Real Analysis Exchange Articles (Project Euclid)
http://projecteuclid.org/euclid.rae
The latest articles from Real Analysis Exchange on Project Euclid, a site for mathematics and statistics resources.en-usCopyright 2010 Cornell University LibraryEuclid-L@cornell.edu (Project Euclid Team)Thu, 05 Aug 2010 15:41 EDTMon, 14 Mar 2011 09:08 EDThttp://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
http://projecteuclid.org/
How to Concentrate Idempotents
http://projecteuclid.org/euclid.rae/1272376220
<strong>J. Marshall Ash</strong><p><strong>Source: </strong>Real Anal. Exchange, Volume 35, Number 1, 1--20.</p><p><strong>Abstract:</strong><br/> Call a sum of exponentials of the form $f(x)=\exp\left( 2\pi iN_{1}x\right) +\exp\left( 2\pi iN_{2}x\right) +\cdot\cdot\cdot+\exp\left( 2\pi iN_{m}x\right) $, where the $N_{k}$ are distinct integers, an \textit{idempotent}. We have $L^{p}$\textit{ interval concentration} if there is a positive constant $a$, depending only on $p$, such that for each interval $I\subset\left[ 0,1\right] $ there is an idempotent $f$ so that $\int _{I}\left\vert f\left( x\right) \right\vert ^{p}dx\diagup\int_{0} ^{1}\left\vert f\left( x\right) \right\vert ^{p}dx>a$. We will explain how to produce such concentration for each $p>0$. The origin of this question and the history of the development of its solution will be surveyed.
</p>projecteuclid.org/euclid.rae/1272376220_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTThe Class of Purely Unrectifiable Sets in \(\ell_2\) is \(\Pii\)-completehttp://projecteuclid.org/euclid.rae/1435669998<strong>Vadim Kulikov</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 39, Number 2, 323--334.</p><p><strong>Abstract:</strong><br/>
The space \(F(\ell_2)\) of all closed subsets of \(\ell_2\) is a Polish space. We show that the subset \(P\subset F(\ell_2)\) consisting of the purely \(1\)-unrectifiable sets is \(\Pii\)-complete.
</p>projecteuclid.org/euclid.rae/1435669998_20150630091318Tue, 30 Jun 2015 09:13 EDTQuasi-Continuity of Horizontally Quasi-Continuous Functionshttp://projecteuclid.org/euclid.rae/1435669999<strong>Alireza Kamel Mirmostafaee</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 39, Number 2, 335--344.</p><p><strong>Abstract:</strong><br/>
Let \(X\) be a Baire space, \(Y\) a topological space, \(Z\) a regular space and \(f:X \times Y \to Z\) be a horizontally quasi-continuous function. We will show that if \(Y\) is first countable and \(f\) is quasi-continuous with respect to the first variable, then every horizontally quasi-continuous function \(f:X \times Y \to Z\) is jointly quasi-continuous. This will extend Martin's Theorem of quasi-continuity of separately quasi-continuous functions for non-metrizable range. Moreover, we will prove quasi-continuity of \(f\) for the case \(Y\) is not necessarily first countable.
</p>projecteuclid.org/euclid.rae/1435669999_20150630091318Tue, 30 Jun 2015 09:13 EDTWeighted a Priori Estimates for the Solution of the Dirichlet Problem in Polygonal Domains in \(\mathbb{R}^2\)http://projecteuclid.org/euclid.rae/1435670000<strong>Marcela Sanmartino</strong>, <strong>Marisa Toschi</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 39, Number 2, 345--362.</p><p><strong>Abstract:</strong><br/>
Let \(\Omega\) be a polygonal domain in \(\mathbb{R}^2\) and let \(U\) be a weak solution of \( -\Delta u=f\) in \( \Omega\) with Dirichlet boundary condition, where \(f\in L^p_\omega(\Omega)\) and \(\omega\) is a weight in \(A_p(\mathbb{R}^2)\), \(1<p<\infty\). We give some estimates of the Green function associated to this problem involving some functions of the distance to the vertices and the angles of \(\Omega\). As a consequence, we can prove an a priori estimate for the solution \(u\) on the weighted Sobolev spaces \(W^{2,p}_\omega(\Omega)\), \(1<p<\infty\).
</p>projecteuclid.org/euclid.rae/1435670000_20150630091318Tue, 30 Jun 2015 09:13 EDTRemarks on a Sum involving Binomial Coefficientshttp://projecteuclid.org/euclid.rae/1435670001<strong>Horst Alzer</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 39, Number 2, 363--366.</p><p><strong>Abstract:</strong><br/>
Let \[ S_p(a,b;t)=\frac{1}{b}\sum_{k=0}^{p} \frac{{p\choose k}}{ {ak+b \choose b} } t^k, \] with \(p\in \mathbf{N}\), \(0<a\in\mathbf{R}\), \(0<b\in\mathbf{R}\), \( t\in\mathbf{R}\). We prove that \( S_p(a,b;t)\) is completely monotonic on \((0,\infty)\) as a function of \(a\) (if \(t>0\)) and as a function of \(b\) (if \(t\geq -1)\). This extends a result of Sofo, who proved that \(a\mapsto S_p(a,b;t)\) is strictly decreasing, convex, and log-convex on \([1,\infty)\).
</p>projecteuclid.org/euclid.rae/1435670001_20150630091318Tue, 30 Jun 2015 09:13 EDTExact Hausdorff Measures of Cantor Setshttp://projecteuclid.org/euclid.rae/1435670002<strong>Malin Palö</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 39, Number 2, 367--384.</p><p><strong>Abstract:</strong><br/>
Cantor sets in \(\mathbb{R}\) are common examples of sets for which Hausdorff measures can be positive and finite. However, there exist Cantor sets for which no Hausdorff measure is supported and finite. The purpose of this paper is to try to resolve this problem by studying an extension of the Hausdorff measures \( \mu_h\) on \(\mathbb{R}\), allowing gauge functions to depend on the midpoint of the covering intervals instead of only on the diameter. As a main result, a theorem about the Hausdorff measure of any regular enough Cantor set, with respect to a chosen gauge function, is obtained.
</p>projecteuclid.org/euclid.rae/1435670002_20150630091318Tue, 30 Jun 2015 09:13 EDTWeighting inequalities for one-sided vector-valued maximal operators with respect to a functionhttp://projecteuclid.org/euclid.rae/1435670003<strong>Álvaro Corvalán</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 39, Number 2, 385--402.</p><p><strong>Abstract:</strong><br/>
The purpose of this paper is to find necessary and sufficient conditions on the weight \(w\) for the weak type \(\left( p,p\right) \) with \(1\leq p<+\infty \) and for the strong type \(\left( p,p\right)\) with \(1<p<+\infty \) respect to a measure \(wdx\) of the vector-valued one-sided maximal operator \(\left(M_{g}^{+}\right) _{r}\).
