We study the $SU$-linear operations in complex cobordism and prove that they are generated by the well-known geometric operations $partial_i$. For the theory $W$ of $c_1$-spherical bordism, we describe all $SU$-linear multiplications on $W$ and projections $MU to W$. We also analyse complex orientations on $W$ and the corresponding formal group laws $F_W$. The relationship between the formal group laws $F_W$ and the coefficient ring $W_*$ of the $W$-theory was studied by Buchstaber in 1972. We extend his results by showing that for any $SU$-linear multiplication and orientation on $W$, the coefficients of the corresponding formal group law $F_W$ do not generate the ring $W_*$, unlike the situation with complex bordism.
{"title":"$SU$-linear operations in complex cobordism and the $c_1$-spherical bordism theory","authors":"Taras Evgenievich Panov, George Chernykh","doi":"10.4213/im9334e","DOIUrl":"https://doi.org/10.4213/im9334e","url":null,"abstract":"We study the $SU$-linear operations in complex cobordism and prove that they are generated by the well-known geometric operations $partial_i$. For the theory $W$ of $c_1$-spherical bordism, we describe all $SU$-linear multiplications on $W$ and projections $MU to W$. We also analyse complex orientations on $W$ and the corresponding formal group laws $F_W$. The relationship between the formal group laws $F_W$ and the coefficient ring $W_*$ of the $W$-theory was studied by Buchstaber in 1972. We extend his results by showing that for any $SU$-linear multiplication and orientation on $W$, the coefficients of the corresponding formal group law $F_W$ do not generate the ring $W_*$, unlike the situation with complex bordism.","PeriodicalId":54910,"journal":{"name":"Izvestiya Mathematics","volume":"23 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":"135007538","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The Michael selection theorem is extended to the case of set-valued mappings with not necessarily convex values. Classical approximation problems on cone-spaces with symmetric and asymmetric seminorms are considered. In particular, conditions for existence of continuous selections for convex subsets of asymmetric spaces are studied. The problem of existence of a Chebyshev centre for a bounded set is solved in a semilinear space consisting of bounded convex sets with Hausdorff semimetric.
{"title":"Continuous selections of set-valued mappings and approximation in asymmetric and semilinear spaces","authors":"Igor' Germanovich Tsar'kov","doi":"10.4213/im9331e","DOIUrl":"https://doi.org/10.4213/im9331e","url":null,"abstract":"The Michael selection theorem is extended to the case of set-valued mappings with not necessarily convex values. Classical approximation problems on cone-spaces with symmetric and asymmetric seminorms are considered. In particular, conditions for existence of continuous selections for convex subsets of asymmetric spaces are studied. The problem of existence of a Chebyshev centre for a bounded set is solved in a semilinear space consisting of bounded convex sets with Hausdorff semimetric.","PeriodicalId":54910,"journal":{"name":"Izvestiya Mathematics","volume":"2 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":"135007551","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Sergei Konstantinovich Vodopyanov, Anastasia Molchanova
This article studies systematically the boundary correspondence problem for $mathcal Q_{p,q}$-homeomorphisms. The presented example demonstrates a deformation of the Euclidean boundary with the weight function degenerating on the boundary.
