A matching in a graph is any set of edges of this graph without common vertices. The number of matchings, also known as the Hosoya index of the graph, is an important parameter, which finds wide applications in mathematical chemistry. Previously, the problem of maximizing the Hosoya index in trees of radius $2$ (that is, diameter $4$) of fixed size was completely solved. This work considers the problem of maximizing the Hosoya index in trees of diameter $5$ on a fixed number $n$ of vertices and solves it completely. It turns out that for any $n$ the extremal tree is unique. Bibliography: 6 titles.
{"title":"On diameter $5$ trees with the maximum number of matchings","authors":"N. A. Kuz’min, D. Malyshev","doi":"10.4213/sm9745e","DOIUrl":"https://doi.org/10.4213/sm9745e","url":null,"abstract":"A matching in a graph is any set of edges of this graph without common vertices. The number of matchings, also known as the Hosoya index of the graph, is an important parameter, which finds wide applications in mathematical chemistry. Previously, the problem of maximizing the Hosoya index in trees of radius $2$ (that is, diameter $4$) of fixed size was completely solved. This work considers the problem of maximizing the Hosoya index in trees of diameter $5$ on a fixed number $n$ of vertices and solves it completely. It turns out that for any $n$ the extremal tree is unique. Bibliography: 6 titles.","PeriodicalId":49573,"journal":{"name":"Sbornik Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74478404","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 problem of the multiplicity of limit cycles appearing after a perturbation of a hyperbolic polycycle with generic set of characteristic numbers is considered. In particular, it is proved that the multiplicity of any limit cycle appearing after a perturbation in a smooth finite-parameter family does not exceed the number of separatrix connections forming the polycycle. Bibliography: 10 titles.
{"title":"Multiplicities of limit cycles appearing after perturbations of hyperbolic polycycles","authors":"A. Dukov","doi":"10.4213/sm9747e","DOIUrl":"https://doi.org/10.4213/sm9747e","url":null,"abstract":"The problem of the multiplicity of limit cycles appearing after a perturbation of a hyperbolic polycycle with generic set of characteristic numbers is considered. In particular, it is proved that the multiplicity of any limit cycle appearing after a perturbation in a smooth finite-parameter family does not exceed the number of separatrix connections forming the polycycle. Bibliography: 10 titles.","PeriodicalId":49573,"journal":{"name":"Sbornik Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79472004","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}
Conditions on spaces $X$ with generalized distance $rho_X$ are investigates under which analogues of Banach's and Nadler's fixed-point theorems and Arutyunov's coincidence-point theorem can be obtained for mappings on such spaces. This is shown to hold if each geometric progression with ratio $<1$ (that is, each sequence ${ x_i}subset X$ satisfying $rho_X(x_{i+1},x_i)leq gamma rho_X(x_i,x_{i-1})$, $ i=1,2,…$, with some $gamma < 1$) is convergent. Examples of spaces with and without this property are given. In particular, the required property holds in a complete $f$-quasimetric space $X$ if the distance $rho_X$ in it satisfies $rho_X(x,z) leq rho_X(x,y)+(rho_X(y,z))^eta$, $x,y,z in X$, for some $etain (0,1)$, that is, if the function $fcolonmathbb{R}_+^{2} to mathbb{R}_+$ is given by $f(r_1,r_2)=r_1 + r_2^{eta}$. Next, for $f(r_1,r_2)=max{ r_1^{eta}, r_2^{eta} }$, where $eta in (0,2^{-1}]$, it is shown that for any $gamma > 0$ there exists an $f$-quasimetric space containing a geometric progression with ratio $gamma$ which is not a Cauchy sequence. The ‘zero-one law’, which means that either each geometric progression with ratio $<1$ is a Cauchy sequence or, for any $gammain (0,1)$, there exists a geometric progression with ratio $gamma$ that is not Cauchy, is discussed for $f$-quasimetric spaces. Bibliography: 29 titles.
