The Dubrovin conjecture predicts a relationship between the monodromy data of the Frobenius manifold associated to the quantum cohomology of a smooth projective variety and the bounded derived category of the same variety. A refinement of this conjecture was given by Cotti, Dubrovin and Guzzetti, which is equivalent to the Gamma conjecture II proposed by Galkin, Golyshev and Iritani. The Gamma conjecture II for quadrics was proved by Hu and Ke. The Lagrangian Grassmanian $LG(2,4)$ is isomorphic to the quadric in $mathbb P^4$. In this paper, we give a new proof of the refined Dubrovin conjecture for the Lagrangian Grassmanian $LG(2,4)$ by explicit computation.
{"title":"Proof of the refined Dubrovin conjecture for the Lagrangian Grassmanian $LG(2,4)$","authors":"Fangze Sheng","doi":"arxiv-2409.03590","DOIUrl":"https://doi.org/arxiv-2409.03590","url":null,"abstract":"The Dubrovin conjecture predicts a relationship between the monodromy data of\u0000the Frobenius manifold associated to the quantum cohomology of a smooth\u0000projective variety and the bounded derived category of the same variety. A\u0000refinement of this conjecture was given by Cotti, Dubrovin and Guzzetti, which\u0000is equivalent to the Gamma conjecture II proposed by Galkin, Golyshev and\u0000Iritani. The Gamma conjecture II for quadrics was proved by Hu and Ke. The\u0000Lagrangian Grassmanian $LG(2,4)$ is isomorphic to the quadric in $mathbb P^4$.\u0000In this paper, we give a new proof of the refined Dubrovin conjecture for the\u0000Lagrangian Grassmanian $LG(2,4)$ by explicit computation.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"47 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142209861","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 decay rate of Riesz capacity as the exponent increases to the dimension of the set is shown to yield Hausdorff measure. The result applies to strongly rectifiable sets, and so in particular to submanifolds of Euclidean space. For strictly self-similar fractals, a one-sided decay estimate is found. Along the way, a purely measure theoretic proof is given for subadditivity of the reciprocal of Riesz energy.
{"title":"Hausdorff measure and decay rate of Riesz capacity","authors":"Qiuling Fan, Richard S. Laugesen","doi":"arxiv-2409.03070","DOIUrl":"https://doi.org/arxiv-2409.03070","url":null,"abstract":"The decay rate of Riesz capacity as the exponent increases to the dimension\u0000of the set is shown to yield Hausdorff measure. The result applies to strongly\u0000rectifiable sets, and so in particular to submanifolds of Euclidean space. For\u0000strictly self-similar fractals, a one-sided decay estimate is found. Along the\u0000way, a purely measure theoretic proof is given for subadditivity of the\u0000reciprocal of Riesz energy.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142209775","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 construct a measure on the well-approximable numbers whose Fourier transform decays at a nearly optimal rate. This gives a logarithmic improvement on a previous construction of Kaufman.
{"title":"Sharp Fourier decay estimates for measures supported on the well-approximable numbers","authors":"Robert Fraser, Thanh Nguyen","doi":"arxiv-2409.02854","DOIUrl":"https://doi.org/arxiv-2409.02854","url":null,"abstract":"We construct a measure on the well-approximable numbers whose Fourier\u0000transform decays at a nearly optimal rate. This gives a logarithmic improvement\u0000on a previous construction of Kaufman.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"36 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142226653","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 establish sharp inequalities involving the incomplete Beta and Gamma functions. These inequalities arise in the approximation of generalized Bernstein functions by higher order Thorin-Bernstein functions. Furthermore, new properties of a related function, namely $x^{lambda}Gamma(x)/Gamma(x+lambda)$ are derived.
{"title":"Approximations of generalized Bernstein functions","authors":"Stamatis Koumandos, Henrik Laurberg Pedersen","doi":"arxiv-2409.02536","DOIUrl":"https://doi.org/arxiv-2409.02536","url":null,"abstract":"We establish sharp inequalities involving the incomplete Beta and Gamma\u0000functions. These inequalities arise in the approximation of generalized\u0000Bernstein functions by higher order Thorin-Bernstein functions. Furthermore,\u0000new properties of a related function, namely\u0000$x^{lambda}Gamma(x)/Gamma(x+lambda)$ are derived.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"267 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142209777","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 provide constructive necessary and sufficient conditions for a family of periodic wavelets to be a Parseval wavelet frame. The criterion generalizes unitary and oblique extension principles. It may be very useful for applications to signal processing because it allows to design any wavelet frame explicitly starting with refinable functions. The practically important case of one wavelet generator and refinable functions being trigonometric polynomials is discussed in details. As an application we study approximation properties of frames and give conditions for a coincidence of approximation orders provided by periodic multiresolution analysis and by a wavelet frame in terms of our criterion.
{"title":"On criteria for periodic wavelet frame","authors":"Anastassia Gorsanova, Elena Lebedeva","doi":"arxiv-2409.01165","DOIUrl":"https://doi.org/arxiv-2409.01165","url":null,"abstract":"We provide constructive necessary and sufficient conditions for a family of\u0000periodic wavelets to be a Parseval wavelet frame. The criterion generalizes\u0000unitary and oblique extension principles. It may be very useful for\u0000applications to signal processing because it allows to design any wavelet frame\u0000explicitly starting with refinable functions. The practically important case of\u0000one wavelet generator and refinable functions being trigonometric polynomials\u0000is discussed in details. As an application we study approximation properties of\u0000frames and give conditions for a coincidence of approximation orders provided\u0000by periodic multiresolution analysis and by a wavelet frame in terms of our\u0000criterion.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"71 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142209758","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 work, we establish $L^{p_1}times cdotstimes L^{p_1}to L^p$ bounds for maximal multi-(sub)linear singular integrals associated with homogeneous kernels $frac{Omega(vec{boldsymbol{y}}')}{|vec{boldsymbol{y}}|^{mn}}$ where $Omega$ is an $L^q$ function on the unit sphere $mathbb{S}^{mn-1}$ with vanishing moment condition and $q>1$. As an application, we obtain almost everywhere convergence results for the associated doubly truncated multilinear singular integrals.