</p>projecteuclid.org/euclid.rae/1435670003_20150630091318Tue, 30 Jun 2015 09:13 EDTRiemann and Riemann-type Integration in Banach Spaceshttp://projecteuclid.org/euclid.rae/1435670004<strong>Sk. Jaker Ali</strong>, <strong>Pratikshan Mondal</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 39, Number 2, 403--440.</p><p><strong>Abstract:</strong><br/>
Riemann, Riemann-Dunford, Riemann-Pettis and Darboux integrable functions with values in a Banach space and Riemann-Gelfand integrable functions with values in the dual of a Banach space are studied in the light of the work of Graves, Alexiewicz and Orlicz, and Gordon. Various properties of these types of integrals and the interrelation between them are established. The Fundamental Theorem of Integral Calculus for these types of integrals is also studied.
</p>projecteuclid.org/euclid.rae/1435670004_20150630091318Tue, 30 Jun 2015 09:13 EDTA Simpler Proof for the \(\epsilon\)-\(\delta\) Characterization of Baire Class One Functionshttp://projecteuclid.org/euclid.rae/1435670005<strong>Jonald P. Fenecios</strong>, <strong>Emmanuel A. Cabral</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 39, Number 2, 441--446.</p><p><strong>Abstract:</strong><br/>
We offer a new and simpler proof of a recent \(\epsilon\)-\(\delta\) characterization of Baire class one functions using a theorem by Henri Lebesgue. The proof is more elementary in the sense that it does not use the Baire Category Theorem. Furthermore, the proof requires only that the domain and range be separable metric spaces instead of Polish spaces.
</p>projecteuclid.org/euclid.rae/1435670005_20150630091318Tue, 30 Jun 2015 09:13 EDTHake's Theorem on Metric Measure Spaceshttp://projecteuclid.org/euclid.rae/1435670006<strong>Surinder Pal Singh</strong>, <strong>Inder K. Rana</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 39, Number 2, 447--458.</p><p><strong>Abstract:</strong><br/>
In this paper, we extend the Hake's theorem over metric measure spaces. We provide its measure theoretic versions in terms of the Henstock variational measure \(V_F\).
</p>projecteuclid.org/euclid.rae/1435670006_20150630091318Tue, 30 Jun 2015 09:13 EDTOn Partitions of the Real Line into Continuum Many Thick Subsetshttp://projecteuclid.org/euclid.rae/1435670007<strong>A. B. Kharazishvili</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 39, Number 2, 459--468.</p><p><strong>Abstract:</strong><br/>
Three classical constructions of Lebesgue nonmeasurable sets on the real line \({\bf R}\) are envisaged from the point of view of the thickness of those sets. It is also shown, within \({\bf ZF}~\&~{\bf DC}\) theory, that the existence of a Lebesgue nonmeasurable subset of \({\bf R}\) implies the existence of a partition of \({\bf R}\) into continuum many thick sets.
</p>projecteuclid.org/euclid.rae/1435670007_20150630091318Tue, 30 Jun 2015 09:13 EDTStrong Derivatives and Integralshttp://projecteuclid.org/euclid.rae/1435670008<strong>Brian S. Thomson</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 39, Number 2, 469--488.</p><p><strong>Abstract:</strong><br/>
The strong derivative is not, without some caution, a useful tool in the study of McShane's (i.e., Lebesgue's) integral. Even so, the underlying structure of that process of derivation is closely connected to the formulation of the Riemann sums definition that McShane gave for his integral. This article discusses some of the features and traps for the study of those connections.
</p>projecteuclid.org/euclid.rae/1435670008_20150630091318Tue, 30 Jun 2015 09:13 EDTThe Denjoy-Young-Saks Theorem in Higher Dimensions: A Surveyhttp://projecteuclid.org/euclid.rae/1435759193<strong>Ákos K. Matszangosz</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 40, Number 1, 1--36.</p><p><strong>Abstract:</strong><br/>
The classical theorem of Denjoy, Young and Saks gives a relation between Dini derivatives of a real variable function that holds almost everywhere. We present what is known in the one and two variable case with an emphasis on the latter. Relations that hold a.e. in both the measure and category sense are considered. Classical and approximate derivatives are both discussed.
</p>projecteuclid.org/euclid.rae/1435759193_20150701100000Wed, 01 Jul 2015 10:00 EDTEquilateral Weights on the Unit Ball of ℝ
nhttp://projecteuclid.org/euclid.rae/1435759194<strong>Emmanuel Chetcuti</strong>, <strong>Joseph Muscat</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 40, Number 1, 37--52.</p><p><strong>Abstract:</strong><br/>
An equilateral set (or regular simplex) in a metric space \(X\) is a sets \(A\) such that the distance between any pair of distinct members of \(A\) is a constant. An equilateral set is standard if the distance between distinct members is equal to \(1\). Motivated by the notion of frame functions, as introduced and characterized by Gleason in [6], we define an equilateral weight on a metric space \(X\) to be a function \(f:X\longrightarrow \mathbb{R}\) such that \(\sum_{i\in I}f(x_i)=W\) for every maximal standard equilateral set \(\{x_i:i\in I\}\) in \(X\), where \(W\in\mathbb{R}\) is the weight of \(f\). In this paper, we characterize the equilateral weights associated with the unit ball \(B^n\) of \(\mathbb{R}^n\) as follows: For \(n\ge 2\), every equilateral weight on \(B^n\) is constant.
</p>projecteuclid.org/euclid.rae/1435759194_20150701100000Wed, 01 Jul 2015 10:00 EDTTangent Measures of Typical Measureshttp://projecteuclid.org/euclid.rae/1435759195<strong>Tuomas Sahlsten</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 40, Number 1, 53--80.</p><p><strong>Abstract:</strong><br/>
We prove that for a typical Radon measure \(\mu\) in \(\mathbb{R}^d\), every non-zero Radon measure is a tangent measure of \(\mu\) at \(\mu\) almost every point. This was already shown by T. O’Neil in his Ph.D. thesis from 1994, but we provide a different self-contained proof for this fact. Moreover, we show that this result is sharp: for any non-zero measure we construct a point in its support where the set of tangent measures does not contain all non-zero measures. We also study a concept similar to tangent measures on trees, micromeasures, and show an analogous typical property for them.
</p>projecteuclid.org/euclid.rae/1435759195_20150701100000Wed, 01 Jul 2015 10:00 EDTThe Equality of Mixed Partial Derivativeshttp://projecteuclid.org/euclid.rae/1435759196<strong>Ettore Minguzzi</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 40, Number 1, 81--98.</p><p><strong>Abstract:</strong><br/>
We review and develop two little known results on the equality of mixed partial derivatives, which can be considered the best results so far available in their respective domains. The former, due to Mikusiński and his school, deals with equality at a given point, while the latter, due to Tolstov, concerns equality almost everywhere. Applications to differential geometry and General Relativity are commented.
