Pub Date : 2021-10-22DOI: 10.14321/realanalexch.48.1.1638769133
Nathan Dalaklis, K. Kawamura, Tobey Mathis, Michalis Paizanis
The differentiability of the one parameter family of Okomoto's functions as functions of $x$ has been analyzed extensively since their introduction in 2005. As an analogue to a similar investigation, in this paper, we consider the partial derivative of Okomoto's functions with respect to the parameter $a$. We place a significant focus on $a = 1/3$ to describe the properties of a nowhere differentiable function $K(x)$ for which the set of points of infinite derivative produces an example of a measure zero set with Hausdorff dimension $1$.
{"title":"The Partial Derivative of Okamoto's Functions with Respect to the Parameter","authors":"Nathan Dalaklis, K. Kawamura, Tobey Mathis, Michalis Paizanis","doi":"10.14321/realanalexch.48.1.1638769133","DOIUrl":"https://doi.org/10.14321/realanalexch.48.1.1638769133","url":null,"abstract":"The differentiability of the one parameter family of Okomoto's functions as functions of $x$ has been analyzed extensively since their introduction in 2005. As an analogue to a similar investigation, in this paper, we consider the partial derivative of Okomoto's functions with respect to the parameter $a$. We place a significant focus on $a = 1/3$ to describe the properties of a nowhere differentiable function $K(x)$ for which the set of points of infinite derivative produces an example of a measure zero set with Hausdorff dimension $1$.","PeriodicalId":44674,"journal":{"name":"Real Analysis Exchange","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2021-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44296262","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 : 2021-10-14DOI: 10.14321/realanalexch.46.1.0099
O. D. de Oliveira
We show an explicit formula, with a quite easy deduction, for the exponential matrix $e^{tA}$ of a real square matrix $A$ of order $ntimes n$. The elementary method developed requires neither Jordan canonical form, nor eigenvectors, nor resolution of linear systems of differential equations, nor resolution of linear systems with constant coefficients, nor matrix inversion, nor complex integration, nor functional analysis. The basic tools are power series and the method of partial fraction decomposition. Two examples are given. A proof of one well-known stability result is given.
{"title":"The exponential matrix: an explicit formula by an elementary method","authors":"O. D. de Oliveira","doi":"10.14321/realanalexch.46.1.0099","DOIUrl":"https://doi.org/10.14321/realanalexch.46.1.0099","url":null,"abstract":"We show an explicit formula, with a quite easy deduction, for the exponential matrix $e^{tA}$ of a real square matrix $A$ of order $ntimes n$. The elementary method developed requires neither Jordan canonical form, nor eigenvectors, nor resolution of linear systems of differential equations, nor resolution of linear systems with constant coefficients, nor matrix inversion, nor complex integration, nor functional analysis. The basic tools are power series and the method of partial fraction decomposition. Two examples are given. A proof of one well-known stability result is given.","PeriodicalId":44674,"journal":{"name":"Real Analysis Exchange","volume":"31 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2021-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84848513","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 : 2021-10-14DOI: 10.14321/realanalexch.46.1.0233
Immanuel D. Calunod, I. Garces
{"title":"Strong derivative and the essentially Riemann integral","authors":"Immanuel D. Calunod, I. Garces","doi":"10.14321/realanalexch.46.1.0233","DOIUrl":"https://doi.org/10.14321/realanalexch.46.1.0233","url":null,"abstract":"","PeriodicalId":44674,"journal":{"name":"Real Analysis Exchange","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2021-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47690486","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 : 2021-10-14DOI: 10.14321/realanalexch.46.1.0247
N. Singh, S. P. S. Kainth
{"title":"Variational measure with respect to measurable gauges","authors":"N. Singh, S. P. S. Kainth","doi":"10.14321/realanalexch.46.1.0247","DOIUrl":"https://doi.org/10.14321/realanalexch.46.1.0247","url":null,"abstract":"","PeriodicalId":44674,"journal":{"name":"Real Analysis Exchange","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2021-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46592971","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 : 2021-10-14DOI: 10.14321/realanalexch.46.1.0037
Igor E. Preobrazhenskii, P. Demidov
{"title":"Sufficient conditions for convergence of riemann sums for function space defined by the $k$-modulus of continuity","authors":"Igor E. Preobrazhenskii, P. Demidov","doi":"10.14321/realanalexch.46.1.0037","DOIUrl":"https://doi.org/10.14321/realanalexch.46.1.0037","url":null,"abstract":"","PeriodicalId":44674,"journal":{"name":"Real Analysis Exchange","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2021-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46000745","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 : 2021-10-14DOI: 10.14321/realanalexch.46.1.0083
Paweł Barbarski
{"title":"Continuous functions in rings generated by a single Darboux function","authors":"Paweł Barbarski","doi":"10.14321/realanalexch.46.1.0083","DOIUrl":"https://doi.org/10.14321/realanalexch.46.1.0083","url":null,"abstract":"","PeriodicalId":44674,"journal":{"name":"Real Analysis Exchange","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2021-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43796622","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 : 2021-07-20DOI: 10.14321/realanalexch.48.1.1626760923
G. David, McKenna Kaczanowski, D. Pinkerton
We study"distance spheres": the set of points lying at constant distance from a fixed arbitrary subset $K$ of $[0,1]^d$. We show that, away from the regions where $K$ is"too dense"and a set of small volume, we can decompose $[0,1]^d$ into a finite number of sets on which the distance spheres can be"straightened"into subsets of parallel $(d-1)$-dimensional planes by a bi-Lipschitz map. Importantly, the number of sets and the bi-Lipschitz constants are independent of the set $K$.
{"title":"Quantitative Straightening of Distance Spheres","authors":"G. David, McKenna Kaczanowski, D. Pinkerton","doi":"10.14321/realanalexch.48.1.1626760923","DOIUrl":"https://doi.org/10.14321/realanalexch.48.1.1626760923","url":null,"abstract":"We study\"distance spheres\": the set of points lying at constant distance from a fixed arbitrary subset $K$ of $[0,1]^d$. We show that, away from the regions where $K$ is\"too dense\"and a set of small volume, we can decompose $[0,1]^d$ into a finite number of sets on which the distance spheres can be\"straightened\"into subsets of parallel $(d-1)$-dimensional planes by a bi-Lipschitz map. Importantly, the number of sets and the bi-Lipschitz constants are independent of the set $K$.","PeriodicalId":44674,"journal":{"name":"Real Analysis Exchange","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2021-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43262151","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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