The autonomous Hietarinta equation is a well-known example of the quad-graph discrete equation which is consistent around the cube. In a recent work, it was conjectured that this equation is Darboux integrable (i.e., for each of two independent discrete variables there exist non-trivial functions that remain unchanged on solutions of the equation after the shift in this discrete variable). We demonstrate that this conjecture is not true for generic values of the equation coefficients. To do this, we employ two-point invertible transformations introduced by R.I.~Yamilov. We prove that an autonomous difference equation on the quad-graph cannot be Darboux integrable if a transformation of the above type maps solutions of this equation into its solutions again. This implies that the generic Hietarinta equation is not Darboux integrable since the Hietarinta equation in the general case possesses the two-point invertible auto-transformations. Along the way, all Darboux integrable subcases of the Hietarinta equation are found. All of them are reduced by point transformations to already known integrable equations. At the end of the article, we also briefly describe another way to prove the Darboux non-integrability of the Hietarinta equation. This alternative way is based on the known fact that a difference substitution relates this equation to a linear one. Thus, the Hietarinta equation gives us an example of a quad-graph equation that is linearizable but not Darboux integrable.
{"title":"On Darboux non-integrability of Hietarinta equation","authors":"S. Startsev","doi":"10.13108/2021-13-2-160","DOIUrl":"https://doi.org/10.13108/2021-13-2-160","url":null,"abstract":"The autonomous Hietarinta equation is a well-known example of the quad-graph discrete equation which is consistent around the cube. In a recent work, it was conjectured that this equation is Darboux integrable (i.e., for each of two independent discrete variables there exist non-trivial functions that remain unchanged on solutions of the equation after the shift in this discrete variable). We demonstrate that this conjecture is not true for generic values of the equation coefficients. To do this, we employ two-point invertible transformations introduced by R.I.~Yamilov. We prove that an autonomous difference equation on the quad-graph cannot be Darboux integrable if a transformation of the above type maps solutions of this equation into its solutions again. This implies that the generic Hietarinta equation is not Darboux integrable since the Hietarinta equation in the general case possesses the two-point invertible auto-transformations. Along the way, all Darboux integrable subcases of the Hietarinta equation are found. All of them are reduced by point transformations to already known integrable equations. At the end of the article, we also briefly describe another way to prove the Darboux non-integrability of the Hietarinta equation. This alternative way is based on the known fact that a difference substitution relates this equation to a linear one. Thus, the Hietarinta equation gives us an example of a quad-graph equation that is linearizable but not Darboux integrable.","PeriodicalId":138462,"journal":{"name":"Ufimskii Matematicheskii Zhurnal","volume":"90 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116528027","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}
. We provide a general solution to a first order ordinary differential equation with a rational right-hand side, which arises in constructing asymptotics for large time of simultaneous solutions of the Korteweg-de Vries equation and the stationary part of its higher non-autonomous symmetry. This symmetry is determined by a linear combination of the first higher autonomous symmetry of the Korteweg-de Vries equation and of its classical Galileo symmetry. This general solution depends on an arbitrary parameter. By the implicit function theorem, locally it is determined by the first integral explicitly written in terms of hypergeometric functions. A particular case of the general solution defines self-similar solutions of the Whitham equations, found earlier by G.V. Potemin in 1988. In the well-known works by A.V. Gurevich and L.P. Pitaevsky in early 1970s, it was estab-lished that these solutions of the Whitham equations describe the origination in the leading term of non-damping oscillating waves in a wide range of problems with a small dispersion. The result of this work supports once again an empirical law saying that under various passages to the limits, integrable equations can produce only integrable, in certain sense, equations. We propose a general conjecture: integrable ordinary differential equations similar to that considered in the present paper should also arise in describing the asymptotics at large times for other symmetry solutions to evolution equations admitting the application of the inverse scattering transform method.
