We consider the generalized Newton’s least resistance problem for convex bodies: minimize the functional ∬Ω(1+|∇u(x,y)|2)−1dxdyiint _Omega (1 + |nabla u(x,y)|^2)^{-1} dx, dy in the class of concave functions u:Ω→[0,M]ucolon Omega to [0,M], where the domain Ω⊂R2
考虑凸体的广义牛顿最小阻力问题:最小化凹函数u类中的泛函∫∫Ω(1 + |)∇u(x,y)| 2) -1 dx dy iint ＿ Omega (1 + | nabla) u(x,y)|²)^{-1} dx， dy:Ω→[0,M] u colonOmegato [0,M]，其中域Ω∧R 2 Omegasubsetmathbb R{^2是凸有界的，M > 0 M > 0。这是已知的(见G. Buttazzo, V. Ferone和B. Kawohl[数学])。Nachr. 173 (1995)， pp. 71-89])，则|∇u(x,y)|≥1 | }nabla u(x,y)| ge 1在所有正则点(x,y) (x,y)使得u(x,y) > M u(x,y) > M。证明如果上水平集L = {(x, y):u(x,y) = M} L = ｛(x,y) colon u(x,y) = M｝具有非空的内部，那么对于其边界(x¯，y¯)∈∂L (bar x{，}bar y{) }inpartial L有lim (x, y)→(x¯，Y¯)
{"title":"On generalized Newton’s aerodynamic problem","authors":"A. Plakhov","doi":"10.1090/mosc/318","DOIUrl":"https://doi.org/10.1090/mosc/318","url":null,"abstract":"<p>We consider the generalized Newton’s least resistance problem for convex bodies: minimize the functional <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-integral Underscript normal upper Omega Endscripts left-parenthesis 1 plus StartAbsoluteValue nabla u left-parenthesis x comma y right-parenthesis EndAbsoluteValue squared right-parenthesis Superscript negative 1 Baseline d x d y\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msub>\u0000 <mml:mo>∬<!-- ∬ --></mml:mo>\u0000 <mml:mi mathvariant=\"normal\">Ω<!-- Ω --></mml:mi>\u0000 </mml:msub>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mn>1</mml:mn>\u0000 <mml:mo>+</mml:mo>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mo stretchy=\"false\">|</mml:mo>\u0000 </mml:mrow>\u0000 <mml:mi mathvariant=\"normal\">∇<!-- ∇ --></mml:mi>\u0000 <mml:mi>u</mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>x</mml:mi>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mi>y</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 <mml:msup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mo stretchy=\"false\">|</mml:mo>\u0000 </mml:mrow>\u0000 <mml:mn>2</mml:mn>\u0000 </mml:msup>\u0000 <mml:msup>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mo>−<!-- − --></mml:mo>\u0000 <mml:mn>1</mml:mn>\u0000 </mml:mrow>\u0000 </mml:msup>\u0000 <mml:mi>d</mml:mi>\u0000 <mml:mi>x</mml:mi>\u0000 <mml:mspace width=\"thinmathspace\" />\u0000 <mml:mi>d</mml:mi>\u0000 <mml:mi>y</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">iint _Omega (1 + |nabla u(x,y)|^2)^{-1} dx, dy</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> in the class of concave functions <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"u colon normal upper Omega right-arrow left-bracket 0 comma upper M right-bracket\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>u</mml:mi>\u0000 <mml:mo>:<!-- : --></mml:mo>\u0000 <mml:mi mathvariant=\"normal\">Ω<!-- Ω --></mml:mi>\u0000 <mml:mo stretchy=\"false\">→<!-- → --></mml:mo>\u0000 <mml:mo stretchy=\"false\">[</mml:mo>\u0000 <mml:mn>0</mml:mn>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mi>M</mml:mi>\u0000 <mml:mo stretchy=\"false\">]</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">ucolon Omega to [0,M]</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, where the domain <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Omega subset-of double-struck upper R squared\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi mathvariant=\"normal\">Ω<!-- Ω --></mml:mi>\u0000 <mml:mo>⊂<!-- ⊂ --></mml:mo>\u0000 <mml:msup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">R</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mn>2</mml:mn>\u0000 </mml:msup>\u0000 </m","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":"3 S4","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41269245","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}
{"title":"Solvability of some nonlinear boundary value problems for singular integral equations of convolution type","authors":"K. Khachatryan","doi":"10.1090/MOSC/306","DOIUrl":"https://doi.org/10.1090/MOSC/306","url":null,"abstract":"","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":"81 1","pages":"1-31"},"PeriodicalIF":0.0,"publicationDate":"2021-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48730299","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}
{"title":"A description of linearly additive metrics on $mathbb {R}^n$","authors":"P. H. Aramyan","doi":"10.1090/MOSC/304","DOIUrl":"https://doi.org/10.1090/MOSC/304","url":null,"abstract":"","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":"81 1","pages":"115-121"},"PeriodicalIF":0.0,"publicationDate":"2021-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41882498","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}
Edmond W. H. Lee, Mikhail Volkov, Edmond W. H. Lee
The non-orthodox 0-simple semigroup A2 of order five admits a unary operation under which it is an involution semigroup. It is known that A2 generates a Specht variety of semigroups. In contrast, it is shown that as an involution semigroup, A2 generates a non-Specht variety.