{"title":"Geometric progressions in distance spaces; applications to fixed points and coincidence points","authors":"E. Zhukovskiy","doi":"10.4213/sm9773e","DOIUrl":"https://doi.org/10.4213/sm9773e","url":null,"abstract":"Conditions on spaces $X$ with generalized distance $rho_X$ are investigates under which analogues of Banach's and Nadler's fixed-point theorems and Arutyunov's coincidence-point theorem can be obtained for mappings on such spaces. This is shown to hold if each geometric progression with ratio $<1$ (that is, each sequence ${ x_i}subset X$ satisfying $rho_X(x_{i+1},x_i)leq gamma rho_X(x_i,x_{i-1})$, $ i=1,2,…$, with some $gamma < 1$) is convergent. Examples of spaces with and without this property are given. In particular, the required property holds in a complete $f$-quasimetric space $X$ if the distance $rho_X$ in it satisfies $rho_X(x,z) leq rho_X(x,y)+(rho_X(y,z))^eta$, $x,y,z in X$, for some $etain (0,1)$, that is, if the function $fcolonmathbb{R}_+^{2} to mathbb{R}_+$ is given by $f(r_1,r_2)=r_1 + r_2^{eta}$. Next, for $f(r_1,r_2)=max{ r_1^{eta}, r_2^{eta} }$, where $eta in (0,2^{-1}]$, it is shown that for any $gamma > 0$ there exists an $f$-quasimetric space containing a geometric progression with ratio $gamma$ which is not a Cauchy sequence. The ‘zero-one law’, which means that either each geometric progression with ratio $<1$ is a Cauchy sequence or, for any $gammain (0,1)$, there exists a geometric progression with ratio $gamma$ that is not Cauchy, is discussed for $f$-quasimetric spaces. Bibliography: 29 titles.","PeriodicalId":49573,"journal":{"name":"Sbornik Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81172393","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}
Let $p_{1},p_{2},…,p_{6}$ be prime numbers. First we show that, with at most $O(N^{1/12+varepsilon})$ exceptions, all even positive integers not exceeding $N$ can be represented in the form $p_{1}^{2}+p_{2}^{2}+p_{3}^{3}+p_{4}^{3}+p_{5}^{3}+p_{6}^{3}$, which improves the previous result $O(N^{1/4+varepsilon})$ obtained by Y. H. Liu. Moreover, we also prove that, with at most $O(N^{5/12+varepsilon})$ exceptions, all even positive integers not exceeding $N$ can be represented in the form $p_{1}^{2}+p_{2}^{3}+p_{3}^{3}+p_{4}^{3}+p_{5}^{3}+p_{6}^{3}$. Bibliography: 21 titles.
{"title":"Slim exceptional sets of Waring-Goldbach problems involving squares and cubes of primes","authors":"Xue Han, Huafeng Liu","doi":"10.4213/sm9689e","DOIUrl":"https://doi.org/10.4213/sm9689e","url":null,"abstract":"Let $p_{1},p_{2},…,p_{6}$ be prime numbers. First we show that, with at most $O(N^{1/12+varepsilon})$ exceptions, all even positive integers not exceeding $N$ can be represented in the form $p_{1}^{2}+p_{2}^{2}+p_{3}^{3}+p_{4}^{3}+p_{5}^{3}+p_{6}^{3}$, which improves the previous result $O(N^{1/4+varepsilon})$ obtained by Y. H. Liu. Moreover, we also prove that, with at most $O(N^{5/12+varepsilon})$ exceptions, all even positive integers not exceeding $N$ can be represented in the form $p_{1}^{2}+p_{2}^{3}+p_{3}^{3}+p_{4}^{3}+p_{5}^{3}+p_{6}^{3}$. Bibliography: 21 titles.","PeriodicalId":49573,"journal":{"name":"Sbornik Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134982656","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}
We present an elementary approach to local combinatorial formulae for the Euler class of a fibre-oriented triangulated spherical fibre bundle. This approach is based on sections averaging technique and very basic knowledge of simplicial (co)homology theory. Our formulae are close relatives of those due to Mnëv. Bibliography: 9 titles.
