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A Burning Problem: Cannabis Lessons Learned from Colorado. 燃烧的问题:从科罗拉多州吸取的大麻教训。
IF 2.9 4区 数学 Q1 MATHEMATICS Pub Date : 2018-01-01 Epub Date: 2017-04-13 DOI: 10.1080/16066359.2017.1315410
Jamie E Parnes, Adrian J Bravo, Bradley T Conner, Matthew R Pearson

With recent increases in cannabis' popularity, including being legalized in several states, new issues have emerged related to use. Increases in the number of users, new products, and home growing all present distinct concerns. In the present review, we explored various cannabis-related concerns (i.e. use, acquiring, growing, and public health/policy) that have arisen in Colorado in order to provide information on emerging issues and future directions to mitigate negative outcomes that could occur in states considering, or that already have implemented, a legalized cannabis market. Specific to Colorado, issues have arisen related to edibles, vaporizers/'e-cannabis', concentrates, growing, quantifying use, intoxicated driving, and arrests. Understanding cannabis dosing (including dose-dependent effects and related consequences), standardizing quantities, evaluating the safety of new products, and developing harm reduction interventions are important next steps for informing public policy and promoting health and well-being. Overall, increasing our knowledge of emerging issues related to cannabis is key to promoting the benefits and combating the potential harms of cannabis, especially for states legalizing medical or recreational cannabis.

随着近来大麻受欢迎程度的提高,包括在几个州的合法化,出现了与大麻使用有关的新问题。使用者数量的增加、新产品和家庭种植都带来了不同的问题。在本综述中,我们探讨了科罗拉多州出现的各种与大麻有关的问题(即使用、获取、种植和公共卫生/政策),以便提供有关新出现问题的信息和未来发展方向,从而减轻正在考虑或已经实施大麻市场合法化的州可能出现的负面结果。具体到科罗拉多州,出现的问题涉及食用、蒸发器/"电子大麻"、浓缩、种植、量化使用、醉酒驾驶和逮捕。了解大麻剂量(包括剂量依赖效应和相关后果)、数量标准化、评估新产品的安全性以及制定减少危害的干预措施,是为公共政策提供信息和促进健康与福祉的重要步骤。总之,增加我们对与大麻有关的新问题的了解是促进大麻的益处和消除其潜在危害的关键,特别是对于医疗或娱乐大麻合法化的州而言。
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引用次数: 0
Spectral Properties of Ruelle Transfer Operators for Regular Gibbs Measures and Decay of Correlations for Contact Anosov Flows 正则Gibbs测度的Ruelle传递算子的谱性质及接触ansov流的相关衰减
IF 1.9 4区 数学 Q1 MATHEMATICS Pub Date : 2017-12-07 DOI: 10.1090/memo/1404
L. Stoyanov
In this work we study strong spectral properties of Ruelle transfer operators related to a large family of Gibbs measures for contact Anosov flows. The ultimate aim is to establish exponential decay of correlations for Hölder observables with respect to a very general class of Gibbs measures. The approach invented in 1997 by Dolgopyat in “On decay of correlations in Anosov flows” and further developed in Stoyanov (2011) is substantially refined here, allowing to deal with much more general situations than before, although we still restrict ourselves to the uniformly hyperbolic case. A rather general procedure is established which produces the desired estimates whenever the Gibbs measure admits a Pesin set with exponentially small tails, that is a Pesin set whose preimages along the flow have measures decaying exponentially fast. We call such Gibbs measures regular. Recent results in Gouëzel and Stoyanov (2019) prove existence of such Pesin sets for hyperbolic diffeomorphisms and flows for a large variety of Gibbs measures determined by Hölder continuous potentials. The strong spectral estimates for Ruelle operators and well-established techniques lead to exponential decay of correlations for Hölder continuous observables, as well as to some other consequences such as: (a) existence of a non-zero analytic continuation of the Ruelle zeta function with a pole at the entropy in a vertical strip containing the entropy in its interior; (b) a Prime Orbit Theorem with an exponentially small error.
在这项工作中,我们研究了与接触Anosov流的一大类Gibbs测度有关的Ruelle转移算子的强谱性质。最终目的是建立Hölder可观察性相对于一类非常一般的吉布斯测度的相关性的指数衰减。Dolgopyat于1997年在“Anosov流中相关性的衰减”中发明并在Stoyanov(2011)中进一步发展的方法在这里得到了实质性的改进,允许处理比以前更普遍的情况,尽管我们仍然将自己限制在一致双曲的情况下。建立了一个相当一般的程序,每当吉布斯测度允许具有指数小尾的Pesin集时,该程序就会产生所需的估计,即Pesin集合的前图像具有指数快速衰减的测度。我们称这种吉布斯测度为正则测度。Gouëzel和Stoyanov(2019)的最新结果证明了由Hölder连续势确定的各种吉布斯测度的双曲微分同胚和流的Pesin集的存在性。Ruelle算子的强谱估计和成熟的技术导致了Hölder连续可观察性的相关性的指数衰减,以及一些其他结果,例如:(a)Ruelle-zeta函数的非零解析连续性的存在,该函数的极点在其内部包含熵的垂直条带中的熵处;(b) 具有指数小误差的素数轨道定理。
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引用次数: 8
Hardy–Littlewood and Ulyanov inequalities Hardy–Littlewood和Ulyanov不等式
IF 1.9 4区 数学 Q1 MATHEMATICS Pub Date : 2017-11-22 DOI: 10.1090/memo/1325
Yurii Kolomoitsev, S. Tikhonov