</p>projecteuclid.org/euclid.rae/1435759196_20150701100000Wed, 01 Jul 2015 10:00 EDTBowen’s Formula for Shift-Generated Finite Conformal Constructionshttp://projecteuclid.org/euclid.rae/1435759197<strong>Andrei E. Ghenciu</strong>, <strong>Mario Roy</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 40, Number 1, 99--112.</p><p><strong>Abstract:</strong><br/>
We study shift-generated finite conformal constructions; i.e. conformal constructions generated by a general shift (shift of finite type, sofic shift and non-sofic shift alike) over a finite alphabet. These constructions are not restricted to shifts of finite type or sofic shifts as in the classical limit set constructions. In particular, we prove that the limit sets of such constructions satisfy Bowen’s formula, which gives the Hausdorff dimension of the limit set as the zero of the topological pressure. We look at several examples, including a one-dimensional construction generated by the so-called context-free shift.
</p>projecteuclid.org/euclid.rae/1435759197_20150701100000Wed, 01 Jul 2015 10:00 EDTHausdorff and Packing Measures of Balanced Cantor Setshttp://projecteuclid.org/euclid.rae/1435759198<strong>Kathryn Hare</strong>, <strong>Ka-Shing Ng</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 40, Number 1, 113--128.</p><p><strong>Abstract:</strong><br/>
We estimate the \(h\)-Hausdorff and \(h\)-packing measures of balanced Cantor sets, and characterize the corresponding dimension partitions. This generalizes results known for Cantor sets associated with positive decreasing summable sequences and central Cantor sets.
</p>projecteuclid.org/euclid.rae/1435759198_20150701100000Wed, 01 Jul 2015 10:00 EDTOn Borel Hull Operationshttp://projecteuclid.org/euclid.rae/1435759199<strong>Tomasz Filipczak</strong>, <strong>Andrzej Rosłanowski</strong>, <strong>Saharon Shelah</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 40, Number 1, 129--140.</p><p><strong>Abstract:</strong><br/>
We show that some set-theoretic assumptions (for example Martin’s Axiom) imply that there is no translation invariant Borel hull operation on the family of Lebesgue null sets and on the family of meager sets (in \(\mathbb{R}^{n}\)). We also prove that if the meager ideal admits a monotone Borel hull operation, then there is also a monotone Borel hull operation on the \(\sigma\)-algebra of sets with the property of Baire.
</p>projecteuclid.org/euclid.rae/1435759199_20150701100000Wed, 01 Jul 2015 10:00 EDTGeneralized Kiesswetter’s Functionshttp://projecteuclid.org/euclid.rae/1435759200<strong>Delong Li</strong>, <strong>Jie Miao</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 40, Number 1, 141--156.</p><p><strong>Abstract:</strong><br/>
In 1966, Kiesswetter found an interesting example of continuous everywhere but differentiable nowhere functions using base-4 expansion of real numbers. In this paper we show how Kiesswetter’s function can be extended to general cases. We also provide an equivalent form for such functions via a recurrence relation.
</p>projecteuclid.org/euclid.rae/1435759200_20150701100000Wed, 01 Jul 2015 10:00 EDTAn Integral on a Complete Metric Measure Spacehttp://projecteuclid.org/euclid.rae/1435759201<strong>Donatella Bongiorno</strong>, <strong>Giuseppa Corrao</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 40, Number 1, 157--178.</p><p><strong>Abstract:</strong><br/>
We study a Henstock-Kurzweil type integral defined on a complete metric measure space \(X\) endowed with a Radon measure \(\mu\) and with a family of “cells” \(\mathcal{F}\) that satisfies the Vitali covering theorem with respect to \(\mu\). This integral encloses, in particular, the classical Henstock-Kurzweil integral on the real line, the dyadic Henstock-Kurzweil integral, the Mawhin’s integral [19], and the \(s\)-HK integral [4]. The main result of this paper is the extension of the usual descriptive characterizations of the Henstock-Kurzweil integral on the real line, in terms of \(ACG^*\) functions (Main Theorem 1) and in terms of
variational measures (Main Theorem 2).
</p>projecteuclid.org/euclid.rae/1435759201_20150701100000Wed, 01 Jul 2015 10:00 EDTA Sufficient Condition for a Bounded Set of Positive Lebesgue Measure in ℝ
2 or ℝ
3 to Contain its Centroidhttp://projecteuclid.org/euclid.rae/1435759202<strong>Eric A. Hintikka</strong>, <strong>Steven G. Krantz</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 40, Number 1, 179--192.</p><p><strong>Abstract:</strong><br/>
In this paper, we give a sufficient condition for a domain in either two- or three-dimensional Euclidean space to contain its centroid. We show that the condition is sharp. The condition is not, however, necessary.
</p>projecteuclid.org/euclid.rae/1435759202_20150701100000Wed, 01 Jul 2015 10:00 EDTExtreme Results on Certain Generalized Riemann Derivativeshttp://projecteuclid.org/euclid.rae/1435759203<strong>John C. Georgiou</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 40, Number 1, 193--208.</p><p><strong>Abstract:</strong><br/>
In this paper the following question is investigated. Given a natural number \(r\) and numbers \(\alpha_j,\beta_j\) for \(j=0,1,\dots,r\) satisfying \( \alpha_0 <\alpha_1 < \dots <lt \alpha_r \) and \begin{equation*} \sum_{j=0}^{r} \beta_j \alpha_j^k= \begin{cases} 0 & \text{if \(k=0,1,\dots,r-1\)}\\ r!& \text{if \(k=r\) } \end{cases} \end{equation*} is there a \( 2\pi\)-periodic, \( r-1\) times continuously differentiable function \( f\) such that \begin{equation*} \limsup_{h \nearrow 0} h^{-r} \Big(\sum_{j=0}^{r} \beta_j f(x+ \alpha_j h)\Big) = \limsup_{h \searrow 0} h^{-r} \Big(\sum_{j=0}^{r} \beta_j f(x+ \alpha_j h)\Big) = \infty, \end{equation*} \begin{equation*} \liminf_{h \nearrow 0} h^{-r} \Big(\sum_{j=0}^{r} \beta_j f(x+ \alpha_j h)\Big) = \liminf_{h \searrow 0} h^{-r} \Big(\sum_{j=0}^{r} \beta_j f(x+ \alpha_j h)\Big) = - \infty \end{equation*} for every \( x \in \mathbb{R} \)?