{"title":"The boundary behavior of $mathcal Q_{p,q}$-homeomorphisms","authors":"Sergei Konstantinovich Vodopyanov, Anastasia Molchanova","doi":"10.4213/im9376e","DOIUrl":"https://doi.org/10.4213/im9376e","url":null,"abstract":"This article studies systematically the boundary correspondence problem for $mathcal Q_{p,q}$-homeomorphisms. The presented example demonstrates a deformation of the Euclidean boundary with the weight function degenerating on the boundary.","PeriodicalId":54910,"journal":{"name":"Izvestiya Mathematics","volume":"144 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":"135009175","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Boris Olegovich Volkov, Alexander Nikolaevich Pechen
In this work, we study the detailed structure of quantum control landscape for the problem of single-qubit phase shift gate generation on the fast time scale. In previous works, the absence of traps for this problem was proved on various time scales. A special critical point which was known to exist in quantum control landscapes was shown to be either a saddle or a global extremum, depending on the parameters of the control system. However, in case of a saddle, the numbers of negative and positive eigenvalues of the Hessian at this point and their magnitudes have not been studied. At the same time, these numbers and magnitudes determine the relative ease or difficulty for practical optimization in a vicinity of the critical point. In this work, we compute the numbers of negative and positive eigenvalues of the Hessian at this saddle point and, moreover, give estimates on magnitude of these eigenvalues. We also significantly simplify our previous proof of the theorem about this saddle point of the Hessian (Theorem 3 in [22]).
{"title":"On the detailed structure of quantum control landscape for fast single qubit phase-shift gate generation","authors":"Boris Olegovich Volkov, Alexander Nikolaevich Pechen","doi":"10.4213/im9364e","DOIUrl":"https://doi.org/10.4213/im9364e","url":null,"abstract":"In this work, we study the detailed structure of quantum control landscape for the problem of single-qubit phase shift gate generation on the fast time scale. In previous works, the absence of traps for this problem was proved on various time scales. A special critical point which was known to exist in quantum control landscapes was shown to be either a saddle or a global extremum, depending on the parameters of the control system. However, in case of a saddle, the numbers of negative and positive eigenvalues of the Hessian at this point and their magnitudes have not been studied. At the same time, these numbers and magnitudes determine the relative ease or difficulty for practical optimization in a vicinity of the critical point. In this work, we compute the numbers of negative and positive eigenvalues of the Hessian at this saddle point and, moreover, give estimates on magnitude of these eigenvalues. We also significantly simplify our previous proof of the theorem about this saddle point of the Hessian (Theorem 3 in [22]).","PeriodicalId":54910,"journal":{"name":"Izvestiya Mathematics","volume":"27 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":"135661671","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Symmetries and conservation laws of the Liouville equation are studied in the frames of the algebra-geometrical approach to partial differential equations.
在偏微分方程的代数-几何方法框架下研究了Liouville方程的对称性和守恒律。
{"title":"Symmetries and conservation laws of the Liouville equation","authors":"Viktor Viktorovich Zharinov","doi":"10.4213/im9356e","DOIUrl":"https://doi.org/10.4213/im9356e","url":null,"abstract":"Symmetries and conservation laws of the Liouville equation are studied in the frames of the algebra-geometrical approach to partial differential equations.","PeriodicalId":54910,"journal":{"name":"Izvestiya Mathematics","volume":"48 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":"135661676","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This article, dedicated to the 100 th anniversary of I. R. Shafarevich, is a survey of techniques of homotopical algebra, applied to the problem of distribution of rational points on algebraic varieties. We due to I. R. Shafarevich, jointly with J. Tate, one of the breakthrough discoveries in this domain: construction of the so-called Shafarevich-Tate groups and the related obstructions to the existence of rational points. Later it evolved into the theory of Brauer-Manin obstructions. Here we focus on some facets of the later developments in Diophantine geometry: the study of the distribution of rational points on them. More precisely, we show how the definition of accumulating subvarieties, based upon counting the number of points whose height is bounded by varying $H$, can be encoded by a special class of categories in such a way that the arithmetical invariants of varieties are translated into homotopical invariants of objects and morphisms of these categories. The central role in this study is played by the structure of an assembler (I. Zakharevich) in general, and a very particular case of it, an assembler on the family of unions of half-open intervals $(a,b]$ with rational ends.
