Pub Date : 2024-07-29DOI: 10.3103/s0027132224700116
M. V. Shamolin
Abstract
Tensor invariants (differential forms) for homogeneous dynamical systems on tangent bundles to smooth two-dimensional manifolds are presented in the paper. The connection between the presence of these invariants and the full set of first integrals necessary for integration of geodesic, potential, and dissipative systems is shown. At the same time, the introduced force fields make the considered systems dissipative with dissipation of different signs and generalize the previously considered ones. We represent the typical examples from rigid body dynamics.
{"title":"Invariants of Systems Having a Small Number of Degrees of Freedom with Dissipation","authors":"M. V. Shamolin","doi":"10.3103/s0027132224700116","DOIUrl":"https://doi.org/10.3103/s0027132224700116","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>Tensor invariants (differential forms) for homogeneous dynamical systems on tangent bundles to smooth two-dimensional manifolds are presented in the paper. The connection between the presence of these invariants and the full set of first integrals necessary for integration of geodesic, potential, and dissipative systems is shown. At the same time, the introduced force fields make the considered systems dissipative with dissipation of different signs and generalize the previously considered ones. We represent the typical examples from rigid body dynamics.</p>","PeriodicalId":42963,"journal":{"name":"Moscow University Mathematics Bulletin","volume":"20 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141862584","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 : 2024-07-29DOI: 10.3103/s002713222470013x
G. V. Fedorov
Abstract
For each (ngeqslant 3), three nonequivalent polynomials (finmathbb{Q}[x]) of degree (n) were previously constructed for which (sqrt{f}) has a periodic continued fraction expansion in the field (mathbb{Q}((x))). In this paper, for each (ngeqslant 5), two new polynomials (fin K[x]) of degree (n) are found, defined over the field (K), ([K:mathbb{Q}]=[(n-1)/2]), for which (sqrt{f}) has a periodic continued fraction expansion in the field (K((x))).
AbstractFor each (ngeqslant 3), three nonequivalent polynomials (finmathbb{Q}[x]) of degree (n) previously been constructed for which (sqrt{f}) has a periodic continued fraction expansion in the field (mathbb{Q}((x))).在本文中，对于每一个 (ngeqslant 5), 都找到了两个新的度（n）的多项式 (fin K[x]), 定义在 (K) 场上，([K:mathbb{Q}]=[(n-1)/2])，对于这些多项式，(sqrt{f}) 在 (K((x))) 场中有一个周期性的连续分数展开。
{"title":"On the Sequences of Polynomials $$boldsymbol{f}$$ with a Periodic Continued Fraction Expansion $$sqrt{boldsymbol{f}}$$","authors":"G. V. Fedorov","doi":"10.3103/s002713222470013x","DOIUrl":"https://doi.org/10.3103/s002713222470013x","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>For each <span>(ngeqslant 3)</span>, three nonequivalent polynomials <span>(finmathbb{Q}[x])</span> of degree <span>(n)</span> were previously constructed for which <span>(sqrt{f})</span> has a periodic continued fraction expansion in the field <span>(mathbb{Q}((x)))</span>. In this paper, for each <span>(ngeqslant 5)</span>, two new polynomials <span>(fin K[x])</span> of degree <span>(n)</span> are found, defined over the field <span>(K)</span>, <span>([K:mathbb{Q}]=[(n-1)/2])</span>, for which <span>(sqrt{f})</span> has a periodic continued fraction expansion in the field <span>(K((x)))</span>.</p>","PeriodicalId":42963,"journal":{"name":"Moscow University Mathematics Bulletin","volume":"74 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141862605","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 : 2024-07-29DOI: 10.3103/s0027132224700128
M. I. Ismailov, I. F. Aliyarova
Abstract
The paper is focused on the basis property of the system of exponentials and trigonometric systems of sine and cosine functions in a separable subspace of the weighted grand Lebesgue space generated by the shift operator. In this paper, with the help of the shift operator, a separable subspace (G_{p),rho}(a,b)) of the weighted space of the grand Lebesgue space (L_{p),rho}(a,b)) is defined. The density in (G_{p),rho}(a,b)) of the set (G_{0}^{infty}([a,b])) of infinitely differentiable functions that are finite on ([a,b]) is studied. It is proved that if the weight function (rho) satisfies the Mackenhoupt condition, then the system of exponentials (left{e^{int}right}_{nin Z}) forms a basis in (G_{p),rho}(-pi,pi)), and trigonometric systems of sine (left{sin ntright}_{ngeqslant 1}) and cosine (left{cos ntright}_{ngeqslant 0}) functions form bases in (G_{p),rho}(0,pi)).
