Alexander Anatol'evich Zlotnik, Boris Nikolaevich Chetverushkin
The Cauchy problems are studied for a first-order multidimensional symmetric linear hyperbolic system of equations with variable coefficients and its singular perturbations that are second-order strongly parabolic and hyperbolic systems of equations with a small parameter $tau>0$ multiplying the second derivatives with respect to $x$ and $t$. The existence and uniqueness of weak solutions of all three systems and $tau$-uniform estimates for solutions of systems with perturbations are established. Estimates for the difference of solutions of the original system and the systems with perturbations are given, including ones of order $O(tau^{alpha/2})$ in the norm of $C(0,T;L^2(mathbb{R}^n))$, for an initial function $mathbf w_0$ in the Sobolev space $H^alpha(mathbb{R}^n)$, $alpha=1,2$, or the Nikolskii space $H_2^{alpha}(mathbb{R}^n)$, $0
{"title":"Properties and errors of second-order parabolic and hyperbolic perturbations of a first-order symmetric hyperbolic system","authors":"Alexander Anatol'evich Zlotnik, Boris Nikolaevich Chetverushkin","doi":"10.4213/sm9800e","DOIUrl":"https://doi.org/10.4213/sm9800e","url":null,"abstract":"The Cauchy problems are studied for a first-order multidimensional symmetric linear hyperbolic system of equations with variable coefficients and its singular perturbations that are second-order strongly parabolic and hyperbolic systems of equations with a small parameter $tau>0$ multiplying the second derivatives with respect to $x$ and $t$. The existence and uniqueness of weak solutions of all three systems and $tau$-uniform estimates for solutions of systems with perturbations are established. Estimates for the difference of solutions of the original system and the systems with perturbations are given, including ones of order $O(tau^{alpha/2})$ in the norm of $C(0,T;L^2(mathbb{R}^n))$, for an initial function $mathbf w_0$ in the Sobolev space $H^alpha(mathbb{R}^n)$, $alpha=1,2$, or the Nikolskii space $H_2^{alpha}(mathbb{R}^n)$, $0<alpha<2$, $alphaneq 1$, and under appropriate assumptions on the free term of the first-order system. For ${alpha=1/2}$ a wide class of discontinuous functions $mathbf w_0$ is covered. Estimates for derivatives of any order with respect to $x$ for solutions and of order $O(tau^{alpha/2})$ for their differences are also deduced. Applications of the results to the first-order system of gas dynamic equations linearized at a constant solution and to its perturbations, namely, the linearized second-order parabolic and hyperbolic quasi-gasdynamic systems of equations, are presented. Bibliography: 34 titles.","PeriodicalId":49573,"journal":{"name":"Sbornik Mathematics","volume":"72 7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136302358","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Sufficient conditions are presented for the construction of a regularization of a distribution in the form $a(sigma)f$, where $f$ is a distribution and $a(sigma)$ is an infinitely differentiable function outside a closed set $N$ which has power-like singularities of derivatives on $N$. Applications of such regularizations to an effective construction of solutions of the equation $Pu=f$, where $P(sigma)$ is a polynomial, are considered. Bibliography: 14 titles.
