Recent zbMATH articles in MSC 28https://zbmath.org/atom/cc/282023-09-22T14:21:46.120933ZWerkzeugAverage Fermat distance of a pseudo-fractal hierarchical scale-free networkhttps://zbmath.org/1517.051632023-09-22T14:21:46.120933Z"Peng, Lulu"https://zbmath.org/authors/?q=ai:peng.lulu"Zeng, Cheng"https://zbmath.org/authors/?q=ai:zeng.cheng"Chen, Dirong"https://zbmath.org/authors/?q=ai:chen.dirong"Xue, Yumei"https://zbmath.org/authors/?q=ai:xue.yumei"Zhao, Zixuan"https://zbmath.org/authors/?q=ai:zhao.zixuan(no abstract)Combinatorial properties for a class of simplicial complexes extended from pseudo-fractal scale-free webhttps://zbmath.org/1517.051842023-09-22T14:21:46.120933Z"Xie, Zixuan"https://zbmath.org/authors/?q=ai:xie.zixuan"Wang, Yucheng"https://zbmath.org/authors/?q=ai:wang.yucheng"Xu, Wanyue"https://zbmath.org/authors/?q=ai:xu.wanyue"Zhu, Liwang"https://zbmath.org/authors/?q=ai:zhu.liwang"Li, Wei"https://zbmath.org/authors/?q=ai:li.wei.258"Zhang, Zhongzhi"https://zbmath.org/authors/?q=ai:zhang.zhongzhi(no abstract)Speed of convergence of Weyl sums over Kronecker sequenceshttps://zbmath.org/1517.111012023-09-22T14:21:46.120933Z"Colzani, Leonardo"https://zbmath.org/authors/?q=ai:colzani.leonardoAuthor's abstract: We study the speed of convergence in the numerical integration with Weyl sums over Kronecker sequences in the torus,
\[
\frac{1}{N} \sum_{n=1}^N f( x+n\alpha) -\int_{\mathbb{T}^d }f(y)dy.
\]
Reviewer: Giovanni Coppola (Napoli)On arithmetic properties of Cantor setshttps://zbmath.org/1517.111262023-09-22T14:21:46.120933Z"Cui, Lu"https://zbmath.org/authors/?q=ai:cui.lu"Ma, Minghui"https://zbmath.org/authors/?q=ai:ma.minghuiSummary: In this paper, we study three types of Cantor sets. For any integer \(m \geqslant 4\), we show that every real number in \([0, k]\) is the sum of at most \textit{k\, m}-th powers of elements in the Cantor ternary set \(C\) for some positive integer \(k\), and the smallest such \(k\) is \(2^m\). Moreover, we generalize this result to the middle-\(\frac{1}{\alpha}\) Cantor set for \(1 < \alpha < 2 + \sqrt{5}\) and \(m\) sufficiently large. For the naturally embedded image \(W\) of the Cantor dust \(C \times C\) into the complex plane \(\mathbb{C}\), we prove that for any integer \(m \geqslant 3\), every element in the closed unit disk in \(\mathbb{C}\) can be written as the sum of at most \(2^{m+8}\; m\)-th powers of elements in \(W\). At last, some similar results on \(p\)-adic Cantor sets are also obtained.Lattice points close to the Heisenberg sphereshttps://zbmath.org/1517.111272023-09-22T14:21:46.120933Z"Campolongo, Elizabeth G."https://zbmath.org/authors/?q=ai:campolongo.elizabeth-g"Taylor, Krystal"https://zbmath.org/authors/?q=ai:taylor.krystalThe authors give an upper bound on the number of lattice points near the surfaces of the Heisenberg norm balls, specifically they give an upper bound on the number of points on and near large dilates of the unit spheres generated by the anisotropic norms \(\|(z,t) \|_\alpha = ( |z|^\alpha + |t|^{\frac{\alpha}{2}})^{\frac{1}{\alpha}}\) where \(\alpha \geq 2\). They establish a bound on the Fourier transform of the surface measures coming from these norms and give a estimate for the number of lattice points in the intersection of two such surfaces.
Reviewer: Steven T. Dougherty (Scranton)Corona decompositions for parabolic uniformly rectifiable setshttps://zbmath.org/1517.280012023-09-22T14:21:46.120933Z"Bortz, S."https://zbmath.org/authors/?q=ai:bortz.simon"Hoffman, J."https://zbmath.org/authors/?q=ai:hoffman.james-j|hoffman.john-r|hoffman.john-w|hoffman.judy|hoffman.jerome-william|hoffman.joe-d|hoffman.james-a|hoffman.josef-d|hoffman.jim|hoffman.johan"Hofmann, S."https://zbmath.org/authors/?q=ai:hofmann.steve"Luna-Garcia, J. L."https://zbmath.org/authors/?q=ai:luna-garcia.jose-luis"Nyström, K."https://zbmath.org/authors/?q=ai:nystrom.kajSummary: We prove that parabolic uniformly rectifiable sets admit (bilateral) corona decompositions with respect to regular \(\mathrm{Lip}(1,1/2)\) graphs. Together with our previous work, this allows us to conclude that if \(\Sigma \subset\mathbb{R}^{n+1}\) is parabolic Ahlfors-David regular, then the following statements are equivalent.
\begin{itemize}
\item[(1)] \(\Sigma\) is parabolic uniformly rectifiable.
\item[(2)] \(\Sigma\) admits a corona decomposition with respect to regular \(\mathrm{Lip}(1,1/2)\) graphs.
\item[(3)] \(\Sigma\) admits a bilateral corona decomposition with respect to regular \(\mathrm{Lip}(1,1/2)\) graphs.
\item[(4)] \(\Sigma\) is big pieces squared of regular \(\mathrm{Lip}(1,1/2)\) graphs.