{"title":"Integrable Abel equation and asymptotics of symmetry solutions of Korteweg-de Vries equation","authors":"B. Suleimanov, A. M. Shavlukov","doi":"10.13108/2021-13-2-99","DOIUrl":"https://doi.org/10.13108/2021-13-2-99","url":null,"abstract":". We provide a general solution to a first order ordinary differential equation with a rational right-hand side, which arises in constructing asymptotics for large time of simultaneous solutions of the Korteweg-de Vries equation and the stationary part of its higher non-autonomous symmetry. This symmetry is determined by a linear combination of the first higher autonomous symmetry of the Korteweg-de Vries equation and of its classical Galileo symmetry. This general solution depends on an arbitrary parameter. By the implicit function theorem, locally it is determined by the first integral explicitly written in terms of hypergeometric functions. A particular case of the general solution defines self-similar solutions of the Whitham equations, found earlier by G.V. Potemin in 1988. In the well-known works by A.V. Gurevich and L.P. Pitaevsky in early 1970s, it was estab-lished that these solutions of the Whitham equations describe the origination in the leading term of non-damping oscillating waves in a wide range of problems with a small dispersion. The result of this work supports once again an empirical law saying that under various passages to the limits, integrable equations can produce only integrable, in certain sense, equations. We propose a general conjecture: integrable ordinary differential equations similar to that considered in the present paper should also arise in describing the asymptotics at large times for other symmetry solutions to evolution equations admitting the application of the inverse scattering transform method.","PeriodicalId":138462,"journal":{"name":"Ufimskii Matematicheskii Zhurnal","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117085605","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}
We construct non-Abelian analogs for some KdV type equations, including the (rational form of) exponential Calogero--Degasperis equation and generalizations of the Schwarzian KdV equation. Equations and differential substitutions under study contain arbitrary non-Abelian parameters.
{"title":"Differential substitutions for non-Abelian equations of KdV type","authors":"Vsevolod Eduardovich Adler","doi":"10.13108/2021-13-2-107","DOIUrl":"https://doi.org/10.13108/2021-13-2-107","url":null,"abstract":"We construct non-Abelian analogs for some KdV type equations, including the (rational form of) exponential Calogero--Degasperis equation and generalizations of the Schwarzian KdV equation. Equations and differential substitutions under study contain arbitrary non-Abelian parameters.","PeriodicalId":138462,"journal":{"name":"Ufimskii Matematicheskii Zhurnal","volume":"77 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122021275","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}
We observe the continuous-time Markov Branching Process without high-order moments and allowing Immigration. Limit properties of transition functions and their convergence to invariant measures are investigated. Main mathematical tool is regularly varying generating functions with remainder.
{"title":"On asymptotic structure of continuous-time Markov branching processes allowing immigration without higher-order moments","authors":"A. Imomov, Abror Khujanazarovich Meyliev","doi":"10.13108/2021-13-1-137","DOIUrl":"https://doi.org/10.13108/2021-13-1-137","url":null,"abstract":"We observe the continuous-time Markov Branching Process without high-order moments and allowing Immigration. Limit properties of transition functions and their convergence to invariant measures are investigated. Main mathematical tool is regularly varying generating functions with remainder.","PeriodicalId":138462,"journal":{"name":"Ufimskii Matematicheskii Zhurnal","volume":"146 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114562525","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}
This paper discusses some regularity of almost periodic solutions of the Poisson's equation $-Delta u = f$ in $mathbb{R}^n$, where $f$ is an almost periodic function. It has been proved by Sibuya [Almost periodic solutions of Poisson's equation. Proc. Amer. Math. Soc., 28:195--198, 1971.] that if $u$ is a bounded continuous function and solves the Poisson's equation in the distribution sense, then $u$ is an almost periodic function. In this work, we relax the assumption of the usual boundedness into boundedness in the sense of distribution which we refer to as a bounded generalized function. The set of bounded generalized functions are wider than the set of usual bounded functions. Then, upon assuming that $u$ is a bounded generalized function and solves the Poisson's equation in the distribution sense, we prove that this solution is bounded in the usual sense, continuous and almost periodic. Moreover, we show that the first partial derivatives of the solution $partial u/ partial x_i$, $i=1, ldots, n$, are also continuous, bounded, and almost periodic functions. The technique is based on extending a representation formula using Green's function for Poisson's equation for solutions in the distribution sense. Some useful properties of distributions are also shown that can be used to study other elliptic problems.
