A sign interlacing phenomenon for Bessel sequences, frames, and Riesz bases (uk)(u_{k}) in L2L^2 spaces over the spaces of homogeneous type Ω=(Ω,ρ,μ)Omega =(Omega ,rho ,mu ) satisfying the doubling/halving conditions is studied. Under some relations among three basic metric-measure parameters of ΩOmega, asymptotics is obtained for the mass moving norms ‖uk‖KR|u_k|_{KR}
研究了满足加倍/减半条件的齐次型空间Ω =(Ω， ρ， μ) {}Omega =(Omega, rho, mu)上l2 L^2空间中Bessel序列、框架和Riesz基(u ) (u k) (u k)的符号交错现象。在Ω Omega的三个基本度量参数之间的一些关系下，得到了{Kantorovich}-Rubinstein意义上的质量移动范数‖u k‖k R | ＿k|＿KR的渐近性，以及Lipschitz嵌入和Hajlasz-Sobolev嵌入的奇异数的渐近性。主要观察结果表明，从数量上讲，收敛速率—u k‖k R→0 |—u k|{＿KR}to 0主要取决于某紧化Lipschitz函数集的Bernstein-Kolmogorov n -宽度，而宽度本身主要取决于几何倍增和测量倍增/减半数值参数之间的相互作用。空间越“均匀”，结果就越清晰。
{"title":"Three dimensions of metric-measure spaces, Sobolev embeddings and optimal sign transport","authors":"N. Nikolski","doi":"10.1090/spmj/1752","DOIUrl":"https://doi.org/10.1090/spmj/1752","url":null,"abstract":"<p>A sign interlacing phenomenon for Bessel sequences, frames, and Riesz bases <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis u Subscript k Baseline right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:msub>\u0000 <mml:mi>u</mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi>k</mml:mi>\u0000 </mml:mrow>\u0000 </mml:msub>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">(u_{k})</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=\"upper L squared\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mi>L</mml:mi>\u0000 <mml:mn>2</mml:mn>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">L^2</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> spaces over the spaces of homogeneous type <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Omega equals left-parenthesis normal upper Omega comma rho comma mu right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi mathvariant=\"normal\">Ω<!-- Ω --></mml:mi>\u0000 <mml:mo>=</mml:mo>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi mathvariant=\"normal\">Ω<!-- Ω --></mml:mi>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mi>ρ<!-- ρ --></mml:mi>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mi>μ<!-- μ --></mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">Omega =(Omega ,rho ,mu )</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> satisfying the doubling/halving conditions is studied. Under some relations among three basic metric-measure parameters of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Omega\">\u0000 <mml:semantics>\u0000 <mml:mi mathvariant=\"normal\">Ω<!-- Ω --></mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">Omega</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, asymptotics is obtained for the mass moving norms <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-vertical-bar u Subscript k Baseline double-vertical-bar Subscript upper K upper R\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mo fence=\"false\" stretchy=\"false\">‖<!-- ‖ --></mml:mo>\u0000 <mml:msub>\u0000 <mml:mi>u</mml:mi>\u0000 <mml:mi>k</mml:mi>\u0000 </mml:msub>\u0000 <mml:msub>\u0000 <mml:mo fence=\"false\" stretchy=\"false\">‖<!-- ‖ --></mml:mo>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi>K</mml:mi>\u0000 <mml:mi>R</mml:mi>\u0000 </mml:mrow>\u0000 </mml:msub>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">|u_k|_{KR}</mml:annotation>\u0000 </","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45377022","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 proved that analogs of the theorems of M. Hall and N. S. Romanovskii are not true in the class of commutative rings. Necessary and sufficient conditions for the local finite separability of monogenic rings are established. As a corollary, it is proved that a finitely generated torsion-free PI-ring is locally finitely separable if and only if its additive group is finitely generated.
