Murat Kazievich Altuev, Vladislav Alexandrovich Kibkalo
An analogue of the Euler top is considered for a pseudo-Euclidean space is under consideration. In the cases when the geometric integral or area integral vanishes the bifurcation diagrams of the moment map are constructed and the homeomorphism class of each leaf of the Liouville foliation is determined. For each arc of the bifurcation diagram, for one of the two possible cases of the mutual arrangement of the moments of inertia, the types of singularities in the preimage of a small neighbourhood of this arc (analogues of Fomenko 3-atoms) are determined, and for nonsingular isoenergy and isointegral surfaces an invariant of rough Liouville equivalence (an analogue of a rough molecule) is constructed. The pseudo-Euclidean Euler system turns out to have noncompact noncritical bifurcations. Bibliography: 23 titles.
{"title":"Topological analysis of pseudo-Euclidean Euler top for special values of the parameters","authors":"Murat Kazievich Altuev, Vladislav Alexandrovich Kibkalo","doi":"10.4213/sm9771e","DOIUrl":"https://doi.org/10.4213/sm9771e","url":null,"abstract":"An analogue of the Euler top is considered for a pseudo-Euclidean space is under consideration. In the cases when the geometric integral or area integral vanishes the bifurcation diagrams of the moment map are constructed and the homeomorphism class of each leaf of the Liouville foliation is determined. For each arc of the bifurcation diagram, for one of the two possible cases of the mutual arrangement of the moments of inertia, the types of singularities in the preimage of a small neighbourhood of this arc (analogues of Fomenko 3-atoms) are determined, and for nonsingular isoenergy and isointegral surfaces an invariant of rough Liouville equivalence (an analogue of a rough molecule) is constructed. The pseudo-Euclidean Euler system turns out to have noncompact noncritical bifurcations. Bibliography: 23 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":"135596978","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}
For any $mathbb{A}^1$-invariant cohomology theory that satisfies Nisnevich excision on the category of smooth schemes over a field $k$ it is proved that the Cousin complex on the complement $U-D$ to the strict normal-crossing divisor $D$ in a local essentially smooth scheme $U$ is acyclic. This claim is also proved for the schemes $(X-D)times(mathbb{A}^1_k-Z_0)times…times(mathbb{A}^1_k-Z_l)$, where $Z_0,…,Z_l$ is a finite family of closed subschemes in the affine line over $k$. Bibliography: 32 titles.
{"title":"Cousin complex on the complement to the strict normal-crossing divisor in a local essentially smooth scheme over a field","authors":"A. Druzhinin","doi":"10.4213/sm9762e","DOIUrl":"https://doi.org/10.4213/sm9762e","url":null,"abstract":"For any $mathbb{A}^1$-invariant cohomology theory that satisfies Nisnevich excision on the category of smooth schemes over a field $k$ it is proved that the Cousin complex on the complement $U-D$ to the strict normal-crossing divisor $D$ in a local essentially smooth scheme $U$ is acyclic. This claim is also proved for the schemes $(X-D)times(mathbb{A}^1_k-Z_0)times…times(mathbb{A}^1_k-Z_l)$, where $Z_0,…,Z_l$ is a finite family of closed subschemes in the affine line over $k$. Bibliography: 32 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":"91338549","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}
Petr Mikhailovich Akhmet'ev, Yury Vladimirovich Muranov
An element $x$ is specified in the Wall group $L_3(D_3)$ of the dihedral group of order $8$ with trivial orientation character, such that $x$ is an element of the third type in the sense of Kharshiladze with respect to any system of one-sided submanifolds of codimension $1$ for which the splitting obstruction group along the first submanifold is isomorphic to $LN_1(mathbb Z/2oplus mathbb Z/2to D_3)$. The element $x$ is not realisable as an obstruction to surgery on a closed $mathrm{PL}$-manifold. It is also proved that the unique nontrivial element of the group $LN_3(mathbb Z/2oplus mathbb Z/2to D_3^-)$ can be detected using the Hasse-Witt $Wh_2$-torsion. Bibliography: 25 titles.
