In this paper, we consider a reduction of a new system of partial difference equations, which was obtained in our previous paper (Joshi and Nakazono, arXiv:1906.06650) and shown to be consistent around a cuboctahedron. We show that this system reduces to $A_2^{(1)ast}$-type discrete Painleve equations by considering a periodic reduction of a three-dimensional lattice constructed from overlapping cuboctahedra.
{"title":"Reduction of quad-equations consistent around a cuboctahedron I: Additive case","authors":"N. Joshi, N. Nakazono","doi":"10.1090/bproc/96","DOIUrl":"https://doi.org/10.1090/bproc/96","url":null,"abstract":"In this paper, we consider a reduction of a new system of partial difference equations, which was obtained in our previous paper (Joshi and Nakazono, arXiv:1906.06650) and shown to be consistent around a cuboctahedron. We show that this system reduces to $A_2^{(1)ast}$-type discrete Painleve equations by considering a periodic reduction of a three-dimensional lattice constructed from overlapping cuboctahedra.","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"52 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129213798","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The isoperimetric problem of maximizing all eigenvalues of the Laplacian on a metric tree graph within the class of trees of a given average edge length is studied. It turns out that, up to rescaling, the unique maximizer of the k k -th positive eigenvalue is the star graph with three edges of lengths 2 k − 1 2 k - 1 , 1 1 and 1 1 . This complements the previously known result that the first nonzero eigenvalue is maximized by all equilateral star graphs and indicates that optimizers of isoperimetric problems for higher eigenvalues may be less balanced in their shape—an observation which is known from numerical results on the optimization of higher eigenvalues of Laplacians on Euclidean domains.
研究了在给定平均边长的树类中，度量树图上拉普拉斯算子的所有特征值最大化的等周问题。结果证明，在重新缩放之前，k k个正特征值的唯一最大化器是具有3条边长度为2 k−1 2 k - 1 1 1和1 1的星图。这补充了先前已知的结果，即第一个非零特征值被所有等边星图最大化，并表明等边问题的高特征值优化器在其形状上可能不太平衡-这一观察从欧几里得域上拉普拉斯算子的高特征值优化的数值结果中已知。
{"title":"Quantum trees which maximize higher eigenvalues are unbalanced","authors":"Jonathan Rohleder","doi":"10.1090/bproc/60","DOIUrl":"https://doi.org/10.1090/bproc/60","url":null,"abstract":"The isoperimetric problem of maximizing all eigenvalues of the Laplacian on a metric tree graph within the class of trees of a given average edge length is studied. It turns out that, up to rescaling, the unique maximizer of the \u0000\u0000 \u0000 k\u0000 k\u0000 \u0000\u0000-th positive eigenvalue is the star graph with three edges of lengths \u0000\u0000 \u0000 \u0000 2\u0000 k\u0000 −\u0000 1\u0000 \u0000 2 k - 1\u0000 \u0000\u0000, \u0000\u0000 \u0000 1\u0000 1\u0000 \u0000\u0000 and \u0000\u0000 \u0000 1\u0000 1\u0000 \u0000\u0000. This complements the previously known result that the first nonzero eigenvalue is maximized by all equilateral star graphs and indicates that optimizers of isoperimetric problems for higher eigenvalues may be less balanced in their shape—an observation which is known from numerical results on the optimization of higher eigenvalues of Laplacians on Euclidean domains.","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"73 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130306829","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Michael A. Brilleslyper, J. Brooks, M. Dorff, Russell W. Howell, Lisbeth E. Schaubroeck
It is well known that complex harmonic polynomials of degree n n may have more than n n zeros. In this paper, we examine a one-parameter family of harmonic trinomials and determine how the number of zeros depends on the parameter. Our proof heavily utilizes the Argument Principle for Harmonic Functions and involves finding the winding numbers about the origin for a family of hypocycloids.