证明了M. Hall定理和N. S. Romanovskii定理的类比在交换环类中是不成立的。建立了单基因环局部有限可分性的充分必要条件。作为推论，证明了有限生成无扭pi环局部有限可分当且仅当其加性群有限生成。
{"title":"On the local finite separability of finitely generated associative rings","authors":"S. Kublanovskiĭ","doi":"10.1090/spmj/1751","DOIUrl":"https://doi.org/10.1090/spmj/1751","url":null,"abstract":"It is proved that analogs of the theorems of M. Hall and N. S. Romanovskii are not true in the class of commutative rings. Necessary and sufficient conditions for the local finite separability of monogenic rings are established. As a corollary, it is proved that a finitely generated torsion-free PI-ring is locally finitely separable if and only if its additive group is finitely generated.","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41659194","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 periodicity is proved for the arithmetic function defined as the quotient of the product of k+1k+1 values (where k≥1k geq 1) of a polynomial f∈Z[x]fin {mathbb Z}[x] at k+1k + 1 consecutive integers f(n)f(n+1)⋯f(n+k){f(n) f(n + 1) cdots f(n + k)}
This is a survey of recent results on multiple transitivity for automorphism groups of affine algebraic varieties. The property of infinite transitivity of the special automorphism group is treated, which is equivalent to the flexibility of the corresponding affine variety. These properties have important algebraic and geometric consequences. At the same time they are fulfilled for wide classes of varieties. Also, the situations are studied where infinite transitivity occurs for automorphism groups generated by finitely many one-parameter subgroups. In the appendices to the paper, the results on infinitely transitive actions in complex analysis and in combinatorial group theory are discussed.
{"title":"Automorphisms of algebraic varieties and infinite transitivity","authors":"I. Arzhantsev","doi":"10.1090/spmj/1749","DOIUrl":"https://doi.org/10.1090/spmj/1749","url":null,"abstract":"This is a survey of recent results on multiple transitivity for automorphism groups of affine algebraic varieties. The property of infinite transitivity of the special automorphism group is treated, which is equivalent to the flexibility of the corresponding affine variety. These properties have important algebraic and geometric consequences. At the same time they are fulfilled for wide classes of varieties. Also, the situations are studied where infinite transitivity occurs for automorphism groups generated by finitely many one-parameter subgroups. In the appendices to the paper, the results on infinitely transitive actions in complex analysis and in combinatorial group theory are discussed.","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44296557","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 Banach space XX defined in terms of a big-OO condition and its subspace x defined by the corresponding little-oo condition, the biduality property (generalizing the concept of reflexivity) asserts that the bidual of x is naturally isometrically isomorphic to XX. The property is known for pairs of many classical function spaces (such as (ℓ∞,c0)(ell _infty , c_0), (BMO, VMO), (Lip, lip), etc.) and plays an important role in the study of their geometric structure. The present paper is devoted to the biduality property for traces to closed subsets S⊂RnSsubset mathbb {R}^n of a generalized Zygmund space
对于由大0条件定义的巴拿赫空间X X及其由相应的小0条件定义的子空间X，其对偶性质(推广自反性概念)断言X的对偶与X X自然是等距同构的。这一性质在许多经典函数空间对(如(r∞，c 0) (ell ＿ infty, c＿0)， (BMO, VMO)， (Lip, Lip)等)中都是已知的，在研究它们的几何结构中起着重要作用。本文研究广义Zygmund空间Z ω (rn) Z^ omega (mathbb R{^n)的闭子集S∧R n S }subsetmathbb R{^n的迹的对偶性。证明方法是基于对迹空间的几何预公数结构的仔细分析，以及迹空间Z ω (R n)| S Z^ }omega (mathbb R{^n)|＿S的一个强有力的有限性定理。}
{"title":"Two stars theorems for traces of the Zygmund space","authors":"A. Brudnyi","doi":"10.1090/spmj/1744","DOIUrl":"https://doi.org/10.1090/spmj/1744","url":null,"abstract":"<p>For a Banach space <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X\">\u0000 <mml:semantics>\u0000 <mml:mi>X</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">X</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> defined in terms of a big-<inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper O\">\u0000 <mml:semantics>\u0000 <mml:mi>O</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">O</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> condition and its subspace <italic>x</italic> defined by the corresponding little-<inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"o\">\u0000 <mml:semantics>\u0000 <mml:mi>o</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">o</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> condition, the biduality property (generalizing the concept of reflexivity) asserts that the bidual of <italic>x</italic> is naturally isometrically isomorphic to <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X\">\u0000 <mml:semantics>\u0000 <mml:mi>X</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">X</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. The property is known for pairs of many classical function spaces (such as <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis script l Subscript normal infinity Baseline comma c 0 right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:msub>\u0000 <mml:mi>ℓ<!