Recent zbMATH articles in MSC 28https://zbmath.org/atom/cc/282021-11-25T18:46:10.358925ZWerkzeugQuantifying patterns in art and naturehttps://zbmath.org/1472.000472021-11-25T18:46:10.358925Z"Balmages, Amanda"https://zbmath.org/authors/?q=ai:balmages.amanda"Schiffman, Lucille"https://zbmath.org/authors/?q=ai:schiffman.lucille"Lyle, Adam"https://zbmath.org/authors/?q=ai:lyle.adam"Lustig, Elijah"https://zbmath.org/authors/?q=ai:lustig.elijah"Narendra-Babu, Kavya"https://zbmath.org/authors/?q=ai:narendra-babu.kavya"Elul, Tamira"https://zbmath.org/authors/?q=ai:elul.tamiraSummary: Many different types of artworks mimic the properties of natural fractal patterns -- in particular, statistical self-similarity at different scales. Here, we describe examples of abstract art created by us and well-known artists such as Ruth Asawa and Sam Francis that evoke the repetition and variability of biological forms. We review the `drip' paintings of Jackson Pollock that display statistical self-similarity at varying scales, and discuss studies that measured the fractal dimension of Pollock's drip paintings. The contemporary environmental artist Edward Burtynsky who captures aerial photographs of man-created and man-altered landscapes that resemble natural patterns is also discussed. We measure fractal dimension and a second shape parameter -- fractional concavity -- for borders in three of Burtynsky's photographs of man-made landscapes and of biological tissues that resemble his compositions. This specifies the complexity of patterns in Burtynsky's photographs of diverse man-impacted landscapes and underscores their similarity to fractal patterns found in nature.Haar null and Haar meager sets: a survey and new resultshttps://zbmath.org/1472.030512021-11-25T18:46:10.358925Z"Elekes, Márton"https://zbmath.org/authors/?q=ai:elekes.marton"Nagy, Donát"https://zbmath.org/authors/?q=ai:nagy.donatThis is a~survey of results about Haar null sets in Polish groups. The notion of a~Haar measure zero set is well defined in locally compact groups although a~Haar measure is not unique. This notion can be generalized for non-locally compact groups although the Haar measure cannot be generalized for groups that are not locally compact. Two such generalizations are considered: the notion of a~Haar null set and the notion of a~generalized Haar null set. Another notion of smallness considered in the paper is the notion of a~Haar meager set. The Haar meager sets were defined in the literature as a~topological counterpart to the Haar null sets. Every Haar meager set is meager but there are Polish groups in which the converse is not true. However, in locally compact Polish groups the Haar meager sets coincide with the meager sets. All these notions define translation invariant \(\sigma\)-ideals of sets. A~number of characterizations of these notions are presented in the survey. The authors present several results by which they discuss the possibility of generalizations of the following theorems for non-locally compact Polish groups: Fubini's theorem, Kuratowski-Ulam theorem, the Steinhaus theorem, the countable chain condition, and a~decomposition into a~Haar null set and a~Haar meager set. They present a~brief outlook of applications of Haar null sets in various fields of mathematics.Discrete encountershttps://zbmath.org/1472.050032021-11-25T18:46:10.358925Z"Bauer, Craig P."https://zbmath.org/authors/?q=ai:bauer.craig-p``This book provides a refreshing approach to discrete mathematics. The author blends traditional course topics and applications with historical context, pop culture reference, and open problems. It focuses on the historical development of the subject and provides fascinating details of the people behind the mathematics, along with their motivations, deepening readers' appreciation of mathematics. This unique book covers many of the same topics found in traditional textbooks, but does so in an alternative, entertaining style that better captures readers' attention. In addition to standard discrete mathematics material, the author shows the interplay between the discrete and the continuous and includes high-interest topics such as fractals, chaos theory, cellular automata, money-saving financial mathematics, and much more. Not only will readers gain a greater understanding of mathematics and its culture, they will also be encouraged to further explore the subject. Long lists of references at the end of each chapter make this easy.'' (from the presentation of the book).
\par Its chapters are the following: Continuous vs. discrete; Logic; Proof techniques; Practice with proofs; Set theory; Venn diagrams; The functional view of mathematics; The multiplication principle; Permutations; Combinations; Pascal and the arithmetic triangle; Stirling and Bell numbers; The basics of probability; The Fibonacci sequence; The tower of Hanoi; Population models; Financial mathematics (and more); More difference equations; Chaos theory and fractals; Cellular automata; Graph theory; Trees; Relations, partial orderings, and partitions; Index. Each of them ends with a set of exercises and references/further reading. The book also includes many illustrations and portraits from the history of mathematics and ends with a series of 39 colored illustrations. The text's narrative style is that of a popular book, not a dry textbook. Its multidisciplinary approach makes this nice book ideal for liberal arts mathematics classes, leisure reading, or as a reference for professors looking to supplement traditional courses.Lyapunov decomposition in \(\mathrm{d}_{0}\)-algebrashttps://zbmath.org/1472.060262021-11-25T18:46:10.358925Z"Avallone, Anna"https://zbmath.org/authors/?q=ai:avallone.anna"Vitolo, Paolo"https://zbmath.org/authors/?q=ai:vitolo.paoloIt is proved that a closed \(d_0\)-measure on a \(d_0\)-algebra can be decomposed into the sum of a Lyapunov \(d_0\)-measure and an anti-Lyapunov \(d_0\)-measure. The same results are obtained for cancellative BCK-algebras.A higher moment formula for the Siegel-Veech transform over quotients by Hecke triangle groupshttps://zbmath.org/1472.112042021-11-25T18:46:10.358925Z"Fairchild, Samantha"https://zbmath.org/authors/?q=ai:fairchild.samanthaOne can begin with author's description of this research:
`` We compute higher moments of the Siegel-Veech transform over quotients of \(\mathrm{SL}(2, \mathbb R)\) by the Hecke triangle groups. After fixing a normalization of the Haar measure on \(\mathrm{SL}(2, \mathbb R)\) we use geometric results and linear algebra to create explicit integration formulas which give information about densities of \(k\)-tuples of vectors in discrete subsets of \(\mathbb R^2\) which arise as orbits of Hecke triangle groups. This generalizes work of W. Schmidt on the variance of the Siegel transform over \(\mathrm{SL}(2, \mathbb R)/ \mathrm{SL}(2, \mathbb Z)\).''
Such notions as the Siegel-Veech transform, the Hecke triangle group, the \(q\)-geometric Euler totient function, a translation surface, etc., are explained. A brief survey is devoted to results in ``the geometry of numbers, followed by background on translation surfaces, Veech groups, and the \(q\)-geometric Euler totient function''.
Several auxiliary statements are proven. Also, the special attention is given to the problem how to interpret the second main theorem of this paper in terms of a counting problem. This discussion is described with explanations and figures.On intrinsic and extrinsic rational approximation to Cantor setshttps://zbmath.org/1472.112052021-11-25T18:46:10.358925Z"Schleischitz, Johannes"https://zbmath.org/authors/?q=ai:schleischitz.johannes\textit{K. Mahler} [Bull. Aust. Math. Soc. 29, 101--108 (1984; Zbl 0517.10001)] raised the question of Diophantine approximation of points in the middle third Cantor set by points inside or outside it (problems of `intrinsic' or `extrinsic' approximation, respectively) and noted that convergents of the continued fraction expansion to numbers in the Cantor set may or may not be in the set. Here new results on a generalized form of these questions are found, applying to more general multi-dimensional fractal sets generated by iterated function systems under some mild hypotheses. For the Cantor set \(C\) itself, non-trivial lower bounds for the distance between a rational outside \(C\) and \(C\) are found (expressed as usual in terms of the size of the denominator). Other properties of rational numbers in fractal sets are discussed, including an upper bound for the number of rational numbers or algebraic numbers in a fractal set of bounded height or degree.On the intersections of exceptional sets in Borel's normal number theorem and Erdös-Rényi limit theoremhttps://zbmath.org/1472.112162021-11-25T18:46:10.358925Z"Zhang, Mengjie"https://zbmath.org/authors/?q=ai:zhang.mengjieFor each number \( x\in [0,1) \), let \(x=\sum_{n=1}^\infty x_n/2^n\) be the unique non-terminating binary expansion of \(x\), where \(x_n \in \{0,1\}, n \geq 1 \) are called the digits of \(x\). Let \(S_n(x) \) be the summation of the fist \(n\) digits of \(x\). The Borel's normal number theorem says that for Lebesgue almost all \(x \in [0,1) \),
\[
\lim_{n \to \infty} \frac{S_n(x)}{n} =\frac{1}{2}.
\]
For any \(x \in [0,1) \), run-length function is defined by
\[
R_n(x)=\max\{k: x_{i+1}=x_{i+2}=\dots=x_{i+k}=1, 0\leq i \leq n-k \}.
\]
Erdös and Rényi proved that for almost all \(x \in [0,1) \),
\[
\lim_{n \to \infty } \frac{R_n(x)}{\log_2 n}=1.
\]
The author considers the sets
\[
B(\alpha_1,\alpha_2)=\Big\{x \in [0,1): \liminf_{n \to \infty} \frac{S_n(x)}{n}=\alpha_1, \liminf_{n \to \infty} \frac{S_n(x)}{n}=\alpha_2 \Big\},
\]
and
\[
E(\beta_1,\beta_2)=\Big\{x \in [0,1): \liminf_{n \to \infty} \frac{R_n(x)}{\log_2 n}=\beta_1, \liminf_{n \to \infty} \frac{R_n(x)}{\log_2n}=\beta_2 \Big\}.
