Pub Date : 2018-01-01Epub Date: 2017-04-13DOI: 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.
{"title":"A Burning Problem: Cannabis Lessons Learned from Colorado.","authors":"Jamie E Parnes, Adrian J Bravo, Bradley T Conner, Matthew R Pearson","doi":"10.1080/16066359.2017.1315410","DOIUrl":"10.1080/16066359.2017.1315410","url":null,"abstract":"<p><p>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.</p>","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":"1 1","pages":"3-10"},"PeriodicalIF":2.9,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10923185/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82597999","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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.
{"title":"Spectral Properties of Ruelle Transfer Operators for Regular Gibbs Measures and Decay of Correlations for Contact Anosov Flows","authors":"L. Stoyanov","doi":"10.1090/memo/1404","DOIUrl":"https://doi.org/10.1090/memo/1404","url":null,"abstract":"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.","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2017-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46346188","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}
We give the full solution of the following problem: obtain sharp inequalities between the moduli of smoothness ωα(f,t)qomega _alpha (f,t)_q and ωβ(f,t)pomega _beta (f,t)_p for 0>p>q≤∞0>p>qle infty. A similar problem for the generalized KK-functionals and their realizations between the couples (Lp,Wp
<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
{"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":"<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\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"epsilon\">\u0000 <mml:semantics>\u0000 <mml:mi>ε<!-- ε --></mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">varepsilon</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=\"ln epsilon equivalent-to negative 1 slash h\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>ln</mml:mi>\u0000 <mml:mo><!-- --></mml:mo>\u0000 <mml:mi>ε<!-- ε --></mml:mi>\u0000 <mml:mo>≍<!-- ≍ --></mml:mo>\u0000 <mml:mo>−<!-- − --></mml:mo>\u0000 <mml:mn>1</mml:mn>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mo>/</mml:mo>\u0000 </mml:mrow>\u0000 <mml:mi>h</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">ln varepsilon asymp -1/h</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, where <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"h\">\u0000 <mml:semantics>\u0000 <mml:mi>h</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">h</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</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\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"epsilon Superscript negative upper N\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mi>ε<!-- ε --></mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mo>−<!-- − --></mml:mo>\u0000 <mml:mi>N</mml:mi>\u0000 </mml:mrow>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">varepsilon ^{-N}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> with an error <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper O left-parenthesis epsilon Superscript upper N Baseline right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">O</mml:mi>\u0000 </mml:mrow>\u0000 </mml:mrow>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:msup>\u0000 <mml:mi>ε<!-- ε --></mml:mi>\u0000 <mml:mi>N</mml:mi>\u0000 </mml:msup>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">{mathcal O}(varepsilon ^N)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. Here <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}
We show that the linear drift of the Brownian motion on the universal cover of a closed connected smooth Riemannian manifold is Ck−2C^{k-2} differentiable along any CkC^{k} curve in the manifold of CkC^k Riemannian metrics with negative sectional curvature. We also show that the stochastic entropy of the Brownian motion is C1C^1 differentiable along any C3C^{3} curve of C3C^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.
{"title":"The Regularity of the Linear Drift in Negatively Curved Spaces","authors":"Franccois Ledrappier, Lin Shu","doi":"10.1090/memo/1387","DOIUrl":"https://doi.org/10.1090/memo/1387","url":null,"abstract":"<p>We show that the linear drift of the Brownian motion on the universal cover of a closed connected smooth Riemannian manifold is <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper C Superscript k minus 2\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mi>C</mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi>k</mml:mi>\u0000 <mml:mo>−<!-- − --></mml:mo>\u0000 <mml:mn>2</mml:mn>\u0000 </mml:mrow>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">C^{k-2}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> differentiable along any <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper C Superscript k\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mi>C</mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi>k</mml:mi>\u0000 </mml:mrow>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">C^{k}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> curve in the manifold of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper C Superscript k\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mi>C</mml:mi>\u0000 <mml:mi>k</mml:mi>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">C^k</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> Riemannian metrics with negative sectional curvature. We also show that the stochastic entropy of the Brownian motion is <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper C Superscript 1\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mi>C</mml:mi>\u0000 <mml:mn>1</mml:mn>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">C^1</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> differentiable along any <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper C cubed\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mi>C</mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mn>3</mml:mn>\u0000 </mml:mrow>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">C^{3}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> curve of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper C cubed\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mi>C</mml:mi>\u0000 <mml:mn>3</mml:mn>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">C^3</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> 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.</p>","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2017-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47176645","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
<p>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>−<!-- − --><
{"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":"<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\">\u0000[\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"white medium square u equals minus u Superscript 5\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mo>◻</mml:mo>\u0000 </mml:mrow>\u0000 <mml:mi>u</mml:mi>\u0000 <mml:mo>=</mml:mo>\u0000 <mml:mo>−<!-- − --></mml:mo>\u0000 <mml:msup>\u0000 <mml:mi>u</mml:mi>\u0000 <mml:mn>5</mml:mn>\u0000 </mml:msup>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">Box u = -u^5</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000]\u0000</disp-formula> on <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper R Superscript 3 plus 1\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">R</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mn>3</mml:mn>\u0000 <mml:mo>+</mml:mo>\u0000 <mml:mn>1</mml:mn>\u0000 </mml:mrow>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">mathbb {R}^{3+1}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> constructed in Krieger, Schlag, and Tartaru (“Slow blow-up solutions for the <inline-formula content-type=\"math/mathml\">\u0000<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\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msup>\u0000 <mml:mi>H</mml:mi>\u0000 <mml:mn>1</mml:mn>\u0000 </mml:msup>\u0000 <mml:mo stretchy=\"false\">(</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:mn>3</mml:mn>\u0000 </mml:msup>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">H^1(mathbb {R}^3)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</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\">\u0000<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\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>λ<!-- λ --></mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>t</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 <mml:mo>=</mml:mo>\u0000 <mml:msup>\u0000 <mml:mi>t</mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mo>−<!-- − --></mml:mo>\u0000 <mml:mn>1</mml:mn>\u0000 <mml:mo>−<!-- − --><","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}