</p>projecteuclid.org/euclid.rae/1435759203_20150701100000Wed, 01 Jul 2015 10:00 EDTAbsolute Continuity in Partial Differential Equationshttp://projecteuclid.org/euclid.rae/1435759204<strong>Amin Farjudian</strong>, <strong>Behrouz Emamizadeh</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 40, Number 1, 209--218.</p><p><strong>Abstract:</strong><br/>
In this note, we study a function which frequently appears in partial differential equations. We prove that this function is absolutely continuous; hence it can be written as a definite integral. As a result, we obtain some estimates regarding solutions of the Hamilton-Jacobi systems.
</p>projecteuclid.org/euclid.rae/1435759204_20150701100000Wed, 01 Jul 2015 10:00 EDTBasic Introduction To Exponential and Logarithmic Functionshttp://projecteuclid.org/euclid.rae/1435759205<strong>Adel B. Badi</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 40, Number 1, 219--226.</p><p><strong>Abstract:</strong><br/>
This article discusses the definitions and properties of exponential and logarithmic functions. The treatment is based on the basic properties of real numbers, sequences and continuous functions. This treatment avoids the use of definite integrals.
</p>projecteuclid.org/euclid.rae/1435759205_20150701100000Wed, 01 Jul 2015 10:00 EDTAddendum to: Some new Types of Filter Limit Theorems for Topological group-valued Measureshttp://projecteuclid.org/euclid.rae/1435759206<strong>Antonio Boccuto</strong>, <strong>Xenofon Dimitriou</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 40, Number 1, 227--232.</p><p><strong>Abstract:</strong><br/>
The purpose of this note is to point out some corrections to the paper: A. Boccuto and X. Dimitriou, “Some new types of filter limit theorems for topological group-valued measures,” \textit{Real Anal. Exchange} \textbf{39} (1) (2014), 139-174.
</p>projecteuclid.org/euclid.rae/1435759206_20150701100000Wed, 01 Jul 2015 10:00 EDTOn Interval Based Generalizations of Absolute Continuity for Functions on \(\mathbb{R}^{n}\)http://projecteuclid.org/euclid.rae/1490580012<strong>Michael Dymond</strong>, <strong>Beata Randrianantoanina</strong>, <strong>Huaqiang Xu</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 42, Number 1, 49--78.</p><p><strong>Abstract:</strong><br/> We study notions of absolute continuity for functions defined on $\mathbb{R}^n$ similar to the notion of $\alpha$-absolute continuity in the sense of Bongiorno. We confirm a conjecture of Malý that 1-absolutely continuous functions do not need to be differentiable a.e., and we show several other pathological examples of functions in this class. We establish some containment relations of the class $1\po AC_{\rm WDN}$ which consits of all functions in $1\po AC$ which are in the Sobolev space $W^{1,2}_{loc}$, are differentiable a.e. and satisfy the Luzin (N) property, with previously studied classes of absolutely continuous functions.
</p>projecteuclid.org/euclid.rae/1490580012_20170326220033Sun, 26 Mar 2017 22:00 EDTA Class of Random Cantor Setshttp://projecteuclid.org/euclid.rae/1490580013<strong>Changhao Chen</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 42, Number 1, 79--120.</p><p><strong>Abstract:</strong><br/> In this paper we study a class of random Cantor sets. We determine their almost sure Hausdorff, packing, box, and Assouad dimensions. From a topological point of view, we also compute their typical dimensions in the sense of Baire category. For the natural random measures on these random Cantor sets, we consider their almost sure lower and upper local dimensions. In the end we study the hitting probabilities of a special subclass of these random Cantor sets.
</p>projecteuclid.org/euclid.rae/1490580013_20170326220033Sun, 26 Mar 2017 22:00 EDTVariance Jensen Type Inequalities for General Lebesgue Integral with Applicationshttp://projecteuclid.org/euclid.rae/1490580014<strong>Silvestru S. Dragomir</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 42, Number 1, 121--148.</p><p><strong>Abstract:</strong><br/> Some inequalities similar to Jensen inequalities for general Lebesgue integral are obtained. Applications for functions of selfadjoint operators and functions of unitary operators on complex Hilbert spaces are provided as well.
</p>projecteuclid.org/euclid.rae/1490580014_20170326220033Sun, 26 Mar 2017 22:00 EDTQuantization for Uniform Distributions on Equilateral Triangleshttp://projecteuclid.org/euclid.rae/1490580015<strong>Carl P. Dettmann</strong>, <strong>Mrinal Kanti Roychowdhury</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 42, Number 1, 149--166.</p><p><strong>Abstract:</strong><br/> We approximate the uniform measure on an equilateral triangle by a measure supported on $n$ points. We find the optimal sets of points ($n$-means) and corresponding approximation (quantization) error for $n\leq4$, give numerical optimization results for $n\leq 21$, and a bound on the quantization error for $n\to\infty$. The equilateral triangle has particularly efficient quantizations due to its connection with the triangular lattice. Our methods can be applied to the uniform distributions on general sets with piecewise smooth boundaries.
</p>projecteuclid.org/euclid.rae/1490580015_20170326220033Sun, 26 Mar 2017 22:00 EDTThe $\ell_1$-Dichotomy Theorem with Respect to a Coidealhttp://projecteuclid.org/euclid.rae/1490580016<strong>Vassiliki Farmaki</strong>, <strong>Andreas Mitropoulos</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 42, Number 1, 167--184.</p><p><strong>Abstract:</strong><br/> In this paper we introduce, for any coideal basis $\B$ on the set $\nat$ of natural numbers, the notions of a $\B$-sequence, a $\B$-subsequence of a $\B$-sequence, and a $\B$-convergent sequence in a metric space. The usual notions of a sequence, subsequence, and convergent sequence obtain for the coideal $\B$ of all the infinite subsets of $\nat$. We first prove a Bolzano-Weierstrass theorem for $\B$-sequences: if $\B$ is a Ramsey coideal basis on $\nat$, then every bounded $\B$-sequence of real numbers has a $\B$-convergent $\B$-subsequence; and next, with the help of this extended Bolzano-Weierstrass theorem, we establish an extension of the fundamental Rosenthal's $\ell_1$-dichotomy theorem: if $\B $ is a semiselective coideal basis on $\nat$, then every bounded $\B$-sequence of real valued functions $(f_n)_{n\in A}$ has a $\B$-subsequence $(f_n)_{n\in B}$, which is either $\B$-convergent or equivalent to the unit vector basis of $\ell_1(B)$.
</p>projecteuclid.org/euclid.rae/1490580016_20170326220033Sun, 26 Mar 2017 22:00 EDTDirectional Differentiability in the Euclidean Planehttp://projecteuclid.org/euclid.rae/1490580017<strong>J. Marshall Ash</strong>, <strong>Stefan Catoiu</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 42, Number 1, 185--192.</p><p><strong>Abstract:</strong><br/> Smoothness conditions on a function $f\:mathbb{R}^{2}\rightarrow \mathbb{R}$ that are weaker than being differentiable or Lipschitz at a point are defined and studied.
