Pub Date : 2020-11-01DOI: 10.14321/REALANALEXCH.45.2.0265
C. Ridolfi
{"title":"Best Local Weighted Approximation. An Approach with Abstract Seminorms","authors":"C. Ridolfi","doi":"10.14321/REALANALEXCH.45.2.0265","DOIUrl":"https://doi.org/10.14321/REALANALEXCH.45.2.0265","url":null,"abstract":"","PeriodicalId":44674,"journal":{"name":"Real Analysis Exchange","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2020-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48446804","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 : 2020-11-01DOI: 10.14321/REALANALEXCH.45.2.0327
K. Ciesielski, P. Jiménez-Rodríguez, G. A. Muñoz-Fernández, J. Seoane-Sepúlveda
{"title":"Non-Differentiability of the Convolution of Differentiable Real Functions","authors":"K. Ciesielski, P. Jiménez-Rodríguez, G. A. Muñoz-Fernández, J. Seoane-Sepúlveda","doi":"10.14321/REALANALEXCH.45.2.0327","DOIUrl":"https://doi.org/10.14321/REALANALEXCH.45.2.0327","url":null,"abstract":"","PeriodicalId":44674,"journal":{"name":"Real Analysis Exchange","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2020-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43593109","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 : 2020-10-29DOI: 10.14321/realanalexch.48.1.1629953964
Antoine Detaille, A. Ponce
We prove that every finite Borel measure $mu$ in $mathbb{R}^N$ that is bounded from above by the Hausdorff measure $mathcal{H}^s$ can be split in countable many parts $mulfloor_{E_k}$ that are bounded from above by the Hausdorff content $mathcal{H}_infty^s$. Such a result generalises a theorem due to R. Delaware that says that any Borel set with finite Hausdorff measure can be decomposed as a countable disjoint union of straight sets. We also investigate the case where $mu$ is not necessarily finite.
{"title":"A Decomposition for Borel Measures (mu le mathcal{H}^{s})","authors":"Antoine Detaille, A. Ponce","doi":"10.14321/realanalexch.48.1.1629953964","DOIUrl":"https://doi.org/10.14321/realanalexch.48.1.1629953964","url":null,"abstract":"We prove that every finite Borel measure $mu$ in $mathbb{R}^N$ that is bounded from above by the Hausdorff measure $mathcal{H}^s$ can be split in countable many parts $mulfloor_{E_k}$ that are bounded from above by the Hausdorff content $mathcal{H}_infty^s$. Such a result generalises a theorem due to R. Delaware that says that any Borel set with finite Hausdorff measure can be decomposed as a countable disjoint union of straight sets. We also investigate the case where $mu$ is not necessarily finite.","PeriodicalId":44674,"journal":{"name":"Real Analysis Exchange","volume":"1 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2020-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42390262","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 : 2020-08-16DOI: 10.14321/realanalexch.47.1.1598582300
Daniel Perry
We study contact 3-manifolds using the techniques of sub-Riemannian geometry and geometric measure theory, in particular establishing properties of their Lipschitz homotopy groups. We prove a biLipschitz version of the Theorem of Darboux: a contact $(2n+1)$-manifold endowed with a sub-Riemannian structure is locally biLipschitz equivalent to the Heisenberg group $mathbb{H}^n$ with its Carnot-Caratheodory metric. Then each contact $(2n+1)$-manifold endowed with a sub-Riemannian structure is purely $k$-unrectifiable for $k>n$. We extend results of Dejarnette et al. (arXiv:1109.4641 [math.FA]) and Wenger and Young (arXiv:1210.6943 [math.GT]) by showing for any purely 2-unrectifiable sub-Riemannian manifold $(M,xi,g)$ that the $n$th Lipschitz homotopy group is trivial for $ngeq2$ and that the set of oriented, horizontal knots in $(M,xi)$ injects into the first Lipschitz homotopy group. Thus, the first Lipschitz homotopy group of any contact 3-manifold is uncountably generated. Therefore, in the sense of Lipschitz homotopy groups, a contact 3-manifold is a $K(pi,1)$-space for an uncountably generated group $pi$. Finally, we prove that each open distributional embedding between purely 2-unrectifiable sub-Riemannian manifolds induces an injective map on the associated first Lipschitz homotopy groups. Therefore, each open subset of a contact 3-manifold determines an uncountable subgroup of the first Lipschitz homotopy group of the contact 3-manifold.