\]
The following theorem is one of the main results of the article.
Theorem. For any \(0\leq \alpha_1 \leq \alpha_2 \leq 1 \) and \( 0 \leq \beta_1 \leq \beta_2 \leq +\infty, \) we have
\[
\dim_H B(\alpha_1,\alpha_2) \cap E(\beta_1,\beta_2)=\min \Big\{\frac{H(\alpha_1)}{\log 2 }, \frac{H(\alpha_2)}{\log 2 } \Big\},
\]
where \(\dim_H\) denotes the Hausdorff dimension and \(H(\cdot)\) is the entropy function defined by \(H(x)=-x \log x+(1-x) \log(1-x), 0\leq x \leq 1. \)On good universality and the Riemann hypothesishttps://zbmath.org/1472.112322021-11-25T18:46:10.358925Z"Nair, Radhakrishnan"https://zbmath.org/authors/?q=ai:nair.radhakrishnan-b"Verger-Gaugry, Jean-Louis"https://zbmath.org/authors/?q=ai:verger-gaugry.jean-louis"Weber, Michel"https://zbmath.org/authors/?q=ai:weber.michel-j-gIn the paper, the authors deal with a sequence \((k_n)_{n \geq 1}\in \mathbb{N}\) which is \(L^p\)-good universal (later, for brevity, LPGU; if for each dynamical system \((X,\beta,\mu,T)\) and for each \(g \in L^p(X,\beta,\mu)\) the limit \(l_{T,g}=\lim_{N \to \infty}\frac{1}{N}\sum_{n=0}^{N-1}g(T^{k_n}x)\) exists \(\mu\) almost everywhere), uniformly distributed modulo one sequence \((x_n)_{n=1}^N\) (if \(\lim_{N\to \infty}\frac{1}{N}\#\{1 \leq n \leq N:\{x_n\}\in I\}=|I|\) for every interval \(I \subseteq [O,1)\)), and a sequence of natural numbers \((k_n)_{n \geq 1}\) which is Hartman uniformly distributed on \(\mathbb{Z}\) (later, for brevity, HUD; if it is uniformly distributed among the residue classes modulo \(m\) for each natural number \(m>1\), and for each irrational number \(\alpha\), the sequence \((\{k_n \alpha\})_{n \geq 1}\) is uniformly distributed modulo 1; here \(\{ a\}\) denotes the fractional part of number \(a\)).
First several average ergodic type theorems are established. For example, if the meromorphic function \(f\) on the half-plane \(\{z \in \mathbb{C}: \Re(z)>c\}\), \(c \in \mathbb{R}\), satisfying certain additional conditions, and \((k_n)_{n\geq 1}\) is LPGU and HUD, for any \(\{z \in \mathbb{C}: \Re(z)>c\}\setminus \{z \in \mathbb{C}: \Re(z)=c\}\), it holds an equality \[ \lim_{N \to \infty}\frac{1}{N}\sum_{n=0}^{N-1}f(s+iT^{k_n}_{\alpha,\beta}(x))=\frac{\alpha}{\pi}\int_{\mathbb{R}}\frac{f(s+i\tau)}{\alpha^2+(\tau-\beta)^2}d\tau \] for almost all \(x \in \mathbb{R}\). Later, using this theorem, new characterisations of the extended Lindelöf hypothesis are obtained for the Riemann zeta-function, Dirichlet \(L\)-functions, Dedekind zeta-functions over number field, Hurwitz zeta-functions. Later, replacing the LPGU and HUD conditions for sequence \((k_n)_{n \geq 1}\) to be Stoltz sequence, similar limit theorems are proved. Also some examples of LPGU sequences and HUD sequences are given.Differentiability of integrable measurable cocycles between nilpotent groupshttps://zbmath.org/1472.200832021-11-25T18:46:10.358925Z"Cantrell, Michael"https://zbmath.org/authors/?q=ai:cantrell.michaelSummary: We prove an analog for integrable measurable cocycles of \textit{P. Pansu}'s differentiation theorem for Lipschitz maps between Carnot-Carathéodory spaces [Ann. Math. (2) 129, No. 1, 1-60 (1989; Zbl 0678.53042)]. This yields an alternative, ergodic theoretic proof of Pansu's quasi-isometric rigidity theorem for nilpotent groups, answers a question of Tim Austin regarding integrable measure equivalence between nilpotent groups, and gives an independent proof and strengthening of \textit{T. Austin}'s result that integrable measure equivalent nilpotent groups have bi-Lipschitz asymptotic cones [Groups Geom. Dyn. 10, No. 1, 117--154 (2016; Zbl 1376.20042)]. Our main tools are a nilpotent-valued cocycle ergodic theorem and a Poincaré recurrence lemma for nilpotent groups.A version of Hake's theorem for Kurzweil-Henstock integral in terms of variational measurehttps://zbmath.org/1472.260022021-11-25T18:46:10.358925Z"Skvortsov, Valentin"https://zbmath.org/authors/?q=ai:skvortsov.valentin-a"Tulone, Francesco"https://zbmath.org/authors/?q=ai:tulone.francescoSummary: We introduce the notion of variational measure with respect to a derivation basis in a topological measure space and consider a Kurzweil-Henstock-type integral related to this basis. We prove a version of Hake's theorem in terms of a variational measure.Generalized \(s\)-convex functions on fractal setshttps://zbmath.org/1472.260032021-11-25T18:46:10.358925Z"Mo, Huixia"https://zbmath.org/authors/?q=ai:mo.huixia"Sui, Xin"https://zbmath.org/authors/?q=ai:sui.xinSummary: We introduce two kinds of generalized \(s\)-convex functions on real linear fractal sets \(\mathbb{R}^\alpha\) (\(0 < \alpha < 1\)). And similar to the class situation, we also study the properties of these two kinds of generalized \(s\)-convex functions and discuss the relationship between them. Furthermore, some applications are given.On projective functions with bad measurability propertieshttps://zbmath.org/1472.280012021-11-25T18:46:10.358925Z"Kharazishvili, Alexander"https://zbmath.org/authors/?q=ai:kharazishvili.alexander-bSummary: Under Martin's axiom, we show the consistency of the existence of a projective function acting from \(\mathbb{R}\) into itself, which has an extremely bad measurability property with respect to a wide class of measures on \(\mathbb{R}\).The behavior of small sets under the product operationhttps://zbmath.org/1472.280022021-11-25T18:46:10.358925Z"Kirtadze, Aleks"https://zbmath.org/authors/?q=ai:kirtadze.aleks-pSummary: For invariant (quasi-invariant) \(\sigma\)-finite measures on an uncountable group \((G,\cdot)\), the behavior of measure zero sets with respect to the product operation is studied.Nonspectrality of certain self-affine measures on \(\mathbb{R}^3\)https://zbmath.org/1472.280032021-11-25T18:46:10.358925Z"Gao, Gui-Bao"https://zbmath.org/authors/?q=ai:gao.guibaoSummary: We will determine the nonspectrality of self-affine measure \(\mu_{B, D}\) corresponding to \(B = \operatorname{diag} [p_1, p_2, p_3]\) (\( p_1 \in(2 \mathbb Z + 1) \smallsetminus \{\pm 1 \}\), \(p_2 \in 2 \mathbb Z \smallsetminus \{0 \}\)), and \(D = \{0, e_1, e_2, e_3 \}\) in the space \(\mathbb{R}^3\) is supported on \(T(B, D)\), where \(e_1, e_2\), and \(e_3\) are the standard basis of unit column vectors in \(\mathbb{R}^3\), and there exist at most 4 mutually orthogonal exponential functions in \(L^2(\mu_{B, D})\), where the number 4 is the best. This generalizes the known results on the spectrality of self-affine measures.The Kakeya needle problem and the existence of Besicovitch and Nikodym sets for rectifiable setshttps://zbmath.org/1472.280042021-11-25T18:46:10.358925Z"Chang, Alan"https://zbmath.org/authors/?q=ai:chang.alan"Csörnyei, Marianna"https://zbmath.org/authors/?q=ai:csornyei.mariannaSummary: We solve the Kakeya needle problem and construct a Besicovitch and a Nikodym set for rectifiable sets.A proof of Carleson's \(\varepsilon^2\)-conjecturehttps://zbmath.org/1472.280052021-11-25T18:46:10.358925Z"Jaye, Benjamin"https://zbmath.org/authors/?q=ai:jaye.benjamin-j"Tolsa, Xavier"https://zbmath.org/authors/?q=ai:tolsa.xavier"Villa, Michele"https://zbmath.org/authors/?q=ai:villa.micheleSummary: In this paper we provide a proof of the Carleson \(\varepsilon^2\)-conjecture. This result yields a characterization (up to exceptional sets of zero length) of the tangent points of a Jordan curve in terms of the finiteness of the associated Carleson \(\varepsilon^2\)-square function.Distance between sets -- a surveyhttps://zbmath.org/1472.280062021-11-25T18:46:10.358925Z"Conci, Aura"https://zbmath.org/authors/?q=ai:conci.aura"Kubrusly, Carlos"https://zbmath.org/authors/?q=ai:kubrusly.carlos-sSummary: The purpose of this paper is to give a survey on the notions of distance between subsets either of a metric space or of a measure space, including definitions, a classification, and a discussion of the best-known distance functions, which is followed by a review on applications used in many areas of knowledge, ranging from theoretical to practical applications.Spectrality of generalized Sierpinski-type self-affine measureshttps://zbmath.org/1472.280072021-11-25T18:46:10.358925Z"Liu, Jing-Cheng"https://zbmath.org/authors/?q=ai:liu.jingcheng"Zhang, Ying"https://zbmath.org/authors/?q=ai:zhang.ying.3|zhang.ying.4|zhang.ying|zhang.ying.1|zhang.ying.5|zhang.ying.2"Wang, Zhi-Yong"https://zbmath.org/authors/?q=ai:wang.zhiyong.2|wang.zhiyong.1"Chen, Ming-Liang"https://zbmath.org/authors/?q=ai:chen.ming-liangSummary: In this work, we study the spectral property of generalized Sierpinski-type self-affine measures \(\mu_{M,D}\) on \(\mathbb{R}^2\) generated by an expanding integer matrix \(M\in M_2(\mathbb{Z})\) with \(\det(M)\in 3\mathbb{Z}\) and a non-collinear integer digit set \(D=\{(0,0)^t,(\alpha_1,\alpha_2)^t,(\beta_1,\beta_2)^t\}\) with \(\alpha_1\beta_2-\alpha_2\beta_1\in 3\mathbb{Z}\). We give the sufficient and necessary conditions for \(\mu_{M,D}\) to be a spectral measure, i.e., there exists a countable subset \(\Lambda\subset \mathbb{R}^2\) such that \(E(\Lambda)=\{e^{2\pi i\langle\lambda,x \rangle}:\lambda\in\Lambda\}\) forms an orthonormal basis for \(L^2(\mu_{M,D})\). This completely settles the spectrality of the self-affine measure \(\mu_{M,D}\).Geometric functionals of fractal percolationhttps://zbmath.org/1472.280082021-11-25T18:46:10.358925Z"Klatt, Michael A."https://zbmath.org/authors/?q=ai:klatt.michael-andreas"Winter, Steffen"https://zbmath.org/authors/?q=ai:winter.steffenSummary: Fractal percolation exhibits a dramatic topological phase transition, changing abruptly from a dust-like set to a system-spanning cluster. The transition points are unknown and difficult to estimate. In many classical percolation models the percolation thresholds have been approximated well using additive geometric functionals, known as intrinsic volumes. Motivated by the question of whether a similar approach is possible for fractal models, we introduce corresponding geometric functionals for the fractal percolation process \(F\). They arise as limits of expected functionals of finite approximations of \(F\). We establish the existence of these limit functionals and obtain explicit formulas for them as well as for their finite approximations.n-fractals in partial metric spaceshttps://zbmath.org/1472.280092021-11-25T18:46:10.358925Z"Minirani, S."https://zbmath.org/authors/?q=ai:minirani.sSummary: A metric space with nonzero self distance gives us a generalization of metric spaces which is coined as a partial metric space. In this paper we discuss the construction of an n-fractal which is the attractor of a collection of n-IFSs in a partial metric space.
