Recent zbMATH articles in MSC 28A80https://www.zbmath.org/atom/cc/28A802021-06-15T18:09:00+00:00WerkzeugMass transference principle: from balls to arbitrary shapes: measure theory.https://www.zbmath.org/1460.280092021-06-15T18:09:00+00:00"Zhong, Wenmin"https://www.zbmath.org/authors/?q=ai:zhong.wenminThe authors -- by providing a further generalization of the singular value function -- extend the Mass Transference Principle as set up by Beresnevich \& Velani to so-called \(\limsup\) sets generated by open sets of arbitrary shapes.
Reviewer: Peter Massopust (München)A weakness measure for GR(1) formulae.https://www.zbmath.org/1460.680542021-06-15T18:09:00+00:00"Cavezza, Davide Giacomo"https://www.zbmath.org/authors/?q=ai:cavezza.davide-giacomo"Alrajeh, Dalal"https://www.zbmath.org/authors/?q=ai:alrajeh.dalal"György, András"https://www.zbmath.org/authors/?q=ai:gyorgy.andrasSummary: In spite of the theoretical and algorithmic developments for system synthesis in recent years, little effort has been dedicated to quantifying the quality of the specifications used for synthesis. When dealing with unrealizable specifications, finding the weakest environment assumptions that would ensure realizability is typically a desirable property; in such context the weakness of the assumptions is a major quality parameter. The question of whether one assumption is weaker than another is commonly interpreted using implication or, equivalently, language inclusion. However, this interpretation does not provide any further insight into the weakness of assumptions when implication does not hold. To our knowledge, the only measure that is capable of comparing two formulae in this case is entropy, but even it fails to provide a sufficiently refined notion of weakness in case of GR(1) formulae, a subset of linear temporal logic formulae which is of particular interest in controller synthesis. In this paper we propose a more refined measure of weakness based on the Hausdorff dimension, a concept that captures the notion of size of the omega-language satisfying a linear temporal logic formula. We identify the conditions under which this measure is guaranteed to distinguish between weaker and stronger GR(1) formulae. We evaluate our proposed weakness measure in the context of computing GR(1) assumptions refinements.
For the entire collection see [Zbl 1391.68007].
Reviewer: Reviewer (Berlin)Editorial on special issue: fractal-based analysis.https://www.zbmath.org/1460.000492021-06-15T18:09:00+00:00"Kunze, Herb (ed.)"https://www.zbmath.org/authors/?q=ai:kunze.herb-e"La Torre, Davide (ed.)"https://www.zbmath.org/authors/?q=ai:la-torre.davide"Mendivil, Franklin (ed.)"https://www.zbmath.org/authors/?q=ai:mendivil.franklinFrom the text: This Special Issue is dedicated to Prof. Edward Robert Vrscay of the Department of Applied Mathematics of the University of Waterloo (Canada) to celebrate his career and well-known contributions to the area of fractals and their applications to image analysis.
Reviewer: Reviewer (Berlin)Dimensions of sets which uniformly avoid arithmetic progressions.https://www.zbmath.org/1460.280082021-06-15T18:09:00+00:00"Fraser, Jonathan M."https://www.zbmath.org/authors/?q=ai:fraser.jonathan-m"Saito, Kota"https://www.zbmath.org/authors/?q=ai:saito.kota"Yu, Han"https://www.zbmath.org/authors/?q=ai:yu.hanThe present paper is devoted to dimensions of sets which uniformly avoid arithmetic progressions (APs) and to an analogous problem in higher dimensions. The main attention is given to quantifying ``how ``small'' a set must be if it uniformly avoids APs''.
Auxiliary notions are explained. The notion of the Assouad dimension and its connections with the Hausdorff and upper box dimensions are recalled. Some attention is given to a brief survey of some connections between dimensions and APs or almost APs.
In this research, obtained results are described with explanations. Also, the special attention is given to a discussion of certain open questions.
Finally, one can note authors' abstract:
``We provide estimates for the dimensions of sets in \(\mathbb R\) which uniformly avoid finite arithmetic progressions (APs). More precisely, we say \(F\) uniformly avoids APs of length \(k \ge 3\) if there is an \(\varepsilon >0\) such that one cannot find an AP of length \(k\) and gap length \(\Delta >0\) inside the \(\varepsilon\Delta\) neighbourhood of F. Our main result is an explicit upper bound for the Assouad (and thus Hausdorff) dimension of such sets in terms of \(k\) and \(\varepsilon\). In the other direction, we provide examples of sets which uniformly avoid APs of a given length but still have relatively large Hausdorff dimension. We also consider higher dimensional analogues of these problems, where APs are replaced with arithmetic patches lying in a hyperplane. As a consequence, we obtain a discretized version of a ``reverse Kakeya problem:'' we show that if the dimension of a set in \(\mathbb R^d\) is sufficiently large, then it closely approximates APs in every direction.''
Reviewer: Symon Serbenyuk (Kyïv)On the Hewitt-Stromberg measure of product sets.https://www.zbmath.org/1460.280052021-06-15T18:09:00+00:00"Guizani, Omrane"https://www.zbmath.org/authors/?q=ai:guizani.omrane"Mahjoub, Amal"https://www.zbmath.org/authors/?q=ai:mahjoub.amal"Attia, Najmeddine"https://www.zbmath.org/authors/?q=ai:attia.najmeddineThe authors are concerned with the evaluation of the Hewitt-Stromberg measure of Cartesian product sets by means of the measure of their components. This is done by the construction of new multifractal measures on Euclidean spaces, in a similar manner to Hewitt-Stromberg measures -- but using the class of all half-open binary cubes of covering sets in the definition rather than the class of all balls.