\end{itemize}Multifractal dimensions and fractional differentiation in automated edge detection on intuitionistic fuzzy enhanced imagehttps://zbmath.org/1517.280022023-09-22T14:21:46.120933Z"Ananthi, V. P."https://zbmath.org/authors/?q=ai:ananthi.v-p"Thangaraj, C."https://zbmath.org/authors/?q=ai:thangaraj.c-j"Easwaramoorthy, D."https://zbmath.org/authors/?q=ai:easwaramoorthy.dSummary: A novel method is introduced to optimize the selection of order of fractional derivative for detecting edges in images. Before commencing the process, images are enhanced using fuzzy enhancement membership function with variable in intuitionistic fuzzy domain. The value of is optimized by comparing visibility of the images by changing. Then the enhanced image is involved in the edge detection process using fractional derivative of variable order q. By changing q, the edge images are detected and are compared using fuzzy divergence operator. The edge image with minimum fuzzy divergence is the final edge image. The results are compared with existing state of the art techniques. Finally, the proposed method is compared graphically with the existing methods with respect to the multifractal dimensional measures. Qualitatively, the proposed technique seems to detect edge in a better way than the compared methods.
For the entire collection see [Zbl 1495.26007].Hausdorff and box dimensions of self-affine sets in a locally compact non-Archimedean fieldhttps://zbmath.org/1517.280032023-09-22T14:21:46.120933Z"Deng, Yang"https://zbmath.org/authors/?q=ai:deng.yang"Li, Bing"https://zbmath.org/authors/?q=ai:li.bing.2|li.bing"Qiu, Hua"https://zbmath.org/authors/?q=ai:qiu.huaThe authors consider affine iterated function systems in a locally compact non-Archimedean field \(\mathbb{F}\). They establish a theory of singular value decomposition in \(\mathbb{F}\) and compute the box and Hausdorff dimensions of self-affine sets in \(\mathbb{F}^n\), in generic sense, which is an analogy of Falconer's result for the real case. In \(\mathbb{R}^n\), the box and Hausdorff dimensions of self-affine sets can be obtained only when the norms of linear parts of affine transformations are strictly less than \(\frac{1}{2}\). However, in a locally compact non-Archimedean field, the same result can be obtained without the restriction of the norms.
Reviewer: Haipeng Chen (Shenzhen)On the Borel regularity of the relative centered multifractal measureshttps://zbmath.org/1517.280042023-09-22T14:21:46.120933Z"Douzi, Zied"https://zbmath.org/authors/?q=ai:douzi.zied"Selmi, Bilel"https://zbmath.org/authors/?q=ai:selmi.bilelSummary: This chapter investigates the Borel regularity of the relative centered Hausdorff and packing measures. It is shown that the relative packing measure is Borel regular. The usual spherical measure is equivalent to the relative Hausdorff measure in the case where the measures and satisfy the doubling condition. Moreover, it is proved that this statement is true even without the doubling condition if and or if and which implies that is a Borel regular measure.
For the entire collection see [Zbl 1495.26007].Fractal calculushttps://zbmath.org/1517.280052023-09-22T14:21:46.120933Z"Golmankhaneh, Alireza Khalili"https://zbmath.org/authors/?q=ai:golmankhaneh.alireza-khalili"Welch, Kern"https://zbmath.org/authors/?q=ai:welch.kern"Priyanka, T. M. C."https://zbmath.org/authors/?q=ai:priyanka.t-m-c"Gowrisankar, A."https://zbmath.org/authors/?q=ai:gowrisankar.aSummary: This chapter the fractal calculus in both its local and non-local form is reviewed. The fractal calculus which is called -calculus, is a generalization of ordinary calculus that adapted to the fractal sets, curves, and spaces. The non-local fractal derivatives are given the analogue of the Riemann-Liouville and Caputo fractional derivatives. The analogue of the Laplace transform and the Fourier transform in the fractal calculus are suggested. Discrete scale invariance is studied by using the scale transform. Finally, the applications of fractal calculus in classical mechanics, quantum mechanics, and optics are presented.
For the entire collection see [Zbl 1495.26007].A mixed multifractal analysis of vector-valued measures: review and extension to densities and regularities of non-necessary Gibbs caseshttps://zbmath.org/1517.280062023-09-22T14:21:46.120933Z"Mabrouk, Anouar Ben"https://zbmath.org/authors/?q=ai:ben-mabrouk.anouar"Selmi, Bilel"https://zbmath.org/authors/?q=ai:selmi.bilelSummary: The multifractal analysis is concerned with the description of irregular measures and functions when the classical analysis does not work. The main goal is the establishment of a multifractal formalism permitting to describe the distribution of the singularities along the support. In its original formulation, the multifractal formalism has been proved to hold for many cases, such as, doubling measures, self-similar functions and measures, and also Gibbs measures. In this chapter, some non-necessary Gibbs vector-valued measures in the framework of the mixed multifractal analysis are concerned. By introducing a gage function in the multifractal formalism, and considering suitable multifractal densities in the framework of relative mixed multifractal analysis, a possible multifractal formalism is proved to hold. Besides, multifractal regularities of the associated multifractal generalizations of the Hausdorff and packing measures are investigated in the new framework.
For the entire collection see [Zbl 1495.26007].Fractal interpolation: from global to local, to nonstationary and quaternionichttps://zbmath.org/1517.280072023-09-22T14:21:46.120933Z"Massopust, Peter R."https://zbmath.org/authors/?q=ai:massopust.peter-rSummary: This chapter presents an introduction to fractal interpolation beginning with a global set-up and then extending to a local, a non-stationary, and finally the novel quaternionic setting. Emphasis is placed on the overall perspective with references given to the more specific questions.