</p>projecteuclid.org/euclid.rae/1490580017_20170326220033Sun, 26 Mar 2017 22:00 EDTBanach Spaces for the Schwartz Distributionshttps://projecteuclid.org/euclid.rae/1525226419<strong>Tepper L. Gill</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 43, Number 1, 1--36.</p><p><strong>Abstract:</strong><br/>
This paper is a survey of a new family of Banach spaces \(\mcB\) that provide the same structure for the Henstock-Kurzweil (HK) integrable functions as the \(L^p\) spaces provide for the Lebesgue integrable functions. These spaces also contain the wide sense Denjoy integrable functions. They were first use to provide the foundations for the Feynman formulation of quantum mechanics. It has recently been observed that these spaces contain the test functions \(\mcD\) as a continuous dense embedding. Thus, by the Hahn-Banach theorem, \(\mcD' \subset \mcB'\). A new family that extend the space of functions of bounded mean oscillation \(BMO[\mathbb{R}^n]\), to include the HK-integrable functions are also introduced.
</p>projecteuclid.org/euclid.rae/1525226419_20180501220024Tue, 01 May 2018 22:00 EDTLinear Subspaces of Hypercyclic Vectorshttps://projecteuclid.org/euclid.rae/1525226420<strong>Juan Bês</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 43, Number 1, 37--42.</p><p><strong>Abstract:</strong><br/>
\noindent In my talk I presented results from previous papers on the existence of hypercyclic algebras for convolution operators acting on the space of entire functions.
</p>projecteuclid.org/euclid.rae/1525226420_20180501220024Tue, 01 May 2018 22:00 EDTSome Results about Big and Little Liphttps://projecteuclid.org/euclid.rae/1525226421<strong>Bruce Hanson</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 43, Number 1, 43--50.</p><p><strong>Abstract:</strong><br/>
Let \(f\:ℝ \to ℝ\) be continuous. We examine the relationship between the so-called “big Lip” and “little lip” functions: Lip \(f\) and lip \(f\).
</p>projecteuclid.org/euclid.rae/1525226421_20180501220024Tue, 01 May 2018 22:00 EDTMeasuring Anisotropy in Planar Setshttps://projecteuclid.org/euclid.rae/1525226422<strong>Toby C. O’Neil</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 43, Number 1, 51--56.</p><p><strong>Abstract:</strong><br/>
We define and discuss a pure mathematics formulation of an approach proposed in the physics literature to analysing anistropy of fractal sets.
</p>projecteuclid.org/euclid.rae/1525226422_20180501220024Tue, 01 May 2018 22:00 EDTContinued Logarithm Representation of Real Numbershttps://projecteuclid.org/euclid.rae/1525226423<strong>Jörg Neunhäuserer</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 43, Number 1, 57--66.</p><p><strong>Abstract:</strong><br/>
We introduce the continued logarithm representation of real numbers and prove results on the occurrence and frequency of digits with respect to this representation.
</p>projecteuclid.org/euclid.rae/1525226423_20180501220024Tue, 01 May 2018 22:00 EDTAn Elementary Proof of an Isoperimetric Inequality for Paths with Finite p-Variationhttps://projecteuclid.org/euclid.rae/1525226424<strong>George Galvin</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 43, Number 1, 67--76.</p><p><strong>Abstract:</strong><br/>
In this article we will prove that if the continuous closed curve \(\gamma : [0, 1] \rightarrow \mathbb{R}^2\) has finite \(p\)-variation with \(p < 2\), then \begin{equation*} (\iint\limits_{\R^2}|\eta(\gamma, (x, y))|^q \,dx \,dy)^{1/q} \le (\frac{1}{2})^\frac{1}{q}(\zeta(\frac{2}{pq})-1)(||\gamma||_{p, [0, 1]})^{\frac{2}{q}} \end{equation*} for all \(q \in [1, \frac{2}{p})\), where \(\eta(\gamma, (x, y))\) is the winding number of \(\gamma\) at \((x, y), \zeta\) is the Reimann zeta function, and \(||\gamma||_{p, [0, 1]}\) is the \(p\)-variation of \(\gamma\) on the interval \([0, 1]\). Our main contribution is that we have explicitly given a bound by known constants, and we have found this by an elementary proof. We are going to be using a method introduced by L.C. Young \cite{young} in 1936.
</p>projecteuclid.org/euclid.rae/1525226424_20180501220024Tue, 01 May 2018 22:00 EDTQuasicontinuous functions with values in Piotrowski spaceshttps://projecteuclid.org/euclid.rae/1525226425<strong>Taras Banakh</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 43, Number 1, 77--104.</p><p><strong>Abstract:</strong><br/>
A topological space \(X\) is called {\em Piotrowski} if every quasicontinuous map \(f:Z\to X\) from a Baire space \(Z\) to \(X\) has a continuity point. In this paper we survey known results on Piotrowski spaces and investigate the relation of Piotrowski spaces to strictly fragmentable, Stegall, and game determined spaces. Also we prove that a Piotrowski Tychonoff space \(X\) contains a dense (completely) metrizable Baire subspace if and only if \(X\) is Baire (Choquet).
</p>projecteuclid.org/euclid.rae/1525226425_20180501220024Tue, 01 May 2018 22:00 EDTOptimal Quantizers for some Absolutely Continuous Probability Measureshttps://projecteuclid.org/euclid.rae/1525226426<strong>Mrinal Kanti Roychowdhury</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 43, Number 1, 105--136.</p><p><strong>Abstract:</strong><br/>
The representation of a given quantity with less information is often referred to as ‘quantization’ and it is an important subject in information theory. In this paper, we have considered absolutely continuous probability measures on unit discs, squares, and the real line. For these probability measures the optimal sets of \(n\)-means and the \(n\)th quantization errors are calculated for some positive integers \(n\).
</p>projecteuclid.org/euclid.rae/1525226426_20180501220024Tue, 01 May 2018 22:00 EDTErgodic Properties of Rational Functions that Preserve Lebesgue Measure on ℝhttps://projecteuclid.org/euclid.rae/1525226427<strong>Rachel L. Bayless</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 43, Number 1, 137--154.</p><p><strong>Abstract:</strong><br/>
We prove that all negative generalized Boole transformations are conservative, exact, pointwise dual ergodic, and quasi-finite with respect to Lebesgue measure on the real line. We then provide a formula for computing the Krengel, Parry, and Poisson entropy of all conservative rational functions that preserve Lebesgue measure on the real line.
