<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
本文研究了半轴上具有紧凑支撑势的狄拉克算子。设 ( k n ) n ≥ 1 (k_n)_{ngeq 1} 为其共振序列，取其乘数并排序，使得| k n | |k_n| 不随 n n 的增长而减小。实验证明，对于任意序列 ( r n ) n ≥ 1 ∈ 1 ℓ 1 (r_n)_{ngeq 1}in ell ^1，使得点 k n + r n k_n + r_n 对于所有 n ≥ 1 ngeq 1 都保持在下半平面，序列 ( k n + r n ) n ≥ 1 (k_n + r_n)_{ngeq 1} 也是类似算子的共振序列。此外，研究还表明，在这种扰动下，狄拉克算子的势会连续变化。
{"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":"42 1","pages":""},"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}
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.
{"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":" 32","pages":"0"},"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}
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.
本文主要研究指数型整函数和正则增长函数。研究了异常集，在异常集之外，这些函数的估计从下面渐近地与它们的估计从上面重合。给出了一个异常集的显式构造，该异常集由以整个函数零点为中心的圆盘组成。引入了适当平衡集的概念，它是B. Ya对正则集概念的自然推广。莱文。证明了一个完整函数的零集是适当平衡的，当且仅当在整个函数的共轭图内部解析且连续到边界的每个函数都用指数为该完整函数的零的指数单项式表示。这一结果推广了a . F. Leont 'ev关于解析函数在凸域上表示的经典结果到多重零集的情况。
{"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":" 11","pages":"0"},"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}
E. Artal Bartolo, Sh. Bannai, T. Shirane, H. Tokunaga
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 NN-tuples are produced.
本文研究了具有光滑不可约分量的可约平面曲线的嵌入拓扑。在以往的研究中，隐式地考虑了光滑分量的零度Picard群中某些扭转类与拓扑之间的关系。本文对这种关系进行了明确的表述，并给出了从扭转类的角度来区分嵌入拓扑的判据。在此基础上，提出了一种系统构造适用于该准则的曲线实例的方法，并给出了Zariski N N元组的新实例。
{"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":" 18","pages":"0"},"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}
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:
众所周知，闵可夫斯基函数的导数?(x) ?(x)(无论何时存在)只能取两个值:0 0和+∞+ infty。设textbfen E＿n{为区间[0;1] [0;1]其部分商(与连分式展开有关)不超过n n。还知道数量是多少?' (x) ? ' (x)在点x = [0;A 1, A 2，…，A t，…]x=[0;a＿1,a＿2, }dots,a＿t, dots]与算术平均值的极限行为(A 1+ A 2+⋯+ A t)/t (a＿1+a＿2+ dots +a＿t)/t有关。特别是a . Dushistova, I. Kan和n . Moshchevitin证明，如果x∈E n x intextbfE＿n{满足1 + a 2 +⋯+ at ＆gt;(κ 1 (n)−ε) t a＿1+a＿2+ }dots +a＿t＆gt;(kappa ^(n{)＿1}- varepsilon) t，其中ε ＆gt;0 varepsilon ＆gt;0和κ 1 (n) kappa ^(n{)＿1}是某个显式常数，则?' (x)=+∞? ' (x)=+ infty。他们还发现，κ 1 (n) kappa ^(n{)＿1}的数量不能增加。本文研究了一个对偶问题:a 1+a 2+⋯+a t−κ 1 (n) t a＿1+a＿2+ dots +a＿t- kappa ^(n{)＿1} t有多小?' (x)=0 ? ' (x)=0 ?找到了该问题的最优估计。
{"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":" 2","pages":"0"},"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}
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 πpi-periodic real-analytic function, then the solution of the Cauchy problem for the Hirota equation is also a real-analytic function in the variable xx; and if the number π/2pi /2 is a period (antiperiod) of the initial function, then the number π/2pi /2 is a period (antiperiod) in the variable xx of the solution of the Cauchy problems for the Hirota equation.
本文利用谱逆问题的方法对一类周期无穷区函数中的非线性Hirota方程进行积分。介绍了周期狄拉克算子谱数据的演化，其中该算子的系数是非线性Hirota方程的解。给出了一类五次连续可微周期无穷带函数的Dubrovin微分方程无穷系的Cauchy问题的可解性。此外，证明了如果初始函数是π π -周期实解析函数，则Hirota方程的Cauchy问题的解也是变量x x上的实解析函数;如果数π /2 pi /2是初始函数的一个周期(反周期)，那么数π /2 pi /2就是变量x x的一个周期(反周期)，这是Hirota方程的柯西问题的解。
{"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":" 31","pages":"0"},"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}
<p>The Hölder classes <inline-formula content-type="math/mathml"> <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"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>α<!-- α --></mml:mi> </mml:mrow> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>L</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">L_p^{alpha } (L)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Subscript p Baseline left-parenthesis upper L right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>L</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">L_p(L)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> norm on a <italic>chord-arc</italic> curve <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R cubed"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">mathbb {R}^3</mml:annotation> </mml:semantics> </mml:math> </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"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript p Baseline left-parenthesis upper L right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>L</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">L^p(L)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> norm, the direct theorem is proved for a certain subclass of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Subscript p Superscri
{"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":" ","pages":""},"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}
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