The combinatorics of the Gelfand–Zetlin polytope is studied. Geometric properties of a linear projection of this polytope onto a cube are employed to derive a recurrence relation for the f f -polynomial of the polytope. This recurrence relation is applied to finding the f f -polynomials and h h -polynomials for one-parameter families of Gelfand–Zetlin polytopes of simplest types.
{"title":"On the number of faces of the Gelfand–Zetlin polytope","authors":"E. Melikhova","doi":"10.1090/spmj/1714","DOIUrl":"https://doi.org/10.1090/spmj/1714","url":null,"abstract":"The combinatorics of the Gelfand–Zetlin polytope is studied. Geometric properties of a linear projection of this polytope onto a cube are employed to derive a recurrence relation for the \u0000\u0000 \u0000 f\u0000 f\u0000 \u0000\u0000-polynomial of the polytope. This recurrence relation is applied to finding the \u0000\u0000 \u0000 f\u0000 f\u0000 \u0000\u0000-polynomials and \u0000\u0000 \u0000 h\u0000 h\u0000 \u0000\u0000-polynomials for one-parameter families of Gelfand–Zetlin polytopes of simplest types.","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49633345","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}
Two constructions are studied that are inspired by the ideas of a recent paper by the authors.
— The diagonal complexDmathcal {D} and its barycentric subdivision BDmathcal {BD} related to an oriented surface of finite type FF equipped with a number of labeled marked points. This time, unlike the paper mentioned above, boundary components without marked points are allowed, called holes.
— The symmetric diagonal complexDinvmathcal {D}^{operatorname {inv}} and its barycentric subdivision BDinvmathcal {BD}^{operatorname {inv}} related to a symmetric (=with an involution) oriented surface
受作者最近一篇论文的启发，研究了两种结构对角线复形Dmathcal｛D｝及其重心细分B Dmathical｛BD｝与配备有多个标记点的有限类型F F的有向表面有关。这一次，与上面提到的论文不同，允许没有标记点的边界组件，称为孔。——对称对角复形D inv mathcal｛D｝^｛operatorname｛inv｝｝及其重心细分B D invmathcal{BD｝^｝operator name｛inv｝与一个对称（=带对合）定向的表面F有关，该表面F配备了许多（对称放置的）标记标记点。对称复形被证明是等价于通过“取”初始对称曲面的一半而获得的曲面的复形的同伦性。
{"title":"Diagonal complexes for surfaces of finite type and surfaces with involution","authors":"G. Panina, J. Gordon","doi":"10.1090/spmj/1709","DOIUrl":"https://doi.org/10.1090/spmj/1709","url":null,"abstract":"<p>Two constructions are studied that are inspired by the ideas of a recent paper by the authors.</p>\u0000\u0000<p>— The <italic> diagonal complex</italic> <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper D\">\u0000 <mml:semantics>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">D</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">mathcal {D}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> and its barycentric subdivision <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper B script upper D\">\u0000 <mml:semantics>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">B</mml:mi>\u0000 <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">D</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">mathcal {BD}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> related to an oriented surface of finite type <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper F\">\u0000 <mml:semantics>\u0000 <mml:mi>F</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">F</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> equipped with a number of labeled marked points. This time, unlike the paper mentioned above, boundary components without marked points are allowed, called <italic>holes</italic>.</p>\u0000\u0000<p>— The <italic>symmetric diagonal complex</italic> <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper D Superscript i n v\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">D</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi>inv</mml:mi>\u0000 </mml:mrow>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">mathcal {D}^{operatorname {inv}}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> and its barycentric subdivision <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper B script upper D Superscript i n v\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">B</mml:mi>\u0000 <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">D</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi>inv</mml:mi>\u0000 </mml:mrow>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">mathcal {BD}^{operatorname {inv}}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> related to a <italic>symmetric</italic> (=with an involution) oriented surface <inline-formula content-type=\"math","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41261484","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 a class of analytic functions in a bounded convex domain GG that admit an exponential series expansion in DD, the behavior of the coefficients of this expansion is studied in terms of the growth order near the boundary ∂Gpartial G. In the case where GG has a smooth boundary, unimprovable two-sided estimates are established for the order via characteristics depending only on the exponents of the exponential series and the support function of GG. As a consequence, a formula is obtained for the growth of the exponential series via the coefficients and the support function of the convergence domain GG.