</p>projecteuclid.org/euclid.rae/1525226427_20180501220024Tue, 01 May 2018 22:00 EDTDivided Differences, Square Functions, and a Law of the Iterated Logarithmhttps://projecteuclid.org/euclid.rae/1525226428<strong>Artur Nicolau</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 43, Number 1, 155--186.</p><p><strong>Abstract:</strong><br/>
The main purpose of the paper is to show that differentiability properties of a measurable function defined in the euclidean space can be described using square functions which involve its second symmetric divided differences. Classical results of Marcinkiewicz, Stein and Zygmund describe, up to sets of Lebesgue measure zero, the set of points where a function~\)f\) is differentiable in terms of a certain square function~\)g(f)\). It is natural to ask for the behavior of the divided differences at the complement of this set, that is, on the set of points where \(f\) is not differentiable. In the nineties, Anderson and Pitt proved that the growth of the divided differences of a function in the Zygmund class obeys a version of the classical Kolmogorov's Law of the Iterated Logarithm (LIL). A square function, which is the conical analogue of \(g(f)\) will be used to state and prove a general version of the LIL of Anderson and Pitt as well as to prove analogues of the classical results of Marcinkiewicz, Stein and Zygmund. Sobolev spaces can also be described using this new square function.
</p>projecteuclid.org/euclid.rae/1525226428_20180501220024Tue, 01 May 2018 22:00 EDTMagic Setshttps://projecteuclid.org/euclid.rae/1525226429<strong>Lorenz Halbeisen</strong>, <strong>Marc Lischka</strong>, <strong>Salome Schumacher</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 43, Number 1, 187--204.</p><p><strong>Abstract:</strong><br/>
\noindent In this paper we study magic sets for certain families \(\mathcal{H}\subseteq {^\mathbb{R}\mathbb{R}}\) which are subsets \(M\subseteq\mathbb{R}\) such that for all functions \(f,g\in\mathcal{H}\) we have that \(g[M]\subseteq f[M]\Rightarrow f=g\). Specifically we are interested in magic sets for the family \(\mathcal{G}\) of all continuous functions that are not constant on any open subset of \(\mathbb{R}\). We will show that these magic sets are stable in the following sense: Adding and removing a countable set does not destroy the property of being a magic set. Moreover, if the union of less than \(\mathfrak{c}\) meager sets is still meager, we can also add and remove sets of cardinality less than \(\mathfrak{c}\) without destroying the magic set. \newline \noindent Then we will enlarge the family \(\mathcal{G}\) to a family \(\mathcal{F}\) by replacing the continuity with symmetry and assuming that the functions are locally bounded. A function \(f\:\mathbb{R}\to\mathbb{R}\) is symmetric iff for every \(x\in\mathbb{R}\) we have that \(\lim_{h\downarrow 0}\frac{1}{2}\left(f(x+h)+f(x-h)\right )=f(x)\). For this family of functions we will construct \(2^\mathfrak{c}\) pairwise different magic sets which cannot be destroyed by adding and removing a set of cardinality less than \(\mathfrak{c}\). We will see that under the continuum hypothesis magic sets and these more stable magic sets for the family \(\mathcal{F}\) are the same. We shall also see that the assumption of local boundedness cannot be omitted. Finally, we will prove that for the existence of a magic set for the family \(\mathcal{F}\) it is sufficient to assume that the union of less than \(\mathfrak{c}\) meager sets is still meager. So for example Martin's axiom for \(\sigma\)-centered partial orders implies the existence of a magic set.
</p>projecteuclid.org/euclid.rae/1525226429_20180501220024Tue, 01 May 2018 22:00 EDTRandom Cutouts of the Unit Cube with I.U.D Centershttps://projecteuclid.org/euclid.rae/1525226430<strong>Z. Y. Zhu</strong>, <strong>E. M. Dong</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 43, Number 1, 205--220.</p><p><strong>Abstract:</strong><br/>
Consider the random open balls \(B_n(\omega):=B(\omega_n,r_n)\) with their centers \(\omega_n\) being i.u.d. on the d-dimensional unit cube \([0,1]^d\) and with their radii \(r_n\sim cn^{-\frac{1}{d}}\) for some constant \(0<c<(\beta(d))^{-\frac{1}{d}}\), where \(\beta(d)\) is the volume of the \(d\) dimensional unit ball. We call \([0,1]^d-\bigcup_{n=1}^{\infty} B_n(\omega)\) a random cutout set. In this paper, we present an exposition of Z\)\ddot{a}\)hle cutout model in \cite{Zahle} by a detailed study of such a random cutout set for the purpose of teaching and learning. We show that with probability one Hausdorff dimension of such random cut-out set is at most \(d(1-\beta(d)c^d)\) and frequently equals \(d(1-\beta(d)c^d)\).
</p>projecteuclid.org/euclid.rae/1525226430_20180501220024Tue, 01 May 2018 22:00 EDTOn the Minkowski Sum of Two Curveshttps://projecteuclid.org/euclid.rae/1525226431<strong>Alan Chang</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 43, Number 1, 221--222.</p><p><strong>Abstract:</strong><br/>
We answer a question posed by Miklós Laczkovich on the Minkowski sum of two curves.
</p>projecteuclid.org/euclid.rae/1525226431_20180501220024Tue, 01 May 2018 22:00 EDTA Note on the Uniqueness Property for Borel G -measureshttps://projecteuclid.org/euclid.rae/1525226432<strong>Alexander Kharazishvili</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 43, Number 1, 223--234.</p><p><strong>Abstract:</strong><br/>
In terms of a group \(G\) of isometries of Euclidean space, it is given a necessary and sufficient condition for the uniqueness of a \(G\)-measure on the Borel \(\sigma\)-algebra of this space.
</p>projecteuclid.org/euclid.rae/1525226432_20180501220024Tue, 01 May 2018 22:00 EDTOn the Minkowski Sum of Two Curveshttps://projecteuclid.org/euclid.rae/1525226433<strong>Andrew M. Bruckner</strong>, <strong>Krzysztof Chris Ciesielski</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 43, Number 1, 235--238.</p><p><strong>Abstract:</strong><br/>
We show that there exists a derivative \(f\colon [0,1]\to[0,1]\) such that the graph of \(f\circ f\) is dense in \([0,1]^2\), so not a \(G_\delta\)-set. In particular, \(f\circ f\) is everywhere discontinuous, so not of Baire class 1, and hence it is not a derivative. %neither of Baire class 1 nor a derivative.
</p>projecteuclid.org/euclid.rae/1525226433_20180501220024Tue, 01 May 2018 22:00 EDTA note on the Luzin-Menchoff theoremhttps://projecteuclid.org/euclid.rae/1525226434<strong>Hajrudin Fejzić</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 43, Number 1, 239--242.</p><p><strong>Abstract:</strong><br/>
A proof of the Luzim-Menchoff theorem.
</p>projecteuclid.org/euclid.rae/1525226434_20180501220024Tue, 01 May 2018 22:00 EDTWhich Integrable Functions Fail to be Absolutely Integrable?https://projecteuclid.org/euclid.rae/1525226435<strong>José Mendoza</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 43, Number 1, 243--248.</p><p><strong>Abstract:</strong><br/>
An answer to the question of the title is given.