-- ℓ --></mml:mi>\u0000 <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi>\u0000 </mml:msub>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:msub>\u0000 <mml:mi>c</mml:mi>\u0000 <mml:mn>0</mml:mn>\u0000 </mml:msub>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">(ell _infty , c_0)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, (BMO, VMO), (Lip, lip), etc.) and plays an important role in the study of their geometric structure. The present paper is devoted to the biduality property for traces to closed subsets <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper S subset-of double-struck upper R Superscript n\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>S</mml:mi>\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\">Ssubset mathbb {R}^n</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> of a generalized Zygmund space <in","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43598591","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}
Consider the equation −Δu+Vu=0-Delta u + Vu = 0 in the half-cylinder [0,∞)×(0,2π)d[0, infty ) times (0,2pi )^d with periodic boundary conditions. Assume that the potential VV is bounded. The possible rate of decay at infinity for a nontrivial solution is studied. It is shown that the fastest rate of decay is e−cxe^{-cx} for d=1d=1 or
考虑半圆柱体[0，∞）×（0，2π）d[0，infty）times（0,2pi）中的方程−Δu+V u=0-Δu+Vu=0^d具有周期性边界条件。假设电势V V是有界的。研究了一个非平凡解在无穷远处的可能衰变率。结果表明，当d＝1 d＝1或2 2时，最快的衰变率是e−c x e ^｛-cx｝；当d≥3时，最慢的衰变率为e−c×第3页；这里x x是轴向变量。
{"title":"On the rate of decay at infinity for solutions to the Schrödinger equation in a half-cylinder","authors":"S. Krymskii, N. Filonov","doi":"10.1090/spmj/1746","DOIUrl":"https://doi.org/10.1090/spmj/1746","url":null,"abstract":"<p>Consider the equation <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"minus normal upper Delta u plus upper V u equals 0\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mo>−<!-- − --></mml:mo>\u0000 <mml:mi mathvariant=\"normal\">Δ<!-- Δ --></mml:mi>\u0000 <mml:mi>u</mml:mi>\u0000 <mml:mo>+</mml:mo>\u0000 <mml:mi>V</mml:mi>\u0000 <mml:mi>u</mml:mi>\u0000 <mml:mo>=</mml:mo>\u0000 <mml:mn>0</mml:mn>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">-Delta u + Vu = 0</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> in the half-cylinder <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-bracket 0 comma normal infinity right-parenthesis times left-parenthesis 0 comma 2 pi right-parenthesis Superscript d\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mo stretchy=\"false\">[</mml:mo>\u0000 <mml:mn>0</mml:mn>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 <mml:mo>×<!-- × --></mml:mo>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mn>0</mml:mn>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mn>2</mml:mn>\u0000 <mml:mi>π<!-- π --></mml:mi>\u0000 <mml:msup>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 <mml:mi>d</mml:mi>\u0000 </mml:msup>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">[0, infty ) times (0,2pi )^d</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> with periodic boundary conditions. Assume that the potential <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper 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> is bounded. The possible rate of decay at infinity for a nontrivial solution is studied. It is shown that the fastest rate of decay is <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"e Superscript minus c x\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mi>e</mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mo>−<!-- − --></mml:mo>\u0000 <mml:mi>c</mml:mi>\u0000 <mml:mi>x</mml:mi>\u0000 </mml:mrow>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">e^{-cx}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> for <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"d equals 1\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>d</mml:mi>\u0000 <mml:mo>=</mml:mo>\u0000 <mml:mn>1</mml:mn>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">d=1</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> or <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"2\">\u0000 <mml:semant","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49192155","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 mutual arrangements of a real algebraic or real pseudoholomorphic plane projective M M -sextic and a line up to isotopy are studied. A complete list of pseudoholomorphic arrangements is obtained. Four of them are proved to be algebraically unrealizable. All the others with two exceptions are algebraically realized.