For the entire collection see [Zbl 1495.26007].Some remarks on multivariate fractal approximationhttps://zbmath.org/1517.280082023-09-22T14:21:46.120933Z"Pandey, Megha"https://zbmath.org/authors/?q=ai:pandey.megha"Agrawal, Vishal"https://zbmath.org/authors/?q=ai:agrawal.vishal"Som, Tanmoy"https://zbmath.org/authors/?q=ai:som.tanmoySummary: Approximation theory encompasses a vast area of mathematics. The current context is primarily concerned with the concept of dimension preserving approximation for real-valued multivariate continuous functions defined on a domain. This chapter establishes quite a few results similar to well-known results of multivariate constrained approximation in terms of dimension preserving approximants. In particular, this chapter gives indication for construction of multivariate dimension preserving approximants using the concept of fractal interpolation functions. In the last part, some multi-valued fractal operators associated with multivariate -fractal functions are defined and studied.
For the entire collection see [Zbl 1495.26007].Perspective of fractal calculus on types of fractal interpolation functionshttps://zbmath.org/1517.280092023-09-22T14:21:46.120933Z"Priyanka, T. M. C."https://zbmath.org/authors/?q=ai:priyanka.t-m-c"Agathiyan, A."https://zbmath.org/authors/?q=ai:agathiyan.a"Gowrisankar, A."https://zbmath.org/authors/?q=ai:gowrisankar.aSummary: Fractal calculus is the calculus involving -integral and -derivative, where in (0,1] is the dimension of the fractals. In defining -integral and -derivative, the mass function and staircase function plays an important role. This chapter discusses the fractal calculus of non-affine fractal interpolation functions namely hidden variable fractal interpolation function and -fractal function. The fractal integral of the hidden variable fractal interpolation function is examined by predefining the initial conditions. Similarly, by predefining the initial conditions and imposing some necessary conditions on the -fractal function, its fractal integral is explored.
For the entire collection see [Zbl 1495.26007].Vector-valued fractal functions: fractal dimension and fractional calculushttps://zbmath.org/1517.280102023-09-22T14:21:46.120933Z"Verma, Manuj"https://zbmath.org/authors/?q=ai:verma.manuj"Priyadarshi, Amit"https://zbmath.org/authors/?q=ai:priyadarshi.amit"Verma, Saurabh"https://zbmath.org/authors/?q=ai:verma.saurabhThe authors investigate the relations between the fractal dimension of a vector-valued function and that of its component functions. They derive bounds for the Hausdorff dimension of a vector-valued fractal interpolation function and also for the Hausdorff dimension of the associated invariant measure supported on the graph of such a function. Additionally, more general upper bounds for the Hausdorff dimension of invariant measures in terms of probability vectors and contraction ratios are derived. Conditions under which a vector-valued fractal interpolation function belongs to a special class of function spaces such as Hölder spaces, the class of functions of bounded variation, and the space of absolutely continuous functions. Moreover, the authors prove that the Riemann-Liouville fractional integral of a vector-valued fractal interpolation function is again a vector-valued fractal interpolation function corresponding to some data set. The Hausdorff and the box counting dimension of the graph of the Riemann-Liouville fractional integral of a vector- valued fractal interpolation function is computed.
Reviewer: Peter Massopust (München)A study on fractal operator corresponding to non-stationary fractal interpolation functionshttps://zbmath.org/1517.280112023-09-22T14:21:46.120933Z"Verma, Saurabh"https://zbmath.org/authors/?q=ai:verma.saurabh"Jha, Sangita"https://zbmath.org/authors/?q=ai:jha.sangitaSummary: This chapter aims to establish the notion of non-stationary-fractal operator and establish some approximations and convergence properties. More specifically, the approximations properties of the non-stationary -fractal polynomials towards a continuous function is discussed. Here a sequence of maps for the non-stationary iterated function systems is used. Further, this chapter shows that the proposed method generalizes the existing stationary interpolant in the sense of iterated function systems. The basic properties of this new notion of interpolant are explored, and its box and Hausdorff dimensions are obtained by comparing it to other well-known results. Additionally, using the method of fractal perturbation of a given function, the associated non-stationary fractal operator is constructed and few of its approximation properties are investigated.
For the entire collection see [Zbl 1495.26007].The Fučík spectrum for one dimensional Kreĭn-Feller operatorshttps://zbmath.org/1517.341142023-09-22T14:21:46.120933Z"Oviedo, Martina"https://zbmath.org/authors/?q=ai:oviedo.martina"Pinasco, Juan Pablo"https://zbmath.org/authors/?q=ai:pinasco.juan-pablo"Scarola, Cristian"https://zbmath.org/authors/?q=ai:scarola.cristianThe authors study the Fučík spectrum of one-dimensional Krein-Feller operators in \([0,1]\), i.e., the pairs \((\alpha,\beta) \in \mathbb{R}^2\) for which there exists a function \(u\) from a certain Sobolev type space \(\mathcal{D}^{1,2}_{0,\mu}\) satisfying
\[
\int_0^1 u'(x) \varphi'(x) \,dx = \int_0^1 m(x) (\alpha u^+(x) - \beta u^-(x)) \varphi(x) d \mu(x)
\]
for any \(\varphi \in C^1_0(0,1)\). Here, \(\mu\) is a Borel, non-atomic, probability measure supported in a closed set \(K_\mu \subset [0, 1]\), and \(m\) is a continuous weight. Using the classical critical point theory, the authors variationally characterize infinitely many curves in the Fučík spectrum, show that they are Lipschitz continuous and decreasing, and provide constructive lowers bounds for them.
