Consider the differential equation ${ mddot{x} +gamma dot{x} -xepsilon cos(omega t) =0}$, $0 leq t leq T$. The form of the fundamental set of solutions are determined by Floquet theory. In the limit as $m to 0$ we can apply WKB theory to get first order approximations of this fundamental set. WKB theory states that this approximation gets better as $m to 0$ in the sense that the difference in sup norm is bounded as function of $m$ for a given $T$. However, convergence of the periodic parts and exponential parts are not addressed. We show that there is convergence to these components. The asymptotic error for the characteristic exponents are $O(m^2)$ and $O(m)$ for the periodic parts.
考虑微分方程${ mddot{x} +gamma dot{x} -xepsilon cos(omega t) =0}$, $0 leq t leq T$。基本解集的形式由Floquet理论决定。在极限为$m to 0$的情况下，我们可以应用WKB理论得到这个基本集的一阶近似。WKB理论指出，这种近似在$m to 0$时变得更好，因为对于给定的$T$, sup范数的差作为$m$的函数有界。然而，周期部分和指数部分的收敛性没有得到解决。我们证明了这些分量是收敛的。周期部分特征指数的渐近误差为$O(m^2)$和$O(m)$。
{"title":"On the convergence of WKB approximations of the damped Mathieu equation","authors":"dwight nwaigwe","doi":"10.1063/1.5145267","DOIUrl":"https://doi.org/10.1063/1.5145267","url":null,"abstract":"Consider the differential equation ${ mddot{x} +gamma dot{x} -xepsilon cos(omega t) =0}$, $0 leq t leq T$. The form of the fundamental set of solutions are determined by Floquet theory. In the limit as $m to 0$ we can apply WKB theory to get first order approximations of this fundamental set. WKB theory states that this approximation gets better as $m to 0$ in the sense that the difference in sup norm is bounded as function of $m$ for a given $T$. However, convergence of the periodic parts and exponential parts are not addressed. We show that there is convergence to these components. The asymptotic error for the characteristic exponents are $O(m^2)$ and $O(m)$ for the periodic parts.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-12-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89278138","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}
Shaoming Guo, Changkeun Oh, J. Roos, Po-Lam Yung, Pavel Zorin-Kranich
We prove sharp decoupling inequalities for all degenerate surfaces of codimension two in $mathbb{R}^5$ given by two quadratic forms in three variables. Together with previous work by Demeter, Guo, and Shi in the non-degenerate case (arXiv:1609.04107), this provides a classification of decoupling inequalities for pairs of quadratic forms in three variables.
{"title":"Decoupling for two quadratic forms in three variables: a complete characterization","authors":"Shaoming Guo, Changkeun Oh, J. Roos, Po-Lam Yung, Pavel Zorin-Kranich","doi":"10.4171/RMI/1332","DOIUrl":"https://doi.org/10.4171/RMI/1332","url":null,"abstract":"We prove sharp decoupling inequalities for all degenerate surfaces of codimension two in $mathbb{R}^5$ given by two quadratic forms in three variables. Together with previous work by Demeter, Guo, and Shi in the non-degenerate case (arXiv:1609.04107), this provides a classification of decoupling inequalities for pairs of quadratic forms in three variables.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81141567","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 investigate the box dimensions of compact sets in $mathbb{R}^n$ that contain a unit distance in every direction (such sets may have zero Hausdorff dimension). Among other results, we show that the lower box dimension must be at least $frac{n^2(n-1)}{2n^2-1}$ and can be at most $frac{n(n-1)}{2n-1}$. This quantifies in a certain sense how far the unit sphere $S^{n-1}$ is from being a difference set.