{"title":"Non-Specht variety generated by an involution semigroup of order five","authors":"Edmond W. H. Lee, Mikhail Volkov, Edmond W. H. Lee","doi":"10.1090/MOSC/302","DOIUrl":"https://doi.org/10.1090/MOSC/302","url":null,"abstract":"The non-orthodox 0-simple semigroup A2 of order five admits a unary operation under which it is an involution semigroup. It is known that A2 generates a Specht variety of semigroups. In contrast, it is shown that as an involution semigroup, A2 generates a non-Specht variety.","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":"81 1","pages":"87-95"},"PeriodicalIF":0.0,"publicationDate":"2021-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41339256","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}
{"title":"Relaxation autowaves in a bi-local neuron model","authors":"S. Glyzin, A. Kolesov, N. Rozov","doi":"10.1090/MOSC/305","DOIUrl":"https://doi.org/10.1090/MOSC/305","url":null,"abstract":"","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":"81 1","pages":"33-70"},"PeriodicalIF":0.0,"publicationDate":"2021-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47864615","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}
{"title":"Applications of noncommutative geometry in function theory and mathematical physics","authors":"A. Sergeev","doi":"10.1090/MOSC/307","DOIUrl":"https://doi.org/10.1090/MOSC/307","url":null,"abstract":"","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":"81 1","pages":"123-167"},"PeriodicalIF":0.0,"publicationDate":"2021-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43646820","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}
Mauro Artigiani, C. Fougeron, P. Hubert, A. Skripchenko
We introduce a new renormalization procedure on double rotations, which is reminiscent of the classical Rauzy induction. Using this renormalization we prove that the set of parameters which induce infinite type double rotations has Hausdorff dimension strictly smaller than 3 3 . Moreover, we construct a natural invariant measure supported on these parameters and show that, with respect to this measure, almost all double rotations are uniquely ergodic.
{"title":"A note on double rotations of infinite type","authors":"Mauro Artigiani, C. Fougeron, P. Hubert, A. Skripchenko","doi":"10.1090/mosc/311","DOIUrl":"https://doi.org/10.1090/mosc/311","url":null,"abstract":"We introduce a new renormalization procedure on double rotations, which is reminiscent of the classical Rauzy induction. Using this renormalization we prove that the set of parameters which induce infinite type double rotations has Hausdorff dimension strictly smaller than \u0000\u0000 \u0000 3\u0000 3\u0000 \u0000\u0000. Moreover, we construct a natural invariant measure supported on these parameters and show that, with respect to this measure, almost all double rotations are uniquely ergodic.","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48326518","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}
Here are two problems. First, understanding the dynamics of a tiling billiard in a cyclic quadrilateral periodic tiling. Second, describing the topology of connected components of plane sections of a centrally symmetric subsurface S ⊂ T 3 S subset mathbb {T}^3 of genus 3 3 . In this paper we show that these two problems are related via a helicoidal construction proposed recently by Ivan Dynnikov. The second problem is a particular case of a classical question formulated by Sergei Novikov. The exploration of the relationship between a large class of tiling billiards (periodic locally foldable tiling billiards) and Novikov’s problem in higher genus seems promising, as we show at the end of this paper.
{"title":"Tiling billiards and Dynnikov’s helicoid","authors":"Olga Paris-Romaskevich","doi":"10.1090/mosc/317","DOIUrl":"https://doi.org/10.1090/mosc/317","url":null,"abstract":"Here are two problems. First, understanding the dynamics of a tiling billiard in a cyclic quadrilateral periodic tiling. Second, describing the topology of connected components of plane sections of a centrally symmetric subsurface \u0000\u0000 \u0000 \u0000 S\u0000 ⊂\u0000 \u0000 \u0000 T\u0000 \u0000 3\u0000 \u0000 \u0000 S subset mathbb {T}^3\u0000 \u0000\u0000 of genus \u0000\u0000 \u0000 3\u0000 3\u0000 \u0000\u0000. In this paper we show that these two problems are related via a helicoidal construction proposed recently by Ivan Dynnikov. The second problem is a particular case of a classical question formulated by Sergei Novikov. The exploration of the relationship between a large class of tiling billiards (periodic locally foldable tiling billiards) and Novikov’s problem in higher genus seems promising, as we show at the end of this paper.","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44242169","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}
For a compact set of actions, an entropy of Kushnirenko type is chosen in such a way that it vanishes on this set but takes infinite values for the typical actions. As a consequence we find that typical measure-preserving transformations are not isomorphic to isometric rearrangements of a finite set of geometric figures.
{"title":"Compact families and typical entropy invariants of measure-preserving actions","authors":"V. Ryzhikov","doi":"10.1090/mosc/321","DOIUrl":"https://doi.org/10.1090/mosc/321","url":null,"abstract":"For a compact set of actions, an entropy of Kushnirenko type is chosen in such a way that it vanishes on this set but takes infinite values for the typical actions. As a consequence we find that typical measure-preserving transformations are not isomorphic to isometric rearrangements of a finite set of geometric figures.","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44740051","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}
We study the images of the Markov measures under transformations generated by the Mealy automata. We find conditions under which the image measure is absolutely continuous or singular relative to the Markov measure. Also, we determine statistical properties of the image of a generic sequence.
{"title":"On Mealy–Moore coding and images of Markov measures","authors":"R. Grigorchuk, R. Kogan, Yaroslav Vorobets","doi":"10.1090/mosc/314","DOIUrl":"https://doi.org/10.1090/mosc/314","url":null,"abstract":"We study the images of the Markov measures under transformations generated by the Mealy automata. We find conditions under which the image measure is absolutely continuous or singular relative to the Markov measure. Also, we determine statistical properties of the image of a generic sequence.","PeriodicalId":37924,"journal":{"name":"Transactions of the Moscow Mathematical Society","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49286567","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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