For the entire collection see [Zbl 1468.65003].Iterated inversion system: an algorithm for efficiently visualizing Kleinian groups and extending the possibilities of fractal arthttps://zbmath.org/1472.280102021-11-25T18:46:10.358925Z"Nakamura, Kento"https://zbmath.org/authors/?q=ai:nakamura.kentoSummary: Kleinian group theory is a branch of mathematics. A visualized Kleinian group often presents a beautiful fractal structure and provides clues for understanding Möbius transformations the mathematical properties of the group. However, it often takes much time to render images of Kleinian groups on a computer. Thus, we propose an efficient algorithm for visualizing some kinds of Kleinian groups: the Iterated Inversion System (IIS), which enables us to render images of Kleinian groups composed of inversions as circles or spheres in real-time. Real-time rendering has various applications; for example, the IIS can be used for experimentation in Kleinian group theory and the creation of mathematical art. The algorithm can also be used to draw both two-dimensional and three-dimensional fractals. The algorithm can extend the possibilities of art originating from Kleinian groups. In this paper, we discuss Kleinian fractals from an artistic viewpoint.A countable fractal interpolation scheme involving Rakotch contractionshttps://zbmath.org/1472.280112021-11-25T18:46:10.358925Z"Pacurar, Cristina Maria"https://zbmath.org/authors/?q=ai:pacurar.cristina-mariaSummary: The main result of this paper states that for a given countable system of data \(\varDelta\), there exists a countable iterated function system consisting of Rakotch contractions, such that its attractor is the graph of a fractal interpolation function corresponding to \(\varDelta\). In this way, on the one hand, we generalize a result due to Secelean (see Univ Beograd Publ Elektrotehn Fak Ser Mat 14:11-19, 2003) by considering countable systems consisting of Rakotch contractions rather than Banach contractions. On the other hand, we generalize a result due to \textit{S. Ri} [Indag. Math., New Ser. 29, No. 3, 962--971 (2018; Zbl 1392.28011)] by considering countable (rather than finite) systems consisting of Rakotch contractions. Some exemplifications are provided.Contractive iterated function systems enriched with nonexpansive mapshttps://zbmath.org/1472.280122021-11-25T18:46:10.358925Z"Strobin, Filip"https://zbmath.org/authors/?q=ai:strobin.filipSummary: Motivated by a recent paper of Leśniak and Snigireva [\textit{Iterated function systems enriched with symmetry}, preprint], we investigate the properties of the semiattractor \(A_{\mathcal{F}\cup\mathcal{G}}^*\) of an IFS \(\mathcal{F}\) enriched by some other IFS \(\mathcal{G}\). We show that in natural cases, the semiattractor \(A_{\mathcal{F}\cup\mathcal{G}}^*\) is in fact the attractor of certain IFSs related naturally with the IFSs \(\mathcal{F}\) and \(\mathcal{G}\). We also give an example when \(A_{\mathcal{F}\cup\mathcal{G}}^*\) is not compact, yet still being the attractor of considered related IFSs. Finally, we use presented machinery to prove that the so called \textit{lower transition attractors} due to Vince are semiattractors of enriched IFSs.Spectrality of a class of self-affine measures and related digit setshttps://zbmath.org/1472.280132021-11-25T18:46:10.358925Z"Yang, Ming-Shu"https://zbmath.org/authors/?q=ai:yang.ming-shuSummary: This work investigates the spectrality of a self-affine measure \(\mu_{M,D}\) and the related digit set \(D\) in the case when \(|\mathrm{det}(M)|=p^{\alpha}\) is a prime power and \(|D|=p\) is a prime, where \(\alpha\in{\mathbb{N}}\), and \(\mu_{M,D}\) is generated by an expanding matrix \(M\in M_n({\mathbb{Z}})\) and a digit set \(D\subset\mathbb{Z}^n\) of cardinality |\(D\)|. We obtain that \(\mu_{M,D}\) is a spectral measure and \(D\) is a spectral set if one nonzero element in \(D\) satisfies certain mild conditions. This is based on the property of vanishing sums of roots of unity and a residue system in number theory. The result here extends the corresponding known results and provides some supportive evidence for a conjecture of Dutkay, Han, and Jorgensen.Application of complex Sugeno type fuzzy set-valued integrals in classifier fusion methodhttps://zbmath.org/1472.280142021-11-25T18:46:10.358925Z"Peng, Dejun"https://zbmath.org/authors/?q=ai:peng.dejun"Ma, Jing"https://zbmath.org/authors/?q=ai:ma.jing"Ma, Shengquan"https://zbmath.org/authors/?q=ai:ma.shengquanSummary: This paper presents the concept of complex Sugeno fuzzy set-valued integrals and reviews its basic properties. We discuss specific details of applying complex Sugeno type fuzzy set-valued integrals in classifier fusion, and we put the operational steps into practice through two examples, through bidirectional membership evaluation and fuzzy inference method. The final classification result reflects the objectivity and comprehensiveness. The results of this study demonstrate that the complex Sugeno fuzzy set-valued integral fusion algorithm is clearly superior to other existing methods.A removability theorem for Sobolev functions and detour setshttps://zbmath.org/1472.300102021-11-25T18:46:10.358925Z"Ntalampekos, Dimitrios"https://zbmath.org/authors/?q=ai:ntalampekos.dimitriosRemovability of compact sets for continuous Sobolev functions is studied, i.e., if \(K\) be a compact set in \(\mathbb{R}^n\) and \(f \in C(\mathbb{R}^n)\cap W^{1,p}(\mathbb{R}^n\setminus K)\), is \(f \in W^{1,p}(\mathbb{R}^n)\)? The problem is intimately connected to the removability problem for quasiconformal maps: If \(f: U \rightarrow\mathbb{R}^n\) is a homeomorphism and \(f|U \setminus K\) is quasiconformal, is \(f\) quasiconformal in \(U\)? The stronger removability, without the continuity assumption, for Sobolev functions has been studied by [\textit{P. Koskela}, Ark. Mat. 37, No. 2, 291--304 (1999; Zbl 1070.46502)]. In the plane it has been shown that boundaries of domains \(\Omega\) satisfying the quasihyperbolic boundary condition and, in particular, boundaries of John domains are removable [\textit{P. Jones} and \textit{S. Smirnov}, Ark. Math. 38, No. 2, 363--379 (2000)]. The author concentrates on sets \(K\) which have infinitely many complementary components. A typical such set is the standard \(1/3\)-Sierpinski carpet \(S\) in the plane which is not \(W^{1,p}\)-removable for any \(p \geq 1\) and so the author focuses on the Sierpinski and Apollonian gaskets. The first is constructed using triangles and the latter using disks. It is shown that the planar Sierpinski and Apollonia gaskets are removable for \(p > 2\). The proof, which holds in \(\mathbb{R}^n\), is based on the result concerning detour sets. This means, roughly speaking, that the set \(K\) has the property that for almost every line \(L\) intersecting \(K\) there is a path which intersects only finitely many complementary components of \(K\) and still remains arbitrarily close to \(L\). In addition to this the complementary components \(D\) of \(K\) need to be uniformly Hölder. It is shown that for a Hölder domain each point \(x \in \partial D\) can be reached from the base point by a quasihyperbolic geodesic, see also [\textit{O. Martio} and \textit{J. Väisälä}, Pure Appl. Math. Q. 7, No. 2, 395--409 (2011; Zbl 1246.30041)]. The results are used to obtain equivalent conditions for a homeomorphism \(f: \mathbb{R}^2 \rightarrow \mathbb{R}^2\) that is quasiconformal in \(\mathbb{R}^2 \setminus K\) where \(K\) is a Sierpinki gasket to be quasiconformal in \(\mathbb{R}^2\). The paper also contains a nice overview of removability results for Sobolev functions outside different Sierpinski-type sets.Potential theory on Sierpiński carpets. With applications to uniformizationhttps://zbmath.org/1472.310012021-11-25T18:46:10.358925Z"Ntalampekos, Dimitrios"https://zbmath.org/authors/?q=ai:ntalampekos.dimitriosThe goal of this book is to establish a uniformization result for planar Sierpiński carpets. In this purpose it continues a tradition of research in the uniformization of metric spaces, as the author expounds in the seven-page introduction (Chapter 1).