{"title":"Lipschitz Homotopy Groups of Contact 3-Manifolds","authors":"Daniel Perry","doi":"10.14321/realanalexch.47.1.1598582300","DOIUrl":"https://doi.org/10.14321/realanalexch.47.1.1598582300","url":null,"abstract":"We study contact 3-manifolds using the techniques of sub-Riemannian geometry and geometric measure theory, in particular establishing properties of their Lipschitz homotopy groups. We prove a biLipschitz version of the Theorem of Darboux: a contact $(2n+1)$-manifold endowed with a sub-Riemannian structure is locally biLipschitz equivalent to the Heisenberg group $mathbb{H}^n$ with its Carnot-Caratheodory metric. Then each contact $(2n+1)$-manifold endowed with a sub-Riemannian structure is purely $k$-unrectifiable for $k>n$. We extend results of Dejarnette et al. (arXiv:1109.4641 [math.FA]) and Wenger and Young (arXiv:1210.6943 [math.GT]) by showing for any purely 2-unrectifiable sub-Riemannian manifold $(M,xi,g)$ that the $n$th Lipschitz homotopy group is trivial for $ngeq2$ and that the set of oriented, horizontal knots in $(M,xi)$ injects into the first Lipschitz homotopy group. Thus, the first Lipschitz homotopy group of any contact 3-manifold is uncountably generated. Therefore, in the sense of Lipschitz homotopy groups, a contact 3-manifold is a $K(pi,1)$-space for an uncountably generated group $pi$. Finally, we prove that each open distributional embedding between purely 2-unrectifiable sub-Riemannian manifolds induces an injective map on the associated first Lipschitz homotopy groups. Therefore, each open subset of a contact 3-manifold determines an uncountable subgroup of the first Lipschitz homotopy group of the contact 3-manifold.","PeriodicalId":44674,"journal":{"name":"Real Analysis Exchange","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2020-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44399263","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}
{"title":"A girsanov result for the pettis integral","authors":"D. Candeloro, A. R. Sambucini, Luca Trastulli","doi":"10.14321/realanalexch.46.1.0175","DOIUrl":"https://doi.org/10.14321/realanalexch.46.1.0175","url":null,"abstract":"A kind of Pettis integral representation for a Banach valued Ito process is given and its drift term is modified using a Girsanov Theorem.","PeriodicalId":44674,"journal":{"name":"Real Analysis Exchange","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2020-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49308766","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 : 2020-06-17DOI: 10.14321/REALANALEXCH.46.2.0523
Haipeng Chen, J. Fraser
Let $p_n$ denote the $n$th prime, and consider the function $1/n mapsto 1/p_n$ which maps the reciprocals of the positive integers bijectively to the reciprocals of the primes. We show that Holder continuity of this function is equivalent to a parameterised family of Cramer type estimates on the gaps between successive primes. Here the parameterisation comes from the Holder exponent. In particular, we show that Cramer's conjecture is equivalent to the map $1/n mapsto 1/p_n$ being Lipschitz. On the other hand, we show that the inverse map $1/p_n mapsto 1/n$ is Holder of all orders but not Lipshitz and this is independent of Cramer's conjecture.
{"title":"ON HÖLDER MAPS AND PRIME GAPS","authors":"Haipeng Chen, J. Fraser","doi":"10.14321/REALANALEXCH.46.2.0523","DOIUrl":"https://doi.org/10.14321/REALANALEXCH.46.2.0523","url":null,"abstract":"Let $p_n$ denote the $n$th prime, and consider the function $1/n mapsto 1/p_n$ which maps the reciprocals of the positive integers bijectively to the reciprocals of the primes. We show that Holder continuity of this function is equivalent to a parameterised family of Cramer type estimates on the gaps between successive primes. Here the parameterisation comes from the Holder exponent. In particular, we show that Cramer's conjecture is equivalent to the map $1/n mapsto 1/p_n$ being Lipschitz. On the other hand, we show that the inverse map $1/p_n mapsto 1/n$ is Holder of all orders but not Lipshitz and this is independent of Cramer's conjecture.","PeriodicalId":44674,"journal":{"name":"Real Analysis Exchange","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2020-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42549915","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 : 2020-05-01DOI: 10.14321/realanalexch.45.1.0101
Ricky F. Rulete, M. A. Labendia
{"title":"Double Lusin Condition and Convergence Theorems for the Backwards Itô-Henstock Integral","authors":"Ricky F. Rulete, M. A. Labendia","doi":"10.14321/realanalexch.45.1.0101","DOIUrl":"https://doi.org/10.14321/realanalexch.45.1.0101","url":null,"abstract":"","PeriodicalId":44674,"journal":{"name":"Real Analysis Exchange","volume":"45 1","pages":"101-126"},"PeriodicalIF":0.2,"publicationDate":"2020-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46109773","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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