<p>We give the full solution of the following problem: obtain sharp inequalities between the moduli of smoothness <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="omega Subscript alpha Baseline left-parenthesis f comma t right-parenthesis Subscript q"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>ω<!-- ω --></mml:mi> <mml:mi>α<!-- α --></mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>f</mml:mi> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mi>q</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">omega _alpha (f,t)_q</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="omega Subscript beta Baseline left-parenthesis f comma t right-parenthesis Subscript p"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>ω<!-- ω --></mml:mi> <mml:mi>β<!-- β --></mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>f</mml:mi> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mi>p</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">omega _beta (f,t)_p</mml:annotation> </mml:semantics></mml:math></inline-formula> for <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0 greater-than p greater-than q less-than-or-equal-to normal infinity"> <mml:semantics> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>></mml:mo> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:mi>q</mml:mi> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mi mathvariant="normal">∞<!-- ∞ --></mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">0>p>qle infty</mml:annotation> </mml:semantics></mml:math></inline-formula>. A similar problem for the generalized <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics></mml:math></inline-formula>-functionals and their realizations between the couples <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper L Subscript p Baseline comma upper W Subscript p Superscript psi Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:msubsup> <mml:mi>W</mml:mi> <mml:mi>p</mml:mi>
我们给出了以下问题的完整解决方案:得到光滑度模ω α (f,t) q omega _ alpha (f,t)_q与ω β (f,t) p omega _ beta (f,t)_p对于0>p>q≤∞0>p>q leinfty之间的明显不等式。求解了广义K泛函在(lp, wp ψ) (L_p, W_p^ psi)和(lq, wq φ) (L_q, W_q^ varphi)对之间的类似问题及其实现。主要的工具是新的Hardy-Littlewood-Nikol 'skii不等式。更准确地说,我们得到了量supt n‖D (ψ) (tn)‖q‖D (φ) (T)的渐近性质n)‖p, 0 > p > q≤∞,begin{equation*} sup _{T_n} frac {Vert mathcal {D}(psi )(T_n)Vert _q}{Vert mathcal {D}({varphi })(T_n)Vert _p},qquad 0>p>qle infty , end{equation*}其中最优取于所有阶数不超过n n的非平凡三角多项式T n T_n和D(ψ), D(φ) mathcal D{(}psi),mathcal D{(}{varphi)是weyl型微分算子。我们还证明了Hardy空间中的Ulyanov和kolyada型不等式。最后,我们将得到的估计应用于Lipschitz和Besov空间的新的嵌入定理。}
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引用次数: 20
Adiabatic Evolution and Shape Resonances 绝热演化与形状共振
IF 1.9 4区 数学 Q1 MATHEMATICS Pub Date : 2017-11-21 DOI: 10.1090/memo/1380
M. Hitrik, A. Mantile, J. Sjoestrand
<p>Motivated by a problem of one mode approximation for a non-linear evolution with charge accumulation in potential wells, we consider a general linear adiabatic evolution problem for a semi-classical Schrödinger operator with a time dependent potential with a well in an island. In particular, we show that we can choose the adiabatic parameter <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="epsilon"> <mml:semantics> <mml:mi>ε<!-- ε --></mml:mi> <mml:annotation encoding="application/x-tex">varepsilon</mml:annotation> </mml:semantics></mml:math></inline-formula> with <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ln epsilon equivalent-to negative 1 slash h"> <mml:semantics> <mml:mrow> <mml:mi>ln</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mi>ε<!-- ε --></mml:mi> <mml:mo>≍<!-- ≍ --></mml:mo> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>h</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">ln varepsilon asymp -1/h</mml:annotation> </mml:semantics></mml:math></inline-formula>, where <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h"> <mml:semantics> <mml:mi>h</mml:mi> <mml:annotation encoding="application/x-tex">h</mml:annotation> </mml:semantics></mml:math></inline-formula> denotes the semi-classical parameter, and get adiabatic approximations of exact solutions over a time interval of length <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="epsilon Superscript negative upper N"> <mml:semantics> <mml:msup> <mml:mi>ε<!-- ε --></mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>−<!-- − --></mml:mo> <mml:mi>N</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">varepsilon ^{-N}</mml:annotation> </mml:semantics></mml:math></inline-formula> with an error <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper O left-parenthesis epsilon Superscript upper N Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">O</mml:mi> </mml:mrow> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>ε<!-- ε --></mml:mi> <mml:mi>N</mml:mi> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{mathcal O}(varepsilon ^N)</mml:annotation> </mml:semantics></mml:math></inline-formula>. Here <inline-formula content
受势阱中电荷累积的非线性演化的单模近似问题的启发,我们考虑了岛上具有时间相关势的半经典薛定谔算子的一般线性绝热演化问题。特别是,我们表明我们可以选择绝热参数εvarepsilon,其中h表示半经典参数,并在长度ε−nvarepsilon^{-n}的时间间隔内获得精确解的绝热近似值,误差为O(εn){mathcal O}(varepsilon^n)。这里n>0 n>0是任意的。Center SummaryEndCenter受势阱中电荷累积演化的单模近似问题的驱动,我们考虑了一个薛定谔算子的线性演化问题,该算子具有岛上阱的时间相关势。特别是,我们表明我们可以选择绝热参数ε≍-1/hlnvarepsilonasymp-1/h,其中h h表示半经典参数,并获得长度为ε−nvarepsilon^{-n}的时间间隔内精确解的绝热近似值,误差为O(εn){mathcal O}(varepsilon^n)。这里n>0 n>0是任意的。
