<p>A complete description is obtained of the Carleman classes on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R Superscript n"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">mathbb {R}^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that every function of bounded type in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper C Subscript plus Superscript n"> <mml:semantics> <mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:mo>+</mml:mo> <mml:mi>n</mml:mi> </mml:msubsup> <mml:annotation encoding="application/x-tex">mathbb {C}^n_+</mml:annotation> </mml:semantics> </mml:math> </inline-formula> whose boundary values belong to the class under study is in fact a member of the corresponding Carleman class in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper C Subscript plus Superscript n Baseline union double-struck upper R Superscript n"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:mo>+</mml:mo> <mml:mi>n</mml:mi> </mml:msubsup> <mml:mo>∪<!-- ∪ --></mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">mathbb {C}^n_+cup mathbb {R}^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Also a refinement of the classical Salinas theorem is obtained, namely: under the conditions of the Salinas theorem on quasi-analyticity, instead of the assumption that a function belongs to the Carleman class in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper C Subscript plus Superscript n Baseline union double-struck upper R Superscript n"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:mo>+</mml:mo> <mml:mi>n</mml:mi> </mml:msubsup> <mml:mo>∪<!-- ∪ --></mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">mat
{"title":"Boundary quasi-analyticity and a Phragmén–Lindelöf type theorem in classes of functions of bounded type in tubular domains","authors":"F. Shamoyan","doi":"10.1090/spmj/1741","DOIUrl":"https://doi.org/10.1090/spmj/1741","url":null,"abstract":"<p>A complete description is obtained of the Carleman classes on <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> such that every function of bounded type in <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper C Subscript plus Superscript n\">\u0000 <mml:semantics>\u0000 <mml:msubsup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">C</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mo>+</mml:mo>\u0000 <mml:mi>n</mml:mi>\u0000 </mml:msubsup>\u0000 <mml:annotation encoding=\"application/x-tex\">mathbb {C}^n_+</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> whose boundary values belong to the class under study is in fact a member of the corresponding Carleman class in <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper C Subscript plus Superscript n Baseline union double-struck upper R Superscript n\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msubsup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">C</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mo>+</mml:mo>\u0000 <mml:mi>n</mml:mi>\u0000 </mml:msubsup>\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\">mathbb {C}^n_+cup mathbb {R}^n</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. Also a refinement of the classical Salinas theorem is obtained, namely: under the conditions of the Salinas theorem on quasi-analyticity, instead of the assumption that a function belongs to the Carleman class in <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper C Subscript plus Superscript n Baseline union double-struck upper R Superscript n\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msubsup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">C</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mo>+</mml:mo>\u0000 <mml:mi>n</mml:mi>\u0000 </mml:msubsup>\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\">mat","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44261564","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 classical theorem about cutting a round pizza into 8 pieces with straight cuts passing through an arbitrary internal point and forming angles of 45 degrees says that the total areas of odd and even pieces are equal if those pieces are ordered around the center of cutting. The current paper proposes a generalization of the Pizza theorem to any dimension and discovers a relationship with the finite reflection group of the series B n B_n .
{"title":"Reflection groups and the pizza theorem","authors":"Yu. Brailov","doi":"10.1090/spmj/1732","DOIUrl":"https://doi.org/10.1090/spmj/1732","url":null,"abstract":"The classical theorem about cutting a round pizza into 8 pieces with straight cuts passing through an arbitrary internal point and forming angles of 45 degrees says that the total areas of odd and even pieces are equal if those pieces are ordered around the center of cutting. The current paper proposes a generalization of the Pizza theorem to any dimension and discovers a relationship with the finite reflection group of the series \u0000\u0000 \u0000 \u0000 B\u0000 n\u0000 \u0000 B_n\u0000 \u0000\u0000.","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43229492","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}
Supercharacter theories are constructed for the finite groups obtained by parabolic contraction from simple groups of A , B , C , D A,B,C,D Lie types. Supercharacters and superclasses are classified in terms of rook placements in root systems.