{"title":"On the Basis Property of the System of Exponentials and Trigonometric Systems of Sine and Cosine Functions in Weighted Grand Lebesgue Spaces","authors":"M. I. Ismailov, I. F. Aliyarova","doi":"10.3103/s0027132224700128","DOIUrl":"https://doi.org/10.3103/s0027132224700128","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>The paper is focused on the basis property of the system of exponentials and trigonometric systems of sine and cosine functions in a separable subspace of the weighted grand Lebesgue space generated by the shift operator. In this paper, with the help of the shift operator, a separable subspace <span>(G_{p),rho}(a,b))</span> of the weighted space of the grand Lebesgue space <span>(L_{p),rho}(a,b))</span> is defined. The density in <span>(G_{p),rho}(a,b))</span> of the set <span>(G_{0}^{infty}([a,b]))</span> of infinitely differentiable functions that are finite on <span>([a,b])</span> is studied. It is proved that if the weight function <span>(rho)</span> satisfies the Mackenhoupt condition, then the system of exponentials <span>(left{e^{int}right}_{nin Z})</span> forms a basis in <span>(G_{p),rho}(-pi,pi))</span>, and trigonometric systems of sine <span>(left{sin ntright}_{ngeqslant 1})</span> and cosine <span>(left{cos ntright}_{ngeqslant 0})</span> functions form bases in <span>(G_{p),rho}(0,pi))</span>.</p>","PeriodicalId":42963,"journal":{"name":"Moscow University Mathematics Bulletin","volume":"22 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141862615","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 : 2024-05-13DOI: 10.3103/s0027132224700074
G. A. Popov, E. B. Yarovaya
Abstract
A time-continuous random walk on a multidimensional lattice which underlies the branching random walk with an infinite number of phase states is considered. The random walk with a countable number of states can be reduced to a system with a finite number of states by aggregating them. The asymptotic behavior of the residence time of the transformed system in each of the states depending on the lattice dimension under the assumption of a finite variance and under the condition leading to an infinite variance of jumps of the original system is studied. It is shown that the aggregation of states in terms of the described process leads to the loss of the Markov property.
{"title":"Phase States Aggregation of Random Walk on a Multidimensional Lattice","authors":"G. A. Popov, E. B. Yarovaya","doi":"10.3103/s0027132224700074","DOIUrl":"https://doi.org/10.3103/s0027132224700074","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>A time-continuous random walk on a multidimensional lattice which underlies the branching random walk with an infinite number of phase states is considered. The random walk with a countable number of states can be reduced to a system with a finite number of states by aggregating them. The asymptotic behavior of the residence time of the transformed system in each of the states depending on the lattice dimension under the assumption of a finite variance and under the condition leading to an infinite variance of jumps of the original system is studied. It is shown that the aggregation of states in terms of the described process leads to the loss of the Markov property.</p>","PeriodicalId":42963,"journal":{"name":"Moscow University Mathematics Bulletin","volume":"22 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2024-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140929965","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 : 2024-05-13DOI: 10.3103/s0027132224700037
T. P. Lukashenko, V. A. Skvortsov, A. P. Solodov
Abstract
Generalizations of construction of Kolmogorov integral to the case of Banach space-valued functions are considered. We demonstrate how the Kolmogorov ideas on integration theory, in particular, the notion of differential equivalence, have been developed in the theory of the Henstock–Kurzweil integral. In this connection, a variational version of a Henstock type integral with respect to a rather general derivation basis is studied. An example of application of this integral to harmonic analysis is given. Some results related to the Kolmogorov (A)-integral are also considered.
{"title":"The Kolmogorov Ideas on the Integration Theory in Modern Research","authors":"T. P. Lukashenko, V. A. Skvortsov, A. P. Solodov","doi":"10.3103/s0027132224700037","DOIUrl":"https://doi.org/10.3103/s0027132224700037","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>Generalizations of construction of Kolmogorov integral to the case of Banach space-valued functions are considered. We demonstrate how the Kolmogorov ideas on integration theory, in particular, the notion of differential equivalence, have been developed in the theory of the Henstock–Kurzweil integral. In this connection, a variational version of a Henstock type integral with respect to a rather general derivation basis is studied. An example of application of this integral to harmonic analysis is given. Some results related to the Kolmogorov <span>(A)</span>-integral are also considered.</p>","PeriodicalId":42963,"journal":{"name":"Moscow University Mathematics Bulletin","volume":"136 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2024-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140937484","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 : 2024-05-13DOI: 10.3103/s0027132224700050
I. V. Astashova, D. A. Lashin, A. V. Filinovskiy
Abstract
We consider an extremum problem associated with a mathematical model of the temperature control. It is based on a one-dimensional non-self-adjoint parabolic equation of general form. Determining the optimal control as a function minimizing the weighted quadratic functional, we prove the existence of a solution to the problem of the double minimum by control and weight functions. We also obtain upper estimates for the norm of the control function in terms of the value of the functional. These estimates are used to prove the existence of the minimizing function for unbounded sets of control functions.