{"title":"Adiabatic Evolution and Shape Resonances","authors":"M. Hitrik, A. Mantile, J. Sjoestrand","doi":"10.1090/memo/1380","DOIUrl":"https://doi.org/10.1090/memo/1380","url":null,"abstract":"&lt;p&gt;Motivated by a problem of one mode approximation for a non-linear evolution with charge accumulation in potential wells, we consider a general linear adiabatic evolution problem for a semi-classical Schrödinger operator with a time dependent potential with a well in an island. In particular, we show that we can choose the adiabatic parameter &lt;inline-formula content-type=\"math/mathml\"&gt;\u0000&lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"epsilon\"&gt;\u0000 &lt;mml:semantics&gt;\u0000 &lt;mml:mi&gt;ε&lt;!-- ε --&gt;&lt;/mml:mi&gt;\u0000 &lt;mml:annotation encoding=\"application/x-tex\"&gt;varepsilon&lt;/mml:annotation&gt;\u0000 &lt;/mml:semantics&gt;\u0000&lt;/mml:math&gt;\u0000&lt;/inline-formula&gt; with &lt;inline-formula content-type=\"math/mathml\"&gt;\u0000&lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"ln epsilon equivalent-to negative 1 slash h\"&gt;\u0000 &lt;mml:semantics&gt;\u0000 &lt;mml:mrow&gt;\u0000 &lt;mml:mi&gt;ln&lt;/mml:mi&gt;\u0000 &lt;mml:mo&gt;⁡&lt;!-- ⁡ --&gt;&lt;/mml:mo&gt;\u0000 &lt;mml:mi&gt;ε&lt;!-- ε --&gt;&lt;/mml:mi&gt;\u0000 &lt;mml:mo&gt;≍&lt;!-- ≍ --&gt;&lt;/mml:mo&gt;\u0000 &lt;mml:mo&gt;−&lt;!-- − --&gt;&lt;/mml:mo&gt;\u0000 &lt;mml:mn&gt;1&lt;/mml:mn&gt;\u0000 &lt;mml:mrow class=\"MJX-TeXAtom-ORD\"&gt;\u0000 &lt;mml:mo&gt;/&lt;/mml:mo&gt;\u0000 &lt;/mml:mrow&gt;\u0000 &lt;mml:mi&gt;h&lt;/mml:mi&gt;\u0000 &lt;/mml:mrow&gt;\u0000 &lt;mml:annotation encoding=\"application/x-tex\"&gt;ln varepsilon asymp -1/h&lt;/mml:annotation&gt;\u0000 &lt;/mml:semantics&gt;\u0000&lt;/mml:math&gt;\u0000&lt;/inline-formula&gt;, where &lt;inline-formula content-type=\"math/mathml\"&gt;\u0000&lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"h\"&gt;\u0000 &lt;mml:semantics&gt;\u0000 &lt;mml:mi&gt;h&lt;/mml:mi&gt;\u0000 &lt;mml:annotation encoding=\"application/x-tex\"&gt;h&lt;/mml:annotation&gt;\u0000 &lt;/mml:semantics&gt;\u0000&lt;/mml:math&gt;\u0000&lt;/inline-formula&gt; denotes the semi-classical parameter, and get adiabatic approximations of exact solutions over a time interval of length &lt;inline-formula content-type=\"math/mathml\"&gt;\u0000&lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"epsilon Superscript negative upper N\"&gt;\u0000 &lt;mml:semantics&gt;\u0000 &lt;mml:msup&gt;\u0000 &lt;mml:mi&gt;ε&lt;!-- ε --&gt;&lt;/mml:mi&gt;\u0000 &lt;mml:mrow class=\"MJX-TeXAtom-ORD\"&gt;\u0000 &lt;mml:mo&gt;−&lt;!-- − --&gt;&lt;/mml:mo&gt;\u0000 &lt;mml:mi&gt;N&lt;/mml:mi&gt;\u0000 &lt;/mml:mrow&gt;\u0000 &lt;/mml:msup&gt;\u0000 &lt;mml:annotation encoding=\"application/x-tex\"&gt;varepsilon ^{-N}&lt;/mml:annotation&gt;\u0000 &lt;/mml:semantics&gt;\u0000&lt;/mml:math&gt;\u0000&lt;/inline-formula&gt; with an error &lt;inline-formula content-type=\"math/mathml\"&gt;\u0000&lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper O left-parenthesis epsilon Superscript upper N Baseline right-parenthesis\"&gt;\u0000 &lt;mml:semantics&gt;\u0000 &lt;mml:mrow&gt;\u0000 &lt;mml:mrow class=\"MJX-TeXAtom-ORD\"&gt;\u0000 &lt;mml:mrow class=\"MJX-TeXAtom-ORD\"&gt;\u0000 &lt;mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\"&gt;O&lt;/mml:mi&gt;\u0000 &lt;/mml:mrow&gt;\u0000 &lt;/mml:mrow&gt;\u0000 &lt;mml:mo stretchy=\"false\"&gt;(&lt;/mml:mo&gt;\u0000 &lt;mml:msup&gt;\u0000 &lt;mml:mi&gt;ε&lt;!-- ε --&gt;&lt;/mml:mi&gt;\u0000 &lt;mml:mi&gt;N&lt;/mml:mi&gt;\u0000 &lt;/mml:msup&gt;\u0000 &lt;mml:mo stretchy=\"false\"&gt;)&lt;/mml:mo&gt;\u0000 &lt;/mml:mrow&gt;\u0000 &lt;mml:annotation encoding=\"application/x-tex\"&gt;{mathcal O}(varepsilon ^N)&lt;/mml:annotation&gt;\u0000 &lt;/mml:semantics&gt;\u0000&lt;/mml:math&gt;\u0000&lt;/inline-formula&gt;. Here &lt;inline-formula content","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2017-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45370536","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}
引用次数: 1
The Regularity of the Linear Drift in Negatively Curved Spaces 负弯曲空间中线性漂移的规律性
IF 1.9 4区 数学 Q1 MATHEMATICS Pub Date : 2017-11-08 DOI: 10.1090/memo/1387
Franccois Ledrappier, Lin Shu

We show that the linear drift of the Brownian motion on the universal cover of a closed connected smooth Riemannian manifold is C k 2 C^{k-2} differentiable along any C k C^{k} curve in the manifold of C k C^k Riemannian metrics with negative sectional curvature. We also show that the stochastic entropy of the Brownian motion is C 1 C^1 differentiable along any C 3 C^{3} curve of C 3 C^3 Riemannian metrics with negative sectional curvature. We formulate the first derivatives of the linear drift and stochastic entropy, respectively, and show they are critical at locally symmetric metrics.