{"title":"Supercharacters for parabolic contractions of finite groups of 𝐴,𝐵,𝐶,𝐷 Lie types","authors":"A. Panov","doi":"10.1090/spmj/1738","DOIUrl":"https://doi.org/10.1090/spmj/1738","url":null,"abstract":"Supercharacter theories are constructed for the finite groups obtained by parabolic contraction from simple groups of \u0000\u0000 \u0000 \u0000 A\u0000 ,\u0000 B\u0000 ,\u0000 C\u0000 ,\u0000 D\u0000 \u0000 A,B,C,D\u0000 \u0000\u0000 Lie types. Supercharacters and superclasses are classified in terms of rook placements in root systems.","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43223022","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 Morse numbers of spaces of matrix elements for real irreducible linear representations of compact connected simple Lie groups are estimate from above in a variety of ways, in terms of the dimension, the Dynkin index of the representation, the eigenvalues of the invariant Laplace operator, and the volume of the group.
{"title":"Upper estimates of the Morse numbers for the matrix elements of real linear irreducible representations of compact connected simple Lie groups","authors":"M. Meshcheryakov","doi":"10.1090/spmj/1737","DOIUrl":"https://doi.org/10.1090/spmj/1737","url":null,"abstract":"The Morse numbers of spaces of matrix elements for real irreducible linear representations of compact connected simple Lie groups are estimate from above in a variety of ways, in terms of the dimension, the Dynkin index of the representation, the eigenvalues of the invariant Laplace operator, and the volume of the group.","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48466929","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 local Lipschitz property is shown for the graphs avoiding multiple point intersection with lines directed in a given cone. The assumption is much stronger than those of Marstrand’s well-known theorem, but the conclusion is much stronger too. Additionally, a continuous curve with a similar property is σ sigma -finite with respect to Hausdorff length and an estimate on the Hausdorff measure of each “piece” is found.
{"title":"Geometry of planar curves intersecting many lines at a few points","authors":"D. Vardakis, A. Volberg","doi":"10.1090/spmj/1742","DOIUrl":"https://doi.org/10.1090/spmj/1742","url":null,"abstract":"The local Lipschitz property is shown for the graphs avoiding multiple point intersection with lines directed in a given cone. The assumption is much stronger than those of Marstrand’s well-known theorem, but the conclusion is much stronger too. Additionally, a continuous curve with a similar property is \u0000\u0000 \u0000 σ\u0000 sigma\u0000 \u0000\u0000-finite with respect to Hausdorff length and an estimate on the Hausdorff measure of each “piece” is found.","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44791503","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 established that the dihedral homologies of involutive A∞A_{infty }-algebras are homotopically invariant with respect to the homotopy equivalences of involutive A∞A_{infty }-algebras. As a consequence, it is shown that over any field, the dihedral homologies of a topological space are isomorphic to the dihedral homologies of the involutive A∞A_{infty }-algebra of homologies for the simplicial group of Kan loops of the original topological space.
