St Petersburg Mathematical Journal最新文献

{"title":"Stability of resonances for the Dirac operator","authors":"D. Mokeev","doi":"10.1090/spmj/1788","DOIUrl":"https://doi.org/10.1090/spmj/1788","url":null,"abstract":"<p>The Dirac operator on the semi-axis with a compactly supported potential is investigated. Let <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis k Subscript n Baseline right-parenthesis Subscript n greater-than-or-equal-to 1\"> <mml:semantics> <mml:mrow> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msub> <mml:mi>k</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:msub> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">(k_n)_{ngeq 1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the sequence of its resonances, taken with multiplicities and ordered so that <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"StartAbsoluteValue k Subscript n Baseline EndAbsoluteValue\"> <mml:semantics> <mml:mrow> <mml:mrow> <mml:mo stretchy=\"false\">|</mml:mo> </mml:mrow> <mml:msub> <mml:mi>k</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mrow> <mml:mo stretchy=\"false\">|</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">|k_n|</mml:annotation> </mml:semantics> </mml:math> </inline-formula> do not decrease as <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n\"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding=\"application/x-tex\">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> grows. It is proved that for any sequence <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis r Subscript n Baseline right-parenthesis Subscript n greater-than-or-equal-to 1 Baseline element-of script l Superscript 1\"> <mml:semantics> <mml:mrow> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msub> <mml:mi>r</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:msub> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:msup> <mml:mi>ℓ<!-- ℓ --></mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">(r_n)_{ngeq 1} in ell ^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that the points <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k Subscript n Baseline plus r Subscript n\"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>k</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>r</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">k_n + r_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> remain in the lower half-plane for all <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/Ma","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140930281","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":"Representation of analytic functions in bounded convex domains on the complex plane","authors":"A. Krivosheev, A. Rafikov","doi":"10.1090/spmj/1779","DOIUrl":"https://doi.org/10.1090/spmj/1779","url":null,"abstract":"The paper is devoted to entire functions of exponential type and regular growth. Exceptional sets are investigated outside of which these functions have estimates from below that asymptotically coincide with their estimates from above. An explicit construction of an exceptional set, which consists of disks with centers at zeros of the entire function, is indicated. The concept of a properly balanced set is introduced, which is a natural generalization of the concept of a regular set by B. Ya. Levin. It is proved that the zero set of an entire function is properly balanced if and only if each function analytic in the interior of the conjugate diagram of the entire function in question and continuous up to the boundary is represented by a series of exponential monomials whose exponents are zeros of this entire function. This result generalizes the classical result of A. F. Leont′ev on the representation of analytic functions in a convex domain to the case of a multiple zero set.","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135291339","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 finite algebras with probability limit laws","authors":"A. Yashunsky","doi":"10.1090/spmj/1782","DOIUrl":"https://doi.org/10.1090/spmj/1782","url":null,"abstract":"An algebraic system has a probability limit law if the values of terms with independent identically distributed random variables have probability distributions that tend to a certain limit (the limit law) as the number of variables in a term grows. For algebraic systems on finite sets, it is shown that, under some geometric conditions on the set of term value distributions, the existence of a limit law strongly restricts the set of possible operations in the algebraic system. In particular, a system that has a limit law without zero components necessarily consists of quasigroup operations (with arbitrary arity), while the limit law is necessarily uniform. Sufficient conditions are also proved for a system to have a probability limit law, which partly match the necessary ones.","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135240833","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":"Sufficient conditions for the minimality of biconcave functions","authors":"M. Novikov","doi":"10.1090/spmj/1781","DOIUrl":"https://doi.org/10.1090/spmj/1781","url":null,"abstract":"This paper describes sufficient conditions under which a biconcave function <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper B colon German upper S equals StartSet left-parenthesis x comma y right-parenthesis element-of double-struck upper R squared colon x minus 2 less-than-or-equal-to y less-than-or-equal-to x plus 2 EndSet right-arrow double-struck upper R\"> <mml:semantics> <mml:mrow> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">B</mml:mi> </mml:mrow> <mml:mo>:<!-- : --></mml:mo> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"fraktur\">S</mml:mi> </mml:mrow> <mml:mo>=</mml:mo> <mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:msup> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">R</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>:<!-- : --></mml:mo> <mml:mi>x</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>2</mml:mn> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mi>y</mml:mi> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mi>x</mml:mi> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> <mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo> <mml:mo stretchy=\"false\">→<!-- → --></mml:mo> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">R</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">mathcal {B}colon mathfrak {S}={ (x,y)in mathbb {R}^2colon x-2le yle x+2 }to mathbb {R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is minimal with respect to an obstacle <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L colon German upper S right-arrow left-bracket negative normal infinity comma plus normal infinity right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mi>L</mml:mi> <mml:mo>:<!-- : --></mml:mo> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"fraktur\">S</mml:mi> </mml:mrow> <mml:mo stretchy=\"false\">→<!