{"title":"Regularization of distributions","authors":"Aleksandr Leonidovich Pavlov","doi":"10.4213/sm9803e","DOIUrl":"https://doi.org/10.4213/sm9803e","url":null,"abstract":"Sufficient conditions are presented for the construction of a regularization of a distribution in the form $a(sigma)f$, where $f$ is a distribution and $a(sigma)$ is an infinitely differentiable function outside a closed set $N$ which has power-like singularities of derivatives on $N$. Applications of such regularizations to an effective construction of solutions of the equation $Pu=f$, where $P(sigma)$ is a polynomial, are considered. Bibliography: 14 titles.","PeriodicalId":49573,"journal":{"name":"Sbornik Mathematics","volume":"128 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136302356","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"‘Far interaction’ of small spectral perturbations of the Neumann boundary conditions for an elliptic system of differential equations in a three-dimensional domain","authors":"S. Nazarov","doi":"10.4213/sm9733e","DOIUrl":"https://doi.org/10.4213/sm9733e","url":null,"abstract":"","PeriodicalId":49573,"journal":{"name":"Sbornik Mathematics","volume":"87 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84200825","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The Bernstein-Szegő inequality for the Weyl derivative of real order $alphage 0$ of trigonometric polynomials of degree $n$ is considered. The aim is to find values of the parameters for which the sharp constant in this inequality is equal to $n^alpha$ (the classical value) in all $L_p$-spaces, $0le pleinfty$. The set of all such $alpha$ is described for some important particular cases of the Weyl-Szegő derivative, namely, for the Riesz derivative and for the conjugate Riesz derivative, for all $ninmathbb N$. Bibliography: 22 titles.
{"title":"Bernstein-Szegő inequality for the Riesz derivative of trigonometric polynomials in $L_p$-spaces, $0le pleinfty$, with classical value of the sharp constant","authors":"Anastasiya Olegovna Leont'eva","doi":"10.4213/sm9822e","DOIUrl":"https://doi.org/10.4213/sm9822e","url":null,"abstract":"The Bernstein-Szegő inequality for the Weyl derivative of real order $alphage 0$ of trigonometric polynomials of degree $n$ is considered. The aim is to find values of the parameters for which the sharp constant in this inequality is equal to $n^alpha$ (the classical value) in all $L_p$-spaces, $0le pleinfty$. The set of all such $alpha$ is described for some important particular cases of the Weyl-Szegő derivative, namely, for the Riesz derivative and for the conjugate Riesz derivative, for all $ninmathbb N$. Bibliography: 22 titles.","PeriodicalId":49573,"journal":{"name":"Sbornik Mathematics","volume":"32 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135596964","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Andrey Dymov, Sergei Kuksin, Alberto Maiocchi, Sergei Vladuts
In his paper from 1996 on quadratic forms Heath-Brown developed a version of the circle method to count points in the intersection of an unbounded quadric with a lattice of small period, when each point is assigned a weight, and approximated this quantity by the integral of the weight function against a measure on the quadric. The weight function is assumed to be $C_0^infty$-smooth and vanish near the singularity of the quadric. In our work we allow the weight function to be finitely smooth, not to vanish at the singularity and have an explicit decay at infinity. The paper uses only elementary number theory and is available to readers with no number-theoretic background. Bibliography: 15 titles.
{"title":"A refinement of Heath-Brown's theorem on quadratic forms","authors":"Andrey Dymov, Sergei Kuksin, Alberto Maiocchi, Sergei Vladuts","doi":"10.4213/sm9711e","DOIUrl":"https://doi.org/10.4213/sm9711e","url":null,"abstract":"In his paper from 1996 on quadratic forms Heath-Brown developed a version of the circle method to count points in the intersection of an unbounded quadric with a lattice of small period, when each point is assigned a weight, and approximated this quantity by the integral of the weight function against a measure on the quadric. The weight function is assumed to be $C_0^infty$-smooth and vanish near the singularity of the quadric. In our work we allow the weight function to be finitely smooth, not to vanish at the singularity and have an explicit decay at infinity. The paper uses only elementary number theory and is available to readers with no number-theoretic background. Bibliography: 15 titles.","PeriodicalId":49573,"journal":{"name":"Sbornik Mathematics","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135181021","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Sergey Vital'evich Zelik, Aleksei Andreevich Ilyin
We prove estimates for the $L^p$-norms of systems of functions and divergence-free vector functions that are orthonormal in the Sobolev space $H^1$ on the 2D sphere. As a corollary, order sharp constants for the embedding $H^1hookrightarrow L^q$, $q
{"title":"On a class of interpolation inequalities on the 2D sphere","authors":"Sergey Vital'evich Zelik, Aleksei Andreevich Ilyin","doi":"10.4213/sm9786e","DOIUrl":"https://doi.org/10.4213/sm9786e","url":null,"abstract":"We prove estimates for the $L^p$-norms of systems of functions and divergence-free vector functions that are orthonormal in the Sobolev space $H^1$ on the 2D sphere. As a corollary, order sharp constants for the embedding $H^1hookrightarrow L^q$, $q<infty$, are obtained in the Gagliardo-Nirenberg interpolation inequalities. Bibliography: 25 titles.","PeriodicalId":49573,"journal":{"name":"Sbornik Mathematics","volume":"31 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135596958","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Structures of cohomogeneity one on $S^6$ are under investigation. Examples of semi-Kähler and quasi-Kähler structures are constructed. Questions concerning the existence of almost Hermitian structures of cohomogeneity one on a round sphere are investigated. Bibliography: 14 titles.