{"title":"The order of growth of an exponential series near the boundary of the convergence domain","authors":"G. Gaisina","doi":"10.1090/spmj/1708","DOIUrl":"https://doi.org/10.1090/spmj/1708","url":null,"abstract":"<p>For a class of analytic functions in a bounded convex domain <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\u0000 <mml:semantics>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> that admit an exponential series expansion in <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper D\">\u0000 <mml:semantics>\u0000 <mml:mi>D</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">D</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, the behavior of the coefficients of this expansion is studied in terms of the growth order near the boundary <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"partial-differential upper G\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi mathvariant=\"normal\">∂<!-- ∂ --></mml:mi>\u0000 <mml:mi>G</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">partial G</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. In the case where <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\u0000 <mml:semantics>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> has a smooth boundary, unimprovable two-sided estimates are established for the order via characteristics depending only on the exponents of the exponential series and the support function of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\u0000 <mml:semantics>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. As a consequence, a formula is obtained for the growth of the exponential series via the coefficients and the support function of the convergence domain <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\u0000 <mml:semantics>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>.</p>","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41505606","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 shown that elliptic solutions to the AKNS hierarchy equations can be obtained by exploring spectral curves that correspond to elliptic solutions of the KdV hierarchy. This also allows one to get the quasirational and trigonometric solutions for AKNS hierarchy equations as a limit case of the elliptic solutions mentioned above.
{"title":"Elliptic solitons and “freak waves”","authors":"V. Matveev, A. Smirnov","doi":"10.1090/spmj/1713","DOIUrl":"https://doi.org/10.1090/spmj/1713","url":null,"abstract":"It is shown that elliptic solutions to the AKNS hierarchy equations can be obtained by exploring spectral curves that correspond to elliptic solutions of the KdV hierarchy. This also allows one to get the quasirational and trigonometric solutions for AKNS hierarchy equations as a limit case of the elliptic solutions mentioned above.","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42432684","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}
Joint weighted universality theorems are proved concerning simultaneous approximation of a collection of analytic functions by a collection of shifts of Hurwitz zeta-functions with parameters α1,…,αralpha _1,dots ,alpha _r. For this, linear independence is required over the field of rational numbers for the set {log(m+αj):m∈N0=N∪{0},j=1,…,r}{log (m+alpha _j),:, min mathbb {N}_0=mathbb {N}cup {0},;j=1,dots ,r}.
The classical theorems of Kronecker, Hartman, Peller, and Adamyan–Arov–Krein are extended to the context of a connected compact Abelian group G G with linearly ordered group of characters, on the basis of a description of the structure of compact Hankel operators on G G . Beurling’s theorem on invariant subspaces is also generalized. Some applications to Hankel operators on discrete groups are given.
{"title":"Compact Hankel operators on compact Abelian groups","authors":"A. Mirotin","doi":"10.1090/spmj/1715","DOIUrl":"https://doi.org/10.1090/spmj/1715","url":null,"abstract":"The classical theorems of Kronecker, Hartman, Peller, and Adamyan–Arov–Krein are extended to the context of a connected compact Abelian group \u0000\u0000 \u0000 G\u0000 G\u0000 \u0000\u0000 with linearly ordered group of characters, on the basis of a description of the structure of compact Hankel operators on \u0000\u0000 \u0000 G\u0000 G\u0000 \u0000\u0000. Beurling’s theorem on invariant subspaces is also generalized. Some applications to Hankel operators on discrete groups are given.","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46261293","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}
On the basis of combinatorial techniques of dihedral modules with ∞infty-simplicial faces, dihedral homology is constructed for involutive A∞A_{infty }-algebras over arbitrary commutative unital rings.