Reviewer: Vladimir Bobkov (Ufa)Entropy estimates for uniform attractors of 2D Navier-Stokes equations with weakly normal measureshttps://zbmath.org/1517.351622023-09-22T14:21:46.120933Z"Xiong, Yangmin"https://zbmath.org/authors/?q=ai:xiong.yangmin"Song, Xiaoya"https://zbmath.org/authors/?q=ai:song.xiaoya"Sun, Chunyou"https://zbmath.org/authors/?q=ai:sun.chunyouSummary: This paper aims at the long-time behavior of non-autonomous 2D Navier-Stokes equations with a class of external forces which are \(H\)-valued measures in time. We first establish the well-posedness of solutions as well as the existence of a strong uniform attractor, and then pay the main attention on the estimation of \(\varepsilon\)-entropy for such uniform attractor in the standard energy phase space.Nondispersive solutions to the mass critical half-wave equation in two dimensionshttps://zbmath.org/1517.352032023-09-22T14:21:46.120933Z"Georgiev, Vladimir"https://zbmath.org/authors/?q=ai:georgiev.vladimir-s"Li, Yuan"https://zbmath.org/authors/?q=ai:li.yuan.8The purpose of this paper is to study two nondispersive phenomena connected with the focusing half-wave equation in two dimensions
\[
\left\{
\begin{array}
[c]{c}
i\partial_{t}u =Du-\left\vert u\right\vert u,\\
u\left( t_{0},x\right) =u_{0}\left( x\right) ,\text{ }u:I\times
\mathbb{R}^{2}\rightarrow\mathbb{C}\text{,}
\end{array}
\right.
\]
where \(t_{0}\in I\subset\mathbb{R}\) is an interval and \(D\) denotes the first-order nonlocal fractional derivative. First, in Theorem 1.1 the authors show the existence of traveling solitary waves of the form
\[
u\left( t,x\right) =e^{it\mu}Q_{\nu}\left( x-\nu t\right)
\]
with a profile \(Q_{\nu}\in H^{\frac{1}{2}}\left(\mathbb{R}\right),\) some \(\mu\in\mathbb{R}\) and traveling velocity \(0<\nu<1.\) Moreover, the authors prove that the mass \(\left\Vert Q_{\nu}\right\Vert _{L^{2}}\ \)tends to the mass of the ground state solution \(Q\) when \(\left\vert \nu\right\vert \rightarrow0\) and it vanishes when \(\left\vert \nu\right\vert \rightarrow1.\) The second result of the paper, given in Theorem 1.2, shows that there is a solution with ground state mass that blows up in finite time. The detailed asymptotics of this solution near the blow-up time is also presented. The delicate proof is based on complicated modulation and energy/virial estimates,
and a bootstrap argument.
Reviewer: Ivan Naumkin (Nice)Directional Kronecker algebra for \(\mathbb{Z}^q\)-actionshttps://zbmath.org/1517.370052023-09-22T14:21:46.120933Z"Liu, Chunlin"https://zbmath.org/authors/?q=ai:liu.chunlin"Xu, Leiye"https://zbmath.org/authors/?q=ai:xu.leiyeIn this paper, the directional sequence entropy and the directional Kronecker algebra for \(\mathbb{Z}^q\)-systems are introduced. The relation between sequence entropy and directional sequence entropy is established. Meanwhile, directional discrete spectrum systems and directional null systems are defined. It is shown that a \(\mathbb{Z}^q\)-system has directional discrete spectrum if and only if it is directional null. Moreover, it turns out that a \(\mathbb{Z}^q\)-system has directional discrete spectrum along \(q\) linearly independent directions if and only if it has discrete spectrum.
Reviewer: Daniele Puglisi (Catania)On the induced measure-theoretic entropy for random dynamical systemshttps://zbmath.org/1517.370282023-09-22T14:21:46.120933Z"Yang, Kexiang"https://zbmath.org/authors/?q=ai:yang.kexiang"Chen, Ercai"https://zbmath.org/authors/?q=ai:chen.ercai"Zhou, Xiaoyao"https://zbmath.org/authors/?q=ai:zhou.xiaoyaoThis paper introduces the concepts of induced topological entropy and induced measure-theoretic entropy for random dynamical systems.
The measure-theoretic entropy of a measurable transformation was presented by \textit{A. N. Kolmogorov} [Dokl. Akad. Nauk SSSR 119, 861--864 (1958; Zbl 0083.10602)] and \textit{Ya. G. Sinai} [Dokl. Akad. Nauk SSSR 124, 768--771 (1959; Zbl 0086.10102)]. The idea of entropy was transferred to the realm of topological dynamical systems by \textit{R. L. Adler} et al. [Trans. Am. Math. Soc. 114, 309--319 (1965; Zbl 0127.13102)] and, using a different (but equivalent) definition, by \textit{R. Bowen} [Trans. Am. Math. Soc. 153, 401--414 (1971; Zbl 0212.29201)]. Then \textit{A. Katok} [Publ. Math., Inst. Hautes Étud. Sci. 51, 137--173 (1980; Zbl 0445.58015)]
provided another equivalent definition of measure-theoretic entropy for an ergodic Borel probability measure and consequently defined a topological version of measure-theoretic entropy of a continuous map on a compact metric space.
Random dynamical systems were described by \textit{L. Arnold} [Random dynamical systems. Berlin: Springer (1998; Zbl 0906.34001)] and a research of entropy in random dynamical systems started afterwords. The notions of measure-theoretic and topological entropy of a continuous bundle random dynamical systems (RDS) appeared.
Inspired by the previous research, this paper proposes the definitions of induced topological entropy and induced measure-theoretic entropy of RDS over \((\Omega,\mathcal{F},\mathbf{P},\vartheta)\). ``Induced'' modifications of entropy are related to \(\varphi\in L^1_{\Omega\times X}(\Omega,C(X))\) with \(\inf \varphi>0\).