我们证明了布朗运动在闭连通光滑黎曼流形的泛盖上的线性漂移是ck−2 C^{k-2}可微沿ck C^{k}曲线在具有负截面曲率的ck C^k黎曼度量流形上的任意ck C^{k}曲线。我们还证明了布朗运动的随机熵是c1 C^1可微沿任何c3c C^3黎曼度量的c3c C^{3}曲线具有负截面曲率。我们分别给出了线性漂移和随机熵的一阶导数,并证明它们在局部对称度量下是临界的。
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引用次数: 1
Type II Blow Up Solutions with Optimal Stability Properties for the Critical Focussing Nonlinear Wave Equation on ℝ³⁺¹ 具有最优稳定性的临界聚焦非线性波动方程的II型爆破解ℝ³⁺cco
IF 1.9 4区 数学 Q1 MATHEMATICS Pub Date : 2017-09-19 DOI: 10.1090/memo/1369
Stefano Burzio, J. Krieger
<p>We show that the finite time type II blow up solutions for the energy critical nonlinear wave equation <disp-formula content-type="math/mathml">[<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="white medium square u equals minus u Superscript 5"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>◻</mml:mo> </mml:mrow> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mo>−<!-- − --></mml:mo> <mml:msup> <mml:mi>u</mml:mi> <mml:mn>5</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">Box u = -u^5</mml:annotation> </mml:semantics></mml:math>]</disp-formula> on <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R Superscript 3 plus 1"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>3</mml:mn> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">mathbb {R}^{3+1}</mml:annotation> </mml:semantics></mml:math></inline-formula> constructed in Krieger, Schlag, and Tartaru (“Slow blow-up solutions for the <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Superscript 1 Baseline left-parenthesis double-struck upper R cubed right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">H^1(mathbb {R}^3)</mml:annotation> </mml:semantics></mml:math></inline-formula> critical focusing semilinear wave equation”, 2009) and Krieger and Schlag (“Full range of blow up exponents for the quintic wave equation in three dimensions”, 2014) are stable along a co-dimension one Lipschitz manifold of data perturbations in a suitable topology, provided the scaling parameter <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="lamda left-parenthesis t right-parenthesis equals t Superscript negative 1 minus nu"> <mml:semantics> <mml:mrow> <mml:mi>λ<!-- λ --></mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>t</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> <mml:mo>−<!-- − --><
我们证明了能量临界非线性波动方程的有限时间II型爆破解◻ 在Krieger,Schlag,和Tartaru(“H1(R3)H^1(mathbb{R}^3)临界聚焦半线性波动方程的慢爆破解”,2009)以及Krieger和Schlag(“三维五次波动方程的全范围爆破指数”,2014)在合适的拓扑中沿着数据扰动的同维Lipschitz流形是稳定的,假设标度参数λ(t)=t−1−Γlambda(t)=t^{-1-nu}足够接近自相似率,即Γ>0nu>0足够小。根据Krieger、Nakamishi和Schlag的结果(“临界波动方程能量空间中基态的中心稳定流形”,2015),该结果在质量上是最优的。本文建立在Krieger和Wong的分析基础上(“关于临界NLW的I型爆破形成”,2014)。
{"title":"Type II Blow Up Solutions with Optimal Stability Properties for the Critical Focussing Nonlinear Wave Equation on ℝ³⁺¹","authors":"Stefano Burzio, J. Krieger","doi":"10.1090/memo/1369","DOIUrl":"https://doi.org/10.1090/memo/1369","url":null,"abstract":"&lt;p&gt;We show that the finite time type II blow up solutions for the energy critical nonlinear wave equation &lt;disp-formula content-type=\"math/mathml\"&gt;\u0000[\u0000&lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"white medium square u equals minus u Superscript 5\"&gt;\u0000 &lt;mml:semantics&gt;\u0000 &lt;mml:mrow&gt;\u0000 &lt;mml:mrow class=\"MJX-TeXAtom-ORD\"&gt;\u0000 &lt;mml:mo&gt;◻&lt;/mml:mo&gt;\u0000 &lt;/mml:mrow&gt;\u0000 &lt;mml:mi&gt;u&lt;/mml:mi&gt;\u0000 &lt;mml:mo&gt;=&lt;/mml:mo&gt;\u0000 &lt;mml:mo&gt;−&lt;!-- − --&gt;&lt;/mml:mo&gt;\u0000 &lt;mml:msup&gt;\u0000 &lt;mml:mi&gt;u&lt;/mml:mi&gt;\u0000 &lt;mml:mn&gt;5&lt;/mml:mn&gt;\u0000 &lt;/mml:msup&gt;\u0000 &lt;/mml:mrow&gt;\u0000 &lt;mml:annotation encoding=\"application/x-tex\"&gt;Box u = -u^5&lt;/mml:annotation&gt;\u0000 &lt;/mml:semantics&gt;\u0000&lt;/mml:math&gt;\u0000]\u0000&lt;/disp-formula&gt; on &lt;inline-formula content-type=\"math/mathml\"&gt;\u0000&lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper R Superscript 3 plus 1\"&gt;\u0000 &lt;mml:semantics&gt;\u0000 &lt;mml:msup&gt;\u0000 &lt;mml:mrow class=\"MJX-TeXAtom-ORD\"&gt;\u0000 &lt;mml:mi mathvariant=\"double-struck\"&gt;R&lt;/mml:mi&gt;\u0000 &lt;/mml:mrow&gt;\u0000 &lt;mml:mrow class=\"MJX-TeXAtom-ORD\"&gt;\u0000 &lt;mml:mn&gt;3&lt;/mml:mn&gt;\u0000 &lt;mml:mo&gt;+&lt;/mml:mo&gt;\u0000 &lt;mml:mn&gt;1&lt;/mml:mn&gt;\u0000 &lt;/mml:mrow&gt;\u0000 &lt;/mml:msup&gt;\u0000 &lt;mml:annotation encoding=\"application/x-tex\"&gt;mathbb {R}^{3+1}&lt;/mml:annotation&gt;\u0000 &lt;/mml:semantics&gt;\u0000&lt;/mml:math&gt;\u0000&lt;/inline-formula&gt; constructed in Krieger, Schlag, and Tartaru (“Slow blow-up solutions for the &lt;inline-formula content-type=\"math/mathml\"&gt;\u0000&lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper H Superscript 1 Baseline left-parenthesis double-struck upper R cubed right-parenthesis\"&gt;\u0000 &lt;mml:semantics&gt;\u0000 &lt;mml:mrow&gt;\u0000 &lt;mml:msup&gt;\u0000 &lt;mml:mi&gt;H&lt;/mml:mi&gt;\u0000 &lt;mml:mn&gt;1&lt;/mml:mn&gt;\u0000 &lt;/mml:msup&gt;\u0000 &lt;mml:mo stretchy=\"false\"&gt;(&lt;/mml:mo&gt;\u0000 &lt;mml:msup&gt;\u0000 &lt;mml:mrow class=\"MJX-TeXAtom-ORD\"&gt;\u0000 &lt;mml:mi mathvariant=\"double-struck\"&gt;R&lt;/mml:mi&gt;\u0000 &lt;/mml:mrow&gt;\u0000 &lt;mml:mn&gt;3&lt;/mml:mn&gt;\u0000 &lt;/mml:msup&gt;\u0000 &lt;mml:mo stretchy=\"false\"&gt;)&lt;/mml:mo&gt;\u0000 &lt;/mml:mrow&gt;\u0000 &lt;mml:annotation encoding=\"application/x-tex\"&gt;H^1(mathbb {R}^3)&lt;/mml:annotation&gt;\u0000 &lt;/mml:semantics&gt;\u0000&lt;/mml:math&gt;\u0000&lt;/inline-formula&gt; critical focusing semilinear wave equation”, 2009) and Krieger and Schlag (“Full range of blow up exponents for the quintic wave equation in three dimensions”, 2014) are stable along a co-dimension one Lipschitz manifold of data perturbations in a suitable topology, provided the scaling parameter &lt;inline-formula content-type=\"math/mathml\"&gt;\u0000&lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"lamda left-parenthesis t right-parenthesis equals t Superscript negative 1 minus nu\"&gt;\u0000 &lt;mml:semantics&gt;\u0000 &lt;mml:mrow&gt;\u0000 &lt;mml:mi&gt;λ&lt;!-- λ --&gt;&lt;/mml:mi&gt;\u0000 &lt;mml:mo stretchy=\"false\"&gt;(&lt;/mml:mo&gt;\u0000 &lt;mml:mi&gt;t&lt;/mml:mi&gt;\u0000 &lt;mml:mo stretchy=\"false\"&gt;)&lt;/mml:mo&gt;\u0000 &lt;mml:mo&gt;=&lt;/mml:mo&gt;\u0000 &lt;mml:msup&gt;\u0000 &lt;mml:mi&gt;t&lt;/mml:mi&gt;\u0000 &lt;mml:mrow class=\"MJX-TeXAtom-ORD\"&gt;\u0000 &lt;mml:mo&gt;−&lt;!-- − --&gt;&lt;/mml:mo&gt;\u0000 &lt;mml:mn&gt;1&lt;/mml:mn&gt;\u0000 &lt;mml:mo&gt;−&lt;!-- − --&gt;&lt;","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2017-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43184618","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}
引用次数: 7
Intense Automorphisms of Finite Groups 有限群的强自同构
IF 1.9 4区 数学 Q1 MATHEMATICS Pub Date : 2017-09-05 DOI: 10.1090/memo/1341
Mima Stanojkovski
<p>Let <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics></mml:math></inline-formula> be a group. An automorphism of <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics></mml:math></inline-formula> is called <italic>intense</italic> if it sends each subgroup of <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics></mml:math></inline-formula> to a conjugate; the collection of such automorphisms is denoted by <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I n t left-parenthesis upper G right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>Int</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">operatorname {Int}(G)</mml:annotation> </mml:semantics></mml:math></inline-formula>. In the special case in which <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics></mml:math></inline-formula> is a prime number and <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics></mml:math></inline-formula> is a finite <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics></mml:math></inline-formula>-group, one can show that <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I n t left-parenthesis upper G right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>Int</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">operatorname {Int}(G)</mml:annotation> </mml:semantics></mml:math></inline-formula> is the semidirect product of a normal
设G G是一个群。G G的自同构称为强自同构,如果它将G G的每个子群发送到共轭;这类自同构的集合用Int表示⁡ (G)运算符名称{Int}(G)。在p p是素数,G G是有限p p群的特殊情况下,可以证明Int⁡ (G)算子名{Int}(G)是正规p-Sylow与除p-1 p-1阶循环子群的半直积。本文对强自同构群本身不是p-群的有限p-群进行了分类。从我们的研究中可以看出,这类群的结构几乎完全由它们的幂零性类决定:对于p>3 p>3,它们与一个唯一确定的无限2-生成的亲p群共享一个商,该商与它们的类一起增长。
{"title":"Intense Automorphisms of Finite Groups","authors":"Mima Stanojkovski","doi":"10.1090/memo/1341","DOIUrl":"https://doi.org/10.1090/memo/1341","url":null,"abstract":"&lt;p&gt;Let &lt;inline-formula content-type=\"math/mathml\"&gt;\u0000&lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\"&gt;\u0000 &lt;mml:semantics&gt;\u0000 &lt;mml:mi&gt;G&lt;/mml:mi&gt;\u0000 &lt;mml:annotation encoding=\"application/x-tex\"&gt;G&lt;/mml:annotation&gt;\u0000 &lt;/mml:semantics&gt;\u0000&lt;/mml:math&gt;\u0000&lt;/inline-formula&gt; be a group. An automorphism of &lt;inline-formula content-type=\"math/mathml\"&gt;\u0000&lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\"&gt;\u0000 &lt;mml:semantics&gt;\u0000 &lt;mml:mi&gt;G&lt;/mml:mi&gt;\u0000 &lt;mml:annotation encoding=\"application/x-tex\"&gt;G&lt;/mml:annotation&gt;\u0000 &lt;/mml:semantics&gt;\u0000&lt;/mml:math&gt;\u0000&lt;/inline-formula&gt; is called &lt;italic&gt;intense&lt;/italic&gt; if it sends each subgroup of &lt;inline-formula content-type=\"math/mathml\"&gt;\u0000&lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\"&gt;\u0000 &lt;mml:semantics&gt;\u0000 &lt;mml:mi&gt;G&lt;/mml:mi&gt;\u0000 &lt;mml:annotation encoding=\"application/x-tex\"&gt;G&lt;/mml:annotation&gt;\u0000 &lt;/mml:semantics&gt;\u0000&lt;/mml:math&gt;\u0000&lt;/inline-formula&gt; to a conjugate; the collection of such automorphisms is denoted by &lt;inline-formula content-type=\"math/mathml\"&gt;\u0000&lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper I n t left-parenthesis upper G right-parenthesis\"&gt;\u0000 &lt;mml:semantics&gt;\u0000 &lt;mml:mrow&gt;\u0000 &lt;mml:mi&gt;Int&lt;/mml:mi&gt;\u0000 &lt;mml:mo&gt;⁡&lt;!-- ⁡ --&gt;&lt;/mml:mo&gt;\u0000 &lt;mml:mo stretchy=\"false\"&gt;(&lt;/mml:mo&gt;\u0000 &lt;mml:mi&gt;G&lt;/mml:mi&gt;\u0000 &lt;mml:mo stretchy=\"false\"&gt;)&lt;/mml:mo&gt;\u0000 &lt;/mml:mrow&gt;\u0000 &lt;mml:annotation encoding=\"application/x-tex\"&gt;operatorname {Int}(G)&lt;/mml:annotation&gt;\u0000 &lt;/mml:semantics&gt;\u0000&lt;/mml:math&gt;\u0000&lt;/inline-formula&gt;. In the special case in which &lt;inline-formula content-type=\"math/mathml\"&gt;\u0000&lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\"&gt;\u0000 &lt;mml:semantics&gt;\u0000 &lt;mml:mi&gt;p&lt;/mml:mi&gt;\u0000 &lt;mml:annotation encoding=\"application/x-tex\"&gt;p&lt;/mml:annotation&gt;\u0000 &lt;/mml:semantics&gt;\u0000&lt;/mml:math&gt;\u0000&lt;/inline-formula&gt; is a prime number and &lt;inline-formula content-type=\"math/mathml\"&gt;\u0000&lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\"&gt;\u0000 &lt;mml:semantics&gt;\u0000 &lt;mml:mi&gt;G&lt;/mml:mi&gt;\u0000 &lt;mml:annotation encoding=\"application/x-tex\"&gt;G&lt;/mml:annotation&gt;\u0000 &lt;/mml:semantics&gt;\u0000&lt;/mml:math&gt;\u0000&lt;/inline-formula&gt; is a finite &lt;inline-formula content-type=\"math/mathml\"&gt;\u0000&lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\"&gt;\u0000 &lt;mml:semantics&gt;\u0000 &lt;mml:mi&gt;p&lt;/mml:mi&gt;\u0000 &lt;mml:annotation encoding=\"application/x-tex\"&gt;p&lt;/mml:annotation&gt;\u0000 &lt;/mml:semantics&gt;\u0000&lt;/mml:math&gt;\u0000&lt;/inline-formula&gt;-group, one can show that &lt;inline-formula content-type=\"math/mathml\"&gt;\u0000&lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper I n t left-parenthesis upper G right-parenthesis\"&gt;\u0000 &lt;mml:semantics&gt;\u0000 &lt;mml:mrow&gt;\u0000 &lt;mml:mi&gt;Int&lt;/mml:mi&gt;\u0000 &lt;mml:mo&gt;⁡&lt;!-- ⁡ --&gt;&lt;/mml:mo&gt;\u0000 &lt;mml:mo stretchy=\"false\"&gt;(&lt;/mml:mo&gt;\u0000 &lt;mml:mi&gt;G&lt;/mml:mi&gt;\u0000 &lt;mml:mo stretchy=\"false\"&gt;)&lt;/mml:mo&gt;\u0000 &lt;/mml:mrow&gt;\u0000 &lt;mml:annotation encoding=\"application/x-tex\"&gt;operatorname {Int}(G)&lt;/mml:annotation&gt;\u0000 &lt;/mml:semantics&gt;\u0000&lt;/mml:math&gt;\u0000&lt;/inline-formula&gt; is the semidirect product of a normal ","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2017-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44652987","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}
引用次数: 3
The Brunn-Minkowski Inequality and A Minkowski Problem for Nonlinear Capacity Brunn—Minkowski不等式与非线性容量的一个Minkowsky问题
IF 1.9 4区 数学 Q1 MATHEMATICS Pub Date : 2017-09-01 DOI: 10.1090/memo/1348
M. Akman, Jasun Gong, Jay Hineman, Johnny M. Lewis, A. Vogel
<p>In this article we study two classical potential-theoretic problems in convex geometry. The first problem is an inequality of Brunn-Minkowski type for a nonlinear capacity, <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C a p Subscript script upper A Baseline comma"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>Cap</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">operatorname {Cap}_{mathcal {A}},</mml:annotation> </mml:semantics></mml:math></inline-formula> where <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper A"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">mathcal {A}</mml:annotation> </mml:semantics></mml:math></inline-formula>-capacity is associated with a nonlinear elliptic PDE whose structure is modeled on the <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics></mml:math></inline-formula>-Laplace equation and whose solutions in an open set are called <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper A"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">mathcal {A}</mml:annotation> </mml:semantics></mml:math></inline-formula>-harmonic.