{"title":"On sets containing a unit distance in every direction","authors":"Pablo Shmerkin, Han Yu","doi":"10.19086/DA.22058","DOIUrl":"https://doi.org/10.19086/DA.22058","url":null,"abstract":"We investigate the box dimensions of compact sets in $mathbb{R}^n$ that contain a unit distance in every direction (such sets may have zero Hausdorff dimension). Among other results, we show that the lower box dimension must be at least $frac{n^2(n-1)}{2n^2-1}$ and can be at most $frac{n(n-1)}{2n-1}$. This quantifies in a certain sense how far the unit sphere $S^{n-1}$ is from being a difference set.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79391673","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 pattern avoidance problem seeks to construct a set with large fractal dimension that avoids a prescribed pattern, such as three term arithmetic progressions, or more general patterns, such as finding a set whose Cartesian product avoids the zero set of a given function. Previous work on the subject has considered patterns described by polynomials, or functions satisfying certain regularity conditions. We provide an exposition of some results in this setting, as well as considering new strategies to avoid what we call `rough patterns'. This thesis contains an expanded description of a method described in a previous paper by the author and his collaborators Malabika Pramanik and Joshua Zahl, as well as new results in the rough pattern avoidance setting. There are several problems that fit into the pattern of rough pattern avoidance. For instance, we prove that for any set $X$ with lower Minkowski dimension $s$, there exists a set $Y$ with Hausdorff dimension $1-s$ such that for any rational numbers $a_1, dots, a_N$, the set $a_1 Y + dots + a_N Y$ is disjoint from $X$, or intersects with $X$ solely at the origin. As a second application, we construct subsets of Lipschitz curves with dimension $1/2$ not containing the vertices of any isosceles triangle.
{"title":"Cartesian products avoiding patterns","authors":"Jacob Denson","doi":"10.14288/1.0387448","DOIUrl":"https://doi.org/10.14288/1.0387448","url":null,"abstract":"The pattern avoidance problem seeks to construct a set with large fractal dimension that avoids a prescribed pattern, such as three term arithmetic progressions, or more general patterns, such as finding a set whose Cartesian product avoids the zero set of a given function. Previous work on the subject has considered patterns described by polynomials, or functions satisfying certain regularity conditions. We provide an exposition of some results in this setting, as well as considering new strategies to avoid what we call `rough patterns'. This thesis contains an expanded description of a method described in a previous paper by the author and his collaborators Malabika Pramanik and Joshua Zahl, as well as new results in the rough pattern avoidance setting. There are several problems that fit into the pattern of rough pattern avoidance. For instance, we prove that for any set $X$ with lower Minkowski dimension $s$, there exists a set $Y$ with Hausdorff dimension $1-s$ such that for any rational numbers $a_1, dots, a_N$, the set $a_1 Y + dots + a_N Y$ is disjoint from $X$, or intersects with $X$ solely at the origin. As a second application, we construct subsets of Lipschitz curves with dimension $1/2$ not containing the vertices of any isosceles triangle.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72792716","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}
Pub Date : 2019-11-29DOI: 10.1016/J.JFA.2021.109203
Wenjuan Li, Huiju Wang
{"title":"A study on a class of generalized Schr\"odinger operators","authors":"Wenjuan Li, Huiju Wang","doi":"10.1016/J.JFA.2021.109203","DOIUrl":"https://doi.org/10.1016/J.JFA.2021.109203","url":null,"abstract":"","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73337796","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}
For $xi = (xi_1, xi_2, ldots, xi_d) in mathbb{R}^d$ let $Q(xi) := sum_{j=1}^d sigma_j xi_j^2$ be a quadratic form with signs $sigma_j in {pm1}$ not all equal. Let $S subset mathbb{R}^{d+1}$ be the hyperbolic paraboloid given by $S = big{(xi, tau) in mathbb{R}^{d}times mathbb{R} : tau = Q(xi)big}$. In this note we prove that Gaussians never extremize an $L^p(mathbb{R}^d) to L^{q}(mathbb{R}^{d+1})$ Fourier extension inequality associated to this surface.