Basic properties of the induced topological entropy are deduced and on this basis, a Katok entropy formula is established for induced measure-theoretic entropy of RDS, which shows the relation between induced measure-theoretic and measure-theoretic entropy of RDS. The obtained formula extends a related expression for deterministic dynamical systems. For an investigation of measure-theoretic entropy, the Shannon-McMillan-Breiman theorem of RDS is extensively used.
Finally, a measure-theoretic entropy for induced pointwise dimensions of RDS is studied and a formula for lower and upper induced pointwise dimensions of RDS is derived. This generalizes a result by \textit{L. Barreira} and \textit{J. Schmeling} [Isr. J. Math. 116, 29--70 (2000; Zbl 0988.37029)]
to the induced version of entropy.
Reviewer: Pavel Ludvík (Olomouc)Entropies for factor maps of amenable group actionshttps://zbmath.org/1517.370292023-09-22T14:21:46.120933Z"Zhang, Guohua"https://zbmath.org/authors/?q=ai:zhang.guohua.2|zhang.guohua.1"Zhu, Lili"https://zbmath.org/authors/?q=ai:zhu.liliSummary: In this paper we study various entropies for factor maps of amenable group actions. We prove firstly theorem inequalities linking relative topological entropy and conditional topological entropy (for factor maps of amenable group actions) without any additional assumption, which strengthens conditional variational principles [Nonlinearity 34, No. 8, 5163--5185 (2021; Zbl 1475.37024); Theorems 2.12 and 3.9] proved by \textit{Z. Changrong} under additional assumptions. Then along the line of [\textit{M. Misiurewicz}, Stud. Math. 55, 175--200 (1976; Zbl 0355.54035)], we introduce a new invariant called relative topological tail entropy and prove a Ledrappier's type variational principle concerning it (for factor maps of amenable group actions); consequently, any factor map with zero relative topological tail entropy admits invariant measures with maximal relative entropy, which provides a nontrivial sufficient condition for the existence of invariant measures with maximal relative entropy in the setting of factor maps of amenable group actions.New approach to weighted topological entropy and pressurehttps://zbmath.org/1517.370392023-09-22T14:21:46.120933Z"Tsukamoto, Masaki"https://zbmath.org/authors/?q=ai:tsukamoto.masakiSummary: Motivated by fractal geometry of self-affine carpets and sponges, \textit{D.-J. Feng} and \textit{W. Huang} [J. Math. Pures Appl. (9) 106, No. 3, 411--452 (2016; Zbl 1360.37080)] introduced weighted topological entropy and pressure for factor maps between dynamical systems, and proved variational principles for them. We introduce a new approach to this theory. Our new definitions of weighted topological entropy and pressure are very different from the original definitions of \textit{D.-J. Feng} and \textit{W. Huang} [loc. cit.]. The equivalence of the two definitions seems highly non-trivial. Their equivalence can be seen as a generalization of the dimension formula for the Bedford-McMullen carpet in purely topological terms.On dynamical gaskets generated by rational maps, Kleinian groups, and Schwarz reflectionshttps://zbmath.org/1517.370552023-09-22T14:21:46.120933Z"Lodge, Russell"https://zbmath.org/authors/?q=ai:lodge.russell"Lyubich, Mikhail"https://zbmath.org/authors/?q=ai:lyubich.mikhail"Merenkov, Sergei"https://zbmath.org/authors/?q=ai:merenkov.sergei"Mukherjee, Sabyasachi"https://zbmath.org/authors/?q=ai:mukherjee.sabyasachiThe authors add a new contribution to the so-called \textit{Fatou-Sullivan dictionary}, which connects two branches of conformal dynamics -- iterations of rational maps and actions of discrete subgroups of Möbius transformations (i.e., Kleinian groups) on \(\hat{\mathbb{C}}\).
In particular, they focus on a family of fractals, named \textit{gaskets}, coming from circle packings and triangulations of the sphere. Given a triangulation \(\mathcal{T}\) of the sphere \(S^2\cong\hat{\mathbb{C}}\), the Circle Packing Theorem of Koebe-Andreev-Thurston (see [\textit{W. P. Thurston}, The geometry and topology of three-manifolds. Providence, RI: American Mathematical Society (AMS) (2022; Zbl 1507.57005)]) guarantees the existence of a circle packing \(\mathcal{C}\) whose \textit{nerve} (a graph encoding the tangency data of the circles) is \(\mathcal{T}\), up to isotopy. Each face \(f\) of \(\mathcal{T}\) gives three mutually tangent circles in \(\mathcal{C}\), and let \(R_f\) be the reflection in the unique circle passing through the three points of tangency. Then the group \(H_\mathcal{T}\) generated by \(\{R_f\}_{f\text{ a face of } \mathcal{T}}\) acts on \(\hat{\mathbb{C}}\) as conformal and anticonformal maps. The limit set of \(H_\mathcal{T}\) is called a \textit{round gasket}. Notice that the complement of a round gasket is a disjoint union of round disks, including the disks bounded by circles in \(\mathcal{C}\). A \textit{gasket} is any subset of \(\hat{\mathbb{C}}\) homeomorphic to a round gasket. The classical Apollonian gasket is an example of a round gasket, associated to the tetrahedral triangulation of the sphere.
By definition, gaskets arise as limit sets of Kleinian groups. They also appear as Julia sets of (anti)rational maps. As a matter of fact, in the paper, for each triangulation \(\mathcal{T}\) of the sphere, the authors construct an antirational map \(g\) whose Julia set is homeomorphic to the limit set of \(H_\mathcal{T}\), by applying Thurston's topological characterization of rational maps [\textit{A. Douady} and \textit{J. H. Hubbard}, Acta Math. 171, No. 2, 263--297 (1993; Zbl 0806.30027)]. In the case of the Apollonian gasket, they also construct an explicit anti-quasiregular model for the map \(g\).