</p><p>In the first part of this article, we prove the Brunn-Minkowski inequality for this capacity: <disp-formula content-type="math/mathml">[<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket upper C a p Subscript script upper A Baseline left-parenthesis lamda upper E 1 plus left-parenthesis 1 minus lamda right-parenthesis upper E 2 right-parenthesis right-bracket Superscript StartFraction 1 Over left-parenthesis n minus p right-parenthesis EndFraction Baseline greater-than-or-equal-to lamda left-bracket upper C a p Subscript script upper A Baseline left-parenthesis upper E 1 right-parenthesis right-bracket Superscript StartFraction 1 Over left-parenthesis n minus p right-parenthesis EndFraction Baseline plus left-parenthesis 1 minus lamda right-parenthesis left-bracket upper C a p Subscript script upper A Baseline left-parenthesis upper E 2 right-parenthesis right-bracket Supe
本文研究了凸几何中的两个经典势论问题。第一个问题是非线性容量Cap a 算子名的Brunn Minkowski型不等式{Cap}_{mathcal{A}},其中Amathcal}-容量与一个非线性椭圆PDE有关,该椭圆PDE的结构以p-p-Laplace方程为模型,其在开集中的解称为Amathical{A〕-harmonic。在本文的第一部分中,我们证明了这个容量的Brun-Minkowski不等式:[[Cap A⁡ (λE1+(1−λ)E2)]1(n−p)≥λ[第A章⁡ (E1)]1(n−p)+(1−λ)[第A章⁡ (E2)]1(n−p)left[operator name{Cap}_mathcal{A}(lambda E_1+(1-lambda)E_2)right]^{frac{1}{(n-p)}}geqlambda,left[operatorname{Cap}_mathcal{A}(E_1)right]^{frac{1}{(n-p)}}+(1-lambda)left[operatorname{Cap}_当1>p>n,0>λ>1,1>p>n,0>lambda>1和E1,E2 E_1,E_2是具有正Amathcal{A}-容量的凸紧集。此外,如果在某些E1 E_1和E2,E_2的上述不等式中成立等式,则在
{"title":"The Brunn-Minkowski Inequality and A Minkowski Problem for Nonlinear Capacity","authors":"M. Akman, Jasun Gong, Jay Hineman, Johnny M. Lewis, A. Vogel","doi":"10.1090/memo/1348","DOIUrl":"https://doi.org/10.1090/memo/1348","url":null,"abstract":"&lt;p&gt;In this article we study two classical potential-theoretic problems in convex geometry. The first problem is an inequality of Brunn-Minkowski type for a nonlinear capacity, &lt;inline-formula content-type=\"math/mathml\"&gt;\u0000&lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper C a p Subscript script upper A Baseline comma\"&gt;\u0000 &lt;mml:semantics&gt;\u0000 &lt;mml:mrow&gt;\u0000 &lt;mml:msub&gt;\u0000 &lt;mml:mi&gt;Cap&lt;/mml:mi&gt;\u0000 &lt;mml:mrow class=\"MJX-TeXAtom-ORD\"&gt;\u0000 &lt;mml:mrow class=\"MJX-TeXAtom-ORD\"&gt;\u0000 &lt;mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\"&gt;A&lt;/mml:mi&gt;\u0000 &lt;/mml:mrow&gt;\u0000 &lt;/mml:mrow&gt;\u0000 &lt;/mml:msub&gt;\u0000 &lt;mml:mo&gt;,&lt;/mml:mo&gt;\u0000 &lt;/mml:mrow&gt;\u0000 &lt;mml:annotation encoding=\"application/x-tex\"&gt;operatorname {Cap}_{mathcal {A}},&lt;/mml:annotation&gt;\u0000 &lt;/mml:semantics&gt;\u0000&lt;/mml:math&gt;\u0000&lt;/inline-formula&gt; where &lt;inline-formula content-type=\"math/mathml\"&gt;\u0000&lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper A\"&gt;\u0000 &lt;mml:semantics&gt;\u0000 &lt;mml:mrow class=\"MJX-TeXAtom-ORD\"&gt;\u0000 &lt;mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\"&gt;A&lt;/mml:mi&gt;\u0000 &lt;/mml:mrow&gt;\u0000 &lt;mml:annotation encoding=\"application/x-tex\"&gt;mathcal {A}&lt;/mml:annotation&gt;\u0000 &lt;/mml:semantics&gt;\u0000&lt;/mml:math&gt;\u0000&lt;/inline-formula&gt;-capacity is associated with a nonlinear elliptic PDE whose structure is modeled on the &lt;inline-formula content-type=\"math/mathml\"&gt;\u0000&lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\"&gt;\u0000 &lt;mml:semantics&gt;\u0000 &lt;mml:mi&gt;p&lt;/mml:mi&gt;\u0000 &lt;mml:annotation encoding=\"application/x-tex\"&gt;p&lt;/mml:annotation&gt;\u0000 &lt;/mml:semantics&gt;\u0000&lt;/mml:math&gt;\u0000&lt;/inline-formula&gt;-Laplace equation and whose solutions in an open set are called &lt;inline-formula content-type=\"math/mathml\"&gt;\u0000&lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper A\"&gt;\u0000 &lt;mml:semantics&gt;\u0000 &lt;mml:mrow class=\"MJX-TeXAtom-ORD\"&gt;\u0000 &lt;mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\"&gt;A&lt;/mml:mi&gt;\u0000 &lt;/mml:mrow&gt;\u0000 &lt;mml:annotation encoding=\"application/x-tex\"&gt;mathcal {A}&lt;/mml:annotation&gt;\u0000 &lt;/mml:semantics&gt;\u0000&lt;/mml:math&gt;\u0000&lt;/inline-formula&gt;-harmonic.