Y. Belousov, M. Karev, A. Malyutin, A. Miller, E. Fominykh
The paper is devoted to a line of the knot theory related to the conjecture on the additivity of the crossing number for knots under connected sum. A series of weak versions of this conjecture are proved. Many of these versions are formulated in terms of the band surgery graph also called the H ( 2 ) H(2) -Gordian graph.
{"title":"Lernaean knots and band surgery","authors":"Y. Belousov, M. Karev, A. Malyutin, A. Miller, E. Fominykh","doi":"10.1090/spmj/1687","DOIUrl":"https://doi.org/10.1090/spmj/1687","url":null,"abstract":"The paper is devoted to a line of the knot theory related to the conjecture on the additivity of the crossing number for knots under connected sum. A series of weak versions of this conjecture are proved. Many of these versions are formulated in terms of the band surgery graph also called the \u0000\u0000 \u0000 \u0000 H\u0000 (\u0000 2\u0000 )\u0000 \u0000 H(2)\u0000 \u0000\u0000-Gordian graph.","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45402357","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}
Orthogonal representations ηn:Sn↷RNeta _ncolon S_ncurvearrowright mathbb {R}^N of the symmetric groups SnS_n, n≥4nge 4, with N=n!/8N=n!/8, emerging from symmetries of double ratios are treated. For n=5n=5, the representation η5eta _5
正交表示ηn:Sn↷ 对称群S N S_N，N≥4nge4，其中N=N！/8 N=N/8，从双重比率的对称性中出现。对于n=5n=5，表示η5eta_5被分解为不可约分量，并表明某个分量产生了描述亚Möbius结构类中Möbius结构的方程的解。在这个意义上，决定Möbius结构的条件已经隐含在二重比的对称性中。
{"title":"Symmetries of double ratios and an equation for Möbius structures","authors":"S. Buyalo","doi":"10.1090/spmj/1688","DOIUrl":"https://doi.org/10.1090/spmj/1688","url":null,"abstract":"<p>Orthogonal representations <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"eta Subscript n Baseline colon upper S Subscript n Baseline clockwise top semicircle arrow double-struck upper R Superscript upper N\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msub>\u0000 <mml:mi>η<!-- η --></mml:mi>\u0000 <mml:mi>n</mml:mi>\u0000 </mml:msub>\u0000 <mml:mo>:<!-- : --></mml:mo>\u0000 <mml:msub>\u0000 <mml:mi>S</mml:mi>\u0000 <mml:mi>n</mml:mi>\u0000 </mml:msub>\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:mi>N</mml:mi>\u0000 </mml:msup>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">eta _ncolon S_ncurvearrowright mathbb {R}^N</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> of the symmetric groups <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper S Subscript n\">\u0000 <mml:semantics>\u0000 <mml:msub>\u0000 <mml:mi>S</mml:mi>\u0000 <mml:mi>n</mml:mi>\u0000 </mml:msub>\u0000 <mml:annotation encoding=\"application/x-tex\">S_n</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n greater-than-or-equal-to 4\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>n</mml:mi>\u0000 <mml:mo>≥<!-- ≥ --></mml:mo>\u0000 <mml:mn>4</mml:mn>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">nge 4</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, with <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper N equals n factorial slash 8\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>N</mml:mi>\u0000 <mml:mo>=</mml:mo>\u0000 <mml:mi>n</mml:mi>\u0000 <mml:mo>!</mml:mo>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mo>/</mml:mo>\u0000 </mml:mrow>\u0000 <mml:mn>8</mml:mn>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">N=n!/8</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, emerging from symmetries of double ratios are treated. For <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n equals 5\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>n</mml:mi>\u0000 <mml:mo>=</mml:mo>\u0000 <mml:mn>5</mml:mn>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">n=5</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, the representation <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"eta 5\">\u0000 <mml:semantics>\u0000 <mml:msub>\u0000 <mml:mi>η<!-- η --></mml:mi>\u0000 <mml:mn>5</mml:mn>\u0000 </mml:msub>\u0000 <mml:annotation encoding=\"application/x-tex\">eta _5</mml:annotation>\u0000 </mml:semantics>\u0000</mml:","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49313998","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 Schrödinger operator −Δ+V(x,y)-Delta + V(x,y) is considered in a cylinder Rm×Umathbb {R}^m times U, where UU is a bounded domain in Rdmathbb {R}^d. The spectrum of such an operator is studied under the assumption that the potential decreases in longitudinal variables, |V(x,y)|≤
考虑Schrödinger算子−Δ + V(x,y) - Delta + V(x,y)在圆柱体R m × U mathbb R{^m }times U中，其中U U是R d mathbb R{^d中的有界域。这种算子的谱是在纵向变量的势减小的假设下研究的，|V(x,y)|≤C⟨x⟩- ρ |V(x,y)| }le C langle x rangle ^{-rho。若ρ > 1}rho > 1，则波算符存在且完备;伯曼不变性原理和极限吸收原理成立;绝对连续光谱填充半轴;奇异连续谱是空的;特征值只能累加到阈值。