{"title":"Zeros of a one-parameter family of harmonic trinomials","authors":"Michael A. Brilleslyper, J. Brooks, M. Dorff, Russell W. Howell, Lisbeth E. Schaubroeck","doi":"10.1090/bproc/51","DOIUrl":"https://doi.org/10.1090/bproc/51","url":null,"abstract":"It is well known that complex harmonic polynomials of degree \u0000\u0000 \u0000 n\u0000 n\u0000 \u0000\u0000 may have more than \u0000\u0000 \u0000 n\u0000 n\u0000 \u0000\u0000 zeros. In this paper, we examine a one-parameter family of harmonic trinomials and determine how the number of zeros depends on the parameter. Our proof heavily utilizes the Argument Principle for Harmonic Functions and involves finding the winding numbers about the origin for a family of hypocycloids.","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"63 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114210575","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Boshernitzan gave a decay condition on the measure of cylinder sets that implies unique ergodicity for minimal subshifts. Interest in the properties of subshifts satisfying this condition has grown recently, due to a connection with discrete Schrödinger operators, and of particular interest is how restrictive the Boshernitzan condition is. While it implies zero topological entropy, our main theorem shows how to construct minimal subshifts satisfying the condition, and whose factor complexity grows faster than any pre-assigned subexponential rate. As an application, via a theorem of Damanik and Lenz, we show that there is no subexponentially growing sequence for which the spectra of all discrete Schrödinger operators associated with subshifts whose complexity grows faster than the given sequence have only finitely many gaps.
{"title":"Boshernitzan’s condition, factor complexity, and an application","authors":"Van Cyr, Bryna Kra","doi":"10.1090/bproc/90","DOIUrl":"https://doi.org/10.1090/bproc/90","url":null,"abstract":"Boshernitzan gave a decay condition on the measure of cylinder sets that implies unique ergodicity for minimal subshifts. Interest in the properties of subshifts satisfying this condition has grown recently, due to a connection with discrete Schrödinger operators, and of particular interest is how restrictive the Boshernitzan condition is. While it implies zero topological entropy, our main theorem shows how to construct minimal subshifts satisfying the condition, and whose factor complexity grows faster than any pre-assigned subexponential rate. As an application, via a theorem of Damanik and Lenz, we show that there is no subexponentially growing sequence for which the spectra of all discrete Schrödinger operators associated with subshifts whose complexity grows faster than the given sequence have only finitely many gaps.","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122335432","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we will prove a new, scale critical regularity criterion for solutions of the Navier–Stokes equation that are sufficiently close to being eigenfunctions of the Laplacian. This estimate improves previous regularity criteria requiring control on the H˙αdot {H}^alpha norm of u,u, with 2≤α>52,2leq alpha >frac {5}{2}, to a regularity criterion requiring control on the H˙αdot {H}^alpha norm multiplied by the deficit in the interpolation inequality for the embedding of H˙
{"title":"Global regularity for solutions of the Navier–Stokes equation sufficiently close to being eigenfunctions of the Laplacian","authors":"E. Miller","doi":"10.1090/bproc/62","DOIUrl":"https://doi.org/10.1090/bproc/62","url":null,"abstract":"<p>In this paper, we will prove a new, scale critical regularity criterion for solutions of the Navier–Stokes equation that are sufficiently close to being eigenfunctions of the Laplacian. This estimate improves previous regularity criteria requiring control on the <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"ModifyingAbove upper H With dot Superscript alpha\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mover>\u0000 <mml:mi>H</mml:mi>\u0000 <mml:mo>˙<!-- ˙ --></mml:mo>\u0000 </mml:mover>\u0000 </mml:mrow>\u0000 <mml:mi>α<!-- α --></mml:mi>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">dot {H}^alpha</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> norm of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"u comma\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>u</mml:mi>\u0000 <mml:mo>,</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">u,</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> with <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"2 less-than-or-equal-to alpha greater-than five halves comma\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mn>2</mml:mn>\u0000 <mml:mo>≤<!-- ≤ --></mml:mo>\u0000 <mml:mi>α<!-- α --></mml:mi>\u0000 <mml:mo>></mml:mo>\u0000 <mml:mfrac>\u0000 <mml:mn>5</mml:mn>\u0000 <mml:mn>2</mml:mn>\u0000 </mml:mfrac>\u0000 <mml:mo>,</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">2leq alpha >frac {5}{2},</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> to a regularity criterion requiring control on the <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"ModifyingAbove upper H With dot Superscript alpha\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mover>\u0000 <mml:mi>H</mml:mi>\u0000 <mml:mo>˙<!-- ˙ --></mml:mo>\u0000 </mml:mover>\u0000 </mml:mrow>\u0000 <mml:mi>α<!