The necessary background is prepared in Chapter 2. Let \(\Omega\subset {\mathbb C}\) be an open Jordan region, \((Q_i)_{i\in {\mathbb N}}\) a family of open Jordan regions compactly contained in \(\Omega\) with disjoint closures. If \(S:=\bar{\Omega}\setminus\cup_{i\in {\mathbb N}}Q_i\) has empty interior and is locally connected, it is called a \textit{Sierpiński carpet}, the \(Q_i\)'s its \textit{peripheral disks}. A key notion for the study is that of a \textit{Sobolev function} on \(S\). Its definition applies the concept of an \textit{upper gradient} for a function. \par Let \(g:S\cap\Omega\rightarrow {\mathbb R}\cup\{-\infty,\infty\}\). A nonnegative sequence \((\rho(Q_i))_{i\in {\mathbb N}}\) is called an \textit{upper gradient} for \(g\) if there exists a family \(\Gamma_0\) of paths in \(\Omega\) with \(\text{mod}(\Gamma_0)=0\) such that for all paths \(\gamma\not\in\Gamma_0\) in \(\Omega\) and \(x, y\in\gamma\cap S\) it holds \(g(x), g(y)\neq\pm\infty\) and \[ |g(x)-g(y)|\le\sum_{i: Q_i\cap\gamma\neq\emptyset}\rho(Q_i)\,. \] Here, the \textit{carpet modulus} \(\text{mod}(\Gamma_0)\) is defined as \(\inf\sum_{i\in {\mathbb N}\cup\{0\}}\sigma(Q_i)^2\) (\(Q_0:={\mathbb C}\setminus \bar{\Omega}\)), where \((\sigma(Q_i))_{i\in {\mathbb N}\cup\{0\}}\) is a nonnegative admissible sequence, meaning that \[ \sum_{i: Q_i\cap\gamma\neq\emptyset}\sigma(Q_i)\ge 1 \] for all \(\gamma\in \Gamma_0\) with \(\mathcal{H}^1 (\gamma\cap S)=0\) (\(\mathcal{H}\) denoting the Hausdorff measure). \par Setting \(M_{Q_i}(g)=\sup_{x\in\partial Q_i}g(x)\), \(m_{Q_i}(g)=\inf_{x\in \partial Q_i}g(x)\), \(\text{osc}_{Q_i}(g)=M_{Q_i}(g)-m_{Q_i}(g)\) (\(i\in {\mathbb N}\)), \(g\) is called a \textit{local Sobolev function} if for every open ball \(B\) relatively compact in \(\Omega\), \[ \sum_{i\in I_B} M_{Q_i}(g)^2 \text{diam}(Q_i)^2 <\infty\;\text{and}\; \sum_{i\in I_B}\text{osc}_{Q_i}(g)^2 <\infty \] (\(I_B:=\{i\in {\mathbb N}: B\cap Q_i\neq\emptyset\}\)) and \((\text{osc}_{Q_i} (g))_{i\in {\mathbb N}}\) is an upper gradient for \(g\). If these inequalities hold for the full sums over \(i\in {\mathbb N}\), the Sobolev property is termed \textit{global}. \par Now, a local Sobolev function \(u\) is called \textit{carpet-harmonic} if for every open set \(V\) relatively compact in \(\Omega\) and every Sobolev function \(\zeta\) supported on \(V\) it holds \[ D_V(u)\le D_V(u+\zeta)\,, \] the so-called Dirichlet energy functional being defined by \(D_V(f)=\sum_{i\in I_V}\text{osc}_{Q_i}(f)^2 \in [0,\infty ]\). The author presents several properties of carpet-harmonic functions, as the Caccioppoli inequality, for instance (Section 8 of Chapter 2).
\par In the sequel we indicate how this notion of harmonicity is applied to obtain the uniformization result. After fixing four points on \(\partial\Omega\), this boundary is decomposed into closed sides \(\Theta_1,\ldots,\Theta_4\), enumerated counterclockwise, where \(\Theta_1\) and \(\Theta_3\) are opposite. Calling a Sobolev function \(g\) on \(S\), continuous up to \(\partial\Omega\), \textit{admissible for the free boundary problem} if \(g|_{\Theta_1}=0\) and \(g|_{\Theta_3}=1\), the author shows that there is a unique carpet-harmonic function \(u\) that minimizes \(D_\Omega (g)\) over all admissible functions \(g\). This \(u\) is continuous up to \(\partial\Omega\), \(u|_{\Theta_1}=0\), and \(u|_{\Theta_3}=1\).
\par By ``integrating'' the ``gradient'' of \(u\) along its level sets, as the author puts it, the \textit{conjugate function} \(v\) of \(u\) is defined (Section 6 of Chapter 3). The function \(v:S \rightarrow [0, D_\Omega (u)]\) is continuous, \(v|_{\Theta_2}=0\), \(v|_{\Theta_4}=D_\Omega (u)\), but it is unclear if this \(v\) is carpet-harmonic. In any case, \(f:=(u,v)\) is the desired uniformization map. It maps \(S\) homeomorphically onto a so-called square Sierpiński carpet, that is, a carpet whose peripheral disks are squares and whose underlying open Jordan region is a rectangle. A condition, however, for this uniformization result is that the peripheral disks \(Q_i\) be uniformly quasiround (there exists \(K_0\ge 1\) such that for each \(Q_i\) there are concentric balls \(B(x,r)\), \(B(x,R)\) with \(B(x,r)\subset Q_i\subset B(x,R)\) and \(\frac{R}{r}\le K_0\)) and uniformly Ahlfors 2-regular (there exists \(K_1>0\) such that for every \(Q_i\) and every \(B(x,r)\) with \(x\in Q_i\) and \(r<\text{diam}(Q_i)\) it holds \(\mathcal{H}^2 (B(x,r)\cap Q_i)\ge K_1 r^2\)). Furthermore, this \(f\) has a property called \textit{carpet-quasiconformality}. Finally, the author presents certain refinements under additional geometric assumptions.Approximate tangents, harmonic measure, and domains with rectifiable boundarieshttps://zbmath.org/1472.310092021-11-25T18:46:10.358925Z"Mourgoglou, Mihalis"https://zbmath.org/authors/?q=ai:mourgoglou.mihalisThis article discusses the connection between approximate tangents, harmonic measures and domains with rectifiable boundaries. In the first result of the paper it is established that if \(E\subset\mathbb{R}^{n+1}\) is closed, \(0<s<1/3\) and \(\mathcal{T}_m(E)\subset E\) be the set of all points \(x\in E\) such that:
(i) there exists an \(s\)-approximate tangent \(m\)-plane \(V_x\) for \(E\) at \(x\);
(ii) \(E\) satisfies the weak lower Ahlfors-David \(m\)-regularity condition at \(x\).
Then, there exists a countable collection of bounded Lipschitz graphs \(\{\Gamma_j\}_{j\geq 1}\) so that \(\mathcal{T}_m(E)\subset \bigcup_{j\geq 1}\Gamma_j\). In particular, \(\mathcal{T}_m(E)\) is \(m\)-rectifiable.
In the second result of the article it is obtained that if \(0<s<1/\sqrt{90}\), then there exist two countable collections of bounded Lipschitz domains \(\{\Omega_j^\pm\}_{j\geq 1}\) such that \(\Omega_j^+\cap \Omega_j^-=\emptyset\), \(\mathcal{T}_n(E)\cap \Omega_j^+=\mathcal{T}_n\cap\Omega_j^-\) and \(\mathcal{T}_m(E)\subset \bigcup_{j\geq 1}\Omega_j^\pm\).