{"title":"Homotopic invariance of dihedral homologies for 𝐴_{∞}-algebras with involution","authors":"S. Lapin","doi":"10.1090/spmj/1736","DOIUrl":"https://doi.org/10.1090/spmj/1736","url":null,"abstract":"<p>It is established that the dihedral homologies of 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 are homotopically invariant with respect to the homotopy equivalences of 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. As a consequence, it is shown that over any field, the dihedral homologies of a topological space are isomorphic to the dihedral homologies of the 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>-algebra of homologies for the simplicial group of Kan loops of the original topological space.</p>","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43089631","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>Consider a system of polynomial equations in <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> variables of degrees at most <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d"> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding="application/x-tex">d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with integer coefficients whose lengths are at most <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. By using a construction close to smooth stratification of algebraic varieties, it is shown that one can construct a positive integer <disp-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Delta greater-than 2 Superscript upper M left-parenthesis n d right-parenthesis Super Superscript c 2 Super Super Superscript n Super Superscript n cubed"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal">Δ<!-- Δ --></mml:mi> <mml:mo>></mml:mo> <mml:msup> <mml:mn>2</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>M</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mi>d</mml:mi> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>c</mml:mi> <mml:mspace width="thinmathspace" /> <mml:msup> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:msup> <mml:msup> <mml:mi>n</mml:mi> <mml:mn>3</mml:mn> </mml:msup> </mml:mrow> </mml:msup> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">begin{equation*} Delta > 2^{M(nd)^{c, 2^n n^3}} end{equation*}</mml:annotation> </mml:semantics> </mml:math> </disp-formula> (here <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c greater-than 0"> <mml:semantics> <mml:mrow> <mml:mi>c</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">c>0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a constant) depending on these polynomials and having the following property. For every prime <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="
考虑一个多项式方程组，其中n个变量的度数最多为d d，其整数系数的长度最多为M M。利用代数变量接近光滑分层的构造，我们可以构造一个正整数Δ >m (nd) c2n3begin{equation*} Delta > 2^{M(nd)^{c, 2^n n^3}} end{equation*}(这里c>0 c>0是一个常数)依赖于这些多项式，并且有以下属性。对于每一个素数p p，所研究的系统在p个p进数环中有解当且仅当它对最小整数N N有模p N p^N的解使得p N p^N不除Δ Delta。这改进了先前已知的，目前由B. J. Birch和K. McCann给出的经典结果。
{"title":"An effective algorithm for deciding the solvability of a system of polynomial equations over 𝑝-adic integers","authors":"A. Chistov","doi":"10.1090/spmj/1740","DOIUrl":"https://doi.org/10.1090/spmj/1740","url":null,"abstract":"<p>Consider a system of polynomial equations in <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n\">\u0000 <mml:semantics>\u0000 <mml:mi>n</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">n</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> variables of degrees at most <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"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> with integer coefficients whose lengths are at most <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper M\">\u0000 <mml:semantics>\u0000 <mml:mi>M</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">M</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. By using a construction close to smooth stratification of algebraic varieties, it is shown that one can construct a positive integer <disp-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Delta greater-than 2 Superscript upper M left-parenthesis n d right-parenthesis Super Superscript c 2 Super Super Superscript n Super Superscript n cubed\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi mathvariant=\"normal\">Δ<!-- Δ --></mml:mi>\u0000 <mml:mo>></mml:mo>\u0000 <mml:msup>\u0000 <mml:mn>2</mml:mn>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi>M</mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>n</mml:mi>\u0000 <mml:mi>d</mml:mi>\u0000 <mml:msup>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi>c</mml:mi>\u0000 <mml:mspace width=\"thinmathspace\" />\u0000 <mml:msup>\u0000 <mml:mn>2</mml:mn>\u0000 <mml:mi>n</mml:mi>\u0000 </mml:msup>\u0000 <mml:msup>\u0000 <mml:mi>n</mml:mi>\u0000 <mml:mn>3</mml:mn>\u0000 </mml:msup>\u0000 </mml:mrow>\u0000 </mml:msup>\u0000 </mml:mrow>\u0000 </mml:msup>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">begin{equation*} Delta > 2^{M(nd)^{c, 2^n n^3}} end{equation*}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</disp-formula>\u0000 (here <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"c greater-than 0\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>c</mml:mi>\u0000 <mml:mo>></mml:mo>\u0000 <mml:mn>0</mml:mn>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">c>0</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is a constant) depending on these polynomials and having the following property. For every prime <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47491456","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}
Several problems are discussed concerning steady-state distribution of heat in domains in R 3 mathbb {R}^3 that are complementary to a finite number of balls. The study of these problems was initiated by M. L. Glasser in 1977. Then, in 1978, M. L. Glasser and S. G. Davison presented numerical evidence that the heat flux from two equal balls in R 3 mathbb {R}^3 decreases when the balls move closer to each other. Those authors interpreted this result in terms of the behavioral habits of sleeping armadillos, the closer animals to each other, the less heat they lose. Much later, in 2003, A. Eremenko proved this monotonicity property rigorously and suggested new questions on the heat fluxes. The goal of this paper is to survey recent developments in this area, provide answers to some open questions, and draw attention to several challenging open problems concerning heat fluxes from configurations consisting of n ≥ 2 nge 2 balls in R 3 mathbb {R}^3 .