&lt;/p&gt;\u0000\u0000&lt;p&gt;In the first part of this article, we prove the Brunn-Minkowski inequality for this capacity: &lt;disp-formula content-type=\"math/mathml\"&gt;\u0000[\u0000&lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-bracket upper C a p Subscript script upper A Baseline left-parenthesis lamda upper E 1 plus left-parenthesis 1 minus lamda right-parenthesis upper E 2 right-parenthesis right-bracket Superscript StartFraction 1 Over left-parenthesis n minus p right-parenthesis EndFraction Baseline greater-than-or-equal-to lamda left-bracket upper C a p Subscript script upper A Baseline left-parenthesis upper E 1 right-parenthesis right-bracket Superscript StartFraction 1 Over left-parenthesis n minus p right-parenthesis EndFraction Baseline plus left-parenthesis 1 minus lamda right-parenthesis left-bracket upper C a p Subscript script upper A Baseline left-parenthesis upper E 2 right-parenthesis right-bracket Supe","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2017-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47809455","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}
引用次数: 28
Purity and Separation for Oriented Matroids 取向拟阵的纯度与分离
IF 1.9 4区 数学 Q1 MATHEMATICS Pub Date : 2017-08-03 DOI: 10.1090/memo/1439
Pavel Galashin, A. Postnikov
Leclerc and Zelevinsky, motivated by the study of quasi-commuting quantum flag minors, introduced the notions of strongly separated and weakly separated collections. These notions are closely related to the theory of cluster algebras, to the combinatorics of the double Bruhat cells, and to the totally positive Grassmannian.A key feature, called the purity phenomenon, is that every maximal by inclusion strongly (resp., weakly) separated collection of subsets in [ n ] [n] has the same cardinality.In this paper, we extend these notions and define M mathcal {M} -separated collections for any oriented matroid M mathcal {M} .We show that maximal by size M mathcal {M} -separated collections are in bijection with fine zonotopal tilings (if M mathcal {M} is a realizable oriented matroid), or with one-element liftings of M mathcal {M} in general position (for an arbitrary oriented matroid).We introduce the class of pure oriented matroids for which the purity phenomenon holds: an oriented matroid M mathcal {M} is pure if M mathcal {M} -separated collections form a pure simplicial complex, i.e., any maximal by inclusion M mathcal {M} -separated collection is also maximal by size.We pay closer attention to several special classes of oriented matroids: oriented matroids of rank 3 3 , graphical oriented matroids, and uniform oriented matroids. We classify pure oriented matroids in these cases. An oriented matroid of rank 3 3 is pure if and only if it is a positroid (up to reorienting and relabeling its ground set). A graphical oriented matroid is pure if and only if its underlying graph is an outerplanar graph, that is, a subgraph of a triangulation of an n n -gon.We give a simple conjectural characterization of pure oriented matroids by forbidden minors and prove it for the above classes of matroids (rank 3 3 , graphical, uniform).
Leclerc和Zelevinsky受到准交换量子子旗研究的启发,引入了强分离集合和弱分离集合的概念。这些概念与簇代数理论、双Bruhat细胞的组合学和完全正的Grassmannian密切相关。一个关键的特征,称为纯度现象,是每一个极大的包含强烈地(对应)。[n] [n]中子集的弱分离集合具有相同的基数。在本文中,我们扩展了这些概念,并定义了M mathcal {M}分离的集合,适用于任何有向矩阵M mathcal {M}。我们证明了M mathcal {M}分离的集合在大小上是最大的,它们具有良好的分区贴图(如果M mathcal {M}是一个可实现的有向矩阵),或者在一般位置上具有M mathcal {M}的单元素提升(对于任意有向矩阵)。我们引入了纯定向矩阵的一类,它的纯粹性现象成立:如果M mathcal {M}分离的集合形成一个纯简单复合体,则M mathcal {M}是纯的,即任何包含M mathcal {M}分离的集合的最大值也是大小的最大值。我们进一步研究了几种特殊的定向拟阵:33阶定向拟阵、图形定向拟阵和均匀定向拟阵。在这些情况下,我们对纯取向拟阵进行分类。秩为33的定向矩阵是纯的,当且仅当它是正极(直到重新定向和重新标记它的基集)。一个面向图形的矩阵是纯的,当且仅当它的底层图是一个外平面图,即一个n - n -gon的三角剖分的子图。我们给出了纯取向拟阵的一个简单的禁忌次元的推测刻划,并证明了上述类的拟阵(秩33,图形,均匀)。
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引用次数: 18
Non-kissing complexes and tau-tilting for gentle algebras 温柔代数的非接吻复形和tau倾斜
IF 1.9 4区 数学 Q1 MATHEMATICS Pub Date : 2017-07-24 DOI: 10.1090/memo/1343
Yann Palu, Vincent Pilaud, Pierre-Guy Plamondon
We interpret the support τ tau -tilting complex of any gentle bound quiver as the non-kissing complex of walks on its blossoming quiver. Particularly relevant examples were previously studied for quivers defined by a subset of the grid or by a dissection of a polygon. We then focus on the case when the non-kissing complex is finite. We show that the graph of increasing flips on its facets is the Hasse diagram of a congruence-uniform lattice. Finally, we study its g mathbf {g} -vector fan and prove that it is the normal fan of a non-kissing associahedron.
我们将任何温和束缚颤振的支持τ tau倾斜复合物解释为在其开花颤振上行走的非亲吻复合物。特别相关的例子以前研究过由网格子集或多边形的解剖定义的颤振。然后我们关注非接吻复合体是有限的情况。我们证明了其面上增加翻转的图是同余一致格的哈塞图。最后,我们研究了它的g mathbf {g}向量扇形,并证明了它是一个非接吻共轭面体的正规扇形。
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引用次数: 30
期刊
Memoirs of the American Mathematical Society
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