Pub Date : 2023-10-01DOI: 10.14321/realanalexch.48.2.1683549275
Udayan Darji
Jack Brown (August 14, 1938- December 10, 2022) had a long, a varied, and a productive career that had an extraordinarily positive influence on several generations of students, the mathematical community of scholars, and the national security of the United States.
{"title":"Jack Brown-In Memoriam","authors":"Udayan Darji","doi":"10.14321/realanalexch.48.2.1683549275","DOIUrl":"https://doi.org/10.14321/realanalexch.48.2.1683549275","url":null,"abstract":"Jack Brown (August 14, 1938- December 10, 2022) had a long, a varied, and a productive career that had an extraordinarily positive influence on several generations of students, the mathematical community of scholars, and the national security of the United States.","PeriodicalId":44674,"journal":{"name":"Real Analysis Exchange","volume":"35 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134979038","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 : 2023-10-01DOI: 10.14321/realanalexch.48.2.1668676378
Alex Burgin, Samuel Goldberg, Tamás Keleti, Connor MacMahon, Xianzhi Wang
We study some variants of the Erdös similarity problem. We pose the question if every measurable subset of the real line with positive measure contains a similar copy of an infinite geometric progression. We construct a compact subset $E$ of the real line such that $0$ is a Lebesgue density point of $E$, but $E$ does not contain any (non-constant) infinite geometric progression. We give a sufficient density type condition that guarantees that a set contains an infinite geometric progression. By slightly improving a recent result of Bradford, Kohut and Mooroo-gen we construct a closed set $Fsubset[0,infty)$ such that the measure of $Fcap[t,t+1]$ tends to $1$ at infinity but $F$ does not contain any infinite arithmetic progression. We also slightly improve a more general recent result by Kolountzakis and Papageorgiou for more general sequences. We give a sufficient condition that guarantees that a given Cantor type set contains at least one infinite geometric progression with any quotient between $0$ and $1$. This can be applied to most symmetric Cantor sets of positive measure.
{"title":"Large Sets Avoiding Infinite Arithmetic / Geometric Progressions","authors":"Alex Burgin, Samuel Goldberg, Tamás Keleti, Connor MacMahon, Xianzhi Wang","doi":"10.14321/realanalexch.48.2.1668676378","DOIUrl":"https://doi.org/10.14321/realanalexch.48.2.1668676378","url":null,"abstract":"We study some variants of the Erdös similarity problem. We pose the question if every measurable subset of the real line with positive measure contains a similar copy of an infinite geometric progression. We construct a compact subset $E$ of the real line such that $0$ is a Lebesgue density point of $E$, but $E$ does not contain any (non-constant) infinite geometric progression. We give a sufficient density type condition that guarantees that a set contains an infinite geometric progression. By slightly improving a recent result of Bradford, Kohut and Mooroo-gen we construct a closed set $Fsubset[0,infty)$ such that the measure of $Fcap[t,t+1]$ tends to $1$ at infinity but $F$ does not contain any infinite arithmetic progression. We also slightly improve a more general recent result by Kolountzakis and Papageorgiou for more general sequences. We give a sufficient condition that guarantees that a given Cantor type set contains at least one infinite geometric progression with any quotient between $0$ and $1$. This can be applied to most symmetric Cantor sets of positive measure.","PeriodicalId":44674,"journal":{"name":"Real Analysis Exchange","volume":"39 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134979041","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 : 2023-10-01DOI: 10.14321/realanalexch.48.2.1671048854
Karel Hrbacek, Mikhail G. Katz
We survey the effective foundations for analysis with infinitesimals developed by Hrbacek and Katz in 2021, and detail some applications. Theories SPOT and SCOT are conservative over respectively ZF and ZF+ADC. The range of applications of these theories illustrates the fact that analysis with infinitesimals requires no more choice than traditional analysis. The theory SCOT incorporates in particular all the axioms of Nelson's Radically Elementary Probability Theory, which is therefore conservative over ZF+ADC.
{"title":"Effective Infinitesimals in $mathbb R$","authors":"Karel Hrbacek, Mikhail G. Katz","doi":"10.14321/realanalexch.48.2.1671048854","DOIUrl":"https://doi.org/10.14321/realanalexch.48.2.1671048854","url":null,"abstract":"We survey the effective foundations for analysis with infinitesimals developed by Hrbacek and Katz in 2021, and detail some applications. Theories SPOT and SCOT are conservative over respectively ZF and ZF+ADC. The range of applications of these theories illustrates the fact that analysis with infinitesimals requires no more choice than traditional analysis. The theory SCOT incorporates in particular all the axioms of Nelson's Radically Elementary Probability Theory, which is therefore conservative over ZF+ADC.","PeriodicalId":44674,"journal":{"name":"Real Analysis Exchange","volume":"57 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134979043","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 : 2023-10-01DOI: 10.14321/realanalexch.48.2.1653627715
Badreddine Meftah, Djamal Foukrach
The aim of this research paper is to establish some new generalized Gamidov type integral inequalities involving a $psi $-fractional operator. We also give two applications to substantiate the validity of our findings.