{"title":"Problems on the loss of heat: herd instinct versus individual feelings","authors":"A. Solynin","doi":"10.1090/spmj/1725","DOIUrl":"https://doi.org/10.1090/spmj/1725","url":null,"abstract":"Several problems are discussed concerning steady-state distribution of heat in domains in \u0000\u0000 \u0000 \u0000 \u0000 R\u0000 \u0000 3\u0000 \u0000 mathbb {R}^3\u0000 \u0000\u0000 that are complementary to a finite number of balls. The study of these problems was initiated by M. L. Glasser in 1977. Then, in 1978, M. L. Glasser and S. G. Davison presented numerical evidence that the heat flux from two equal balls in \u0000\u0000 \u0000 \u0000 \u0000 R\u0000 \u0000 3\u0000 \u0000 mathbb {R}^3\u0000 \u0000\u0000 decreases when the balls move closer to each other. Those authors interpreted this result in terms of the behavioral habits of sleeping armadillos, the closer animals to each other, the less heat they lose. Much later, in 2003, A. Eremenko proved this monotonicity property rigorously and suggested new questions on the heat fluxes.\u0000\u0000The goal of this paper is to survey recent developments in this area, provide answers to some open questions, and draw attention to several challenging open problems concerning heat fluxes from configurations consisting of \u0000\u0000 \u0000 \u0000 n\u0000 ≥\u0000 2\u0000 \u0000 nge 2\u0000 \u0000\u0000 balls in \u0000\u0000 \u0000 \u0000 \u0000 R\u0000 \u0000 3\u0000 \u0000 mathbb {R}^3\u0000 \u0000\u0000.","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45127126","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 problem of approximation of functions on plane compact sets by polyanalytic functions of order higher than two in the Hölder spaces C m C^m , m ∈ ( 0 , 1 ) min (0,1) , is significantly more complicated than the well-studied problem of approximation by analytic functions. In particular, the fundamental solutions of the corresponding operators belong to all the indicated Hölder spaces, but this does not lead to the triviality of the approximation conditions. In the model case of polyanalytic functions of order 3, approximation conditions and a constructive approximation method generalizing the Vitushkin localization method are studied. Some unsolved problems are formulated.