{"title":"Arf invariants of codimension one in a Wall group of the dihedral group","authors":"Petr Mikhailovich Akhmet'ev, Yury Vladimirovich Muranov","doi":"10.4213/sm9716e","DOIUrl":"https://doi.org/10.4213/sm9716e","url":null,"abstract":"An element $x$ is specified in the Wall group $L_3(D_3)$ of the dihedral group of order $8$ with trivial orientation character, such that $x$ is an element of the third type in the sense of Kharshiladze with respect to any system of one-sided submanifolds of codimension $1$ for which the splitting obstruction group along the first submanifold is isomorphic to $LN_1(mathbb Z/2oplus mathbb Z/2to D_3)$. The element $x$ is not realisable as an obstruction to surgery on a closed $mathrm{PL}$-manifold. It is also proved that the unique nontrivial element of the group $LN_3(mathbb Z/2oplus mathbb Z/2to D_3^-)$ can be detected using the Hasse-Witt $Wh_2$-torsion. Bibliography: 25 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":"134982660","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}
Questions relating to theorems of Levinson-Sjöberg-Wolf type in complex and harmonic analysis are explored. The well-known Dyn'kin problem of effective estimation of the growth majorant of an analytic function in a neighbourhood of its set of singularities is discussed, together with the problem, dual to it in certain sense, on the rate of convergence to zero of the extremal function in a nonquasianalytic Carleman class in a neighbourhood of a point at which all the derivatives of functions in this class vanish. The first problem was solved by Matsaev and Sodin. Here the second Dyn'kin problem, going back to Bang, is fully solved. As an application, a sharp asymptotic estimate is given for the distance between the imaginary exponentials and the algebraic polynomials in a weighted space of continuous functions on the real line. Bibliography: 24 titles.
{"title":"Levinson-type theorem and Dyn'kin problems","authors":"Ahtyar Magazovich Gaisin, Rashit Akhtyarovich Gaisin","doi":"10.4213/sm9802e","DOIUrl":"https://doi.org/10.4213/sm9802e","url":null,"abstract":"Questions relating to theorems of Levinson-Sjöberg-Wolf type in complex and harmonic analysis are explored. The well-known Dyn'kin problem of effective estimation of the growth majorant of an analytic function in a neighbourhood of its set of singularities is discussed, together with the problem, dual to it in certain sense, on the rate of convergence to zero of the extremal function in a nonquasianalytic Carleman class in a neighbourhood of a point at which all the derivatives of functions in this class vanish. The first problem was solved by Matsaev and Sodin. Here the second Dyn'kin problem, going back to Bang, is fully solved. As an application, a sharp asymptotic estimate is given for the distance between the imaginary exponentials and the algebraic polynomials in a weighted space of continuous functions on the real line. Bibliography: 24 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":"134982661","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}
Vyacheslav Zigmuntovich Grines, Elena Yakovlevna Gurevich
We obtain necessary and sufficient conditions for the topological equivalence of gradient-like flows without heteroclinic intersections defined on the connected sum of a finite number of manifolds homeomorphic to $mathbb{S}^{n-1}times mathbb{S}^1$, $ngeq 3$. For $n>3$, this result extends substantially the class of manifolds such that structurally stable systems on these manifolds admit a topological classification. Bibliography: 36 titles.