{"title":"Regularity of almost periodic solutions of Poisson equation","authors":"'Ergash Muhamadiev, Murtazo Nazarov","doi":"10.13108/2020-12-2-97","DOIUrl":"https://doi.org/10.13108/2020-12-2-97","url":null,"abstract":"This paper discusses some regularity of almost periodic solutions of the Poisson's equation $-Delta u = f$ in $mathbb{R}^n$, where $f$ is an almost periodic function. It has been proved by Sibuya [Almost periodic solutions of Poisson's equation. Proc. Amer. Math. Soc., 28:195--198, 1971.] that if $u$ is a bounded continuous function and solves the Poisson's equation in the distribution sense, then $u$ is an almost periodic function. In this work, we relax the assumption of the usual boundedness into boundedness in the sense of distribution which we refer to as a bounded generalized function. The set of bounded generalized functions are wider than the set of usual bounded functions. Then, upon assuming that $u$ is a bounded generalized function and solves the Poisson's equation in the distribution sense, we prove that this solution is bounded in the usual sense, continuous and almost periodic. Moreover, we show that the first partial derivatives of the solution $partial u/ partial x_i$, $i=1, ldots, n$, are also continuous, bounded, and almost periodic functions. The technique is based on extending a representation formula using Green's function for Poisson's equation for solutions in the distribution sense. Some useful properties of distributions are also shown that can be used to study other elliptic problems.","PeriodicalId":138462,"journal":{"name":"Ufimskii Matematicheskii Zhurnal","volume":"228 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122485994","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}
In this article, we show constructively that Morrey spaces are not uniformly non-$ell^1_n$ for any $nge 2$. This result is sharper than those previously obtained in cite{GKSS, MG}, which show that Morrey spaces are not uniformly non-square and also not uniformly non-octahedral. We also discuss the $n$-th James constant $C_{rm J}^{(n)}(X)$ and the $n$-th Von Neumann-Jordan constant $C_{rm NJ}^{(n)}(X)$ for a Banach space $X$, and obtain that both constants for any Morrey space $mathcal{M}^p_q(mathbb{R}^d)$ with $1le p
在这篇文章中，我们建设性地证明了Morrey空间对于任意$n 2$不是一致非$ n$的。这一结果比先前在cite{GKSS, MG}中得到的结果更清晰，表明Morrey空间不是均匀非正方形的，也不是均匀非八面体的。我们还讨论了Banach空间$X$的$n$- James常数$C_{rm J}^{(n)}(X)$和$n$- Von Neumann-Jordan常数$C_{rm NJ}^{(n)}(X)$，并得到了这两个常数对于任意Morrey空间$mathcal{M}^p_q(mathbb{R}^d)$具有$1le p
{"title":"On geometric properties of Morrey spaces","authors":"H. Gunawan, D. Hakim, A. S. Putri","doi":"10.13108/2021-13-1-131","DOIUrl":"https://doi.org/10.13108/2021-13-1-131","url":null,"abstract":"In this article, we show constructively that Morrey spaces are not uniformly non-$ell^1_n$ for any $nge 2$. This result is sharper than those previously obtained in cite{GKSS, MG}, which show that Morrey spaces are not uniformly non-square and also not uniformly non-octahedral. We also discuss the $n$-th James constant $C_{rm J}^{(n)}(X)$ and the $n$-th Von Neumann-Jordan constant $C_{rm NJ}^{(n)}(X)$ for a Banach space $X$, and obtain that both constants for any Morrey space $mathcal{M}^p_q(mathbb{R}^d)$ with $1le p","PeriodicalId":138462,"journal":{"name":"Ufimskii Matematicheskii Zhurnal","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131988038","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}
We present a series of Darboux integrable discrete equations on the square lattice. Equations of the series are numbered with natural numbers $M$. All the equations have a first integral of the first order in one of directions of the two-dimensional lattice. The minimal order of a first integral in the other direction is equal to $3M$ for an equation with the number $M$. In the cases $M=1, 2, 3$ we show that those equations are integrable in quadratures. More precisely, we construct their general solutions in terms of the discrete integrals. We also construct a modified series of Darboux integrable discrete equations which have in different directions the first integrals of the orders $2$ and $3M-1$, where $M$ is the equation number in series. Both first integrals are unobvious in this case.
{"title":"On series of Darboux integrable discrete equations on square lattice","authors":"R. Garifullin, R. Yamilov","doi":"10.13108/2019-11-3-99","DOIUrl":"https://doi.org/10.13108/2019-11-3-99","url":null,"abstract":"We present a series of Darboux integrable discrete equations on the square lattice. Equations of the series are numbered with natural numbers $M$. All the equations have a first integral of the first order in one of directions of the two-dimensional lattice. The minimal order of a first integral in the other direction is equal to $3M$ for an equation with the number $M$. \u0000In the cases $M=1, 2, 3$ we show that those equations are integrable in quadratures. More precisely, we construct their general solutions in terms of the discrete integrals. \u0000We also construct a modified series of Darboux integrable discrete equations which have in different directions the first integrals of the orders $2$ and $3M-1$, where $M$ is the equation number in series. Both first integrals are unobvious in this case.","PeriodicalId":138462,"journal":{"name":"Ufimskii Matematicheskii Zhurnal","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133996732","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}
We study the class of Hankel matrices for which the $ktimes k$-block-matrices are positive semi-definite. We prove that a $ktimes k$-block-matrix has non zero determinant if and only if all $ktimes k$-block matrices have non zero determinant. We use this result to extend the notion of propagation phenomena to $k$-hyponormal weighted shifts. Finally we give a study on invariance of $k$-hyponormal weighted shifts under one rank perturbation.