</p>projecteuclid.org/euclid.rae/1525226435_20180501220024Tue, 01 May 2018 22:00 EDTMycielski-Regularity of Gibbs Measures on Cookie-Cutter Setshttps://projecteuclid.org/euclid.rae/1530064959<strong>Jeremiah J. Bass</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 43, Number 2, 249--262.</p><p><strong>Abstract:</strong><br/>
It has been shown that all Radon probability measures on \(\mathbbm{R}\) are Mycielski-regular, as well as Lebesgue measure on the unit cube and certain self-similar measures. In this paper, these results are extended to Gibbs measures on cookie-cutter sets.
</p>projecteuclid.org/euclid.rae/1530064959_20180626220252Tue, 26 Jun 2018 22:02 EDTThe Choquet Integral in Capacityhttps://projecteuclid.org/euclid.rae/1530064960<strong>Sorin G. Gal</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 43, Number 2, 263--280.</p><p><strong>Abstract:</strong><br/>
In this paper we introduce and study the new concept of Choquet integral in capacity, which generalizes the Riemann integral in probability and the classical Choquet integral. Properties of this new integral are proved and some applications are presented.
</p>projecteuclid.org/euclid.rae/1530064960_20180626220252Tue, 26 Jun 2018 22:02 EDTMinimal Degrees of Genocchi-Peano Functions: Calculus Motivated Number Theoretical Estimateshttps://projecteuclid.org/euclid.rae/1530064961<strong>Krzysztof Chris Ciesielski</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 43, Number 2, 281--292.</p><p><strong>Abstract:</strong><br/>
A rational function of the form \(\frac{x_1^{\alpha_1} x_2^{\alpha_2} \cdots x_n^{\alpha_n}}{x_1^{\beta_1} + x_2^{\beta_2}+\cdots + x_n^{\beta_n}}\) is a >Genocchi-Peano example, GPE , provided it is discontinuous, but its restriction to any hyperplane is continuous. We show that the minimal degree \(D(n)\) of a GPE of \(n\)-variables equals \(2\left\lfloor \frac{e^2}{e^2-1} n \right\rfloor+2i\) for some \(i\in\{0,1,2\}\). We also investigate the minimal degree \(D_b(n)\) of a bounded GPE of \(n\)-variables and note that \(D(n)\leq D_b(n)\leq n(n+1)\). Finding better bounds for the numbers \(D_b(n)\) remains an open problem.
</p>projecteuclid.org/euclid.rae/1530064961_20180626220252Tue, 26 Jun 2018 22:02 EDTLipschitz Restrictions of Continuous Functions and a Simple Construction of Ulam-Zahorski \(C^1\) Interpolationhttps://projecteuclid.org/euclid.rae/1530064962<strong>Krzysztof Chris Ciesielski</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 43, Number 2, 293--300.</p><p><strong>Abstract:</strong><br/>
We present a simple argument showing that for every continuous function \(f\colon\mathbb{R}\to\mathbb{R}\), its restriction to some perfect set is Lipschitz. We will use this result to provide an elementary proof of the \(C^1\) free interpolation theorem, that for every continuous function \(f\colon\mathbb{R}\to\mathbb{R}\) there exists a continuously differentiable function \(g\colon\mathbb{R}\to\mathbb{R}\) which agrees with \(f\) on an uncountable set. The key novelty of our presentation is that no part of it, including the cited results, requires from the reader any prior familiarity with Lebesgue measure theory.
</p>projecteuclid.org/euclid.rae/1530064962_20180626220252Tue, 26 Jun 2018 22:02 EDTEqui-Riemann and Equi-Riemann Type Integrable Functions with Values in a Banach Spacehttps://projecteuclid.org/euclid.rae/1530064963<strong>Pratikshan Mondal</strong>, <strong>Lakshmi Kanta Dey</strong>, <strong>Sk. Jaker Ali</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 43, Number 2, 301--324.</p><p><strong>Abstract:</strong><br/>
In this paper we study equi-Riemann and equi-Riemann-type integrability of a collection of functions defined on a closed interval of \(\mathbb{R}\) with values in a Banach space. We obtain some properties of such collections and interrelations among them. Moreover we establish equi-integrability of different types of collections of functions. Finally, we obtain relations among equi-Riemann integrability with other properties of a collection of functions.
</p>projecteuclid.org/euclid.rae/1530064963_20180626220252Tue, 26 Jun 2018 22:02 EDTThe Baire Classification of Strongly Separately Continuous Functions on \(\ell_\infty\)https://projecteuclid.org/euclid.rae/1530064964<strong>Olena Karlova</strong>, <strong>Tomá\v{s} Visnyai</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 43, Number 2, 325--332.</p><p><strong>Abstract:</strong><br/>
We prove that for any \(\alpha\in[0,\omega_1)\) there exists a strongly separately continuous function \(f:\ell_\infty\rightarrow [0,1]\) such that \(f\) belongs to the \((\alpha+1)\)'th \) /(\alpha+2)\)'th/ Baire class and does not belong to the \(\alpha\)'th Baire class if \(\alpha\) is finite /infinite/.
</p>projecteuclid.org/euclid.rae/1530064964_20180626220252Tue, 26 Jun 2018 22:02 EDTOn the Growth of Real Functions and their Derivativeshttps://projecteuclid.org/euclid.rae/1530064965<strong>J\"urgen Grahl</strong>, <strong>Shahar Nevo</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 43, Number 2, 333--346.</p><p><strong>Abstract:</strong><br/>
We show that for any \(k\)-times differentiable function \(f:[a,\infty)\to\mathbb{R}\), any integer \(q\ge 0\) and any \(\alpha>1\) the inequality \[ \liminf_{x\to\infty} \frac{x^k \cdot\log x\cdot \log_2 x\cdot\ldots\cdot \log_q x \cdot |f^{(k)}(x)|}{1+|f(x)|^\alpha}= 0 \] holds and that this result is best possible in the sense that \(\log_q x\) cannot be replaced by \((\log_q x)^\beta\) with any \(\beta>1\).
</p>projecteuclid.org/euclid.rae/1530064965_20180626220252Tue, 26 Jun 2018 22:02 EDTRestricted Families of Projections and Random Subspaceshttps://projecteuclid.org/euclid.rae/1530064966<strong>Changhao Chen</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 43, Number 2, 347--358.</p><p><strong>Abstract:</strong><br/>
We study the restricted families of orthogonal projections in \(\mathbb{R}^{3}\). We show that there are families of random subspaces which admit a Marstrand-Mattila type projection theorem.