{"title":"On Problems of Extremum and Estimates of Control Function for Parabolic Equation","authors":"I. V. Astashova, D. A. Lashin, A. V. Filinovskiy","doi":"10.3103/s0027132224700050","DOIUrl":"https://doi.org/10.3103/s0027132224700050","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>We consider an extremum problem associated with a mathematical model of the temperature control. It is based on a one-dimensional non-self-adjoint parabolic equation of general form. Determining the optimal control as a function minimizing the weighted quadratic functional, we prove the existence of a solution to the problem of the double minimum by control and weight functions. We also obtain upper estimates for the norm of the control function in terms of the value of the functional. These estimates are used to prove the existence of the minimizing function for unbounded sets of control functions.</p>","PeriodicalId":42963,"journal":{"name":"Moscow University Mathematics Bulletin","volume":"28 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2024-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140930089","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 : 2024-05-13DOI: 10.3103/s0027132224700013
T. A. Garmanova, I. A. Sheipak
Abstract
The paper describes the splines (Q_{n,k}(x,a)), which define the relations (y^{(k)}(a)=intlimits_{0}^{1}y^{(n)}(x)Q^{(n)}_{n,k}(x,a)dx) for an arbitrary point (ain(0;1)) and an arbitrary function (yinmathring{W}^{n}_{p}[0;1]). The connection of the minimization of the norm (|Q^{(n)}_{n,k}|_{L_{p^{prime}}[0;1]}) ((1/p+1/p^{prime}=1)) by parameter (a) with the problem of best estimates for derivatives (|y^{(k)}(a)|leqslant A_{n,k,p}(a)|y^{(n)}|_{L_{p}[0;1]}), and also with the problem of finding the exact embedding constants of the Sobolev space (mathring{W}^{n}_{p}[0;1]) into the space (mathring{W}^{k}_{infty}[0;1]), (ninmathbb{N}), (0leqslant kleqslant n-1). Exact embedding constants are found for all (ninmathbb{N}), (k=n-1) for (p=1) and for (p=infty).
Abstract The paper describes the splines (Q_{n,k}(x,a)), which define the relations(y^{(k)}(a)=intlimits_{0}^{1}y^{(n)}(x)Q^{(n)}_{n,k}(x,a)dx) for an arbitrary point (ain(0. 1))和 an arbitrary function (yinmathring{W}^{n}_{p}[0;1])；1)）和任意函数（yinmathring{W}^{n}_{p}[0;1]).最小化规范 (|Q^{(n)}_{n,k}|_{L_{p^{prime}}[0；参数 (a) 的 ((1/p+1/p^{prime}=1)) 与导数 (|y^{(k)}(a)|leqslant A_{n,k,p}(a)|y^{(n)}|{L_{p}[0；1]}）的问题，以及找到索波列夫空间 (mathring{W}^{n}_{p}[0;1]) 到空间 (mathring{W}^{k}_{infty}[0;1]), (ninmathbb{N}), (0leqslant kleqslant n-1) 的精确嵌入常数的问题。对于所有的（ninmathbb{N}）、（k=n-1）的（p=1）和（p=infty），都可以找到精确的嵌入常数。
{"title":"Sharp Estimates of High-Order Derivatives in Sobolev Spaces","authors":"T. A. Garmanova, I. A. Sheipak","doi":"10.3103/s0027132224700013","DOIUrl":"https://doi.org/10.3103/s0027132224700013","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>The paper describes the splines <span>(Q_{n,k}(x,a))</span>, which define the relations <span>(y^{(k)}(a)=intlimits_{0}^{1}y^{(n)}(x)Q^{(n)}_{n,k}(x,a)dx)</span> for an arbitrary point <span>(ain(0;1))</span> and an arbitrary function <span>(yinmathring{W}^{n}_{p}[0;1])</span>. The connection of the minimization of the norm <span>(|Q^{(n)}_{n,k}|_{L_{p^{prime}}[0;1]})</span> (<span>(1/p+1/p^{prime}=1)</span>) by parameter <span>(a)</span> with the problem of best estimates for derivatives <span>(|y^{(k)}(a)|leqslant A_{n,k,p}(a)|y^{(n)}|_{L_{p}[0;1]})</span>, and also with the problem of finding the exact embedding constants of the Sobolev space <span>(mathring{W}^{n}_{p}[0;1])</span> into the space <span>(mathring{W}^{k}_{infty}[0;1])</span>, <span>(ninmathbb{N})</span>, <span>(0leqslant kleqslant n-1)</span>. Exact embedding constants are found for all <span>(ninmathbb{N})</span>, <span>(k=n-1)</span> for <span>(p=1)</span> and for <span>(p=infty)</span>.</p>","PeriodicalId":42963,"journal":{"name":"Moscow University Mathematics Bulletin","volume":"20 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2024-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140937096","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 : 2024-05-13DOI: 10.3103/s0027132224700062
I. N. Sergeev
Abstract
New characteristics of differential systems are studied, which meaningfully develop the concepts of Lyapunov, Perron, and upper limit stability or instability of the zero solution of a differential system from the standpoint of probability theory. Examples of autonomous systems are proposed for which these characteristics take opposite values in a certain sense.