{"title":"Rational points of algebraic varieties: a homotopical approach","authors":"Yuri Ivanovich Manin","doi":"10.4213/im9315e","DOIUrl":"https://doi.org/10.4213/im9315e","url":null,"abstract":"This article, dedicated to the 100 th anniversary of I. R. Shafarevich, is a survey of techniques of homotopical algebra, applied to the problem of distribution of rational points on algebraic varieties. We due to I. R. Shafarevich, jointly with J. Tate, one of the breakthrough discoveries in this domain: construction of the so-called Shafarevich-Tate groups and the related obstructions to the existence of rational points. Later it evolved into the theory of Brauer-Manin obstructions. Here we focus on some facets of the later developments in Diophantine geometry: the study of the distribution of rational points on them. More precisely, we show how the definition of accumulating subvarieties, based upon counting the number of points whose height is bounded by varying $H$, can be encoded by a special class of categories in such a way that the arithmetical invariants of varieties are translated into homotopical invariants of objects and morphisms of these categories. The central role in this study is played by the structure of an assembler (I. Zakharevich) in general, and a very particular case of it, an assembler on the family of unions of half-open intervals $(a,b]$ with rational ends.","PeriodicalId":54910,"journal":{"name":"Izvestiya Mathematics","volume":"17 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":"135440254","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We construct nonisogenous simple ordinary abelian varieties over an algebraic closure of a finite field with isomorphic endomorphism algebras.
在具有同构自同态代数的有限域的代数闭包上构造非同构简单普通阿贝尔变。
{"title":"Isogeny classes and endomorphism algebras of abelian varieties over finite fields","authors":"Yuri Gennad'evich Zarhin","doi":"10.4213/im9332e","DOIUrl":"https://doi.org/10.4213/im9332e","url":null,"abstract":"We construct nonisogenous simple ordinary abelian varieties over an algebraic closure of a finite field with isomorphic endomorphism algebras.","PeriodicalId":54910,"journal":{"name":"Izvestiya Mathematics","volume":"30 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":"135440263","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we are concerned with the following Schrödinger-Poisson system $$ begin{cases} -Delta u+u+lambdaphi u= Q(x)|u|^{4}u+mu dfrac{|x|^beta}{1+|x|^beta}|u|^{q-2}u&in mathbb{R}^3, -Delta phi=u^{2} &in mathbb{R}^3, end{cases} $$ where $0< beta<3$, $60$ are real parameters. By the variational method and the Nehari method, we obtain that the system has $k$ positive solutions.
{"title":"Multiple positive solutions for a Schrödinger-Poisson system with critical and supercritical growths","authors":"Jun Lei, Hong-Min Suo","doi":"10.4213/im9244e","DOIUrl":"https://doi.org/10.4213/im9244e","url":null,"abstract":"In this paper, we are concerned with the following Schrödinger-Poisson system $$ begin{cases} -Delta u+u+lambdaphi u= Q(x)|u|^{4}u+mu dfrac{|x|^beta}{1+|x|^beta}|u|^{q-2}u&amp;in mathbb{R}^3, -Delta phi=u^{2} &amp;in mathbb{R}^3, end{cases} $$ where $0< beta<3$, $6<q<6+2beta$, $Q(x)$ is a positive continuous function on $mathbb{R}^3$, $lambda,mu>0$ are real parameters. By the variational method and the Nehari method, we obtain that the system has $k$ positive solutions.","PeriodicalId":54910,"journal":{"name":"Izvestiya Mathematics","volume":"141 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":"136079306","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the eigenvalues of the Neumann and Dirichlet boundary-value problems in a two-dimensional domain containing several small, of diameter $O(varepsilon)$, inclusions of large "density" $O(varepsilon^{-gamma})$, $gammageq2$, that is, the "mass" $O(varepsilon^{2-gamma})$ of each of them is comparable ($gamma=2$) or much bigger ($gamma>2$) than that of the embracing material. We construct a model of such spectral problems on concentrated masses which (the model) provides an asymptotic expansions of the eigenvalues with remainders of power-law smallness order $O(varepsilon^{vartheta})$ as $varepsilonto+0$ and $varthetain(0,1)$. Besides, the correction terms are real analytic functions of the parameter $|{ln varepsilon}|^{-1}$. A "far-field interaction" of the inclusions is observed at the levels $|{ln varepsilon}|^{-1}$ or $|{ln varepsilon}|^{-2}$. The results are obtained with the help of the machinery of weighted spaces with detached asymptotics and also by using weighted estimates of solutions to limit problems in a bounded punctured domain and in the intact plane.