{"title":"Some New Gamidov Type Integral Inequalities Associated with $psi$-Fractional Operators","authors":"Badreddine Meftah, Djamal Foukrach","doi":"10.14321/realanalexch.48.2.1653627715","DOIUrl":"https://doi.org/10.14321/realanalexch.48.2.1653627715","url":null,"abstract":"The aim of this research paper is to establish some new generalized Gamidov type integral inequalities involving a $psi $-fractional operator. We also give two applications to substantiate the validity of our findings.","PeriodicalId":44674,"journal":{"name":"Real Analysis Exchange","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134979044","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 : 2023-10-01DOI: 10.14321/realanalexch.48.2.1665637941
Morgan O'Brien
For a sequence in a Banach space $mathcal{X}$, it is known that the set of subsequential limits of the sequence forms a closed subset of $mathcal{X}$. Similarly, if the sequence is convergent, then the sequence of its Cesàro averages also converge to the same value. In this article, we study the properties of the set of Cesàro limits of subsequences of a given sequence in a Banach space using techniques from ergodic theory.
{"title":"On Subsequential Averages of Sequences in Banach Spaces","authors":"Morgan O'Brien","doi":"10.14321/realanalexch.48.2.1665637941","DOIUrl":"https://doi.org/10.14321/realanalexch.48.2.1665637941","url":null,"abstract":"For a sequence in a Banach space $mathcal{X}$, it is known that the set of subsequential limits of the sequence forms a closed subset of $mathcal{X}$. Similarly, if the sequence is convergent, then the sequence of its Cesàro averages also converge to the same value. In this article, we study the properties of the set of Cesàro limits of subsequences of a given sequence in a Banach space using techniques from ergodic theory.","PeriodicalId":44674,"journal":{"name":"Real Analysis Exchange","volume":"43 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134979035","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 : 2023-10-01DOI: 10.14321/realanalexch.48.2.1676962925
Martina Maiuriello
Motivated by recent studies on the notions of lineability and spaceability in the context of linear dynamics, we investigate the existence of infinite-dimensional closed subspaces of frequently hypercyclic vectors for frequently hypercyclic composition operators, known in the literature as Koopman operators and extensively used in many applications (like, for instance, the analysis of the dynamics of economic models formulated in terms of dynamical systems). All the results are obtained on $L^p$ spaces, $1 leq p < infty$, and in the dissipative setting with the extra hypothesis of bounded distortion. This allows us, as a consequence, to deduce analogous conclusions for fundamental mathematical objects: bilateral weighted backward shifts on $ell^p$ spaces.
在线性动力学背景下对线性性和空间性概念的最新研究的激励下，我们研究了频繁超循环组合算子的频繁超循环向量的无限维闭子空间的存在性，这些算子在文献中被称为Koopman算子，并广泛用于许多应用(例如，用动力系统表述的经济模型的动力学分析)。所有结果都是在$L^p$空间、$1 leq p < infty$和附加有界畸变假设的耗散设置下得到的。因此，这使我们能够为基本的数学对象推导出类似的结论:$ell^p$空间上的双边加权后移。
{"title":"On the Existence of Infinite-Dimensional Closed Subspaces of Frequently Hypercyclic Vectors for $T_f$","authors":"Martina Maiuriello","doi":"10.14321/realanalexch.48.2.1676962925","DOIUrl":"https://doi.org/10.14321/realanalexch.48.2.1676962925","url":null,"abstract":"Motivated by recent studies on the notions of lineability and spaceability in the context of linear dynamics, we investigate the existence of infinite-dimensional closed subspaces of frequently hypercyclic vectors for frequently hypercyclic composition operators, known in the literature as Koopman operators and extensively used in many applications (like, for instance, the analysis of the dynamics of economic models formulated in terms of dynamical systems). All the results are obtained on $L^p$ spaces, $1 leq p < infty$, and in the dissipative setting with the extra hypothesis of bounded distortion. This allows us, as a consequence, to deduce analogous conclusions for fundamental mathematical objects: bilateral weighted backward shifts on $ell^p$ spaces.","PeriodicalId":44674,"journal":{"name":"Real Analysis Exchange","volume":"89 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134979039","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 : 2023-10-01DOI: 10.14321/realanalexch.48.2.1683015838
Gregory V. Cox, John T. Walsh, Patrick Reardon
Jack Brown (August 14, 1938 -- December 10, 2022). Three former Ph.D. students of Jack Brown reflect on the magnificent ways that he challenged them to think and shaped their careers through his careful guiding touch.