在Hölder空间C m C^m，m∈（0，1）min（0，0）中，用二阶以上的多解析函数逼近平面紧集上的函数的问题比用解析函数逼近的问题要复杂得多。特别地，对应算子的基本解属于所有指示的Hölder空间，但这并不导致近似条件的平凡性。在3阶多解析函数的模型情况下，研究了逼近条件和推广Vitushkin局部化方法的构造逼近方法。提出了一些尚未解决的问题。
{"title":"Approximation by polyanalytic functions in Hölder spaces","authors":"M. Mazalov","doi":"10.1090/spmj/1728","DOIUrl":"https://doi.org/10.1090/spmj/1728","url":null,"abstract":"The problem of approximation of functions on plane compact sets by polyanalytic functions of order higher than two in the Hölder spaces \u0000\u0000 \u0000 \u0000 C\u0000 m\u0000 \u0000 C^m\u0000 \u0000\u0000, \u0000\u0000 \u0000 \u0000 m\u0000 ∈\u0000 (\u0000 0\u0000 ,\u0000 1\u0000 )\u0000 \u0000 min (0,1)\u0000 \u0000\u0000, is significantly more complicated than the well-studied problem of approximation by analytic functions. In particular, the fundamental solutions of the corresponding operators belong to all the indicated Hölder spaces, but this does not lead to the triviality of the approximation conditions.\u0000\u0000In the model case of polyanalytic functions of order 3, approximation conditions and a constructive approximation method generalizing the Vitushkin localization method are studied. Some unsolved problems are formulated.","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42728931","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>By using the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="v"> <mml:semantics> <mml:mi>v</mml:mi> <mml:annotation encoding="application/x-tex">v</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-operation, a new characterization of the property for a power series ring to be a GCD domain is discussed. It is shown that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D"> <mml:semantics> <mml:mi>D</mml:mi> <mml:annotation encoding="application/x-tex">D</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U upper F upper D"> <mml:semantics> <mml:mi>UFD</mml:mi> <mml:annotation encoding="application/x-tex">operatorname {UFD}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, then <inline-formula content-type="math/tex"> <tex-math> DlBrack XrBrack </tex-math></inline-formula> is a GCD domain if and only if for any two integral <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="v"> <mml:semantics> <mml:mi>v</mml:mi> <mml:annotation encoding="application/x-tex">v</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-invertible <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="v"> <mml:semantics> <mml:mi>v</mml:mi> <mml:annotation encoding="application/x-tex">v</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-ideals <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I"> <mml:semantics> <mml:mi>I</mml:mi> <mml:annotation encoding="application/x-tex">I</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="upper J"> <mml:semantics> <mml:mi>J</mml:mi> <mml:annotation encoding="application/x-tex">J</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type="math/tex"> <tex-math> DlBrack XrBrack </tex-math></inline-formula> such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper I upper J right-parenthesis Subscript 0 Baseline not-equals left-parenthesis 0 right-parenthesis comma"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>I</mml:mi> <mml:mi>J</mml:mi> <mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> <mml:mo>≠<!-- ≠ --></mml:mo> <mml:mo stretchy="false">(<
{"title":"A new characterization of GCD domains of formal power series","authors":"A. Hamed","doi":"10.1090/spmj/1731","DOIUrl":"https://doi.org/10.1090/spmj/1731","url":null,"abstract":"<p>By using the <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"v\">\u0000 <mml:semantics>\u0000 <mml:mi>v</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">v</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-operation, a new characterization of the property for a power series ring to be a GCD domain is discussed. It is shown that if <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> is a <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper U upper F upper D\">\u0000 <mml:semantics>\u0000 <mml:mi>UFD</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">operatorname {UFD}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, then <inline-formula content-type=\"math/tex\">\u0000<tex-math>\u0000DlBrack XrBrack </tex-math></inline-formula> is a GCD domain if and only if for any two integral <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"v\">\u0000 <mml:semantics>\u0000 <mml:mi>v</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">v</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-invertible <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"v\">\u0000 <mml:semantics>\u0000 <mml:mi>v</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">v</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-ideals <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper I\">\u0000 <mml:semantics>\u0000 <mml:mi>I</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">I</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> and <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper J\">\u0000 <mml:semantics>\u0000 <mml:mi>J</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">J</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> of <inline-formula content-type=\"math/tex\">\u0000<tex-math>\u0000DlBrack XrBrack </tex-math></inline-formula> such that <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis upper I upper J right-parenthesis Subscript 0 Baseline not-equals left-parenthesis 0 right-parenthesis comma\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>I</mml:mi>\u0000 <mml:mi>J</mml:mi>\u0000 <mml:msub>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mn>0</mml:mn>\u0000 </mml:mrow>\u0000 </mml:msub>\u0000 <mml:mo>≠<!-- ≠ --></mml:mo>\u0000 <mml:mo stretchy=\"false\">(<","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48505408","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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