{"title":"Schrödinger operator with decreasing potential in a cylinder","authors":"N. Filonov","doi":"10.1090/spmj/1694","DOIUrl":"https://doi.org/10.1090/spmj/1694","url":null,"abstract":"<p>The Schrödinger operator <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"negative normal upper Delta plus upper V left-parenthesis x comma y right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mo>−<!-- − --></mml:mo>\u0000 <mml:mi mathvariant=\"normal\">Δ<!-- Δ --></mml:mi>\u0000 <mml:mo>+</mml:mo>\u0000 <mml:mi>V</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:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">-Delta + V(x,y)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is considered in a cylinder <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper R Superscript m Baseline times upper U\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">R</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mi>m</mml:mi>\u0000 </mml:msup>\u0000 <mml:mo>×<!-- × --></mml:mo>\u0000 <mml:mi>U</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">mathbb {R}^m times U</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, where <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper U\">\u0000 <mml:semantics>\u0000 <mml:mi>U</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">U</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is a bounded domain in <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper R Superscript d\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">R</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mi>d</mml:mi>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">mathbb {R}^d</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. The spectrum of such an operator is studied under the assumption that the potential decreases in longitudinal variables, <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"StartAbsoluteValue upper V left-parenthesis x comma y right-parenthesis EndAbsoluteValue less-than-or-equal-to upper C mathematical left-angle x mathematical right-angle Superscript negative rho\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mo stretchy=\"false\">|</mml:mo>\u0000 </mml:mrow>\u0000 <mml:mi>V</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:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mo stretchy=\"false\">|</mml:mo>\u0000 </mml:mrow>\u0000 <mml:mo>≤<!-- ≤ --></mml:mo>\u0000 <mm","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48177839","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 Federer theorem deals with the “massiveness” of the set of critical values for a tt-smooth map acting from Rmmathbb R^m to Rnmathbb R^n: it claims that the Hausdorff pp-measure of this set is zero for certain pp. If n≥mnge m, it has long been known that the assumption of that theorem relating the parameters m,n,t,pm,n,t,p is sharp. Here it is shown by an example that this assumption is also sharp for
费德勒定理处理的是从R m mathbb R^m到R n mathbb R^n的光滑映射的临界值集合的“海量性”:它声称该集合的Hausdorff p p -测度在特定的p p下为零。如果n≥m nge m，我们早就知道关于参数m,n,t,p m,n,t,p的定理的假设是尖锐的。这里的一个例子表明，这个假设对于n b>00 m和n bb11m也是尖锐的。
{"title":"On the sharpness of assumptions in the Federer theorem","authors":"B. Makarov, A. Podkorytov","doi":"10.1090/spmj/1691","DOIUrl":"https://doi.org/10.1090/spmj/1691","url":null,"abstract":"<p>The Federer theorem deals with the “massiveness” of the set of critical values for a <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"t\">\u0000 <mml:semantics>\u0000 <mml:mi>t</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">t</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-smooth map acting from <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper R Superscript m\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">R</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mi>m</mml:mi>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">mathbb R^m</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> to <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper R Superscript n\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">R</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mi>n</mml:mi>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">mathbb R^n</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>: it claims that the Hausdorff <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\">\u0000 <mml:semantics>\u0000 <mml:mi>p</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">p</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-measure of this set is zero for certain <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\">\u0000 <mml:semantics>\u0000 <mml:mi>p</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">p</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. If <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n greater-than-or-equal-to m\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>n</mml:mi>\u0000 <mml:mo>≥<!-- ≥ --></mml:mo>\u0000 <mml:mi>m</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">nge m</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, it has long been known that the assumption of that theorem relating the parameters <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"m comma n comma t comma p\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>m</mml:mi>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mi>n</mml:mi>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mi>t</mml:mi>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mi>p</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">m,n,t,p</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is sharp. Here it is shown by an example that this assumption is also sharp for <inline-formula content-typ","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45598824","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 Clebsh–Gordan coefficients for the Lie algebra g l 3 mathfrak {gl}_3 in the Gelfand–Tsetlin base are calculated. In contrast to previous papers, the result is given as an explicit formula. To obtain the result, a realization of a representation in the space of functions on the group G L 3 GL_3 is used. The keystone fact that allows one to carry the calculation of Clebsh–Gordan coefficients is the theorem that says that functions corresponding to the Gelfand–Tsetlin base vectors can be expressed in terms of generalized hypergeometric functions.