The connection between the group \(H\) and the map \(g\) is dynamical. In fact, the authors show that they can be mated via David surgery and Schwartz reflection to produce a hybrid dynamical system.
One motivation of the paper is to study the group of quasisymmetric homeomorphisms of a fractal. Previously, this group was determined for a class of Sierpiński carpet Julia sets [\textit{M. Bonk} et al., Adv. Math. 301, 383--422 (2016; Zbl 1358.37083)] and Basilica [\textit{M. Lyubich} and \textit{S. Merenkov}, Geom. Funct. Anal. 28, No. 3, 727--754 (2018; Zbl 1499.30238)]. Here the authors consider this problem for a large family of dynamical gaskets (limit sets and Julia sets).
More specifically, under a weak assumption on \(\mathcal{T}\), they show the following:
\begin{itemize}
\item For the limit set \(\Lambda_H\) of \(H\), the group of homeomorphisms of \(\Lambda_H\) is the semidirect product of the (finite) group of symmetries of \(\mathcal{T}\) with \(H\). In particular, every homeomorphism is conformal or anticonformal;
\item For the Julia set \(\mathcal{J}_g\) of \(g\), the group of homeomorphisms coincide with the group of quasisymmetries;
\item There exists a dynamically defined homeomorphism \(h:\Lambda_H\to\mathcal{J}_g\), so that it induces an isomorphism between the groups of homeomorphisms (and hence quasisymmetries).
\end{itemize}
The homeomorphism \(h\) mentioned above is \emph{not} quasiconformal, so the fact that it induces an isomorphism between groups of quasisymmetries is not automatic. This also means that the group of quasisymmetries does not distinguish the quasisymmetric type of \(\Lambda_H\) and \(\mathcal{J}_g\).
Reviewer: Yongquan Zhang (Stony Brook)A non-singular version of the Oseledeč ergodic theoremhttps://zbmath.org/1517.370562023-09-22T14:21:46.120933Z"Dooley, Anthony H."https://zbmath.org/authors/?q=ai:dooley.anthony-haynes"Jin, Jie"https://zbmath.org/authors/?q=ai:jin.jieSummary: Kingman's subadditive ergodic theorem is traditionally proved in the setting of a measure-preserving invertible transformation \(T\) of a measure space \((X,\mu)\). We use a theorem of \textit{C. E. Silva} and \textit{P. Thieullen} [J. Math. Anal. Appl. 154, No. 1, 83--99 (1991; Zbl 0719.28008)]
to extend the theorem to the setting of a not necessarily invertible transformation, which is non-singular under the assumption that \(\mu\) and \(\mu\circ T\) have the same null sets. Using this, we are able to produce versions of the Furstenberg-Kesten theorem and the Oseledeč ergodic theorem for products of random matrices without the assumption that the transformation is either invertible or measure-preserving.Polynomial Roth theorems on sets of fractional dimensionshttps://zbmath.org/1517.420042023-09-22T14:21:46.120933Z"Fraser, Robert"https://zbmath.org/authors/?q=ai:fraser.robert"Guo, Shaoming"https://zbmath.org/authors/?q=ai:guo.shaoming"Pramanik, Malabika"https://zbmath.org/authors/?q=ai:pramanik.malabikaSummary: Let \(E\subset\mathbb{R}\) be a closed set of Hausdorff dimension \(\alpha\in(0,1)\). Let \(P: \mathbb{R}\to\mathbb{R}\) be a polynomial without a constant term whose degree is bigger than one. We prove that if \(E\) supports a probability measure satisfying certain dimension condition and Fourier decay condition, then \(E\) contains three points \(x\), \(x+t\), \(x+P(t)\) for some \(t>0\). Our result extends the one of \textit{I. Łaba} and \textit{M. Pramanik} [Geom. Funct. Anal. 19, No. 2, 429--456 (2009; Zbl 1184.28010)] to the polynomial setting, under the same assumption. It also gives an affirmative answer to a question in [\textit{K. Henriot} et al., Anal. PDE 9, No. 5, 1153--1184 (2016; Zbl 1420.28003)].Talagrand's influence inequality revisitedhttps://zbmath.org/1517.420262023-09-22T14:21:46.120933Z"Cordero-Erausquin, Dario"https://zbmath.org/authors/?q=ai:cordero-erausquin.dario"Eskenazis, Alexandros"https://zbmath.org/authors/?q=ai:eskenazis.alexandrosSummary: Let \(\mathscr{C}_n=\{-1,1\}^n\) be the discrete hypercube equipped with the uniform probability measure \(\sigma_n\). Talagrand's influence inequality [\textit{M. Talagrand}, Ann. Probab. 22, No. 3, 1576--1587 (1994; Zbl 0819.28002)], also known as the \(L_1-L_2\) inequality, asserts that there exists \(C\in (0,\infty)\) such that for every \(n\in\mathbb{N} \), every function \(f:\mathscr{C}_n\to\mathbb{C}\) satisfies
\[
\operatorname{Var}_{\sigma_n}(f) \leqslant C \sum_{i=1}^n \frac{\|\partial_if\|_{L_2(\sigma_n)}^2}{1+\log(\|\partial_if\|_{L_2(\sigma_n)}/\|\partial_i f\|_{L_1(\sigma_n)})}.