<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
{"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":"<p>Let <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\u0000 <mml:semantics>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> be a group. An automorphism of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\u0000 <mml:semantics>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is called <italic>intense</italic> if it sends each subgroup of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\u0000 <mml:semantics>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> to a conjugate; the collection of such automorphisms is denoted by <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper I n t left-parenthesis upper G right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>Int</mml:mi>\u0000 <mml:mo><!-- --></mml:mo>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">operatorname {Int}(G)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. In the special case in which <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\">\u0000 <mml:semantics>\u0000 <mml:mi>p</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">p</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is a prime number and <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\u0000 <mml:semantics>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is a finite <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\">\u0000 <mml:semantics>\u0000 <mml:mi>p</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">p</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-group, one can show that <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper I n t left-parenthesis upper G right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>Int</mml:mi>\u0000 <mml:mo><!-- --></mml:mo>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">operatorname {Int}(G)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> 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}
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
{"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":"<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\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper C a p Subscript script upper A Baseline comma\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msub>\u0000 <mml:mi>Cap</mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">A</mml:mi>\u0000 </mml:mrow>\u0000 </mml:mrow>\u0000 </mml:msub>\u0000 <mml:mo>,</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">operatorname {Cap}_{mathcal {A}},</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> where <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper A\">\u0000 <mml:semantics>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">A</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">mathcal {A}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-capacity is associated with a nonlinear elliptic PDE whose structure is modeled on the <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\">\u0000 <mml:semantics>\u0000 <mml:mi>p</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">p</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-Laplace equation and whose solutions in an open set are called <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper A\">\u0000 <mml:semantics>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">A</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">mathcal {A}</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-harmonic.</p>\u0000\u0000<p>In the first part of this article, we prove the Brunn-Minkowski inequality for this capacity: <disp-formula content-type=\"math/mathml\">\u0000[\u0000<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}
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).
{"title":"Purity and Separation for Oriented Matroids","authors":"Pavel Galashin, A. Postnikov","doi":"10.1090/memo/1439","DOIUrl":"https://doi.org/10.1090/memo/1439","url":null,"abstract":"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.\u0000\u0000A key feature, called the purity phenomenon, is that every maximal by inclusion strongly (resp., weakly) separated collection of subsets in \u0000\u0000 \u0000 \u0000 [\u0000 n\u0000 ]\u0000 \u0000 [n]\u0000 \u0000\u0000 has the same cardinality.\u0000\u0000In this paper, we extend these notions and define \u0000\u0000 \u0000 \u0000 M\u0000 \u0000 mathcal {M}\u0000 \u0000\u0000-separated collections for any oriented matroid \u0000\u0000 \u0000 \u0000 M\u0000 \u0000 mathcal {M}\u0000 \u0000\u0000.\u0000\u0000We show that maximal by size \u0000\u0000 \u0000 \u0000 M\u0000 \u0000 mathcal {M}\u0000 \u0000\u0000-separated collections are in bijection with fine zonotopal tilings (if \u0000\u0000 \u0000 \u0000 M\u0000 \u0000 mathcal {M}\u0000 \u0000\u0000 is a realizable oriented matroid), or with one-element liftings of \u0000\u0000 \u0000 \u0000 M\u0000 \u0000 mathcal {M}\u0000 \u0000\u0000 in general position (for an arbitrary oriented matroid).\u0000\u0000We introduce the class of pure oriented matroids for which the purity phenomenon holds: an oriented matroid \u0000\u0000 \u0000 \u0000 M\u0000 \u0000 mathcal {M}\u0000 \u0000\u0000 is pure if \u0000\u0000 \u0000 \u0000 M\u0000 \u0000 mathcal {M}\u0000 \u0000\u0000-separated collections form a pure simplicial complex, i.e., any maximal by inclusion \u0000\u0000 \u0000 \u0000 M\u0000 \u0000 mathcal {M}\u0000 \u0000\u0000-separated collection is also maximal by size.\u0000\u0000We pay closer attention to several special classes of oriented matroids: oriented matroids of rank \u0000\u0000 \u0000 3\u0000 3\u0000 \u0000\u0000, graphical oriented matroids, and uniform oriented matroids. We classify pure oriented matroids in these cases. An oriented matroid of rank \u0000\u0000 \u0000 3\u0000 3\u0000 \u0000\u0000 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 \u0000\u0000 \u0000 n\u0000 n\u0000 \u0000\u0000-gon.\u0000\u0000We give a simple conjectural characterization of pure oriented matroids by forbidden minors and prove it for the above classes of matroids (rank \u0000\u0000 \u0000 3\u0000 3\u0000 \u0000\u0000, graphical, uniform).","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2017-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48849532","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}
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.
{"title":"Non-kissing complexes and tau-tilting for gentle algebras","authors":"Yann Palu, Vincent Pilaud, Pierre-Guy Plamondon","doi":"10.1090/memo/1343","DOIUrl":"https://doi.org/10.1090/memo/1343","url":null,"abstract":"We interpret the support \u0000\u0000 \u0000 τ\u0000 tau\u0000 \u0000\u0000-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 \u0000\u0000 \u0000 \u0000 g\u0000 \u0000 mathbf {g}\u0000 \u0000\u0000-vector fan and prove that it is the normal fan of a non-kissing associahedron.","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2017-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43026040","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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