基于具有∞单纯形面的二面体模的组合技术，构造了任意交换酉环上对合A∞A_代数的二面体同调。
{"title":"Dihedral modules with ∞-simplicial faces and dihedral homology for involutive 𝐴_{∞}-algebras over rings","authors":"S. Lapin","doi":"10.1090/spmj/1711","DOIUrl":"https://doi.org/10.1090/spmj/1711","url":null,"abstract":"<p>On the basis of combinatorial techniques of dihedral modules with <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal infinity\">\u0000 <mml:semantics>\u0000 <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">infty</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-simplicial faces, dihedral homology is constructed for involutive <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A Subscript normal infinity\">\u0000 <mml:semantics>\u0000 <mml:msub>\u0000 <mml:mi>A</mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi>\u0000 </mml:mrow>\u0000 </mml:msub>\u0000 <mml:annotation encoding=\"application/x-tex\">A_{infty }</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-algebras over arbitrary commutative unital rings.</p>","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42773224","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 tail of distribution (i.e., the measure of the set { f ≥ x } {fge x} ) is estimated for those functions f f whose dyadic square function is bounded by a given constant. In particular, an estimate following from the Chang–Wilson–Wolf theorem is slightly improved. The study of the Bellman function corresponding to the problem reveals a curious structure of this function: it has jumps of the first derivative at a dense subset of the interval [ 0 , 1 ] [0,1] (where it is calculated exactly), but it is of C ∞ C^infty -class for x > 3 x>sqrt 3 (where it is calculated up to a multiplicative constant). An unusual feature of the paper consists of the usage of computer calculations in the proof. Nevertheless, all the proofs are quite rigorous, since only the integer arithmetic was assigned to a computer.
分布的尾部（即集合｛f≥x｝｛f ge x｝的测度）是为那些二进平方函数由给定常数定界的函数f f估计的。特别是，根据Chang–Wilson–Wolf定理得出的估计略有改进。对与该问题相对应的Bellman函数的研究揭示了该函数的一个奇怪结构：它在区间[0，1][0,1]的稠密子集上有一阶导数的跳跃（在这里它是精确计算的），但对于x>3x>sqrt3（其中它被计算到乘法常数），它是C∞C^infty类的。本文的一个不同寻常的特点是在证明中使用了计算机计算。尽管如此，所有的证明都相当严格，因为只有整数运算被分配给了计算机。
{"title":"On a Bellman function associated with the Chang–Wilson–Wolff theorem: a case study","authors":"F. Nazarov, V. Vasyunin, A. Volberg","doi":"10.1090/spmj/1719","DOIUrl":"https://doi.org/10.1090/spmj/1719","url":null,"abstract":"The tail of distribution (i.e., the measure of the set \u0000\u0000 \u0000 \u0000 {\u0000 f\u0000 ≥\u0000 x\u0000 }\u0000 \u0000 {fge x}\u0000 \u0000\u0000) is estimated for those functions \u0000\u0000 \u0000 f\u0000 f\u0000 \u0000\u0000 whose dyadic square function is bounded by a given constant. In particular, an estimate following from the Chang–Wilson–Wolf theorem is slightly improved. The study of the Bellman function corresponding to the problem reveals a curious structure of this function: it has jumps of the first derivative at a dense subset of the interval \u0000\u0000 \u0000 \u0000 [\u0000 0\u0000 ,\u0000 1\u0000 ]\u0000 \u0000 [0,1]\u0000 \u0000\u0000 (where it is calculated exactly), but it is of \u0000\u0000 \u0000 \u0000 C\u0000 ∞\u0000 \u0000 C^infty\u0000 \u0000\u0000-class for \u0000\u0000 \u0000 \u0000 x\u0000 >\u0000 \u0000 3\u0000 \u0000 \u0000 x>sqrt 3\u0000 \u0000\u0000 (where it is calculated up to a multiplicative constant).\u0000\u0000An unusual feature of the paper consists of the usage of computer calculations in the proof. Nevertheless, all the proofs are quite rigorous, since only the integer arithmetic was assigned to a computer.","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46830079","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 a Hilbert space Hmathfrak {H}, a family of operators A(t)A(t), t∈Rtin mathbb {R}, is treated admitting a factorization of the form A(t)=X(t)∗X(t)A(t) = X(t)^* X(t), where X(t)=X0+