-- α --></mml:mi>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">dot {H}^alpha</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> norm multiplied by the deficit in the interpolation inequality for the embedding of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"ModifyingAbove upper H With dot Superscript alpha minus 2 Baseline intersection ModifyingAbove upper H With dot Superscript alpha Baseline right-arrow with hook ModifyingAbove upper H With dot Superscript alpha minus 1 Baseline period\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mover>\u0000 <mml:mi>H</mml:mi>\u0000 <mml:mo>˙<!-- ˙ --></mml:mo>\u0000 </mml:mover>\u0000 ","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117106214","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show a number of Toda brackets in the homotopy of the motivic bordism spectrum MGLMGL and of the Real bordism spectrum MURMU_{mathbb R}. These brackets are “red-shifting” in the sense that while the terms in the bracket will be of some chromatic height nn, the bracket itself will be of chromatic height (n+1)(n+1). Using these, we deduce a family of exotic multiplications in the π(∗,∗)MGLpi _{(ast ,ast )}MGL-module structure of the motivic Morava
我们给出了动力谱MGL MGL和实谱MU R MU_{mathbb R}同伦中的若干Toda括号。这些括号是“红移”的意思是，虽然括号中的项的色高是n n，但括号本身的色高是(n+1) (n+1)。利用这些，我们推导出了在动机Morava K -理论的π(∗，∗)MGL pi _{(ast，ast)}MGL -模结构中的一类奇异乘法，包括非平凡乘2 2。这些反过来又暗示了在真实Morava K -理论上π - - - U R pi _{星}MU_mathbb R - -模结构中的类似奇异乘法族。
{"title":"Transchromatic extensions in motivic bordism","authors":"A. Beaudry, M. Hill, Xiaolin Shi, Mingcong Zeng","doi":"10.1090/bproc/108","DOIUrl":"https://doi.org/10.1090/bproc/108","url":null,"abstract":"<p>We show a number of Toda brackets in the homotopy of the motivic bordism spectrum <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper M upper G upper L\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>M</mml:mi>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:mi>L</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">MGL</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> and of the Real bordism spectrum <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper M upper U Subscript double-struck upper R\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>M</mml:mi>\u0000 <mml:msub>\u0000 <mml:mi>U</mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">R</mml:mi>\u0000 </mml:mrow>\u0000 </mml:mrow>\u0000 </mml:msub>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">MU_{mathbb R}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. These brackets are “red-shifting” in the sense that while the terms in the bracket will be of some chromatic height <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>, the bracket itself will be of chromatic height <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis n plus 1 right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>n</mml:mi>\u0000 <mml:mo>+</mml:mo>\u0000 <mml:mn>1</mml:mn>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">(n+1)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. Using these, we deduce a family of exotic multiplications in the <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"pi Subscript left-parenthesis asterisk comma asterisk right-parenthesis Baseline upper M upper G upper L\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msub>\u0000 <mml:mi>π<!-- π --></mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mo>∗<!-- ∗ --></mml:mo>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mo>∗<!-- ∗ --></mml:mo>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 </mml:msub>\u0000 <mml:mi>M</mml:mi>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:mi>L</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">pi _{(ast ,ast )}MGL</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-module structure of the motivic Morava <inline-formula content-type=\"math/mathml\">\u0000<mml:ma","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"97 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116100155","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A subfield KK of Q¯bar {mathbb {Q}} is large if every smooth curve CC over KK with a KK-rational point has infinitely many KK-rational points. A subfield KK of Q¯bar {mathbb {Q}} is big if for every positive integer
Q¯bar {mathbb {Q}}的子域K K是大的，如果每条光滑曲线C C / K K有K个K个有理点有无限多个K个K个有理点。Q¯bar {mathbb {Q}}的子域K K是大的，如果对于每一个正整数n n, K K包含一个数字域F F，且[F: Q] [F:mathbb {Q}]能被n n整除。是否所有的大领域都是大的问题似乎已经流传了一段时间，尽管我们一直无法找到它的起源。在本文中，我们证明了存在一些不大的大场。
{"title":"Big fields that are not large","authors":"B. Mazur, K. Rubin","doi":"10.1090/bproc/57","DOIUrl":"https://doi.org/10.1090/bproc/57","url":null,"abstract":"<p>A subfield <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K\">\u0000 <mml:semantics>\u0000 <mml:mi>K</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">K</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper Q overbar\">\u0000 <mml:semantics>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mover>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">Q</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mo stretchy=\"false\">¯<!