{"title":"Multilinear estimates for maximal rough singular integrals","authors":"Bae Jun Park","doi":"arxiv-2409.00357","DOIUrl":"https://doi.org/arxiv-2409.00357","url":null,"abstract":"In this work, we establish $L^{p_1}times cdotstimes L^{p_1}to L^p$ bounds\u0000for maximal multi-(sub)linear singular integrals associated with homogeneous\u0000kernels $frac{Omega(vec{boldsymbol{y}}')}{|vec{boldsymbol{y}}|^{mn}}$ where $Omega$ is an $L^q$ function on the unit sphere $mathbb{S}^{mn-1}$\u0000with vanishing moment condition and $q>1$. As an application, we obtain almost everywhere convergence results for the\u0000associated doubly truncated multilinear singular integrals.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"48 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142209912","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}
There are several questions one may ask about polynomials $q_m(x)=q_m(x;t)=sum_{n=0}^mt^mp_n(x)$ attached to a family of orthogonal polynomials ${p_n(x)}_{nge0}$. In this note we draw attention to the naturalness of this partial-sum deformation and related beautiful structures. In particular, we investigate the location and distribution of zeros of $q_m(x;t)$ in the case of varying real parameter $t$.
{"title":"A partial-sum deformation for a family of orthogonal polynomials","authors":"Erik Koelink, Pablo Román, Wadim Zudilin","doi":"arxiv-2409.00261","DOIUrl":"https://doi.org/arxiv-2409.00261","url":null,"abstract":"There are several questions one may ask about polynomials\u0000$q_m(x)=q_m(x;t)=sum_{n=0}^mt^mp_n(x)$ attached to a family of orthogonal\u0000polynomials ${p_n(x)}_{nge0}$. In this note we draw attention to the\u0000naturalness of this partial-sum deformation and related beautiful structures.\u0000In particular, we investigate the location and distribution of zeros of\u0000$q_m(x;t)$ in the case of varying real parameter $t$.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"55 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142209757","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 have investigated $L^{p}$ boundedness of the multilinear maximal Bochner--Riesz means and the corresponding square function. We have exploited the ideas given in the paper "Maximal estimates for bilinear Bochner--Riesz means" (Adv. Math. 395(2022) 108100) by Jotsaroop and Shrivastava, in order to prove our results.
{"title":"$L^{p}$ estimates for multilinear maximal Bochner--Riesz means and square function","authors":"Kalachand Shuin","doi":"arxiv-2408.17069","DOIUrl":"https://doi.org/arxiv-2408.17069","url":null,"abstract":"In this article we have investigated $L^{p}$ boundedness of the multilinear\u0000maximal Bochner--Riesz means and the corresponding square function. We have\u0000exploited the ideas given in the paper \"Maximal estimates for bilinear\u0000Bochner--Riesz means\" (Adv. Math. 395(2022) 108100) by Jotsaroop and\u0000Shrivastava, in order to prove our results.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142209756","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 2007, 'A. Baricz put forward a conjecture concerning Tur'an-type inequalities for Gaussian hypergeometric functions (see Conjecture ref{ConjA} in Section ref{Sec1}). In this paper, the authors disprove this conjecture with several methods, and present Tur'an-type double inequalities for Gaussian hypergeometric functions, and sharp bounds for complete and generalized elliptic integrals of the first kind.
{"title":"Turán-Type Inequalities for Gaussian Hypergeometric Functions, and Baricz's Conjecture","authors":"Song-Liang Qiu, Xiao-Yan Ma, Xue-Yan Xiang","doi":"arxiv-2408.15723","DOIUrl":"https://doi.org/arxiv-2408.15723","url":null,"abstract":"In 2007, 'A. Baricz put forward a conjecture concerning Tur'an-type\u0000inequalities for Gaussian hypergeometric functions (see Conjecture ref{ConjA}\u0000in Section ref{Sec1}). In this paper, the authors disprove this conjecture\u0000with several methods, and present Tur'an-type double inequalities for Gaussian\u0000hypergeometric functions, and sharp bounds for complete and generalized\u0000elliptic integrals of the first kind.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142209790","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 series expansion of $x^m (-log x)^l$ in terms of the shifted Chebyshev Polynomials $T_n^*(x)$ requires evaluation of the integral family $int_0^1 x^m (-log x)^l dx / sqrt{x-x^2}$. We demonstrate that these can be reduced by partial integration to sums over integrals with exponent $m=0$ which have known representations as finite sums over polygamma functions.
{"title":"Chebyshev approximation of $x^m (-log x)^l$ in the interval $0le x le 1$","authors":"Richard J. Mathar","doi":"arxiv-2408.15212","DOIUrl":"https://doi.org/arxiv-2408.15212","url":null,"abstract":"The series expansion of $x^m (-log x)^l$ in terms of the shifted Chebyshev\u0000Polynomials $T_n^*(x)$ requires evaluation of the integral family $int_0^1 x^m\u0000(-log x)^l dx / sqrt{x-x^2}$. We demonstrate that these can be reduced by\u0000partial integration to sums over integrals with exponent $m=0$ which have known\u0000representations as finite sums over polygamma functions.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"64 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142209860","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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