-- → --></mml:mo> <mml:mo stretchy=\"false\">[</mml:mo> <mml:mo>−<!-- − --></mml:mo> <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi> <mml:mo>,</mml:mo> <mml:mo>+</mml:mo> <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">Lcolon mathfrak {S}to [-infty ,+infty )</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, that is, it is the pointwise minimal among all biconcave functions <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper B colon German upper S right-arrow double-struck upper R\"> <mml:semantics> <mml:mrow> <mml:mi>B</mml:mi> <mml:mo>:<!-- : --></mml:mo> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"fraktur\">S</mml:mi> </mml:mrow> <mml:mo stretchy=\"fals","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135291341","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":"Torsion divisors of plane curves and Zariski pairs","authors":"E. Artal Bartolo, Sh. Bannai, T. Shirane, H. Tokunaga","doi":"10.1090/spmj/1776","DOIUrl":"https://doi.org/10.1090/spmj/1776","url":null,"abstract":"This paper is devoted to the study of the embedded topology of reducible plane curves having a smooth irreducible component. In previous studies, the relationship between the topology and certain torsion classes in the Picard group of degree zero of the smooth component was implicitly considered. Here this relationship is formulated clearly and a criterion is given for distinguishing the embedded topology in terms of torsion classes. Furthermore, a method is presented for systematically constructing examples of curves where this criterion is applicable, and new examples of Zariski <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper N\"> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding=\"application/x-tex\">N</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-tuples are produced.","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135240953","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":"Derivative of the Minkowski function for numbers with bounded partial quotients","authors":"D. Gayfulin","doi":"10.1090/spmj/1777","DOIUrl":"https://doi.org/10.1090/spmj/1777","url":null,"abstract":"It is well known that the derivative of the Minkowski function <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"question-mark left-parenthesis x right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mo>?</mml:mo> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">?(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (whenever exists) may take only two values: <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"0\"> <mml:semantics> <mml:mn>0</mml:mn> <mml:annotation encoding=\"application/x-tex\">0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"plus normal infinity\"> <mml:semantics> <mml:mrow> <mml:mo>+</mml:mo> <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">+infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Let <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"bold upper E Subscript n\"> <mml:semantics> <mml:msub> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mtext mathvariant=\"bold\">E</mml:mtext> </mml:mrow> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding=\"application/x-tex\">textbf {E}_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the set of irrational numbers on the interval <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-bracket 0 semicolon 1 right-bracket\"> <mml:semantics> <mml:mrow> <mml:mo stretchy=\"false\">[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>;</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy=\"false\">]</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">[0; 1]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> whose partial quotients (related to the continued fraction expansion) do not exceed <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n\"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding=\"application/x-tex\">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. It is also known that the quantity <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"question-mark Superscript prime Baseline left-parenthesis x right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mo>?</mml:mo> <mml:mo>′</mml:mo> </mml:msup> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">?’(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> at a point <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135241092","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":"Cauchy problem for the nonlinear Hirota equation in the class of periodic infinite-zone functions","authors":"G. Mannonov, A. Khasanov","doi":"10.1090/spmj/1780","DOIUrl":"https://doi.org/10.1090/spmj/1780","url":null,"abstract":"In this paper, the method of inverse spectral problem is used to integrate the nonlinear Hirota equation in the class of periodic infinite-zone functions. An evolution of the spectral data of the periodic Dirac operator is introduced, where the coefficient of this operator is the solution of the nonlinear Hirota equation. The solvability of the Cauchy problem for an infinite system of Dubrovin differential equations in the class of five times continuously differentiable periodic infinite-zone functions is shown. In addition, it is proved that if the initial function is a <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"pi\"> <mml:semantics> <mml:mi>π<!-- π --></mml:mi> <mml:annotation encoding=\"application/x-tex\">pi</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-periodic real-analytic function, then the solution of the Cauchy problem for the Hirota equation is also a real-analytic function in the variable <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"x\"> <mml:semantics> <mml:mi>x</mml:mi> <mml:annotation encoding=\"application/x-tex\">x</mml:annotation> </mml:semantics> </mml:math> </inline-formula>; and if the number <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"pi slash 2\"> <mml:semantics> <mml:mrow> <mml:mi>π<!-- π --></mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">pi /2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a period (antiperiod) of the initial function, then the number <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"pi slash 2\"> <mml:semantics> <mml:mrow> <mml:mi>π<!-- π --></mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">pi /2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a period (antiperiod) in the variable <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"x\"> <mml:semantics> <mml:mi>x</mml:mi> <mml:annotation encoding=\"application/x-tex\">x</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the solution of the Cauchy problems for the Hirota equation.","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135240834","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 electric impedance tomography problem for nonorientable surfaces with internal holes","authors":"D. Korikov","doi":"10.1090/spmj/1778","DOIUrl":"https://doi.org/10.1090/spmj/1778","url":null,"abstract":"Let <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis upper M comma g right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>M</mml:mi> <mml:mo>,</mml:mo> <mml:mi>g</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">(M,g)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a compact (in general, nonorientable) surface with boundary <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"partial-differential upper M\"> <mml:semantics> <mml:mrow> <mml:mi mathvariant=\"normal\">∂<!