-- ¯ --></mml:mo>\u0000 </mml:mover>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">bar {mathbb {Q}}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is <italic>large</italic> if every smooth curve <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper C\">\u0000 <mml:semantics>\u0000 <mml:mi>C</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">C</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> over <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K\">\u0000 <mml:semantics>\u0000 <mml:mi>K</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">K</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> with a <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K\">\u0000 <mml:semantics>\u0000 <mml:mi>K</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">K</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-rational point has infinitely many <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K\">\u0000 <mml:semantics>\u0000 <mml:mi>K</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">K</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-rational points. A subfield <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K\">\u0000 <mml:semantics>\u0000 <mml:mi>K</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">K</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper Q overbar\">\u0000 <mml:semantics>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mover>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">Q</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mo stretchy=\"false\">¯<!-- ¯ --></mml:mo>\u0000 </mml:mover>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">bar {mathbb {Q}}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is <italic>big</italic> if for every positive integer <inline-formula con","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133524467","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we will prove a regularity criterion that guarantees solutions of the Navier–Stokes equation must remain smooth so long as the vorticity restricted to a plane remains bounded in the scale critical space L t 4 L x 2 L^4_t L^2_x , where the plane may vary in space and time as long as the gradient of the unit vector orthogonal to the plane remains bounded. This extends previous work by Chae and Choe that guaranteed that solutions of the Navier–Stokes equation must remain smooth as long as the vorticity restricted to a fixed plane remains bounded in a family of scale critical spaces. This regularity criterion also can be seen as interpolating between Chae and Choe’s regularity criterion in terms of two vorticity components and Beirão da Veiga and Berselli’s regularity criterion in terms of the gradient of vorticity direction. In physical terms, this regularity criterion is consistent with key aspects of the Kolmogorov theory of turbulence, because it requires that finite-time blowup for solutions of the Navier–Stokes equation must be fully three dimensional at all length scales.
在本文中，我们将证明一个正则性准则，保证Navier-Stokes方程的解必须保持光滑，只要限制在一个平面上的涡量在尺度临界空间L ~ 4l × 2l ^4_t L^2_x中保持有界，其中平面可以在空间和时间上变化，只要与平面正交的单位向量的梯度保持有界。这扩展了Chae和Choe之前的工作，保证了Navier-Stokes方程的解必须保持光滑，只要限制在固定平面上的涡度在一系列尺度临界空间中保持有界。该正则性判据也可以看作是Chae和Choe的两个涡度分量正则性判据和beir o da Veiga和Berselli的涡度方向梯度正则性判据之间的插值。在物理方面，这个规则准则与Kolmogorov湍流理论的关键方面是一致的，因为它要求在所有长度尺度上，Navier-Stokes方程解的有限时间爆破必须是完全三维的。
{"title":"A locally anisotropic regularity criterion for the Navier–Stokes equation in terms of vorticity","authors":"E. Miller","doi":"10.1090/bproc/74","DOIUrl":"https://doi.org/10.1090/bproc/74","url":null,"abstract":"In this paper, we will prove a regularity criterion that guarantees solutions of the Navier–Stokes equation must remain smooth so long as the vorticity restricted to a plane remains bounded in the scale critical space \u0000\u0000 \u0000 \u0000 \u0000 L\u0000 t\u0000 4\u0000 \u0000 \u0000 L\u0000 x\u0000 2\u0000 \u0000 \u0000 L^4_t L^2_x\u0000 \u0000\u0000, where the plane may vary in space and time as long as the gradient of the unit vector orthogonal to the plane remains bounded. This extends previous work by Chae and Choe that guaranteed that solutions of the Navier–Stokes equation must remain smooth as long as the vorticity restricted to a fixed plane remains bounded in a family of scale critical spaces. This regularity criterion also can be seen as interpolating between Chae and Choe’s regularity criterion in terms of two vorticity components and Beirão da Veiga and Berselli’s regularity criterion in terms of the gradient of vorticity direction. In physical terms, this regularity criterion is consistent with key aspects of the Kolmogorov theory of turbulence, because it requires that finite-time blowup for solutions of the Navier–Stokes equation must be fully three dimensional at all length scales.","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"32 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128271436","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let AA be a principally polarized abelian variety of dimension gg over a number field KK. Assume that the image of the adelic Galois representation of AA is an open subgroup of GSp2g(Z^)GSp_{2g}(hat {mathbb {Z}}). Then there exists a positive integer mm so that the Galois image of AA is the full preimage of its reduction modulo
设A A是数域K K上的维g g的主极化阿贝尔变换。假设A的亚历伽罗瓦表示的象是GSp 2g (Z ^) GSp_{2g}(hat {mathbb {Z}})的开子群。则存在正整数m m，使得a a的伽罗瓦像是其约化模m m的完整原像。具有这种性质的最小的m m记为m A m_A，称为A A的像导体。Jones [Pacific J. Math. 308 (2020)， pp. 307-331]最近在A A为椭圆曲线且没有复数乘法的情况下，根据A A的标准不变量建立了m A m_A的上界。在本文中，我们推广了上述结果，给出了任意维上的一个类似界。