{"title":"Weak positive matrices and hyponormal weighted shifts","authors":"Hamza El-Azhar, K. Idrissi, E. Zerouali","doi":"10.13108/2019-11-3-88","DOIUrl":"https://doi.org/10.13108/2019-11-3-88","url":null,"abstract":"We study the class of Hankel matrices for which the $ktimes k$-block-matrices are positive semi-definite. We prove that a $ktimes k$-block-matrix has non zero determinant if and only if all $ktimes k$-block matrices have non zero determinant. We use this result to extend the notion of propagation phenomena to $k$-hyponormal weighted shifts. Finally we give a study on invariance of $k$-hyponormal weighted shifts under one rank perturbation.","PeriodicalId":138462,"journal":{"name":"Ufimskii Matematicheskii Zhurnal","volume":"32 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125113893","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}
. We consider and study the notions of a random operator, random operator-valued function and a random semigroup defined on a Hilbert space as well as their averagings. We obtain conditions under which the averaging of a random strongly continuous function is also strongly continuous. In particular, we show that each random strongly continuous contractive operator-valued function possesses a strongly continuous contractive averaging. We consider two particular random semigroups: a matrix semigroup of random orthogonal transformations of Euclidean space and a semigroup of operators defined on the Hilbert space of functions square integrable on the sphere in the Euclidean space such that these operators describe random orthogonal transformations of the domain these functions. The latter semigroup is called a random rotation semigroup; it can be interpreted as a random walk on the sphere. We prove the existence of the averaging for both random semigroups. We study an operator-valued function obtained by replacing the time variable 𝑡 by √ 𝑡 in averaging of the random rotation semigroup. By means of Chernoff theorem, under some conditions, we prove the convergence of the sequence of Feynman–Chernoff iterations of this function to a strongly continuous semigroup describing the diffusion on the sphere in the Euclidean space. In order to do this, we first find and study the derivative of this operator-valued function at zero being at the same time the generator of the limiting semigroup. We obtain a simple divergence form of this generator. By means of this form we obtain conditions ensuring that this generator is a second order elliptic operator; under these conditions we prove that it is essentially self-adjoint.
{"title":"Averaging of random orthogonal transformations of domain of functions","authors":"Konstantin Yurievich Zamana","doi":"10.13108/2021-13-4-23","DOIUrl":"https://doi.org/10.13108/2021-13-4-23","url":null,"abstract":". We consider and study the notions of a random operator, random operator-valued function and a random semigroup defined on a Hilbert space as well as their averagings. We obtain conditions under which the averaging of a random strongly continuous function is also strongly continuous. In particular, we show that each random strongly continuous contractive operator-valued function possesses a strongly continuous contractive averaging. We consider two particular random semigroups: a matrix semigroup of random orthogonal transformations of Euclidean space and a semigroup of operators defined on the Hilbert space of functions square integrable on the sphere in the Euclidean space such that these operators describe random orthogonal transformations of the domain these functions. The latter semigroup is called a random rotation semigroup; it can be interpreted as a random walk on the sphere. We prove the existence of the averaging for both random semigroups. We study an operator-valued function obtained by replacing the time variable 𝑡 by √ 𝑡 in averaging of the random rotation semigroup. By means of Chernoff theorem, under some conditions, we prove the convergence of the sequence of Feynman–Chernoff iterations of this function to a strongly continuous semigroup describing the diffusion on the sphere in the Euclidean space. In order to do this, we first find and study the derivative of this operator-valued function at zero being at the same time the generator of the limiting semigroup. We obtain a simple divergence form of this generator. By means of this form we obtain conditions ensuring that this generator is a second order elliptic operator; under these conditions we prove that it is essentially self-adjoint.","PeriodicalId":138462,"journal":{"name":"Ufimskii Matematicheskii Zhurnal","volume":"31 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125880511","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}
{"title":"Law of large numbers for weakly dependent random variables with values in $D[0,1]$","authors":"Olimjon Shukurovich Sharipov, A. F. Norjigitov","doi":"10.13108/2021-13-4-123","DOIUrl":"https://doi.org/10.13108/2021-13-4-123","url":null,"abstract":"","PeriodicalId":138462,"journal":{"name":"Ufimskii Matematicheskii Zhurnal","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125302670","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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