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Conformally Natural Families of Probability Distributions on Hyperbolic Disc with a View on Geometric Deep Learning 双曲圆盘上概率分布的共形自然族与几何深度学习视角
arXiv - MATH - Complex Variables Pub Date : 2024-07-23 DOI: arxiv-2407.16733
Vladimir Jacimovic, Marijan Markovic
{"title":"Conformally Natural Families of Probability Distributions on Hyperbolic Disc with a View on Geometric Deep Learning","authors":"Vladimir Jacimovic, Marijan Markovic","doi":"arxiv-2407.16733","DOIUrl":"https://doi.org/arxiv-2407.16733","url":null,"abstract":"We introduce the novel family of probability distributions on hyperbolic\u0000disc. The distinctive property of the proposed family is invariance under the\u0000actions of the group of disc-preserving conformal mappings. The\u0000group-invariance property renders it a convenient and tractable model for\u0000encoding uncertainties in hyperbolic data. Potential applications in Geometric\u0000Deep Learning and bioinformatics are numerous, some of them are briefly\u0000discussed. We also emphasize analogies with hyperbolic coherent states in\u0000quantum physics.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"59 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141772602","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}
引用次数: 0
Kobayashi hyperbolicity in Riemannian manifolds 黎曼流形中的小林双曲性
arXiv - MATH - Complex Variables Pub Date : 2024-07-22 DOI: arxiv-2407.15976
Hervé Gaussier, Alexandre Sukhov
{"title":"Kobayashi hyperbolicity in Riemannian manifolds","authors":"Hervé Gaussier, Alexandre Sukhov","doi":"arxiv-2407.15976","DOIUrl":"https://doi.org/arxiv-2407.15976","url":null,"abstract":"We study the boundary behavior of the Kobayashi-Royden metric and the\u0000Kobayashi hyperbolicity of domains in Riemannian manifolds.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"101 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141772648","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}
引用次数: 0
Unbounded operators and the uncertainty principle 无界算子和不确定性原理
arXiv - MATH - Complex Variables Pub Date : 2024-07-22 DOI: arxiv-2407.15803
Friedrich Haslinger
{"title":"Unbounded operators and the uncertainty principle","authors":"Friedrich Haslinger","doi":"arxiv-2407.15803","DOIUrl":"https://doi.org/arxiv-2407.15803","url":null,"abstract":"We study a variant of the uncertainty principle in terms of the annihilation\u0000and creation operator on generalized Segal Bargmann spaces, which are used for\u0000the FBI-Bargmann transform. In addition, we compute the Berezin transform of\u0000these operators and indicate how to use spaces of entire functions in one\u0000variable to study the SzegH{o} kernel for hypersurfaces in $mathbb C^2.$","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141772593","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}
引用次数: 0
Geometric subfamily of functions convex in some direction and Blaschke products 沿某一方向凸的函数几何亚族和布拉什克积
arXiv - MATH - Complex Variables Pub Date : 2024-07-20 DOI: arxiv-2407.14922
Liulan Li, Saminthan Ponnusamy
{"title":"Geometric subfamily of functions convex in some direction and Blaschke products","authors":"Liulan Li, Saminthan Ponnusamy","doi":"arxiv-2407.14922","DOIUrl":"https://doi.org/arxiv-2407.14922","url":null,"abstract":"Consider the family of locally univalent analytic functions $h$ in the unit\u0000disk $|z|<1$ with the normalization $h(0)=0$, $h'(0)=1$ and satisfying the\u0000condition $${real} left( frac{z h''(z)}{alpha h'(z)}right) <frac{1}{2}\u0000~mbox{ for $zin ID$,} $$ where $0<alphaleq1$. The aim of this article is\u0000to show that this family has several elegant properties such as involving\u0000Blaschke products, Schwarzian derivative and univalent harmonic mappings.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"46 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141772587","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}
引用次数: 0
Lipschitz geometry of complex surface germs via inner rates of primary ideals 通过主理想的内率实现复曲面胚芽的 Lipschitz 几何学
arXiv - MATH - Complex Variables Pub Date : 2024-07-19 DOI: arxiv-2407.14265
Yenni Cherik
{"title":"Lipschitz geometry of complex surface germs via inner rates of primary ideals","authors":"Yenni Cherik","doi":"arxiv-2407.14265","DOIUrl":"https://doi.org/arxiv-2407.14265","url":null,"abstract":"Let $(X, 0)$ be a normal complex surface germ embedded in $(mathbb{C}^n,\u00000)$, and denote by $mathfrak{m}$ the maximal ideal of the local ring\u0000$mathcal{O}_{X,0}$. In this paper, we associate to each $mathfrak{m}$-primary\u0000ideal $I$ of $mathcal{O}_{X,0}$ a continuous function $mathcal{I}_I$ defined\u0000on the set of positive (suitably normalized) semivaluations of\u0000$mathcal{O}_{X,0}$. We prove that the function $mathcal{I}_{mathfrak{m}}$ is\u0000determined by the outer Lipschitz geometry of the surface $(X, 0)$. We further\u0000demonstrate that for each $mathfrak{m}$-primary ideal $I$, there exists a\u0000complex surface germ $(X_I, 0)$ with an isolated singularity whose\u0000normalization is isomorphic to $(X, 0)$ and $mathcal{I}_I =\u0000mathcal{I}_{mathfrak{m}_I}$, where $mathfrak{m}_I$ is the maximal ideal of\u0000$mathcal{O}_{X_I,0}$. Subsequently, we construct an infinite family of complex\u0000surface germs with isolated singularities, whose normalizations are isomorphic\u0000to $(X,0)$ (in particular, they are homeomorphic to $(X,0)$) but have distinct\u0000outer Lipschitz types.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"61 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141737315","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}