{"title":"A bound for the image conductor of a principally polarized abelian variety with open Galois image","authors":"Jacob Mayle","doi":"10.1090/bproc/131","DOIUrl":"https://doi.org/10.1090/bproc/131","url":null,"abstract":"<p>Let <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A\">\u0000 <mml:semantics>\u0000 <mml:mi>A</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">A</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> be a principally polarized abelian variety of dimension <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"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> over a number field <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K\">\u0000 <mml:semantics>\u0000 <mml:mi>K</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">K</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. Assume that the image of the adelic Galois representation of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A\">\u0000 <mml:semantics>\u0000 <mml:mi>A</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">A</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is an open subgroup of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G upper S p Subscript 2 g Baseline left-parenthesis ModifyingAbove double-struck upper Z With caret right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:mi>S</mml:mi>\u0000 <mml:msub>\u0000 <mml:mi>p</mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mn>2</mml:mn>\u0000 <mml:mi>g</mml:mi>\u0000 </mml:mrow>\u0000 </mml:msub>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mover>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">Z</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mo stretchy=\"false\">^<!-- ^ --></mml:mo>\u0000 </mml:mover>\u0000 </mml:mrow>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">GSp_{2g}(hat {mathbb {Z}})</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. Then there exists a positive integer <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"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> so that the Galois image of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A\">\u0000 <mml:semantics>\u0000 <mml:mi>A</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">A</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is the full preimage of its reduction modulo <inline-formula content-type=\"math/math","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123155889","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper completes a classification of the types of orientable and non-orientable cusps that can arise in the quotients of hyperbolic knot complements. In particular, S2(2,4,4)S^2(2,4,4) cannot be the cusp cross-section of any orbifold quotient of a hyperbolic knot complement. Furthermore, if a knot complement covers an orbifold with a S2(2,3,6)S^2(2,3,6) cusp, it also covers an orbifold with a S2(3,3,3)S^2(3,3,3) cusp. We end with a discussion that shows all cusp types arise in the quotients of link complements.
{"title":"Cusp types of quotients of hyperbolic knot complements","authors":"Neil R. Hoffman","doi":"10.1090/bproc/104","DOIUrl":"https://doi.org/10.1090/bproc/104","url":null,"abstract":"<p>This paper completes a classification of the types of orientable and non-orientable cusps that can arise in the quotients of hyperbolic knot complements. In particular, <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper S squared left-parenthesis 2 comma 4 comma 4 right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msup>\u0000 <mml:mi>S</mml:mi>\u0000 <mml:mn>2</mml:mn>\u0000 </mml:msup>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mn>2</mml:mn>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mn>4</mml:mn>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mn>4</mml:mn>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">S^2(2,4,4)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> cannot be the cusp cross-section of any orbifold quotient of a hyperbolic knot complement. Furthermore, if a knot complement covers an orbifold with a <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper S squared left-parenthesis 2 comma 3 comma 6 right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msup>\u0000 <mml:mi>S</mml:mi>\u0000 <mml:mn>2</mml:mn>\u0000 </mml:msup>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mn>2</mml:mn>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mn>3</mml:mn>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mn>6</mml:mn>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">S^2(2,3,6)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> cusp, it also covers an orbifold with a <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper S squared left-parenthesis 3 comma 3 comma 3 right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msup>\u0000 <mml:mi>S</mml:mi>\u0000 <mml:mn>2</mml:mn>\u0000 </mml:msup>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mn>3</mml:mn>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mn>3</mml:mn>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mn>3</mml:mn>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">S^2(3,3,3)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> cusp. We end with a discussion that shows all cusp types arise in the quotients of link complements.</p>","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126407385","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：