{"title":"Clebsh–Gordan coefficients for the algebra 𝔤𝔩₃ and hypergeometric functions","authors":"D. Artamonov","doi":"10.1090/spmj/1686","DOIUrl":"https://doi.org/10.1090/spmj/1686","url":null,"abstract":"The Clebsh–Gordan coefficients for the Lie algebra \u0000\u0000 \u0000 \u0000 \u0000 g\u0000 l\u0000 \u0000 3\u0000 \u0000 mathfrak {gl}_3\u0000 \u0000\u0000 in the Gelfand–Tsetlin base are calculated. In contrast to previous papers, the result is given as an explicit formula. To obtain the result, a realization of a representation in the space of functions on the group \u0000\u0000 \u0000 \u0000 G\u0000 \u0000 L\u0000 3\u0000 \u0000 \u0000 GL_3\u0000 \u0000\u0000 is used. The keystone fact that allows one to carry the calculation of Clebsh–Gordan coefficients is the theorem that says that functions corresponding to the Gelfand–Tsetlin base vectors can be expressed in terms of generalized hypergeometric functions.","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42555184","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 an ample Hausdorff groupoid Gmathcal {G}, and the Steinberg algebra AR(G)A_R(mathcal {G}) with coefficients in the commutative ring RR with unit, the centralizer is described for the subalgebra AR(U)A_R(U) with UU an open closed invariant subset of the unit space of Gmathcal {G}. In particular, it is shown that the algebra of the interior of the isotropy is indeed the centralizer of the diagonal subalgebra of the Steinberg algebra. This will unify seve
{"title":"A note on the centralizer of a subalgebra of the Steinberg algebra","authors":"R. Hazrat, Huanhuan Li","doi":"10.1090/spmj/1695","DOIUrl":"https://doi.org/10.1090/spmj/1695","url":null,"abstract":"<p>For an ample Hausdorff groupoid <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper G\">\u0000 <mml:semantics>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">G</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">mathcal {G}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, and the Steinberg algebra <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A Subscript upper R Baseline left-parenthesis script upper G right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msub>\u0000 <mml:mi>A</mml:mi>\u0000 <mml:mi>R</mml:mi>\u0000 </mml:msub>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">G</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">A_R(mathcal {G})</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> with coefficients in the commutative ring <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper R\">\u0000 <mml:semantics>\u0000 <mml:mi>R</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">R</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> with unit, the centralizer is described for the subalgebra <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A Subscript upper R Baseline left-parenthesis upper U right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msub>\u0000 <mml:mi>A</mml:mi>\u0000 <mml:mi>R</mml:mi>\u0000 </mml:msub>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>U</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">A_R(U)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> with <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper U\">\u0000 <mml:semantics>\u0000 <mml:mi>U</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">U</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> an open closed invariant subset of the unit space of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper G\">\u0000 <mml:semantics>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">G</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">mathcal {G}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. In particular, it is shown that the algebra of the interior of the isotropy is indeed the centralizer of the diagonal subalgebra of the Steinberg algebra. This will unify seve","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43792544","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}
A summation procedure is described for the construction of the optimal solution in the null controllability problem for a differential equation with distributed delay.
描述了一个求和过程，用于构造具有分布时滞的微分方程零可控性问题的最优解。
{"title":"Summation method in an optimal control problem with delay","authors":"P. Barkhayev, Yu.Lyubarskii","doi":"10.1090/spmj/1759","DOIUrl":"https://doi.org/10.1090/spmj/1759","url":null,"abstract":"A summation procedure is described for the construction of the optimal solution in the null controllability problem for a differential equation with distributed delay.","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43343749","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 maximal ideal space of ^{∞} on the countable disjoint union of open disks","authors":"A. Brudnyi","doi":"10.1090/SPMJ/1681","DOIUrl":"https://doi.org/10.1090/SPMJ/1681","url":null,"abstract":"","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83739357","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}
This survey of the spectral properties of substitution dynamical systems is devoted to primitive aperiodic substitutions and associated dynamical systems: Z mathbb {Z} -actions and R mathbb {R} -actions, the latter viewed as tiling flows. The focus is on the continuous part of the spectrum. For Z mathbb {Z} -actions the maximal spectral type can be represented in terms of matrix Riesz products, whereas for tiling flows, the local dimension of the spectral measure is governed by the spectral cocycle. References are given to complete proofs and emphasize ideas and various links.
{"title":"Self-similarity and spectral theory: on the spectrum of substitutions","authors":"A. Bufetov, B. Solomyak","doi":"10.1090/spmj/1756","DOIUrl":"https://doi.org/10.1090/spmj/1756","url":null,"abstract":"This survey of the spectral properties of substitution dynamical systems is devoted to primitive aperiodic substitutions and associated dynamical systems: \u0000\u0000 \u0000 \u0000 Z\u0000 \u0000 mathbb {Z}\u0000 \u0000\u0000-actions and \u0000\u0000 \u0000 \u0000 R\u0000 \u0000 mathbb {R}\u0000 \u0000\u0000-actions, the latter viewed as tiling flows. The focus is on the continuous part of the spectrum. For \u0000\u0000 \u0000 \u0000 Z\u0000 \u0000 mathbb {Z}\u0000 \u0000\u0000-actions the maximal spectral type can be represented in terms of matrix Riesz products, whereas for tiling flows, the local dimension of the spectral measure is governed by the spectral cocycle. References are given to complete proofs and emphasize ideas and various links.","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44659221","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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