对于$xi = (xi_1, xi_2, ldots, xi_d) in mathbb{R}^d$，设$Q(xi) := sum_{j=1}^d sigma_j xi_j^2$为二次型，其符号$sigma_j in {pm1}$不都相等。设$S subset mathbb{R}^{d+1}$为$S = big{(xi, tau) in mathbb{R}^{d}times mathbb{R} : tau = Q(xi)big}$给出的双曲抛物面。在这篇笔记中，我们证明高斯函数从不极化与这个曲面相关的$L^p(mathbb{R}^d) to L^{q}(mathbb{R}^{d+1})$傅立叶扩展不等式。
{"title":"Gaussians never extremize Strichartz inequalities for hyperbolic paraboloids","authors":"E. Carneiro, L. Oliveira, Mateus Sousa","doi":"10.1090/proc/15782","DOIUrl":"https://doi.org/10.1090/proc/15782","url":null,"abstract":"For $xi = (xi_1, xi_2, ldots, xi_d) in mathbb{R}^d$ let $Q(xi) := sum_{j=1}^d sigma_j xi_j^2$ be a quadratic form with signs $sigma_j in {pm1}$ not all equal. Let $S subset mathbb{R}^{d+1}$ be the hyperbolic paraboloid given by $S = big{(xi, tau) in mathbb{R}^{d}times mathbb{R} : tau = Q(xi)big}$. In this note we prove that Gaussians never extremize an $L^p(mathbb{R}^d) to L^{q}(mathbb{R}^{d+1})$ Fourier extension inequality associated to this surface.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80844615","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 article we prove a multidimensional version of the Nazarov lemma. The proof is based on an appropriate generalisation of the regularised system of intervals introduced by Havin, Nazarov and Mashreghi to several dimensions.
{"title":"On the multidimensional Nazarov lemma","authors":"I. Vasilyev","doi":"10.1090/proc/15805","DOIUrl":"https://doi.org/10.1090/proc/15805","url":null,"abstract":"In this article we prove a multidimensional version of the Nazarov lemma. The proof is based on an appropriate generalisation of the regularised system of intervals introduced by Havin, Nazarov and Mashreghi to several dimensions.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82741229","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}
Pub Date : 2019-11-14DOI: 10.1007/978-3-030-56190-1_6
J. F. van Diejen
{"title":"Stable Equilibria for the Roots of the Symmetric Continuous Hahn and Wilson Polynomials","authors":"J. F. van Diejen","doi":"10.1007/978-3-030-56190-1_6","DOIUrl":"https://doi.org/10.1007/978-3-030-56190-1_6","url":null,"abstract":"","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88722370","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}
Pub Date : 2019-11-08DOI: 10.1016/J.MATPUR.2021.07.004
Katrin Fassler, Tuomas Orponen
{"title":"Singular integrals on regular curves in the Heisenberg group","authors":"Katrin Fassler, Tuomas Orponen","doi":"10.1016/J.MATPUR.2021.07.004","DOIUrl":"https://doi.org/10.1016/J.MATPUR.2021.07.004","url":null,"abstract":"","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88586847","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 that discrete singular Radon transforms along a certain class of polynomial mappings $P:mathbb{Z}^dto mathbb{Z}^n$ satisfy sparse bounds. For $n=d=1$ we can handle all polynomials. In higher dimensions, we pose restrictions on the admissible polynomial mappings stemming from a combination of interacting geometric, analytic and number-theoretic obstacles.
{"title":"Sparse bounds for\u0000discrete singular Radon transforms","authors":"T. Anderson, Bingyang Hu, J. Roos","doi":"10.4064/cm8296-8-2020","DOIUrl":"https://doi.org/10.4064/cm8296-8-2020","url":null,"abstract":"We show that discrete singular Radon transforms along a certain class of polynomial mappings $P:mathbb{Z}^dto mathbb{Z}^n$ satisfy sparse bounds. For $n=d=1$ we can handle all polynomials. In higher dimensions, we pose restrictions on the admissible polynomial mappings stemming from a combination of interacting geometric, analytic and number-theoretic obstacles.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73372576","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}
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