在Hilbert空间Hmathfrak｛H｝中，一个算子族a（t）a（t，其中，X（t）=X 0+X 1 t+…+X p t p X（t）=X_0+X_1t+cdots+X_pt^p，p≥2 pge 2。假定点λ0=0λ0=0是A（0）A（0。设F（t）F（t。对于|t|≤t0|t|le t^0，在Hmathfrak｛H｝中，得到了具有误差O（t2 p）O（t^｛2p｝）的投影F（t）F（t带有错误O（t4p）O（t^｛4p｝）
{"title":"Threshold approximations for the resolvent of a polynomial nonnegative operator pencil","authors":"V. Sloushch, T. Suslina","doi":"10.1090/spmj/1704","DOIUrl":"https://doi.org/10.1090/spmj/1704","url":null,"abstract":"<p>In a Hilbert space <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"German upper H\">\u0000 <mml:semantics>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"fraktur\">H</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">mathfrak {H}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, a family of operators <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A left-parenthesis t right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>A</mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>t</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">A(t)</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=\"t element-of double-struck upper R\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>t</mml:mi>\u0000 <mml:mo>∈<!-- ∈ --></mml:mo>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">R</mml:mi>\u0000 </mml:mrow>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">tin mathbb {R}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, is treated admitting a factorization of the form <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A left-parenthesis t right-parenthesis equals upper X left-parenthesis t right-parenthesis Superscript asterisk Baseline upper X left-parenthesis t right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>A</mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>t</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 <mml:mo>=</mml:mo>\u0000 <mml:mi>X</mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>t</mml:mi>\u0000 <mml:msup>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 <mml:mo>∗<!-- ∗ --></mml:mo>\u0000 </mml:msup>\u0000 <mml:mi>X</mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>t</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">A(t) = X(t)^* X(t)</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 X left-parenthesis t right-parenthesis equals upper X 0 plus upper X 1 t plus midline-horizontal-ellipsis plus upper X Subscript p Baseline t Superscript p\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>X</mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>t</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 <mml:mo>=</mml:mo>\u0000 <mml:msub>\u0000 <mml:mi>X</mml:mi>\u0000 <mml:mn>0</mml:mn>\u0000 </mml:msub>\u0000 <mml:mo>+</mml:mo>\u0000 <mml:msu","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49483908","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 system of two nonlinear differential equations with slowly varying coefficients is treated. The asymptotics in the small parameter for the solutions that have a narrow transition layer is studied. Such a layer occurs near the moment where the number of roots of the corresponding algebraic system of equations changes. To construct the asymptotics, the matching method involving three scales is used.
{"title":"Solution asymptotics for the system of Landau–Lifshitz equations under a saddle-node dynamical bifurcation","authors":"L. Kalyakin","doi":"10.1090/spmj/1698","DOIUrl":"https://doi.org/10.1090/spmj/1698","url":null,"abstract":"A system of two nonlinear differential equations with slowly varying coefficients is treated. The asymptotics in the small parameter for the solutions that have a narrow transition layer is studied. Such a layer occurs near the moment where the number of roots of the corresponding algebraic system of equations changes. To construct the asymptotics, the matching method involving three scales is used.","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47708952","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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