{"title":"Examples of Autonomous Differential Systems with Contrasting Combinations of Lyapunov, Perron, and Upper-Limit Stability Measures","authors":"I. N. Sergeev","doi":"10.3103/s0027132224700062","DOIUrl":"https://doi.org/10.3103/s0027132224700062","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>New characteristics of differential systems are studied, which meaningfully develop the concepts of Lyapunov, Perron, and upper limit stability or instability of the zero solution of a differential system from the standpoint of probability theory. Examples of autonomous systems are proposed for which these characteristics take opposite values in a certain sense.</p>","PeriodicalId":42963,"journal":{"name":"Moscow University Mathematics Bulletin","volume":"65 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2024-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140930375","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 : 2024-05-13DOI: 10.3103/s0027132224700049
E. Vl. Bulinskaya
Abstract
We consider the model of branching random walk on an integer lattice (mathbb{Z}^{d}) with periodic sources of branching. It is supposed that the regime of branching is supercritical and the Cramér condition is satisfied for a jump of the random walk. The theorem established describes the rate of front propagation for particles population over the lattice as the time increases unboundedly. The proofs are based on fundamental results related to the spatial spread of general branching random walk.
{"title":"Propagation of the Front of Random Walk with Periodic Branching Sources","authors":"E. Vl. Bulinskaya","doi":"10.3103/s0027132224700049","DOIUrl":"https://doi.org/10.3103/s0027132224700049","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>We consider the model of branching random walk on an integer lattice <span>(mathbb{Z}^{d})</span> with periodic sources of branching. It is supposed that the regime of branching is supercritical and the Cramér condition is satisfied for a jump of the random walk. The theorem established describes the rate of front propagation for particles population over the lattice as the time increases unboundedly. The proofs are based on fundamental results related to the spatial spread of general branching random walk.</p>","PeriodicalId":42963,"journal":{"name":"Moscow University Mathematics Bulletin","volume":"28 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2024-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140930373","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 : 2024-05-13DOI: 10.3103/s0027132224700025
V. N. Samokhin, G. A. Chechkin
Abstract
Nonclassical problems in mathematical hydrodynamics arise when studying the motion of rheologically complex media, as well as under boundary conditions different from classical ones. In this paper, existence and uniqueness theorems are established for the classical solution to the problem of a stationary boundary layer of a liquid with the rheological law of Ladyzhenskaya near a solid wall with given conditions characterizing the force of surface tension and the phenomenon of slipping near this wall.
{"title":"Nonclassical Problems of the Mathematical Theory of Hydrodynamic Boundary Layer","authors":"V. N. Samokhin, G. A. Chechkin","doi":"10.3103/s0027132224700025","DOIUrl":"https://doi.org/10.3103/s0027132224700025","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>Nonclassical problems in mathematical hydrodynamics arise when studying the motion of rheologically complex media, as well as under boundary conditions different from classical ones. In this paper, existence and uniqueness theorems are established for the classical solution to the problem of a stationary boundary layer of a liquid with the rheological law of Ladyzhenskaya near a solid wall with given conditions characterizing the force of surface tension and the phenomenon of slipping near this wall.</p>","PeriodicalId":42963,"journal":{"name":"Moscow University Mathematics Bulletin","volume":"100 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2024-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140937168","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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