-- ∂ --></mml:mi> <mml:mi>M</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">partial M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and let <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Gamma 0\"> <mml:semantics> <mml:msub> <mml:mi mathvariant=\"normal\">Γ<!-- Γ --></mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:annotation encoding=\"application/x-tex\">Gamma _0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, …, <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Gamma Subscript m minus 1\"> <mml:semantics> <mml:msub> <mml:mi mathvariant=\"normal\">Γ<!-- Γ --></mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>m</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:annotation encoding=\"application/x-tex\">Gamma _{m-1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be connected components of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"partial-differential upper M\"> <mml:semantics> <mml:mrow> <mml:mi mathvariant=\"normal\">∂<!-- ∂ --></mml:mi> <mml:mi>M</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">partial M</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Let <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"u equals u Superscript f Baseline left-parenthesis x right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>u</mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>f</mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">u=u^{f}(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a solution to the problem <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Delta Subscript g Baseline u equals 0\"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi mathvariant=\"normal\">Δ<!-- Δ --></mml:mi> <mml:mrow cla","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135241223","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":"Improved resolvent 𝐿²-approximations in homogenization of fourth order operators","authors":"S. Pastukhova","doi":"10.1090/spmj/1772","DOIUrl":"https://doi.org/10.1090/spmj/1772","url":null,"abstract":"<p>A divergent elliptic operator <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A Subscript epsilon\">\u0000 <mml:semantics>\u0000 <mml:msub>\u0000 <mml:mi>A</mml:mi>\u0000 <mml:mi>ε<!-- ε --></mml:mi>\u0000 </mml:msub>\u0000 <mml:annotation encoding=\"application/x-tex\">A_varepsilon</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> of the fourth order with <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"epsilon\">\u0000 <mml:semantics>\u0000 <mml:mi>ε<!-- ε --></mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">varepsilon</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-periodic coefficients acting in the space <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> is treated, where <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"epsilon\">\u0000 <mml:semantics>\u0000 <mml:mi>ε<!-- ε --></mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">varepsilon</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is a small parameter. For the resolvent <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis upper A Subscript epsilon Baseline plus 1 right-parenthesis Superscript negative 1\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:msub>\u0000 <mml:mi>A</mml:mi>\u0000 <mml:mi>ε<!-- ε --></mml:mi>\u0000 </mml:msub>\u0000 <mml:mo>+</mml:mo>\u0000 <mml:mn>1</mml:mn>\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:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">(A_varepsilon +1)^{-1}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, approximations are constructed in the operator <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis upper L squared right-arrow upper L squared right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:msup>\u0000 <mml:mi>L</mml:mi>\u0000 <mml:mn>2</mml:mn>\u0000 </mml:msup>\u0000 <mml:mo stretchy=\"false\">→<!-- → --></mml:mo>\u0000 <mml:msup>\u0000 <mml:mi>L</mml:mi>\u0000 <mml:mn>2</mml:mn>\u0000 </mml:msup>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45665651","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":"Hölder classes in the 𝐿^{𝑝} norm on a chord-arc curve in ℝ³","authors":"T. Alexeeva, N. Shirokov","doi":"10.1090/spmj/1769","DOIUrl":"https://doi.org/10.1090/spmj/1769","url":null,"abstract":"<p>The Hölder classes <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L Subscript p Superscript alpha Baseline left-parenthesis upper L right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msubsup>\u0000 <mml:mi>L</mml:mi>\u0000 <mml:mi>p</mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi>α<!-- α --></mml:mi>\u0000 </mml:mrow>\u0000 </mml:msubsup>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>L</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">L_p^{alpha } (L)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> in the <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L Subscript p Baseline left-parenthesis upper L right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msub>\u0000 <mml:mi>L</mml:mi>\u0000 <mml:mi>p</mml:mi>\u0000 </mml:msub>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>L</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">L_p(L)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> norm on a <italic>chord-arc</italic> curve <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L\">\u0000 <mml:semantics>\u0000 <mml:mi>L</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">L</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> in <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper R cubed\">\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:mn>3</mml:mn>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">mathbb {R}^3</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> are defined and direct and inverse approximation theorems are proved for functions from these classes by functions harmonic in a neighborhood of the curve. The approximation is estimated in the <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L Superscript p Baseline left-parenthesis upper L right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msup>\u0000 <mml:mi>L</mml:mi>\u0000 <mml:mi>p</mml:mi>\u0000 </mml:msup>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>L</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">L^p(L)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> norm, the direct theorem is proved for a certain subclass of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L Subscript p Superscri","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47050295","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}