{"title":"Remembering Jack","authors":"Gregory V. Cox, John T. Walsh, Patrick Reardon","doi":"10.14321/realanalexch.48.2.1683015838","DOIUrl":"https://doi.org/10.14321/realanalexch.48.2.1683015838","url":null,"abstract":"Jack Brown (August 14, 1938 -- December 10, 2022). Three former Ph.D. students of Jack Brown reflect on the magnificent ways that he challenged them to think and shaped their careers through his careful guiding touch.","PeriodicalId":44674,"journal":{"name":"Real Analysis Exchange","volume":"41 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134979046","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 : 2023-10-01DOI: 10.14321/realanalexch.48.2.1680708522
Gady Kozma, Alexander M. Olevskiĭ
We survey our recent result that for every continuous function there is an absolutely continuous homeomorphism such that the composition has a uniformly converging Fourier expansion. We mention the history of the problem, orginally stated by Luzin, and some details of the proof.
{"title":"Homeomorphisms and Fourier Expansion","authors":"Gady Kozma, Alexander M. Olevskiĭ","doi":"10.14321/realanalexch.48.2.1680708522","DOIUrl":"https://doi.org/10.14321/realanalexch.48.2.1680708522","url":null,"abstract":"We survey our recent result that for every continuous function there is an absolutely continuous homeomorphism such that the composition has a uniformly converging Fourier expansion. We mention the history of the problem, orginally stated by Luzin, and some details of the proof.","PeriodicalId":44674,"journal":{"name":"Real Analysis Exchange","volume":"39 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135606026","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 : 2023-10-01DOI: 10.14321/realanalexch.48.2.1681628520
Angel D. Cruz, Chun-Kit Lai, Malabika Pramanik
A conjecture of Erdös states that for any infinite set $A subseteq mathbb R$, there exists a Borel set $E subseteq mathbb R$ of positive Lebesgue measure that does not contain any non-trivial affine copy of $A$. The conjecture remains open for most fast-decaying sequences, including the geometric sequence $A = {2^{-k} : k geq 1}$. In this article, we consider infinite decreasing sequences $A = {a_k: k geq 1}$ in $R$ that converge to zero at a prescribed rate; namely $log (a_n/a_{n+1}) = e^{varphi(n)} $, where $varphi(n)/nto 0$ as $ntoinfty$. This condition is satisfied by sequences whose logarithm has polynomial decay, and in particular by the geometric sequence. For any such sequence $A$, we construct a Cantor set $K subseteq mathbb [0,1]$ with measure arbitrarily close to 1, such that the set of Erdös points $mathcal{E}subseteq K$ has Hasudorff dimension 1.
{"title":"Large Sets Avoiding Affine Copies of Infinite Sequences","authors":"Angel D. Cruz, Chun-Kit Lai, Malabika Pramanik","doi":"10.14321/realanalexch.48.2.1681628520","DOIUrl":"https://doi.org/10.14321/realanalexch.48.2.1681628520","url":null,"abstract":"A conjecture of Erdös states that for any infinite set $A subseteq mathbb R$, there exists a Borel set $E subseteq mathbb R$ of positive Lebesgue measure that does not contain any non-trivial affine copy of $A$. The conjecture remains open for most fast-decaying sequences, including the geometric sequence $A = {2^{-k} : k geq 1}$. In this article, we consider infinite decreasing sequences $A = {a_k: k geq 1}$ in $R$ that converge to zero at a prescribed rate; namely $log (a_n/a_{n+1}) = e^{varphi(n)} $, where $varphi(n)/nto 0$ as $ntoinfty$. This condition is satisfied by sequences whose logarithm has polynomial decay, and in particular by the geometric sequence. For any such sequence $A$, we construct a Cantor set $K subseteq mathbb [0,1]$ with measure arbitrarily close to 1, such that the set of Erdös points $mathcal{E}subseteq K$ has Hasudorff dimension 1.","PeriodicalId":44674,"journal":{"name":"Real Analysis Exchange","volume":"33 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134979034","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 : 2023-10-01DOI: 10.14321/realanalexch.48.2.1659596483
Jin Hatomoto
We study a diffeomorphism which admits a weak hyperbolic product structure region, which is the intersection of two transversal families of weak stable and weak unstable disks, with countably many branches and integrable return times. We show that for such maps the distributions of the number of visits to a ball $B(x, r)$ converges to a Poisson distributions as the radius $r to 0$. Applications of our resutls are some partially hyperbolic diffeomorphisms of which restriction on one dimensional unstable direction behaves as Manneville-Pomeau maps.
{"title":"Poisson Limit Distribution for Diffeomorphisms with Weak Hyperbolic Product Structure","authors":"Jin Hatomoto","doi":"10.14321/realanalexch.48.2.1659596483","DOIUrl":"https://doi.org/10.14321/realanalexch.48.2.1659596483","url":null,"abstract":"We study a diffeomorphism which admits a weak hyperbolic product structure region, which is the intersection of two transversal families of weak stable and weak unstable disks, with countably many branches and integrable return times. We show that for such maps the distributions of the number of visits to a ball $B(x, r)$ converges to a Poisson distributions as the radius $r to 0$. Applications of our resutls are some partially hyperbolic diffeomorphisms of which restriction on one dimensional unstable direction behaves as Manneville-Pomeau maps.","PeriodicalId":44674,"journal":{"name":"Real Analysis Exchange","volume":"54 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134979040","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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