{"title":"An elementary approach to local combinatorial formulae for the Euler class of a PL spherical fibre bundle","authors":"Gaiane Yur'evna Panina","doi":"10.4213/sm9737e","DOIUrl":"https://doi.org/10.4213/sm9737e","url":null,"abstract":"We present an elementary approach to local combinatorial formulae for the Euler class of a fibre-oriented triangulated spherical fibre bundle. This approach is based on sections averaging technique and very basic knowledge of simplicial (co)homology theory. Our formulae are close relatives of those due to Mnëv. Bibliography: 9 titles.","PeriodicalId":49573,"journal":{"name":"Sbornik Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135596971","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":"On the sharp Baer-Suzuki theorem for the $pi$-radical of a finite group","authors":"N. Yang, Zhen-Lin Wu, D. Revin, E. Vdovin","doi":"10.4213/sm9698e","DOIUrl":"https://doi.org/10.4213/sm9698e","url":null,"abstract":"","PeriodicalId":49573,"journal":{"name":"Sbornik Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87987221","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}
Mohammad Soud Alkousa, Alexander Vladimirovich Gasnikov, Egor Leonidovich Gladin, Ilya Alekseevich Kuruzov, Dmitry Arkad'evich Pasechnyuk, Fedor Sergeevich Stonyakin
Algorithmic methods are developed that guarantee efficient complexity estimates for strongly convex-concave saddle-point problems in the case when one group of variables has a high dimension, while another has a rather low dimension (up to 100). These methods are based on reducing problems of this type to the minimization (maximization) problem for a convex (concave) functional with respect to one of the variables such that an approximate value of the gradient at an arbitrary point can be obtained with the required accuracy using an auxiliary optimization subproblem with respect to the other variable. It is proposed to use the ellipsoid method and Vaidya's method for low-dimensional problems and accelerated gradient methods with inexact information about the gradient or subgradient for high-dimensional problems. In the case when one group of variables, ranging over a hypercube, has a very low dimension (up to five), another proposed approach to strongly convex-concave saddle-point problems is rather efficient. This approach is based on a new version of a multidimensional analogue of Nesterov's method on a square (the multidimensional dichotomy method) with the possibility to use inexact values of the gradient of the objective functional. Bibliography: 28 titles.
{"title":"Solving strongly convex-concave composite saddle-point problems with low dimension of one group of variable","authors":"Mohammad Soud Alkousa, Alexander Vladimirovich Gasnikov, Egor Leonidovich Gladin, Ilya Alekseevich Kuruzov, Dmitry Arkad'evich Pasechnyuk, Fedor Sergeevich Stonyakin","doi":"10.4213/sm9700e","DOIUrl":"https://doi.org/10.4213/sm9700e","url":null,"abstract":"Algorithmic methods are developed that guarantee efficient complexity estimates for strongly convex-concave saddle-point problems in the case when one group of variables has a high dimension, while another has a rather low dimension (up to 100). These methods are based on reducing problems of this type to the minimization (maximization) problem for a convex (concave) functional with respect to one of the variables such that an approximate value of the gradient at an arbitrary point can be obtained with the required accuracy using an auxiliary optimization subproblem with respect to the other variable. It is proposed to use the ellipsoid method and Vaidya's method for low-dimensional problems and accelerated gradient methods with inexact information about the gradient or subgradient for high-dimensional problems. In the case when one group of variables, ranging over a hypercube, has a very low dimension (up to five), another proposed approach to strongly convex-concave saddle-point problems is rather efficient. This approach is based on a new version of a multidimensional analogue of Nesterov's method on a square (the multidimensional dichotomy method) with the possibility to use inexact values of the gradient of the objective functional. Bibliography: 28 titles.","PeriodicalId":49573,"journal":{"name":"Sbornik Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135596980","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":"On uniqueness for Franklin series with a convergent subsequence of partial sums","authors":"G. Gevorkyan","doi":"10.4213/sm9741e","DOIUrl":"https://doi.org/10.4213/sm9741e","url":null,"abstract":"We show that if the partial sums $S_{n_i}(x)=sum_{k=0}^{n_i}a_kf_k(x)$ of a Franklin series $sum_{k=0}^{infty}a_kf_k(x)$, where $sup_i{n_i}/(n_{i-1})<infty$, converge in measure to a bounded function $f$ and $sup_i|S_{n_i}(x)|<infty$ for $ xnotin B$, where $B$ is some countable set, then this series is the Fourier-Franklin series of $f$. Bibliography: 24 titles.","PeriodicalId":49573,"journal":{"name":"Sbornik Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78757190","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":"Structure of the spectrum of a nonselfadjoint Dirac operator","authors":"A. Makin","doi":"10.4213/sm9709e","DOIUrl":"https://doi.org/10.4213/sm9709e","url":null,"abstract":"","PeriodicalId":49573,"journal":{"name":"Sbornik Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77746366","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：