture coincide. Simultaneous multiple capture of the whole group of evaders $(m geqslant 2)$ occurs if, as a result of pursuit, each evader is repeatedly caught simultaneously, and at the same time. In terms of the initial positions of the participants, necessary and sufficient conditions for the simultaneous multiple capture of the whole group of evaders are obtained.","PeriodicalId":42053,"journal":{"name":"Izvestiya Instituta Matematiki i Informatiki-Udmurtskogo Gosudarstvennogo Universiteta","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87498781","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-05-01DOI: 10.35634/2226-3594-2023-61-02
M. S. Woldeab, L. I. Rodina
We consider a population whose dynamics in the absence of exploitation is given by a system of linear homogeneous differential equations, and some random shares of the resource of each species at fixed times, are extracted from this population. We assume that the harvesting process can be controlled in such a way as to limit the amount of the extracted resource in order to increase the size of the next harvesting. A method for harvesting a resource is described, in which the largest value of the average time benefit is reached with a probability of one, provided that the initial amount of the population is constantly maintained or periodically restored. The harvesting modes are also considered in which the average time benefit is infinite. To prove the main assertions, we use the corollary of the law of large numbers proved by A.N. Kolmogorov. The results on the optimal resource extraction for systems of linear difference equations, a particular case of which are Leslie and Lefkovich population dynamics models, are given.
{"title":"On the exploitation of a population given by a system of linear equations with random parameters","authors":"M. S. Woldeab, L. I. Rodina","doi":"10.35634/2226-3594-2023-61-02","DOIUrl":"https://doi.org/10.35634/2226-3594-2023-61-02","url":null,"abstract":"We consider a population whose dynamics in the absence of exploitation is given by a system of linear homogeneous differential equations, and some random shares of the resource of each species at fixed times, are extracted from this population. We assume that the harvesting process can be controlled in such a way as to limit the amount of the extracted resource in order to increase the size of the next harvesting. A method for harvesting a resource is described, in which the largest value of the average time benefit is reached with a probability of one, provided that the initial amount of the population is constantly maintained or periodically restored. The harvesting modes are also considered in which the average time benefit is infinite. To prove the main assertions, we use the corollary of the law of large numbers proved by A.N. Kolmogorov. The results on the optimal resource extraction for systems of linear difference equations, a particular case of which are Leslie and Lefkovich population dynamics models, are given.","PeriodicalId":42053,"journal":{"name":"Izvestiya Instituta Matematiki i Informatiki-Udmurtskogo Gosudarstvennogo Universiteta","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74951021","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-11-01DOI: 10.35634/2226-3594-2022-60-03
O. Kuvshinov
The paper studies the geometry of a closed nonconvex smooth simply connected curve on a plane — of the Cassini oval, as well as the geometry of the $varepsilon$-layer around the set whose boundary is the Cassini oval. Various analytical representations of the $varepsilon$-layer boundary are formed, and special points of this boundary are described. The measure of nonconvexity $alpha$ of a simply connected set whose boundary is the Cassini oval is determined, and the angular characteristic of nonconvexity of its $varepsilon$-neighborhood is defined.
{"title":"About the geometry of the Cassini oval, its non-convexity degree and $varepsilon$-offset layer","authors":"O. Kuvshinov","doi":"10.35634/2226-3594-2022-60-03","DOIUrl":"https://doi.org/10.35634/2226-3594-2022-60-03","url":null,"abstract":"The paper studies the geometry of a closed nonconvex smooth simply connected curve on a plane — of the Cassini oval, as well as the geometry of the $varepsilon$-layer around the set whose boundary is the Cassini oval. Various analytical representations of the $varepsilon$-layer boundary are formed, and special points of this boundary are described. The measure of nonconvexity $alpha$ of a simply connected set whose boundary is the Cassini oval is determined, and the angular characteristic of nonconvexity of its $varepsilon$-neighborhood is defined.","PeriodicalId":42053,"journal":{"name":"Izvestiya Instituta Matematiki i Informatiki-Udmurtskogo Gosudarstvennogo Universiteta","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81934464","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-11-01DOI: 10.35634/2226-3594-2022-60-07
V. Ushakov, A. Ushakov
A conflict-controlled system in a finite-dimensional Euclidean space is considered. We study the game problem of approaching the system to the goal set in the phase space over a finite time interval. The study of the problem is based on methods developed in the theory of positional differential games. Within the framework of this theory, an approach to constructing approximate solutions to the approach problem is presented.
{"title":"On the approach problem for a control system on a finite time interval","authors":"V. Ushakov, A. Ushakov","doi":"10.35634/2226-3594-2022-60-07","DOIUrl":"https://doi.org/10.35634/2226-3594-2022-60-07","url":null,"abstract":"A conflict-controlled system in a finite-dimensional Euclidean space is considered. We study the game problem of approaching the system to the goal set in the phase space over a finite time interval. The study of the problem is based on methods developed in the theory of positional differential games. Within the framework of this theory, an approach to constructing approximate solutions to the approach problem is presented.","PeriodicalId":42053,"journal":{"name":"Izvestiya Instituta Matematiki i Informatiki-Udmurtskogo Gosudarstvennogo Universiteta","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85468523","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-11-01DOI: 10.35634/2226-3594-2022-60-01
B. Bayraktar, J. N. Nápoles Valdés
In this article, we establish several inequalities for $(h,m)$-convex maps, related to weighted integrals, used in previous works. Throughout the work, we show that our results generalize several of the integral inequalities known from the literature.
{"title":"New generalized integral inequalities via $(h,m)$-convex modified functions","authors":"B. Bayraktar, J. N. Nápoles Valdés","doi":"10.35634/2226-3594-2022-60-01","DOIUrl":"https://doi.org/10.35634/2226-3594-2022-60-01","url":null,"abstract":"In this article, we establish several inequalities for $(h,m)$-convex maps, related to weighted integrals, used in previous works. Throughout the work, we show that our results generalize several of the integral inequalities known from the literature.","PeriodicalId":42053,"journal":{"name":"Izvestiya Instituta Matematiki i Informatiki-Udmurtskogo Gosudarstvennogo Universiteta","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76130332","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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