Pub Date : 2022-06-01DOI: 10.14321/realanalexch.47.1.1590599512
G. Ivanova, Irena Domnik
{"title":"On Density and Sigma-Porosity in Some Families of Darboux Functions","authors":"G. Ivanova, Irena Domnik","doi":"10.14321/realanalexch.47.1.1590599512","DOIUrl":"https://doi.org/10.14321/realanalexch.47.1.1590599512","url":null,"abstract":"","PeriodicalId":44674,"journal":{"name":"Real Analysis Exchange","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49244400","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 : 2022-06-01DOI: 10.14321/realanalexch.47.1.1610537860
P. Viswanathan
{"title":"A Brief Exposition of the Space of Relatively Bounded Nonlinear Operators","authors":"P. Viswanathan","doi":"10.14321/realanalexch.47.1.1610537860","DOIUrl":"https://doi.org/10.14321/realanalexch.47.1.1610537860","url":null,"abstract":"","PeriodicalId":44674,"journal":{"name":"Real Analysis Exchange","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43317551","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 : 2022-06-01DOI: 10.14321/realanalexch.47.1.1596546460
Martin E. Price
{"title":"Approximating the Decreasing Rearrangement","authors":"Martin E. Price","doi":"10.14321/realanalexch.47.1.1596546460","DOIUrl":"https://doi.org/10.14321/realanalexch.47.1.1596546460","url":null,"abstract":"","PeriodicalId":44674,"journal":{"name":"Real Analysis Exchange","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47233762","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 : 2022-06-01DOI: 10.14321/realanalexch.47.1.1600145293
K. B. Bhaskara Rao
{"title":"Comments on Charges on the Boolean Algebra of Regular Open Sets","authors":"K. B. Bhaskara Rao","doi":"10.14321/realanalexch.47.1.1600145293","DOIUrl":"https://doi.org/10.14321/realanalexch.47.1.1600145293","url":null,"abstract":"","PeriodicalId":44674,"journal":{"name":"Real Analysis Exchange","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49062513","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 : 2021-11-29DOI: 10.14321/realanalexch.47.2.1638769133
Marta Kossaczká, L. Zajícek
Sanchez, Viader, Paradis and Carrillo (2016) proved that there exists an increasing continuous singular function $f$ on $[0,1]$ such that the set $A_f$ of points where $f$ has a nonzero finite derivative has Hausdorff dimension 1 in each subinterval of $[0,1]$. We prove a stronger (and optimal) result showing that a set $A_f$ as above can contain any prescribed $F_{sigma}$ null subset of $[0,1]$.
{"title":"On the Set of Points at which an Increasing Continuous Singular Function has a Nonzero Finite Derivative","authors":"Marta Kossaczká, L. Zajícek","doi":"10.14321/realanalexch.47.2.1638769133","DOIUrl":"https://doi.org/10.14321/realanalexch.47.2.1638769133","url":null,"abstract":"Sanchez, Viader, Paradis and Carrillo (2016) proved that there exists an increasing continuous singular function $f$ on $[0,1]$ such that the set $A_f$ of points where $f$ has a nonzero finite derivative has Hausdorff dimension 1 in each subinterval of $[0,1]$. We prove a stronger (and optimal) result showing that a set $A_f$ as above can contain any prescribed $F_{sigma}$ null subset of $[0,1]$.","PeriodicalId":44674,"journal":{"name":"Real Analysis Exchange","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2021-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45668327","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 : 2021-11-01DOI: 10.14321/REALANALEXCH.46.2.0423
Nicholas P. M. Kayban, Xianfu Wang
We show that a few basic classes of lower semicontinuous functions on ℝn are densely recoverable. Specifically, we show that the sum of a convex and a continuous function, the difference of two convex and lower semicontinuous functions, a K-increasing function (where K is a cone of nonempty interior), and differences of K-increasing functions are all functions uniquely determined by their values on a dense set in ℝn. Thus, sets of such functions of each type are densely recoverable sets. In general, the sum and difference of two densely recoverable sets of functions is shown to not be densely recoverable.
{"title":"ON FUNCTIONS DETERMINED BY DENSE SETS","authors":"Nicholas P. M. Kayban, Xianfu Wang","doi":"10.14321/REALANALEXCH.46.2.0423","DOIUrl":"https://doi.org/10.14321/REALANALEXCH.46.2.0423","url":null,"abstract":"We show that a few basic classes of lower semicontinuous functions on ℝn are densely recoverable. Specifically, we show that the sum of a convex and a continuous function, the difference of two convex and lower semicontinuous functions, a K-increasing function (where K is a cone of nonempty interior), and differences of K-increasing functions are all functions uniquely determined by their values on a dense set in ℝn. Thus, sets of such functions of each type are densely recoverable sets. In general, the sum and difference of two densely recoverable sets of functions is shown to not be densely recoverable.","PeriodicalId":44674,"journal":{"name":"Real Analysis Exchange","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2021-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48717237","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 : 2021-11-01DOI: 10.14321/REALANALEXCH.46.2.0297
G. Gruenhage
Washek Pfeffer's contributions in general topology are discussed, especially his paper with van Douwen on $D$-spaces and his work with Prikry on small spaces.
{"title":"SOME REMARKS ON WASHEK PFEFFER'S CONTRIBUTIONS IN GENERAL TOPOLOGY","authors":"G. Gruenhage","doi":"10.14321/REALANALEXCH.46.2.0297","DOIUrl":"https://doi.org/10.14321/REALANALEXCH.46.2.0297","url":null,"abstract":"Washek Pfeffer's contributions in general topology are discussed, especially his paper with van Douwen on $D$-spaces and his work with Prikry on small spaces.","PeriodicalId":44674,"journal":{"name":"Real Analysis Exchange","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2021-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46769933","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 : 2021-11-01DOI: 10.14321/REALANALEXCH.46.2.0451
M. Roychowdhury
The basic goal of quantization for probability distribution is to reduce the number of values, which is typically uncountable, describing a probability distribution to some finite set and thus approximation of a continuous probability distribution by a discrete distribution. Mixed distributions are an exciting new area for optimal quantization. In this paper, we have determined the optimal sets of n-means, the nth quantization errors, and the quantization dimensions of different mixed distributions. Besides, we have discussed whether the quantization coefficients for the mixed distributions exist. The results in this paper will give a motivation and insight into more general problems in quantization for mixed distributions.
{"title":"OPTIMAL QUANTIZATION FOR MIXED DISTRIBUTIONS","authors":"M. Roychowdhury","doi":"10.14321/REALANALEXCH.46.2.0451","DOIUrl":"https://doi.org/10.14321/REALANALEXCH.46.2.0451","url":null,"abstract":"The basic goal of quantization for probability distribution is to reduce the number of values, which is typically uncountable, describing a probability distribution to some finite set and thus approximation of a continuous probability distribution by a discrete distribution. Mixed distributions are an exciting new area for optimal quantization. In this paper, we have determined the optimal sets of n-means, the nth quantization errors, and the quantization dimensions of different mixed distributions. Besides, we have discussed whether the quantization coefficients for the mixed distributions exist. The results in this paper will give a motivation and insight into more general problems in quantization for mixed distributions.","PeriodicalId":44674,"journal":{"name":"Real Analysis Exchange","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2021-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45607039","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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