引用次数: 0
Slope-semistability and moduli of coherent sheaves: a survey 相干剪切的斜向可分性与模态：概览
arXiv - MATH - Complex Variables Pub Date : 2024-07-18 DOI: arxiv-2407.13485
Mihai Pavel, Matei Toma
{"title":"Slope-semistability and moduli of coherent sheaves: a survey","authors":"Mihai Pavel, Matei Toma","doi":"arxiv-2407.13485","DOIUrl":"https://doi.org/arxiv-2407.13485","url":null,"abstract":"We survey old and new results on the existence of moduli spaces of semistable\u0000coherent sheaves both in algebraic and in complex geometry.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"43 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141737317","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}
引用次数: 0
A remark on the Hölder regularity of solutions to the complex Hessian equation 关于复赫西恩方程解的赫尔德正则性的评论
arXiv - MATH - Complex Variables Pub Date : 2024-07-18 DOI: arxiv-2407.13130
Slawomir Kolodziej, Ngoc Cuong Nguyen
{"title":"A remark on the Hölder regularity of solutions to the complex Hessian equation","authors":"Slawomir Kolodziej, Ngoc Cuong Nguyen","doi":"arxiv-2407.13130","DOIUrl":"https://doi.org/arxiv-2407.13130","url":null,"abstract":"We prove that the Dirichlet problem for the complex Hessian equation has the\u0000H\"older continuous solution provided it has a subsolution with this property.\u0000Compared to the previous result of Benali-Zeriahi and Charabati-Zeriahi we\u0000remove the assumption on the finite total mass of the measure on the right hand\u0000side.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"36 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141737316","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}
引用次数: 0
Cauchy transforms and Szegő projections in dual Hardy spaces: inequalities and Möbius invariance 对偶哈代空间中的考奇变换和斯格ő投影：不等式和莫比乌斯不变性
arXiv - MATH - Complex Variables Pub Date : 2024-07-17 DOI: arxiv-2407.13033
David E. Barrett, Luke D. Edholm
{"title":"Cauchy transforms and Szegő projections in dual Hardy spaces: inequalities and Möbius invariance","authors":"David E. Barrett, Luke D. Edholm","doi":"arxiv-2407.13033","DOIUrl":"https://doi.org/arxiv-2407.13033","url":null,"abstract":"Dual pairs of interior and exterior Hardy spaces associated to a simple\u0000closed Lipschitz planar curve are considered, leading to a M\"obius invariant\u0000function bounding the norm of the Cauchy transform $bf{C}$ from below. This\u0000function is shown to satisfy strong rigidity properties and is closely\u0000connected via the Berezin transform to the square of the Kerzman-Stein\u0000operator. Explicit example calculations are presented. For ellipses, a new\u0000asymptotically sharp lower bound on the norm of $bf{C}$ is produced.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"78 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141745485","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}
引用次数: 0
Quasi-coherent sheaves on complex analytic spaces 复解析空间上的准相干剪切
arXiv - MATH - Complex Variables Pub Date : 2024-07-16 DOI: arxiv-2407.11656
Haohao Liu
{"title":"Quasi-coherent sheaves on complex analytic spaces","authors":"Haohao Liu","doi":"arxiv-2407.11656","DOIUrl":"https://doi.org/arxiv-2407.11656","url":null,"abstract":"We show that in the category of analytic sheaves on a complex analytic space,\u0000the full subcategory of quasi-coherent sheaves is an abelian subcategory.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"7 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141721448","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}
引用次数: 0
On the connectedness of the boundary of $q$-complete domains 论$q$完整域边界的连通性
arXiv - MATH - Complex Variables Pub Date : 2024-07-16 DOI: arxiv-2407.11897
Rafael B. Andrist
{"title":"On the connectedness of the boundary of $q$-complete domains","authors":"Rafael B. Andrist","doi":"arxiv-2407.11897","DOIUrl":"https://doi.org/arxiv-2407.11897","url":null,"abstract":"The boundary of every relatively compact Stein domain in a complex manifold\u0000of dimension at least two is connected. No assumptions on the boundary\u0000regularity are necessary. The same proofs hold also for $q$-complete domains,\u0000and in the context of almost complex manifolds as well.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"32 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141722489","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}
引用次数: 0
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