{"title":"Arrangements of a plane 𝑀-sextic with respect to a line","authors":"S. Orevkov","doi":"10.1090/spmj/1747","DOIUrl":"https://doi.org/10.1090/spmj/1747","url":null,"abstract":"The mutual arrangements of a real algebraic or real pseudoholomorphic plane projective \u0000\u0000 \u0000 M\u0000 M\u0000 \u0000\u0000-sextic and a line up to isotopy are studied. A complete list of pseudoholomorphic arrangements is obtained. Four of them are proved to be algebraically unrealizable. All the others with two exceptions are algebraically realized.","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45832975","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 algebraic cobordism spectra MSLmathbf {MSL} and MSpmathbf {MSp} are constructed. They are commutative monoids in the category of symmetric T∧2T^{wedge 2}-spectra. The spectrum MSpmathbf {MSp} comes with a natural symplectic orientation given either by a tautological Thom class thMSp∈M
构造了代数同基谱MSLmathbf{MSL}和MSpmathbf{MSp}。它们是对称T∧2 T^｛wedge 2｝-谱范畴中的交换幺群。谱M S p mathbf｛MSp｝具有一个自然辛定向，该定向由一个重言托姆类t h M S p∈M S p 4，2（M S p 2）th ^｛mathbf｛MSp｝｝ in mathbf｛MSp｝^｛4,2｝{MSp}_2)或一个重言的Borel类b1M S p∈M S p 4，2（HP∞）b_｛1｝^｛mathbf｛MSp｝｝inmathbf{MSp｝^｛4,2｝（HP^｛infty｝），或其他六种等效结构中的任何一种。对于范畴S H（S）{SH}（S）中的一个交换幺半群E E，证明了赋值φ↦ φ（t h M S p）varphimapstovarphi（th ^｛mathbf｛MSp｝｝）确定了么半群的同态集φ：M S p→ Evarphicolonmathbf｛MSp｝ to E在运动稳定的同伦学范畴SH（S）SH（S。对于M S L mathbf｛MSL｝和特殊的线性取向，得到了一个较弱的普适性结果。的普遍性
{"title":"On the algebraic cobordism spectra 𝐌𝐒𝐋 and 𝐌𝐒𝐩","authors":"I. Panin, C. Walter","doi":"10.1090/spmj/1748","DOIUrl":"https://doi.org/10.1090/spmj/1748","url":null,"abstract":"<p>The algebraic cobordism spectra <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"bold upper M bold upper S bold upper L\">\u0000 <mml:semantics>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"bold\">M</mml:mi>\u0000 <mml:mi mathvariant=\"bold\">S</mml:mi>\u0000 <mml:mi mathvariant=\"bold\">L</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">mathbf {MSL}</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=\"bold upper M bold upper S bold p\">\u0000 <mml:semantics>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"bold\">M</mml:mi>\u0000 <mml:mi mathvariant=\"bold\">S</mml:mi>\u0000 <mml:mi mathvariant=\"bold\">p</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">mathbf {MSp}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> are constructed. They are commutative monoids in the category of symmetric <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper T Superscript logical-and 2\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mi>T</mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mo>∧<!-- ∧ --></mml:mo>\u0000 <mml:mn>2</mml:mn>\u0000 </mml:mrow>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">T^{wedge 2}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-spectra. The spectrum <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"bold upper M bold upper S bold p\">\u0000 <mml:semantics>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"bold\">M</mml:mi>\u0000 <mml:mi mathvariant=\"bold\">S</mml:mi>\u0000 <mml:mi mathvariant=\"bold\">p</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">mathbf {MSp}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> comes with a natural symplectic orientation given either by a tautological Thom class <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"t h Superscript bold upper M bold upper S bold p element-of bold upper M bold upper S bold p Superscript 4 comma 2 Baseline left-parenthesis bold upper M bold upper S bold p Subscript 2 Baseline right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>t</mml:mi>\u0000 <mml:msup>\u0000 <mml:mi>h</mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"bold\">M</mml:mi>\u0000 <mml:mi mathvariant=\"bold\">S</mml:mi>\u0000 <mml:mi mathvariant=\"bold\">p</mml:mi>\u0000 </mml:mrow>\u0000 </mml:mrow>\u0000 </mml:msup>\u0000 <mml:mo>∈<!-- ∈ --></mml:mo>\u0000 <mml:msup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"bold\">M</mml:mi>\u0000","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47168639","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 the present work some Jackson Stechkin type direct theorems of trigonometric approximation are proved in Orlicz spaces with weights satisfying some Muckenhoupt A p A_p condition. To obtain a refined version of the Jackson type inequality, an extrapolation theorem, Marcinkiewicz multiplier theorem, and Littlewood–Paley type results are proved. As a consequence, refined inverse Marchaud type inequalities are obtained. By means of a realization result, an equivalence is found between the fractional order weighted modulus of smoothness and Peetre’s classical weighted K K -functional.