Further characterizations of the countable collections of bounded Lipschitz domains \(\{\Omega_j^\pm\}_{j\geq 1}\) are provided in the article.Pseudo \(S\)-asymptotically Bloch type periodicity with applications to some evolution equationshttps://zbmath.org/1472.341162021-11-25T18:46:10.358925Z"Chang, Yong-Kui"https://zbmath.org/authors/?q=ai:chang.yong-kui"Wei, Yanyan"https://zbmath.org/authors/?q=ai:wei.yanyanSummary: This paper is mainly focused upon the pseudo \(S\)-asymptotically Bloch type periodicity and its applications. Firstly, a new notion of pseudo \(S\)-asymptotically Bloch type periodic functions is introduced, and some fundamental properties on pseudo \(S\)-asymptotically Bloch type periodic functions are established. Then, the notion and properties of weighted pseudo \(S\)-asymptotically Bloch type periodic functions are similarly presented. Finally, the obtained results are applied to investigate the existence and uniqueness of pseudo \(S\)-asymptotically Bloch type periodic mild solutions for some semi-linear evolution equations in Banach spaces.On uniqueness of solutions to the boundary value problems on the Sierpiński gaskethttps://zbmath.org/1472.351992021-11-25T18:46:10.358925Z"Stegliński, Robert"https://zbmath.org/authors/?q=ai:steglinski.robertSummary: Using the monotonicity methods, we obtain conditions for the existence of the unique weak solution of Dirichlet problem
\[
\begin{cases}
\Delta u(x)+a(x)u(x)=f(x,u(x))\quad x\in V\backslash V_0\\
u_{|V_0}=0,
\end{cases}
\]
considered on the Sierpiński gasket. We argue for the optimality of some assumptions. Some Lyapunov-type inequalities are also given. We also consider a problem with parameters and we prove the theorem about continuous dependence on parameters.Asymptotic behavior of density in the boundary-driven exclusion process on the Sierpinski gaskethttps://zbmath.org/1472.352082021-11-25T18:46:10.358925Z"Chen, Joe P."https://zbmath.org/authors/?q=ai:chen.joe-p-j|chen.joe-p"Gonçalves, Patrícia"https://zbmath.org/authors/?q=ai:goncalves.patricia-cSummary: We derive the macroscopic laws that govern the evolution of the density of particles in the exclusion process on the Sierpinski gasket in the presence of a variable speed boundary. We obtain, at the hydrodynamics level, the heat equation evolving on the Sierpinski gasket with either Dirichlet or Neumann boundary conditions, depending on whether the reservoirs are fast or slow. For a particular strength of the boundary dynamics we obtain linear Robin boundary conditions. As for the fluctuations, we prove that, when starting from the stationary measure, namely the product Bernoulli measure in the equilibrium setting, they are governed by Ornstein-Uhlenbeck processes with the respective boundary conditions.Well-posedness of Weinberger's center of mass by Euclidean energy minimizationhttps://zbmath.org/1472.352562021-11-25T18:46:10.358925Z"Laugesen, R. S."https://zbmath.org/authors/?q=ai:laugesen.richard-snyderSummary: The center of mass of a finite measure with respect to a radially increasing weight is shown to exist, be unique, and depend continuously on the measure.Application of local fractional series expansion method to solve Klein-Gordon equations on Cantor setshttps://zbmath.org/1472.353422021-11-25T18:46:10.358925Z"Yang, Ai-Min"https://zbmath.org/authors/?q=ai:yang.aimin"Zhang, Yu-Zhu"https://zbmath.org/authors/?q=ai:zhang.yuzhu"Cattani, Carlo"https://zbmath.org/authors/?q=ai:cattani.carlo"Xie, Gong-Nan"https://zbmath.org/authors/?q=ai:xie.gongnan"Rashidi, Mohammad Mehdi"https://zbmath.org/authors/?q=ai:rashidi.mohammad-mehdi"Zhou, Yi-Jun"https://zbmath.org/authors/?q=ai:zhou.yijun"Yang, Xiao-Jun"https://zbmath.org/authors/?q=ai:yang.xiao-junSummary: We use the local fractional series expansion method to solve the Klein-Gordon equations on Cantor sets within the local fractional derivatives. The analytical solutions within the nondifferential terms are discussed. The obtained results show the simplicity and efficiency of the present technique with application to the problems of the liner differential equations on Cantor sets.Mild solutions are weak solutions in a class of (non)linear measure-valued evolution equations on a bounded domainhttps://zbmath.org/1472.354212021-11-25T18:46:10.358925Z"Evers, Joep H. M."https://zbmath.org/authors/?q=ai:evers.joep-h-mSummary: We study the connection between mild and weak solutions for a class of measure-valued evolution equations on the bounded domain \([0,1]\). Mass moves, driven by a velocity field that is either a function of the spatial variable only, \(v=v(x0\), or depends on the solution \(\mu\) itself: \(v=v[\mu](x)\). The flow is stopped at the boundaries of \([0,1]\), while mass is gated away by a certain right-hand side. In previous works \textit{J. H. M. Evers} et al. [J. Differ. Equations 259, No. 3, 1068--1097 (2015; Zbl 1315.35057); SIAM J. Math. Anal. 48, No. 3, 1929--1953 (2016; Zbl 1342.28004)], we showed the existence and uniqueness of appropriately defined mild solutions for \(v=v(x)\) and \(v=v[\mu](x0\), respectively. In the current paper we define weak solutions (by specifying the weak formulation and the space of test functions). The main result is that the aforementioned mild solutions are weak solutions, both when \(v=v(x)\) and when \(v=v[\mu](x)\).Configuration sets with nonempty interiorhttps://zbmath.org/1472.354702021-11-25T18:46:10.358925Z"Greenleaf, Allan"https://zbmath.org/authors/?q=ai:greenleaf.allan"Iosevich, Alex"https://zbmath.org/authors/?q=ai:iosevich.alex"Taylor, Krystal"https://zbmath.org/authors/?q=ai:taylor.krystalSummary: A theorem of Steinhaus states that if \(E\subset\mathbb{R}^d\) has positive Lebesgue measure, then the difference set \(E-E\) contains a neighborhood of 0. Similarly, if \(E\) merely has Hausdorff dimension \(\dim_{\mathcal{H}}(E)>(d+1)/2\), a result of Mattila and Sjölin states that the distance set \(\varDelta(E)\subset\mathbb{R}\) contains an open interval. In this work, we study such results from a general viewpoint, replacing \(E-E\) or \(\varDelta (E)\) with more general \(\varPhi\)-configurations for a class of \(\varPhi:\mathbb{R}^d\times\mathbb{R}^d\rightarrow\mathbb{R}^k\), and showing that, under suitable lower bounds on \(\dim_{\mathcal{H}}(E)\) and a regularity assumption on the family of generalized Radon transforms associated with \(\varPhi\), it follows that the set \(\varDelta_\varPhi(E)\) of \(\varPhi\)-configurations in \(E\) has nonempty interior in \(\mathbb{R}^k\). Further extensions hold for \(\varPhi\)-configurations generated by two sets, \(E\) and \(F\), in spaces of possibly different dimensions and with suitable lower bounds on \(\dim_{\mathcal{H}}(E)+\dim_{\mathcal{H}}(F)\).A nonsingular action of the full symmetric group admits an equivalent invariant measurehttps://zbmath.org/1472.370022021-11-25T18:46:10.358925Z"Nessonov, Nilolay"https://zbmath.org/authors/?q=ai:nessonov.nilolaySummary: Let \(\overline{\mathfrak{S}}_\infty\) denote the set of all bijections of natural numbers. Consider an action of \(\overline{\mathfrak{S}}_\infty\) on a measure space \((X,\mathfrak{M},\mu)\), where \(\mu\) is an \(\overline{\mathfrak{S}}_\infty\)-quasi-invariant measure. We prove that there exists an \(\overline{\mathfrak{S}}_\infty\)-invariant measure equivalent to \(\mu\).A theorem of Besicovitch and a generalization of the Birkhoff ergodic theoremhttps://zbmath.org/1472.370052021-11-25T18:46:10.358925Z"Hagelstein, Paul"https://zbmath.org/authors/?q=ai:hagelstein.paul-alton"Herden, Daniel"https://zbmath.org/authors/?q=ai:herden.daniel"Stokolos, Alexander"https://zbmath.org/authors/?q=ai:stokolos.alexander-mThe Lebesgue differentiation theorem states that for a.e. \(x\in \mathbb{R}^2\) the averages of an integrable function \(f\) over discs shrinking to \(x\) tend to \(f(x)\). Instead of discs one may use rectangles. If we restrict the class of rectangles to \(B_2\), i.e., the family of rectangles whose sides are parallel to the coordinate axes, then one can obtain a similar result. \textit{A. S. Besicovitch} [Fundam. Math. 25, 209--216 (1935; JFM 61.0256.01)]
proved that if \(f\) is integrable on \(\mathbb{R}^2\) and its associated strong maximal function \[M_Sf=\sup_{x\in R\in B_2}\frac{1}{|R|} \int_R |f|\] is finite a.e. then for a.e. \(x\) we have \[\lim_{j\to \infty} \frac{1}{|R_j|}\int_{R_j} f=f(x),\] where \(\{R_j\}\) is a sequence of rectangles from \(B_2\) shrinking to \(x\).