{"title":"A combinatorial invariant of gradient-like flows on a connected sum of $mathbb{S}^{n-1}timesmathbb{S}^1$","authors":"Vyacheslav Zigmuntovich Grines, Elena Yakovlevna Gurevich","doi":"10.4213/sm9761e","DOIUrl":"https://doi.org/10.4213/sm9761e","url":null,"abstract":"We obtain necessary and sufficient conditions for the topological equivalence of gradient-like flows without heteroclinic intersections defined on the connected sum of a finite number of manifolds homeomorphic to $mathbb{S}^{n-1}times mathbb{S}^1$, $ngeq 3$. For $n>3$, this result extends substantially the class of manifolds such that structurally stable systems on these manifolds admit a topological classification. Bibliography: 36 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":"134982663","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 consider the cell decomposition of the moduli space of real genus $2$ curves with marked point on the unique real oval. The cells are enumerated by certain graphs, whose weights describe the complex structure on the curve. We show that the collapse of an edge in a graph results in a root-like singularity of the natural map from the weights on graphs to the moduli space of curves. Bibliography: 24 titles.
{"title":"Degeneration of a graph describing conformal structure","authors":"Andrei Borisovich Bogatyrev","doi":"10.4213/sm9703e","DOIUrl":"https://doi.org/10.4213/sm9703e","url":null,"abstract":"We consider the cell decomposition of the moduli space of real genus $2$ curves with marked point on the unique real oval. The cells are enumerated by certain graphs, whose weights describe the complex structure on the curve. We show that the collapse of an edge in a graph results in a root-like singularity of the natural map from the weights on graphs to the moduli space of curves. Bibliography: 24 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":"135596952","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 find the convergence rates of the collocation approximation by deep ReLU neural networks of solutions to elliptic PDEs with lognormal inputs, parametrized by $boldsymbol{y}$ in the noncompact set ${mathbb R}^infty$. The approximation error is measured in the norm of the Bochner space $L_2({mathbb R}^infty, V, gamma)$, where $gamma$ is the infinite tensor-product standard Gaussian probability measure on ${mathbb R}^infty$ and $V$ is the energy space. We also obtain similar dimension-independent results in the case when the lognormal inputs are parametrized by ${mathbb R}^M$ of very large dimension $M$, and the approximation error is measured in the $sqrt{g_M}$-weighted uniform norm of the Bochner space $L_infty^{sqrt{g}}({mathbb R}^M, V)$, where $g_M$ is the density function of the standard Gaussian probability measure on ${mathbb R}^M$. Bibliography: 62 titles.
{"title":"Collocation approximation by deep neural ReLU networks for parametric and stochastic PDEs with lognormal inputs","authors":"Dung Dinh","doi":"10.4213/sm9791e","DOIUrl":"https://doi.org/10.4213/sm9791e","url":null,"abstract":"We find the convergence rates of the collocation approximation by deep ReLU neural networks of solutions to elliptic PDEs with lognormal inputs, parametrized by $boldsymbol{y}$ in the noncompact set ${mathbb R}^infty$. The approximation error is measured in the norm of the Bochner space $L_2({mathbb R}^infty, V, gamma)$, where $gamma$ is the infinite tensor-product standard Gaussian probability measure on ${mathbb R}^infty$ and $V$ is the energy space. We also obtain similar dimension-independent results in the case when the lognormal inputs are parametrized by ${mathbb R}^M$ of very large dimension $M$, and the approximation error is measured in the $sqrt{g_M}$-weighted uniform norm of the Bochner space $L_infty^{sqrt{g}}({mathbb R}^M, V)$, where $g_M$ is the density function of the standard Gaussian probability measure on ${mathbb R}^M$. Bibliography: 62 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":"136302353","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 topology of the Liouville foliations of integrable magnetic topological billiards, systems in which a ball moves on piecewise smooth two-dimensional surfaces in a constant magnetic field, is considered. The Fomenko-Zieschang invariants of Liouville equivalence are calculated for the Hamiltonian systems arising, and the topology of invariant 3-manifolds, isointegral and isoenergy ones, is investigated. The Liouville equivalence of such billiards to some known Hamiltonian systems is discovered, for instance, to the geodesic flows on 2-surfaces and to systems of rigid body dynamics. In particular, peculiar saddle singularities are discovered in which singular circles have different orientations - such systems were also previously encountered in mechanical systems in a magnetic field on surfaces of revolution homeomorphic to a 2-sphere. Bibliography: 13 titles.