</p>projecteuclid.org/euclid.rae/1530064966_20180626220252Tue, 26 Jun 2018 22:02 EDTSimultaneous Small Coverings by Smooth Functions Under the Covering Property Axiomhttps://projecteuclid.org/euclid.rae/1530064967<strong>Krzysztof C. Ciesielski</strong>, <strong>Juan B. Seoane--Sepúlveda</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 43, Number 2, 359--386.</p><p><strong>Abstract:</strong><br/>
The covering property axiom CPA is consistent with ZFC: it is satisfied in the iterated perfect set model. We show that CPA implies that for every \(\nu\in\omega\cup\{\infty\}\) there exists a family \(\mathcal{F}_\nu\subset C^\nu(\mathbb{R})\) of cardinality \(\omega_1<\mathfrak{c}\) such that for every \(g\in D^\nu(\mathbb{R})\) the set \(g\setminus \bigcup \mathcal{F}_\nu\) has cardinality \(\leq\omega_1\). Moreover, we show that this result remains true for partial functions \(g\) (i.e., \(g\in D^\nu(X)\) for some \(X\subset\mathbb{R}\)) if, and only if, \(\nu \in\{0,1\}\). The proof of this result is based on the following theorem of independent interest (which, for \(\nu\neq 0\), seems to have been previously unnoticed): for every \(X\subset\mathbb{R}\) with no isolated points, every \(\nu\)-times differentiable function \(g\colon X\to\mathbb{R}\) admits a \(\nu\)-times differentiable extension \(\bar g\colon B\to\mathbb{R}\), where \(B \supset X\) is a Borel subset of \(\mathbb{R}\). The presented arguments rely heavily on a Whitney’s Extension Theorem for the functions defined on perfect subsets of \(\mathbb{R}\), which short but fully detailed proof is included. Some open questions are also posed.
</p>projecteuclid.org/euclid.rae/1530064967_20180626220252Tue, 26 Jun 2018 22:02 EDTA Note on Level Sets of Differentiable Functions f(x,y) with Non-Vanishing Gradienthttps://projecteuclid.org/euclid.rae/1530064968<strong>Anna K. Savvopoulou</strong>, <strong>Christopher M. Wedrychowcz</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 43, Number 2, 387--392.</p><p><strong>Abstract:</strong><br/>
The purpose of this note is to give an alternate proof of a result of M. Elekes. We show that if \( f\:\mathbb{R}^2\rightarrow \mathbb{R}\) is a differentiable function with everywhere non-zero gradient, then for every point \(x\in \mathbb{R}^2\) in the level set \(\{x\: \:: f(x)=c\}\) there is a neighborhood \(V\) of \(x\) such that \(\{f=c\}\cap V\) is homeomorphic to an open interval or the union of finitely many open segments passing through a point.
</p>projecteuclid.org/euclid.rae/1530064968_20180626220252Tue, 26 Jun 2018 22:02 EDTS-Limited Shiftshttps://projecteuclid.org/euclid.rae/1530064969<strong>Benjamin Matson</strong>, <strong>Elizabeth Sattler</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 43, Number 2, 393--416.</p><p><strong>Abstract:</strong><br/>
In this paper, we explore the construction and dynamical properties of \(\mathcal{S}\)-limited shifts. An \(S\)-limited shift is a subshift defined on a finite alphabet \(\mathcal{A} = \{1, \ldots,p\}\) by a set \(\mathcal{S} = \{S_1, \ldots, S_p\}\), where \(S_i \subseteq \mathbb{N}\) describes the allowable lengths of blocks in which the corresponding letter may appear. We give conditions for which an \(\mathcal{S}\)-limited shift is a subshift of finite type or sofic. We give an exact formula for finding the entropy of such a shift and show that an \(\mathcal{S}\)-limited shift and its factors must be intrinsically ergodic. Finally, we give some conditions for which two such shifts can be conjugate, and additional information about conjugate \(\mathcal{S}\)-limited shifts.
</p>projecteuclid.org/euclid.rae/1530064969_20180626220252Tue, 26 Jun 2018 22:02 EDTSome Applications of Order-Embeddings of Countable Ordinals into the Real Linehttps://projecteuclid.org/euclid.rae/1530064970<strong>Leonard Huang</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 43, Number 2, 417--428.</p><p><strong>Abstract:</strong><br/>
It is a well-known fact that an ordinal \( \alpha \) can be embedded into the real line \( \mathbb{R} \) in an order-preserving manner if and only if \( \alpha \) is countable. However, it would seem that outside of set theory, this fact has not yet found any concrete applications. The goal of this paper is to present some applications. More precisely, we show how two classical results, one in point-set topology and the other in real analysis, can be proven by defining specific order-embeddings of countable ordinals into \( \mathbb{R} \).
</p>projecteuclid.org/euclid.rae/1530064970_20180626220252Tue, 26 Jun 2018 22:02 EDTThe Implicit Function Theorem for Maps that are Only Differentiable: An Elementary Proofhttps://projecteuclid.org/euclid.rae/1530064971<strong>Oswaldo de Oliveira</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 43, Number 2, 429--444.</p><p><strong>Abstract:</strong><br/>
This article shows a very elementary and straightforward proof of the Implicit Function Theorem for differentiable maps \(F(x,y)\) defined on a finite-dimensional Euclidean space. There are no hypotheses on the continuity of the partial derivatives of \(F\). The proof employs the mean-value theorem, the intermediate-value theorem, Darboux’s property (the intermediate-value property for derivatives), and determinants theory. The proof avoids compactness arguments, fixed-point theorems, and Lebesgue’s measure. A stronger than the classical version of the Inverse Function Theorem is also shown. Two illustrative examples are given.
</p>projecteuclid.org/euclid.rae/1530064971_20180626220252Tue, 26 Jun 2018 22:02 EDTUniqueness Properties of Harmonic Functionshttps://projecteuclid.org/euclid.rae/1530064972<strong>Steven G. Krantz</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 43, Number 2, 445--450.</p><p><strong>Abstract:</strong><br/>
We study the zero set of a harmonic function of several real variables. Using the theory of real analytic functions, we analyze such sets. We generalize these results to solutions of elliptic partial differential equations with constant coefficients.
</p>projecteuclid.org/euclid.rae/1530064972_20180626220252Tue, 26 Jun 2018 22:02 EDTAn Earlier Fractal Graphhttps://projecteuclid.org/euclid.rae/1530064973<strong>Harvey Rosen</strong>. <p><strong>Source: </strong>Real Analysis Exchange, Volume 43, Number 2, 451--454.</p><p><strong>Abstract:</strong><br/>
A function \(f:\mathbb{R}\to \mathbb{R}\) is additive if \( f(x+y)=f(x)+f(y)\) for all real numbers \(x\) and \(y\). We give examples of an additive function whose graph is fractal.
</p>projecteuclid.org/euclid.rae/1530064973_20180626220252Tue, 26 Jun 2018 22:02 EDT