Reviewer: George Stoica (Saint John)Graphs whose independence fractals are line segments.https://www.zbmath.org/1460.051422021-06-15T18:09:00+00:00"Barik, Sasmita"https://www.zbmath.org/authors/?q=ai:barik.sasmita"Nayak, Tarakanta"https://www.zbmath.org/authors/?q=ai:nayak.tarakanta"Pradhan, Ankit"https://www.zbmath.org/authors/?q=ai:pradhan.ankitSummary: Let \(G\) be a simple graph. By an independent set in \(G\), we mean a set of pairwise non-adjacent vertices in \(G\). The independence polynomial of \(G\) is defined as \(I_G(z)=i_0+i_1z+i_2 z^2+\cdots+i_\alpha z^{\alpha}\), where \(i_m=i_m(G)\) is the number of independent sets in \(G\) with cardinality \(m\) and \(\alpha=\alpha(G)\) denotes the cardinality of a largest independent set in \(G\) (known as the independence number of \(G)\). Let \(G^k\) denote the \(k\)-times lexicographic product of \(G\) with itself. The set of roots of \(I_{G^k}\) is known to converge as \(k\) tends to \(\infty\), with respect to the Hausdorff metric, and the limiting set is known as the independence attractor. The independence fractal of a graph is the limiting set of roots of the reduced independence polynomial \(I_{G^k}-1\) of \(G^k\) as \(k\) tends to \(\infty\). In this article, we consider the independence fractals of graphs with independence number 3. We attempt to find all such graphs whose independence fractal is a line segment. It is shown that the independence fractal and the independence attractor coincide when the earlier is a line segment. The line segment turns out to be an interval \([-\frac{4}{k}, 0]\) for \(k\in\{1,2,3,4\}\). It is found that each of these graphs have 9 vertices and there are exactly 13 such disconnected graphs. We show that there does not exist any connected graph for \(k=4\). For \(k=1\), there are 17 such connected graphs and for \(k=2,3\) the number is quite large.
Reviewer: Reviewer (Berlin)Affine embeddings of Cantor sets in the plane.https://www.zbmath.org/1460.280062021-06-15T18:09:00+00:00"Algom, Amir"https://www.zbmath.org/authors/?q=ai:algom.amirGiven a nonempty self-similar set \(F\subseteq\mathbb{R}^2\) generated by an iterated function system \(\Phi\) satisfying the strong separation condition, the author characterizes those affine maps \(g:\mathbb{R}^2\to\mathbb{R}^2\) for which \(g(F)\subseteq F\). In case that the group \(G_\Phi\) generated by the orthogonal parts of the maps in \(\Phi\) has infinite cardinality \(|G_\Phi|\), it is shown that the self-embedding \(g\) is necessarily a similitude. On the other hand, if the cardinality of \(G_\Phi\) is finite and \(\Phi\) is uniformly contracting with contractivity constant \(\lambda\), then the linear part of the embedding \(g\) is diagonalizable and the norm of each of its eigenvalues is a rational power of \(\lambda\).
In addition, the author investigates the existence and properties of affine maps \(g\) with the property that \(g(F)\subseteq E\), where \(E\) is another self-similar subset of \(\mathbb{R}^2\) generated by another iterated function system \(\Psi\). More credence is given to the conjecture by Feng, Huang \& Rao, that such an embedding exists only if the contraction ratios of the maps in \(\Phi\) are algebraically dependent on the contraction ratios of the maps in \(\Psi\). Moreover, it is proved that under some mild conditions \(|G_\Phi| = \infty\) implies \(|G_\Psi| = \infty\) and \(|G_\Phi| < \infty\) implies \(|G_\Psi| < \infty\).
Reviewer: Peter Massopust (München)Fixed point results for locally contraction with applications to fractals.https://www.zbmath.org/1460.540532021-06-15T18:09:00+00:00"Petruşel, Adrian"https://www.zbmath.org/authors/?q=ai:petrusel.adrian"Petruşel, Gabriela"https://www.zbmath.org/authors/?q=ai:petrusel.gabriela"Wong, Mu-Ming"https://www.zbmath.org/authors/?q=ai:wong.mu-mingSummary: The purpose of this paper is to connect the fixed point results for uniformly locally contractions in complete metric and \(\varepsilon\)-chainable metric spaces to coupled fixed point problems and to fractal and coupled fractal theory.
Reviewer: Reviewer (Berlin)A sequential view of self-similar measures; or, what the ghosts of Mahler and Cantor can teach us about dimension.https://www.zbmath.org/1460.280072021-06-15T18:09:00+00:00"Coons, Michael"https://www.zbmath.org/authors/?q=ai:coons.michael"Evans, James"https://www.zbmath.org/authors/?q=ai:evans.james-w|evans.james-a|evans.james-b|evans.james-d|evans.james-r|evans.james-eAfter a very well thought introduction into the subject, the authors show that a missing \(q\)-ary digit set \(F\subseteq [0, 1]\) has a corresponding naturally associated countable binary \(q\)-automatic sequence \(f\). In addition, they show that the Hausdorff dimension of \(F\) is equal to the base-\(q\) logarithm of the Mahler eigenvalue of \(f\); and that the standard mass distribution supported on \(F\) is equal to the ghost measure of \(f\).
Reviewer: George Stoica (Saint John)