The present article contains an analogue of Besicovich's result in the context of ergodic theory. The strong maximal operator \(M_S\) is replaced by an ergodic maximal operator \(T^*f\).Markov random walks on homogeneous spaces and Diophantine approximation on fractalshttps://zbmath.org/1472.370082021-11-25T18:46:10.358925Z"Prohaska, Roland"https://zbmath.org/authors/?q=ai:prohaska.roland"Sert, Cagri"https://zbmath.org/authors/?q=ai:sert.cagriAuthors' abstract: In the first part, using the recent measure classification results of Eskin-Lindenstrauss, we give a criterion to ensure a.s. equidistribution of empirical measures of an i.i.d. random walk on a homogeneous space \(G/\Gamma \). Employing renewal and joint equidistribution arguments, this result is generalized in the second part to random walks with Markovian dependence. Finally, following a strategy of \textit{D. Simmons} and \textit{B. Weiss} [Invent. Math. 216, No. 2, 337--394 (2019; Zbl 1454.22009)], we apply these results to Diophantine approximation problems on fractals and show that almost every point with respect to Hausdorff measure on a graph directed self-similar set is of generic type, so, in particular, well approximable.Extreme partitions of a Lebesgue space and their application in topological dynamicshttps://zbmath.org/1472.370152021-11-25T18:46:10.358925Z"Bułatek, Wojciech"https://zbmath.org/authors/?q=ai:bulatek.wojciech"Kamiński, Brunon"https://zbmath.org/authors/?q=ai:kaminski.brunon"Szymański, Jerzy"https://zbmath.org/authors/?q=ai:szymanski.jerzySummary: It is shown that any topological action \(\Phi\) of a countable orderable and amenable group \(G\) on a compact metric space \(X\) and every \(\Phi \)-invariant probability Borel measure \(\mu\) admit an extreme partition \(\zeta\) of \(X\) such that the equivalence relation \(R_{\zeta}\) associated with \(\zeta\) contains the asymptotic relation \(A(\Phi)\) of \(\Phi . As\) an application of this result and the generalized Glasner theorem it is proved that \(A(\Phi)\) is dense for the set \(E_{\mu}(\Phi)\) of entropy pairs.Hausdorff dimension of frequency sets of univoque sequenceshttps://zbmath.org/1472.370162021-11-25T18:46:10.358925Z"Li, Yao-Qiang"https://zbmath.org/authors/?q=ai:li.yaoqiangSummary: We study the set \(\Gamma\) consisting of univoque sequences, the set \(\Lambda\) consisting of sequences in which the lengths of consecutive zeros and consecutive ones are bounded, and their frequency subsets \(\Gamma_a\), \(\underline{\Gamma}_a\), \(\overline{\Gamma}_a\) and \(\Lambda_a\), \(\underline{\Lambda}_a\), \(\overline{\Lambda}_a\) consisting of sequences respectively in \(\Gamma\) and \(\Lambda\) with frequency, lower frequency and upper frequency of zeros equal to some \(a\in[0,1]\). The Hausdorff dimension of all these sets are obtained by studying the dynamical system \((\Lambda^{(m)},\sigma)\) where \(\sigma\) is the shift map and \(\Lambda^{(m)}=\left\{w\in\{0,1\}^{\mathbb{N}}:w\text{ does not contain }0^m\text{ or }1^m\right\}\) for integer \(m\geq 3\), studying the Bernoulli-type measures on \(\Lambda^{(m)}\) and finding out the unique equivalent \(\sigma\)-invariant ergodic probability measure.The topological entropy of stable sets for bi-orderable amenable groupshttps://zbmath.org/1472.370202021-11-25T18:46:10.358925Z"Li, Jie"https://zbmath.org/authors/?q=ai:li.jie"Yu, Tao"https://zbmath.org/authors/?q=ai:yu.taoThe authors deal with the topological entropy of stable sets for countable discrete infinite bi-orderable amenable group actions. Let \((X,d)\) be a compact metric space and bi-orderable group \(G\) continuously act on \(X\). Then \((X, G)\) is called a \(G\)-system. Let \(S\) be an infinite subset of \(G\). A pair \((x, y)\in X\times X\) is called an \(S\)-asymptotic pair if for each \(\varepsilon > 0\), there are only finitely many elements \(s\in S\) with \(d(sx, sy) >\varepsilon\). Let \((X, G)\) be a \(G\)-system, \(\mu\) be a \(G\)-invariant ergodic measure with \(h_{\mu}(G) > 0\) and \(\Phi\) the algebraic past of \(G\). The authors prove that if \(S\) is a \(\Phi\)-admissible subsemigroup of \(G\) and \(\cup_{n=1}^{\infty}F_n \subset (\Phi\cap\{e_G\})\), then for \(\mu\)-a.e. \(x\in X\), there exists a closed subset \(A(x)\subseteq W_S(x, G)\) such that \(h_{\text{top}}(A(x), {F_n})\geq h_{\mu}(G)\), where \(h_{\mu}(G)\) and \(h_{\text{top}}(A(x),{F_n})\) denote the measure-theoretic entropy of group \(G\) and the Bowen entropy of \(A(x)\) with respect to the two-sided Folner sequence \(\{Fn\}\) respectively. Then they prove the following:
Theorem. Let \((X, G)\) be a \(G\)-system and \(\mu\) be a \(G\)-invariant ergodic measure with \(h_{\mu}(G) > 0\). If \(S\) is an infinite subset of \(G\) with \(|\{s\in S : s <_{\Phi} g\}| <\infty\) for each \(g\in G\), then for \(\mu\)-a.e. \(x\in X\), there exists a closed subset \(E(x)\subseteq \overline{W_S(x, G)}\) such that
(1) \(E(x)\) is a weakly mixing set;
(2) \(h_{\text{top}}(A(x), {F_n})\geq h_{\mu}(G)\).
Let \(G\) be a finitely generated torsion-free nilpotent group. It is proved that there exist algebraic \(\Phi\) and \(\Phi\)-admissible subsemigroup \(S\) of \(G\) such that \(|\{s\in S : s <_{\Phi} g\}| <\infty\) for each \(g\in G\).Almost additive multifractal analysis for flowshttps://zbmath.org/1472.370372021-11-25T18:46:10.358925Z"Barreira, Luis"https://zbmath.org/authors/?q=ai:barreira.luis-m"Holanda, Carllos"https://zbmath.org/authors/?q=ai:holanda.carllosSemi-fast convergent sequences and \(k\)-sums of central Cantor setshttps://zbmath.org/1472.400012021-11-25T18:46:10.358925Z"Bartoszewicz, Artur"https://zbmath.org/authors/?q=ai:bartoszewicz.artur"Filipczak, Małgorzata"https://zbmath.org/authors/?q=ai:filipczak.malgorzata"Prus-Wiśniowski, Franciszek"https://zbmath.org/authors/?q=ai:prus-wisniowski.franciszekSummary: We study \(k\)-sums of central Cantor sets and present a full characterization of the case of positive measure for such \(k\)-sums. This generalizes the celebrated Sannami's counterexample for the Palis hypothesis [\textit{A. Sannami}, Hokkaido Math. J. 21, No. 1, 7--24 (1992; Zbl 0787.58028); \textit{J. Palis}, Contemp. Math. 58/III, 203--216 (1987; Zbl 0617.58027)]. Moreover, we find a characterization of all semi-fast convergent sequences generating central Cantor sets.Evaluation formula and approximation for Wiener integrals via the Fourier-type functionalhttps://zbmath.org/1472.420062021-11-25T18:46:10.358925Z"Chung, Hyun Soo"https://zbmath.org/authors/?q=ai:chung.hyun-soo"Lee, Un Gi"https://zbmath.org/authors/?q=ai:lee.un-giSummary: In order to calculate the Wiener integrals for functionals on Wiener space, one can usually apply the change of variable theorem. But, there are many functionals that are difficult or impossible to calculate even when using the change of variable formula. In order to solve this problem, we establish an evaluation formula via the Fourier-type functionals on Wiener space. We then present various examples to which our evaluation formula can be applied and with the corresponding numerical approximations.\(L^2\)-bounded singular integrals on a purely unrectifiable set in \(R^d\)https://zbmath.org/1472.420222021-11-25T18:46:10.358925Z"Mateu, Joan"https://zbmath.org/authors/?q=ai:mateu.joan"Prat, Laura"https://zbmath.org/authors/?q=ai:prat.lauraSummary: We construct an example of a purely unrectifiable measure \(\mu\) in \(\mathbb{R}^d\) for which the singular integrals associated to the kernels \(K(x)=P_{2k+1}(x)/|x|^{2k+d} \), with \(k\geq 1\) and \(P_{2k+1}\) a homogeneous harmonic polynomial of degree \(2k+1\), are bounded in \(L^2(\mu)\). This contrasts starkly with the results concerning the Riesz kernel \(x/|x|^d\) in \(\mathbb{R}^d\).Variation inequalities for rough singular integrals and their commutators on Morrey spaces and Besov spaceshttps://zbmath.org/1472.420262021-11-25T18:46:10.358925Z"Zhang, Xiao"https://zbmath.org/authors/?q=ai:zhang.xiao"Liu, Feng"https://zbmath.org/authors/?q=ai:liu.feng.4|liu.feng|liu.feng.3|liu.feng.2|liu.feng.5|liu.feng.1"Zhang, Huiyun"https://zbmath.org/authors/?q=ai:zhang.huiyunSummary: This paper is devoted to investigating the boundedness, continuity and compactness for variation operators of singular integrals and their commutators on Morrey spaces and Besov spaces. More precisely, we establish the boundedness for the variation operators of singular integrals with rough kernels \(\varOmega \in L^q (S^{n-1})\) \((q > 1)\) and their commutators on Morrey spaces as well as