{"title":"Classification of Liouville foliations of integrable topological billiards in magnetic fields","authors":"V. V. Vedyushkina, S. Pustovoitov","doi":"10.4213/sm9770e","DOIUrl":"https://doi.org/10.4213/sm9770e","url":null,"abstract":"The topology of the Liouville foliations of integrable magnetic topological billiards, systems in which a ball moves on piecewise smooth two-dimensional surfaces in a constant magnetic field, is considered. The Fomenko-Zieschang invariants of Liouville equivalence are calculated for the Hamiltonian systems arising, and the topology of invariant 3-manifolds, isointegral and isoenergy ones, is investigated. The Liouville equivalence of such billiards to some known Hamiltonian systems is discovered, for instance, to the geodesic flows on 2-surfaces and to systems of rigid body dynamics. In particular, peculiar saddle singularities are discovered in which singular circles have different orientations - such systems were also previously encountered in mechanical systems in a magnetic field on surfaces of revolution homeomorphic to a 2-sphere. Bibliography: 13 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":"78644310","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}
Petr Vladimirovich Paramonov, Konstantin Yurievich Fedorovskiy
The main aim of this paper is to study the geometric and metric properties of $B$- and $C$-capacities related to problems of uniform approximation of functions by solutions of homogeneous second-order elliptic equations with constant complex coefficients on compact subsets of Euclidean spaces. In the harmonic case this problem is well known, and it was studied in detail in the framework of classical potential theory in the first half of the 20th century. For a wide class of equations mentioned above, we obtain two-sided estimates between the corresponding $B_+$- and $C_+$-capacities (defined in terms of potentials of positive measures) and the harmonic capacity in the same dimension. Our research method is based on new simple explicit formulae obtained for the fundamental solutions of the equations under consideration. Bibliography: 12 titles.
{"title":"Explicit form of fundamental solutions to certain elliptic equations and associated $B$- and $C$-capacities","authors":"Petr Vladimirovich Paramonov, Konstantin Yurievich Fedorovskiy","doi":"10.4213/sm9807e","DOIUrl":"https://doi.org/10.4213/sm9807e","url":null,"abstract":"The main aim of this paper is to study the geometric and metric properties of $B$- and $C$-capacities related to problems of uniform approximation of functions by solutions of homogeneous second-order elliptic equations with constant complex coefficients on compact subsets of Euclidean spaces. In the harmonic case this problem is well known, and it was studied in detail in the framework of classical potential theory in the first half of the 20th century. For a wide class of equations mentioned above, we obtain two-sided estimates between the corresponding $B_+$- and $C_+$-capacities (defined in terms of potentials of positive measures) and the harmonic capacity in the same dimension. Our research method is based on new simple explicit formulae obtained for the fundamental solutions of the equations under consideration. Bibliography: 12 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":"136301577","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}
In Vinberg's works certain non-Abelian gradings of simple Lie algebras were introduced and investigated, namely, short $mathrm{SO}_3$- and $mathrm{SL}_3$-structures. We investigate a different kind of these, short $mathrm{SL}_2$-structures. The main results refer to the one-to-one correspondence between such structures and certain special Jordan algebras. Bibliography: 8 titles.
{"title":"Short $mathrm{SL}_2$-structures on simple Lie algebras","authors":"Roman Olegovich Stasenko","doi":"10.4213/sm9788e","DOIUrl":"https://doi.org/10.4213/sm9788e","url":null,"abstract":"In Vinberg's works certain non-Abelian gradings of simple Lie algebras were introduced and investigated, namely, short $mathrm{SO}_3$- and $mathrm{SL}_3$-structures. We investigate a different kind of these, short $mathrm{SL}_2$-structures. The main results refer to the one-to-one correspondence between such structures and certain special Jordan algebras. Bibliography: 8 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":"136302014","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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