\]
We undertake a systematic investigation of this and related inequalities via harmonic analytic and stochastic techniques and derive applications to metric embeddings. We prove that Talagrand's inequality extends, up to an additional doubly logarithmic factor, to Banach space-valued functions under the necessary assumption that the target space has Rademacher type 2 and that this doubly logarithmic term can be omitted if the target space admits an equivalent 2-uniformly smooth norm. These are the first vector-valued extensions of Talagrand's influence inequality. Moreover, our proof implies vector-valued versions of a general family of \(L_1-L_p\) inequalities, each refining the dimension independent \(L_p\)-Poincaré inequality on \((\mathscr{C}_n,\sigma_n)\). We also obtain a joint strengthening of results of \textit{D. Bakry} and \textit{P. A. Meyer} [Lect. Notes Math. 920, 138--145 (1982); ibid. 920, 146--150 (1982)] and \textit{A. Naor} and \textit{G. Schechtman} [J. Reine Angew. Math. 552, 213--236 (2002; Zbl 1033.46013)] on the action of negative powers of the hypercube Laplacian on functions \(f:\mathscr{C}_n\to E\), whose target space \((E,\|\cdot\|_E)\) has nontrivial Rademacher type via a new vector-valued version of Meyer's multiplier theorem [\textit{P. A. Meyer}, Lect. Notes Math., 179--193 (1984; Zbl 0543.60078)]. Inspired by Talagrand's influence inequality, we introduce a new metric invariant called Talagrand type and estimate it for Banach spaces with prescribed Rademacher or martingale type, Gromov hyperbolic groups and simply connected Riemannian manifolds of pinched negative curvature. Finally, we prove that Talagrand type is an obstruction to the bi-Lipschitz embeddability of nonlinear quotients of the hypercube \( \mathscr{C}_n\) equipped with the Hamming metric, thus deriving new nonembeddability results for these finite metrics. Our proofs make use of Banach space-valued Itô calculus, Riesz transform inequalities, Littlewood-Paley-Stein theory and hypercontractivity.Strictly singular non-compact operators between \(L_p\) spaceshttps://zbmath.org/1517.470602023-09-22T14:21:46.120933Z"Hernández, Francisco L."https://zbmath.org/authors/?q=ai:hernandez.francisco-l"Semenov, Evgeny M."https://zbmath.org/authors/?q=ai:semenov.evgenii-m"Tradacete, Pedro"https://zbmath.org/authors/?q=ai:tradacete-perez.pedroSummary: We study the structure of strictly singular non-compact operators between \(L_p\) spaces. Answering a question raised in earlier work on interpolation properties of strictly singular operators [Adv. Math. 316, 667--690 (2017; Zbl 1394.46014)], it is shown that there exist operators \(T\), for which the set of points \((1/p, 1/q) \in (0, 1) \times (0, 1)\) such that \(T : L_p \to L_q\) is strictly singular but not compact contains a line segment in the triangle \(\{(1/p, 1/q) : 1 < p < q < \infty\}\) of any positive slope. This will be achieved by means of Riesz potential operators between metric measure spaces with different Hausdorff dimension. The relation between compactness and strict singularity of regular (i.e., difference of positive) operators defined on subspaces of \(L_p\) is also explored.Strong laws of large numbers for double arrays of blockwise \(M\)-dependent random setshttps://zbmath.org/1517.600342023-09-22T14:21:46.120933Z"Castaing, Charles"https://zbmath.org/authors/?q=ai:castaing.charles"Duyen, Hoang Thi"https://zbmath.org/authors/?q=ai:duyen.hoang-thi"Van Quang, Nguyen"https://zbmath.org/authors/?q=ai:quang.nguyen-vanSummary: In this paper we study the concept of blockwise \(M\)-dependence for double arrays of closed valued random sets and then prove several results of Wijsman convergence, weak compact slice convergence, slice convergence and Mosco convergence for double arrays of blockwise \(M\)-dependent closed valued random sets. Some well-known strong laws of large numbers for double arrays of random variables or random sets are extended.A reduced basis method for Darcy flow systems that ensures local mass conservation by using exact discrete complexeshttps://zbmath.org/1517.651022023-09-22T14:21:46.120933Z"Boon, Wietse M."https://zbmath.org/authors/?q=ai:boon.wietse-m"Fumagalli, Alessio"https://zbmath.org/authors/?q=ai:fumagalli.alessioIn this paper, a three-step solution procedure is proposed for Darcy flow systems based on the exact de Rham complex. The mass conservation equation is first solved and the flux field is subsequently corrected by adding a solenoidal vector field. The computational cost in constructing the correction is reduced by applying reduced basis methods based on proper orthogonal decomposition. In the third step, the pressure field is constructed with discretization methods capable of conserving mass locally. The procedure was extended to the setting of Darcy flow in fractured porous media by employing mixed-dimensional differential operators.
Reviewer: Bülent Karasözen (Ankara)Synthetic topology in homotopy type theory for probabilistic programminghttps://zbmath.org/1517.680722023-09-22T14:21:46.120933Z"Bidlingmaier, Martin E."https://zbmath.org/authors/?q=ai:bidlingmaier.martin-e"Faissole, Florian"https://zbmath.org/authors/?q=ai:faissole.florian"Spitters, Bas"https://zbmath.org/authors/?q=ai:spitters.basSummary: The ALEA Coq library formalizes measure theory based on a variant of the Giry monad on the category of sets. This enables the interpretation of a probabilistic programming language with primitives for sampling from discrete distributions. However, continuous distributions have to be discretized because the corresponding measures cannot be defined on all subsets of their carriers. This paper proposes the use of synthetic topology to model continuous distributions for probabilistic computations in type theory. We study the initial \(\sigma\)-frame and the corresponding induced topology on arbitrary sets. Based on these intrinsic topologies, we define valuations and lower integrals on sets and prove versions of the Riesz and Fubini theorems. We then show how the Lebesgue valuation, and hence continuous distributions, can be constructed.Event-based transformations of set functions and the consensus requirementhttps://zbmath.org/1517.683702023-09-22T14:21:46.120933Z"Bronevich, Andrey G."https://zbmath.org/authors/?q=ai:bronevich.andrey-g"Rozenberg, Igor N."https://zbmath.org/authors/?q=ai:rozenberg.igor-nSummary: Non-additive measures, capacities or generally set functions are widely used in decision models, data processing and game theory. In these applications we can find many structures identified as linear transformations or linear operators. The most remarkable of them are Choquet integral, Möbius transform, interaction transform, Shapley value. The main goal of the presented paper is to study some of them recently called event-based linear transformations. We describe them considering the set of all possible linear operators as a linear space w.r.t. their linear combinations and compute the dimensions of its some subspaces. We also study the consensus requirement, i.e. we analyze the condition when the linear operator maps one family of non-additive measures to other family.