the compactness for the above commutators on Lebesgue spaces and Morrey spaces. In addition, we present a criterion on the boundedness and continuity for a class of variation operators of singular integrals and their commutators on Besov spaces. As applications, we obtain the boundedness and continuity for the variation operators of Hilbert transform, Hermit Riesz transform, Riesz transforms and rough singular integrals as well as their commutators on Besov spaces.Maximal operators and decoupling for \(\Lambda (p)\) Cantor measureshttps://zbmath.org/1472.420302021-11-25T18:46:10.358925Z"Łaba, Isabella"https://zbmath.org/authors/?q=ai:laba.isabellaSummary: For \(2\leq p<\infty\), \(\alpha^\prime>2/p\), and \(\delta>0\), we construct Cantor-type measures on \(\mathbb{R}\) supported on sets of Hausdorff dimension \(\alpha<\alpha'\) for which the associated maximal operator is bounded from \(L^p_\delta (\mathbb{R})\) to \(L^p(\mathbb{R})\). Maximal theorems for fractal measures on the line were previously obtained by \textit{M. Pramanik} and \textit{I. Łaba} [Duke Math. J. 158, No. 3, 347--411 (2011; Zbl 1242.42011)]. The result here is weaker in that we are not able to obtain \(L^p\) estimates; on the other hand, our approach allows Cantor measures that are self-similar, have arbitrarily low dimension \(\alpha>0\), and have no Fourier decay. The proof is based on a decoupling inequality similar to that of \textit{I. Łaba} and \textit{H. Wang} [Int. Math. Res. Not. 2018, No. 9, 2944--2966 (2018; Zbl 1442.42031)].Full proof of Kwapień's theorem on representing bounded mean zero functions on \([0,1]\)https://zbmath.org/1472.460272021-11-25T18:46:10.358925Z"Ber, Aleksei"https://zbmath.org/authors/?q=ai:ber.aleksey"Borst, Matthijs"https://zbmath.org/authors/?q=ai:borst.matthijs"Sukochev, Fedor"https://zbmath.org/authors/?q=ai:sukochev.fedor-aThe authors are able to fill a gap in the proof of a theorem in [\textit{S.~Kwapień}, Math. Nachr. 119, 175--179 (1984; Zbl 0575.46003)]. Whereas Kwapień's original proof holds for continuous functions, a gap appears for functions with discontinuities. Indeed, a proof of the following theorem in full generality is given. For every mean zero function \(f \in L_\infty[0, 1]\), there exists \(g \in L_\infty[0,1]\) and a mod \(0\) measure preserving transformation \(T\) of \([0, 1]\) such that \(f=g\circ T - g\). The original gap is discussed and a counterexample is also given.An invitation to optimal transport, Wasserstein distances, and gradient flowshttps://zbmath.org/1472.490012021-11-25T18:46:10.358925Z"Figalli, Alessio"https://zbmath.org/authors/?q=ai:figalli.alessio"Glaudo, Federico"https://zbmath.org/authors/?q=ai:glaudo.federicoThis graduate text offers a relatively self-contained introduction to the optimal transport theory. It consists of five chapters and two appendices.
Chapter 1 gives a brief review of the optimal transport theory, recalls certain of basics of measure theory and Riemannian geometry, and shows three typical examples of the transport maps in connection to the classical isoperimetry.
Chapter 2 presents the so-called core of the optimal transport theory: the solution to Kantorovich's problem for general costs; the duality theory; the solution to Monge's problem for suitable costs.
Chapter 3 utilizes the \([1,\infty)\ni p\)-Wasserstein distances to handle an essential relationship among the optimal transport theory, gradient flows in the Hilbert spaces, and partial differential equations.
Chapter 4 shows a differential viewpoint of the optimal transport theory via studying Benamou-Brenier's and Otto's formulas based on the probability measures.
Chapter 5 suggests several applied topics of the optimal transport theory.
Appendix A includes a set of some interesting exercises and their solutions.
Appendix B outlines a proof of the disintegration theorem.Inscribed rectangles in a smooth Jordan curve attain at least one third of all aspect ratioshttps://zbmath.org/1472.510112021-11-25T18:46:10.358925Z"Hugelmeyer, Cole"https://zbmath.org/authors/?q=ai:hugelmeyer.coleSummary: We prove that for every smooth Jordan curve \(\gamma\), if \(X\) is the set of all \(r \in [0,1]\) so that there is an inscribed rectangle in \(\gamma\) of aspect ratio \(\tan(r\cdot\pi/4)\), then the Lebesgue measure of \(X\) is at least \(1/3\). To do this, we study sets of disjoint homologically nontrivial projective planes smoothly embedded in \(\mathbb{R}\times\mathbb{R}P^3\). We prove that any such set of projective planes can be equipped with a natural total ordering. We then combine this total ordering with Kemperman's theorem in \(S^1\) to prove that \(1/3\) is a sharp lower bound on the probability that a Möbius strip filling the \((2,1)\)-torus knot in the solid torus times an interval will intersect its rotation by a uniformly random angle.Leaves decompositions in Euclidean spaceshttps://zbmath.org/1472.520052021-11-25T18:46:10.358925Z"Ciosmak, Krzysztof J."https://zbmath.org/authors/?q=ai:ciosmak.krzysztof-jSummary: We partly extend the localisation technique from convex geometry to the multiple constraints setting.
For a given 1-Lipschitz map \(u:\mathbb{R}^n\to\mathbb{R}^m\), \(m\leq n\), we define and prove the existence of a partition of \(\mathbb{R}^n\), up to a set of Lebesgue measure zero, into maximal closed convex sets such that restriction of \(u\) is an isometry on these sets.
We consider a disintegration, with respect to this partition, of a log-concave measure. We prove that for almost every set of the partition of dimension \(m\), the associated conditional measure is log-concave. This result is proven also in the context of the curvature-dimension condition for weighted Riemannian manifolds. This partially confirms a conjecture of Klartag.The space consisting of uniformly continuous functions on a metric measure space with the \(L^p\) normhttps://zbmath.org/1472.540062021-11-25T18:46:10.358925Z"Koshino, Katsuhisa"https://zbmath.org/authors/?q=ai:koshino.katsuhisaLet \(\mathbf{s} = (-1,1)^\mathbb{N}\) be a countable infinite product of lines endowed with the product topology and let \(c_0\) be the subspace of \(\mathbf{s}\) consisting of those sequences converging to zero. Kadec, Bessaga, Pelczynski and other well-known mathematicians studied homeomorphisms between various infinite dimensional Banach and Fréchet spaces motivated by several questions posed by Fréchet and Banach. A classical celebrated result due to \textit{R. D. Anderson} [Bull. Am. Math. Soc. 72, 515--519 (1966; Zbl 0137.09703)] and \textit{M. I. Kadets} [Funkts. Anal. Prilozh. 1, No. 1, 61--70 (1967; Zbl 0166.10603)] states that every separable infinite dimensional Banach space or Fréchet space is homeomorphic to \(\mathbf{s}\).
In contrast with this result, \textit{R. Cauty} showed in [Fundam. Math. 139, No. 1, 23--36 (1991; Zbl 0793.46008)] that the subspace of \(L^p[0,1]\) consisting of those elements having a representative which is continuous is homeomorphic to \(c_0\) for any \(1\leq p < \infty\). In this paper the author generalizes the aforementioned result of Cauty showing that if \(X\) is a metric measure space satisfying some natural conditions then the subspace \(C_u(X)\) of \(L^p(X)\) consisting of those elements having a representative which is uniformly continuous is homeomorphic to \(c_0\) for any \(1\leq p < \infty\).Toward a quasi-Möbius characterization of invertible homogeneous metric spaceshttps://zbmath.org/1472.540142021-11-25T18:46:10.358925Z"Freeman, David"https://zbmath.org/authors/?q=ai:freeman.david-mandell|freeman.david-f|freeman.david-l|freeman.david-j"Le Donne, Enrico"https://zbmath.org/authors/?q=ai:le-donne.enricoThe paper under review contributes to the ongoing metric characterization of boundaries of rank-one symmetric spaces of non-compact type. These spaces posses boundaries at infinity, metric spaces equipped with visual distances. In the following a boundary of a rank-one symmetric space is abbreviated BROSS.
In the present paper it is conjectured that:
Conjecture 1.1. A metric space is bi-Lipschitz equivalent to some BROSS if and only if it is locally compact, connected, uniformly bi-Lipschitz homogeneous and quasi-invertible.
The article presents a series of theorems that work towards resolving this conjecture. First in terms of Möbius homogenity, then coarse versions in terms of uniformly strongly quasi-Möbius homogenity, and finally two results pertaining to unbounded, proper and disconnected metric spaces are presented.