{"title":"Some classes of almost Hermitian structures that can be realized on $S^6$","authors":"Nataliya Aleksandrovna Daurtseva","doi":"10.4213/sm9830e","DOIUrl":"https://doi.org/10.4213/sm9830e","url":null,"abstract":"Structures of cohomogeneity one on $S^6$ are under investigation. Examples of semi-Kähler and quasi-Kähler structures are constructed. Questions concerning the existence of almost Hermitian structures of cohomogeneity one on a round sphere are investigated. Bibliography: 14 titles.","PeriodicalId":49573,"journal":{"name":"Sbornik Mathematics","volume":"44 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134982655","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Elmir Rufkatovich Bigushev, Oleg Nikolaevich German
A three-dimensional analogue of the connection between the exponent of the irrationality of a real number and the growth of the partial quotients of its expansion in a simple continued fraction is investigated. As a multidimensional generalization of continued fractions, Klein polyhedra are considered. Bibliography: 12 titles.
{"title":"Diophantine exponents of lattices and the growth of higher-dimensional analogues of partial quotients","authors":"Elmir Rufkatovich Bigushev, Oleg Nikolaevich German","doi":"10.4213/sm9746e","DOIUrl":"https://doi.org/10.4213/sm9746e","url":null,"abstract":"A three-dimensional analogue of the connection between the exponent of the irrationality of a real number and the growth of the partial quotients of its expansion in a simple continued fraction is investigated. As a multidimensional generalization of continued fractions, Klein polyhedra are considered. Bibliography: 12 titles.","PeriodicalId":49573,"journal":{"name":"Sbornik Mathematics","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135596974","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
So-called normalized metrics are considered on the set of elements of a geometric progression. A full description of normalized metrics with maximal stabilizer, which is the group of integer degrees of the common ratio of the progression, is presented. Previously, it was known that this group is the stabilizer for the minimal normalized metric (inherited from the real line) and the maximal normalized metric (an intrinsic metric all paths in which pass through zero). The stabilizer of a metric space is understood as the set of positive numbers such that multiplying the metric by this number produces a metric space lying at a finite Gromov-Hausdorff distance from the original space. Bibliography: 5 titles.
{"title":"Geometric progression stabilizer in a general metric","authors":"Semeon Antonovich Bogatyi","doi":"10.4213/sm9782e","DOIUrl":"https://doi.org/10.4213/sm9782e","url":null,"abstract":"So-called normalized metrics are considered on the set of elements of a geometric progression. A full description of normalized metrics with maximal stabilizer, which is the group of integer degrees of the common ratio of the progression, is presented. Previously, it was known that this group is the stabilizer for the minimal normalized metric (inherited from the real line) and the maximal normalized metric (an intrinsic metric all paths in which pass through zero). The stabilizer of a metric space is understood as the set of positive numbers such that multiplying the metric by this number produces a metric space lying at a finite Gromov-Hausdorff distance from the original space. Bibliography: 5 titles.","PeriodicalId":49573,"journal":{"name":"Sbornik Mathematics","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135596976","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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