For the entire collection see [Zbl 1398.68040].Integral representations of a coherent upper conditional prevision by the symmetric Choquet integral and the asymmetric Choquet integral with respect to Hausdorff outer measureshttps://zbmath.org/1517.683732023-09-22T14:21:46.120933Z"Doria, Serena"https://zbmath.org/authors/?q=ai:doria.serenaSummary: Complex decisions in human decision-making may arise when the Emotional Intelligence and Rational Reasoning produce different preference ordering between alternatives. From a mathematical point of view, complex decisions can be defined as decisions where a preference ordering between random variables cannot be represented by a linear functional. The Asymmetric and the Symmetric Choquet integrals with respect to non additive-measures have been defined as aggregation operators of data sets and as a tool to assess an ordering between random variables. They could be considered to represent preference orderings of the conscious and unconscious mind when a human being make decision. Sufficient conditions are given such that the two integral representations of a coherent upper conditional prevision by the Asymmetric Choquet integral and the Symmetric Choquet integral with respect to Hausdorff outer measures coincide and linearity holds.
For the entire collection see [Zbl 1396.68010].Sugeno integrals and the commutation problemhttps://zbmath.org/1517.683742023-09-22T14:21:46.120933Z"Dubois, Didier"https://zbmath.org/authors/?q=ai:dubois.didier"Fargier, Hélène"https://zbmath.org/authors/?q=ai:fargier.helene"Rico, Agnès"https://zbmath.org/authors/?q=ai:rico.agnesSummary: In decision problems involving two dimensions (like several agents and several criteria) the properties of expected utility ensure that the result of a multicriteria multiperson evaluation does not depend on the order with which the aggregations of local evaluations are performed (agents first, criteria next, or the converse). We say that the aggregations on each dimension \textit{commute}. Ben Amor, Essghaier and Fargier have shown that this property holds when using pessimistic possibilistic integrals on each dimension, or optimistic ones, while it fails when using a pessimistic possibilistic integral on one dimension and an optimistic one on the other. This paper studies and completely solves this problem when Sugeno integrals are used in place of possibilistic integrals, indicating that there are capacities other than possibility and necessity measures that ensure commutation of Sugeno integrals.
For the entire collection see [Zbl 1398.68040].Characterization of \(k\)-Choquet integralshttps://zbmath.org/1517.683782023-09-22T14:21:46.120933Z"Horanská, L'ubomíra"https://zbmath.org/authors/?q=ai:horanska.lubomira"Takáč, Zdenko"https://zbmath.org/authors/?q=ai:takac.zdenkoSummary: In the present paper we characterize the class of all \(n\)-ary \(k\)-Choquet integrals and we find a minimal subset of points in the unit hypercube, the values on which fully determine the \(k\)-Choquet integral.
For the entire collection see [Zbl 1398.68040].Discrete Sugeno integrals and their applicationshttps://zbmath.org/1517.683862023-09-22T14:21:46.120933Z"Rico, Agnès"https://zbmath.org/authors/?q=ai:rico.agnesSummary: This paper is an overview of the discrete Sugeno integrals and their applications when the evaluation scale is a totally ordered set. The various expressions of the Sugeno integrals are presented. Some major characterisation results are recalled: results based on characteristic properties and act-based axiomatisation. We discuss the properties of a preference relation modelling by a Sugeno integral. We also present its power expression to represent a dataset and its interpretation with a set of if-then rules.
For the entire collection see [Zbl 1396.68010].Insights on entanglement entropy in \(1 + 1\) dimensional causal setshttps://zbmath.org/1517.830072023-09-22T14:21:46.120933Z"Keseman, Théo"https://zbmath.org/authors/?q=ai:keseman.theo"Muneesamy, Hans J."https://zbmath.org/authors/?q=ai:muneesamy.hans-j"Yazdi, Yasaman K."https://zbmath.org/authors/?q=ai:yazdi.yasaman-kSummary: Entanglement entropy in causal sets offers a fundamentally covariant characterisation of quantum field degrees of freedom. A known result in this context is that the degrees of freedom consist of a number of contributions that have continuum-like analogues, in addition to a number of contributions that do not. The latter exhibit features below the discreteness scale and are excluded from the entanglement entropy using a `truncation scheme'. This truncation is necessary to recover the standard spatial area law of entanglement entropy. In this paper we build on previous work on the entanglement entropy of a massless scalar field on a causal set approximated by a \(1 + 1\)D causal diamond in Minkowski spacetime. We present new insights into the truncated contributions, including evidence that they behave as fluctuations and encode features specific to a particular causal set sprinkling. We extend previous results in the massless theory to include Rényi entropies and include new results for the massive theory. We also discuss the implications of our work for the treatment of entanglement entropy in causal sets in more general settings.