本文在权值满足Muckenhoupt A p A p条件的Orlicz空间中证明了几个Jackson Stechkin型的三角逼近直接定理。为了得到Jackson型不等式的一个改进版本，证明了外推定理、Marcinkiewicz乘数定理和Littlewood-Paley型结果。因此，得到了精细的逆Marchaud型不等式。通过一个实现结果，发现了分数阶加权光滑模与Peetre经典加权kk泛函之间的等价性。
{"title":"Jackson type inequalities for differentiable functions in weighted Orlicz spaces","authors":"R. Akgün","doi":"10.1090/spmj/1743","DOIUrl":"https://doi.org/10.1090/spmj/1743","url":null,"abstract":"In the present work some Jackson Stechkin type direct theorems of trigonometric approximation are proved in Orlicz spaces with weights satisfying some Muckenhoupt \u0000\u0000 \u0000 \u0000 A\u0000 p\u0000 \u0000 A_p\u0000 \u0000\u0000 condition. To obtain a refined version of the Jackson type inequality, an extrapolation theorem, Marcinkiewicz multiplier theorem, and Littlewood–Paley type results are proved. As a consequence, refined inverse Marchaud type inequalities are obtained. By means of a realization result, an equivalence is found between the fractional order weighted modulus of smoothness and Peetre’s classical weighted \u0000\u0000 \u0000 K\u0000 K\u0000 \u0000\u0000-functional.","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42305178","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 the three nonstationary Schrödinger equations i ℏ Ψ τ = H ( x , y , − i ℏ ∂ ∂ x , − i ℏ ∂ ∂ y ) Ψ , begin{equation*} ihbar Psi _{tau }=H(x,y,-ihbar frac {partial }{partial x},-ihbar frac {partial }{partial y})Psi , end{equation*} solutions are constructed that correspond to conservative Hamiltonian systems with two degrees of freedom whose general solutions can be represented by those of the second Painlevé equation. These solutions of the Schrödinger equations are expressed via fundamental solutions of systems of linear equations arising in the isomonodromic deformations method, the compatibility condition of which is the second Painlevé equation. The constructed solutions of two nonstationary Schrödinger equations are globally smooth. Some of the smooth solutions in question of one of these two equations exponentially tend to zero as x 2 + y 2 → ∞ x^2+y^2to infty if the corresponding solutions of linear systems that are used in the method of isomonodromic deformations are compatible on the so-called 1-tronquée solutions of the second Painlevé equation.
{"title":"Isomonodromic quantization of the second Painlevé equation by means of conservative Hamiltonian systems with two degrees of freedom","authors":"B. Suleimanov","doi":"10.1090/spmj/1739","DOIUrl":"https://doi.org/10.1090/spmj/1739","url":null,"abstract":"For the three nonstationary Schrödinger equations \u0000\u0000 \u0000 \u0000 i\u0000 ℏ\u0000 \u0000 Ψ\u0000 \u0000 τ\u0000 \u0000 \u0000 =\u0000 H\u0000 (\u0000 x\u0000 ,\u0000 y\u0000 ,\u0000 −\u0000 i\u0000 ℏ\u0000 \u0000 ∂\u0000 \u0000 ∂\u0000 x\u0000 \u0000 \u0000 ,\u0000 −\u0000 i\u0000 ℏ\u0000 \u0000 ∂\u0000 \u0000 ∂\u0000 y\u0000 \u0000 \u0000 )\u0000 Ψ\u0000 ,\u0000 \u0000 begin{equation*} ihbar Psi _{tau }=H(x,y,-ihbar frac {partial }{partial x},-ihbar frac {partial }{partial y})Psi , end{equation*}\u0000 \u0000\u0000\u0000 solutions are constructed that correspond to conservative Hamiltonian systems with two degrees of freedom whose general solutions can be represented by those of the second Painlevé equation. These solutions of the Schrödinger equations are expressed via fundamental solutions of systems of linear equations arising in the isomonodromic deformations method, the compatibility condition of which is the second Painlevé equation. The constructed solutions of two nonstationary Schrödinger equations are globally smooth. Some of the smooth solutions in question of one of these two equations exponentially tend to zero as \u0000\u0000 \u0000 \u0000 \u0000 x\u0000 2\u0000 \u0000 +\u0000 \u0000 y\u0000 2\u0000 \u0000 →\u0000 ∞\u0000 \u0000 x^2+y^2to infty\u0000 \u0000\u0000 if the corresponding solutions of linear systems that are used in the method of isomonodromic deformations are compatible on the so-called 1-tronquée solutions of the second Painlevé equation.","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42734995","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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