{"title":"\"Far-field interaction\" of concentrated masses in two-dimensional Neumann and Dirichlet problems","authors":"S. Nazarov","doi":"10.4213/im9262e","DOIUrl":"https://doi.org/10.4213/im9262e","url":null,"abstract":"We study the eigenvalues of the Neumann and Dirichlet boundary-value problems in a two-dimensional domain containing several small, of diameter $O(varepsilon)$, inclusions of large \"density\" $O(varepsilon^{-gamma})$, $gammageq2$, that is, the \"mass\" $O(varepsilon^{2-gamma})$ of each of them is comparable ($gamma=2$) or much bigger ($gamma>2$) than that of the embracing material. We construct a model of such spectral problems on concentrated masses which (the model) provides an asymptotic expansions of the eigenvalues with remainders of power-law smallness order $O(varepsilon^{vartheta})$ as $varepsilonto+0$ and $varthetain(0,1)$. Besides, the correction terms are real analytic functions of the parameter $|{ln varepsilon}|^{-1}$. A \"far-field interaction\" of the inclusions is observed at the levels $|{ln varepsilon}|^{-1}$ or $|{ln varepsilon}|^{-2}$. The results are obtained with the help of the machinery of weighted spaces with detached asymptotics and also by using weighted estimates of solutions to limit problems in a bounded punctured domain and in the intact plane.","PeriodicalId":54910,"journal":{"name":"Izvestiya Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70326946","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Tile $mathrm{B}$-splines in $mathbb R^d$ are defined as autoconvolutions of indicators of tiles, which are special self-similar compact sets whose integer translates tile the space $mathbb R^d$. These functions are not piecewise-polynomial, however, being direct generalizations of the classical $mathrm{B}$-splines, they enjoy many of their properties and have some advantages. In particular, exact values of the Hölder exponents of tile $mathrm{B}$-splines are evaluated and are shown, in some cases, to exceed those of the classical $mathrm{B}$-splines. Orthonormal systems of wavelets based on tile B-splines are constructed, and estimates of their exponential decay are obtained. Efficiency in applications of tile $mathrm{B}$-splines is demonstrated on an example of subdivision schemes of surfaces. This efficiency is achieved due to high regularity, fast convergence, and small number of coefficients in the corresponding refinement equation.
{"title":"Multivariate tile $mathrm{B}$-splines","authors":"T. Zaitseva","doi":"10.4213/im9296e","DOIUrl":"https://doi.org/10.4213/im9296e","url":null,"abstract":"Tile $mathrm{B}$-splines in $mathbb R^d$ are defined as autoconvolutions of indicators of tiles, which are special self-similar compact sets whose integer translates tile the space $mathbb R^d$. These functions are not piecewise-polynomial, however, being direct generalizations of the classical $mathrm{B}$-splines, they enjoy many of their properties and have some advantages. In particular, exact values of the Hölder exponents of tile $mathrm{B}$-splines are evaluated and are shown, in some cases, to exceed those of the classical $mathrm{B}$-splines. Orthonormal systems of wavelets based on tile B-splines are constructed, and estimates of their exponential decay are obtained. Efficiency in applications of tile $mathrm{B}$-splines is demonstrated on an example of subdivision schemes of surfaces. This efficiency is achieved due to high regularity, fast convergence, and small number of coefficients in the corresponding refinement equation.","PeriodicalId":54910,"journal":{"name":"Izvestiya Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70326993","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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