The following terminology is used to state the theorems. For a set \(X\) and \(p \in X\) let
\[
X_p := X \setminus \{p\} \quad\text{and}\quad \hat{X}:=X\cup\{\infty\}.
\]
A metric space \((X,d)\) is called \emph{invertible} if it is unbounded and admits a homeomorphism \(\tau_p:X_p\to X_p\) (called \emph{inversion at p}) such that for \(x,y \in X_p\):
\[
d(\tau_p(x), \tau_p(y)) = \frac{d(x,y)}{d(x,p)d(y,p)},
\]
and \(\tau_p\) admits a continuous extension to \(\infty \in \hat{X}\). Furthermore define \(\text{Inv}_p(X) := (\hat{X},i_p)\) and \(\text{Sph}_p(X):=(\hat{X},s_p)\), where
\[
i_p(x,y):= \frac{d(x,y)}{d(x,p)d(y,p)},\quad i_p(x,\infty):= \frac{1}{d(x,p)},
\]
\[
s_p(x,y):= \frac{d(x,y)}{(1+d(x,p))(1+d(y,p))},\quad s_p(x,\infty):= \frac{1}{1+d(x,p)}.
\]
\(i_p\) respectively \(s_p\) are metrics if and only if \(X\) is a Ptolemy space. Furthermore a space admits a metric inversion if and only if \(\text{Inv}_p(X)\) is isometric to \(X\).
A homeomorphism \(f:X \to Y\) between (quasi-)metric spaces is called \emph{Möbius} if it preserves the cross-ratio for any quadruples \((a,b,c,d)\) of distinct points in \(X\):
\[
\frac{d(f(a),f(c))d(f(b),f(d))}{d(f(a),f(d))d(f(b),f(c))} = \frac{d(a,c)d(b,d)}{d(a,d)d(b,c)}.
\]
The group of all Möbius self-homeomorphisms of \(X\) is denoted \(\text{Möb}(X)\). A metric space \(X\) is called \emph{\(2\)-point Möbius homogeneous} if for every two pairs \(\{x,y\}, \{a,b\}\) of distinct points in \(X\), there exists an \(f \in \text{Möb}(X)\) with \(f(x) = a\) and \(f(y) = b\).
Coarse versions in the definitions (quasi-invertible, quasi-Möbius, etc.) are established by requiring only bi-Lipschitz equivalence instead of equality in the defining statements.
Given a metric space \((X,d)\) and \(\alpha \in \ ]0,1]\), \((X,d^\alpha)\) is called the \emph{\(\alpha\)-snowflake of \((X,d)\)}.
The main results then are:
Theorem 1.2. Suppose \(X\) is an unbounded, locally compact, complete, and connected metric space. The following statements are equivalent:
\begin{enumerate}
\item \(X\) is Möbius homeomorphic to some BROSS.
\item \(X\) is isometric to some BROSS.
\item \(X\) is isometrically homogeneous and invertible.
\item The sphericalized space \(\text{Sph}_p(X)\) is \(2\)-point Möbius homogeneous, for some (and hence any) \(p \in X\).
\end{enumerate}
This is similar to the main result in [\textit{S. Buyalo} and \textit{V. Schroeder}, Geom. Dedicata 172, 1--45 (2014; Zbl 1362.53061)] but in the present theorem the involution is not required to be fixed point free and does not assume the presence of a Ptolemy circle.
Theorem 1.2. results in
Corollary 1.3. Suppose \(X\) is an unbounded, locally compact, and connected metric space. There exists \(n \in \mathbb{N}\) and \(\alpha \in \ ]0,1]\) such that \(X\) is isometric to \((\mathbb{R}^n, |\cdot|^\alpha)\) if and only if the space \(\text{Sph}_p(X)\) is \(3\)-point Möbius homogeneous, for some/any \(p \in X\).
The consequences of (\(1\)-point) Möbius homogenity are also explored:
Theorem 1.4. Let \(X\) be a compact and quasi-convex metric space of finite topological dimension. If \(X\) is Möbius homogeneous, then \(X\) is bi-Lipschitz homeomorphic to a sub-Riemannian manifold.
Corollary 1.5. Let \(X\) be the boundary of a CAT(\(-1\))-space. If \(X\) is Möbius homogeneous, of finite topological dimension, and connected by Möbius circles, then \(X\) is bi-Lipschitz homeomorphic to a sub-Riemannian manifold.
Coarse versions of the results are then presented:
Proposition 1.6. A proper and unbounded metric space \(X\) is uniformly bi-Lipschitz homogeneous and quasi-invertible if and only if, for some \(p \in X\), the quasi-sphericalized space \(\text{sph}_p(X)\) (the metric space bi-Lipschitz to \(\text{Sph}_p(X)\)) is \(2\)-point uniformly strongly quasi-Möbius homogeneous.
Proposition 1.7. A homeomorphism \(f:X \to Y\) between proper and unbounded metric spaces is strongly quasi-Möbius if and only if it is bi-Lipschitz. Furthermore, \(f\) is Möbius if and only if \(f\) is a similarity.
The article's main contribution towards Conjecture 1.1. is the following
Theorem 1.8. If \(X\) is an unbounded locally compact metric space that is uniformly bi-Lipschitz homogeneous, quasi-invertible, and contains an non-degenerate curve, then \(X\) is path connected, locally path connected, proper, and Ahlfors regular. Furthermore,
\begin{enumerate}
\item if in addition \(X\) contains a cut point, then \(X\) is bi-Lipschitz homeomorphic to \((\mathbb{R}, |\cdot|^\alpha)\), for some \(\alpha \in \ ]0,1]\);
\item if instead \(X\) contains no cut points, then \(X\) is linearly locally connected. Moreover,
\begin{enumerate}
\item if in addition \(X\) contains a non-degenerate rectifiable curve, then \(X\) is annularly quasi-convex;
\item if instead all rectifiable curves in \(X\) are degenerate, then, for some \(\alpha \in \ ]0,1[\), the space \(X\) is bi-Lipschitz homeomorphic to an \(\alpha\)-snowflake.
\end{enumerate}
\end{enumerate}
There is no quasi-Möbius analogue to Corollary 1.3.:
Proposition 1.9. The sphericalized Heisenberg group \(\text{Sph}_e(\mathbb{H}_{\mathbb{C}}^1)\) is \(3\)-point uniformly strongly quasi-Möbius homogeneous. Equivalently, there exists \(L \geq 1\) such that, given any \(x,y \in \mathbb{H}_{\mathbb{C}}^1 \setminus \{e\}\), there exists a \((\lambda, L)\)-quasi-dilation \(f: \mathbb{H}_{\mathbb{C}}^1 \to \mathbb{H}_{\mathbb{C}}^1\) such that \(f(e) = e, f(x)=y\), and \(\lambda = \rho(e,y)/\rho(e,x)\).
Results pertaining to unbounded, proper, and disconnected metric spaces are presented:
Theorem 1.10. Suppose \(X\) is disconnected, unbounded, locally compact, and isometrically homogeneous. If \(X\) is invertible, then there exists \(s>1\) and a positive integer \(N \geq 2\) such that \(X\) is bi-Lipschitz homeomorphic to \((C_N, \rho_s)\).
Here \(C_N\) is the parabolic visual boundary of the \((N+1)\)-regular tree equipped with the path distance with edge length \(1\). \(\rho_s\) is the parabolic visual distance with parameter \(s\).
Theorem 1.11. Suppose \(X\) is a disconnected, unbounded, and locally compact metric space. There exists \(s > 1\) and a positive integer \(N \geq 2\) such that \(X\) is isometric to \((C_N, \rho_s)\) if and only if \(\hat{X}\) is \(3\)-point Möbius homogeneous.
Theorem 1.12. Suppose \(X\) is a disconnected, unbounded, locally compact, and uniformly bi-Lipschitz homogeneous metric space. If \(X\) is quasi-invertible, then \(X\) is quasi-Möbius homeomorphic to \((C_2, \rho_2)\).Hausdorff, large deviation and Legendre multifractal spectra of Lévy multistable processeshttps://zbmath.org/1472.600682021-11-25T18:46:10.358925Z"Le Guével, R."https://zbmath.org/authors/?q=ai:le-guevel.ronan"Lévy Véhel, J."https://zbmath.org/authors/?q=ai:vehel.j-levy|levy-vehel.jacquesIn the article, the authors compute the Hausdorff multifractal spectrum of two versions of multistable Lévy processes. These processes are extensions of classical Lévy processes motion to the case in which the stability exponent \(\alpha\) evolves in time. The spectrum provides a decomposition of the unit interval \([0, 1]\) into an uncountable disjoint union of sets of the Hausdorff dimension one. The authors also compute the increments-based large deviations multifractal spectrum of the multistable Lévy processes with independent increments. It is shown that this spectrum is concave, and thus, it coincides with the Legendre multifractal spectrum, but it is different from the Hausdorff multifractal spectrum. In this view, the multistable Lévy process with independent increments provides an example in which the strong multifractal formalism does not hold.A multi-parameter family of self-avoiding walks on the Sierpiński gaskethttps://zbmath.org/1472.600802021-11-25T18:46:10.358925Z"Otsuka, Takafumi"https://zbmath.org/authors/?q=ai:otsuka.takafumiSummary: In this paper, we construct a multi-parameter family of self-avoiding walks on the Sierpiński gasket. It includes the branching model, the loop-erased random walk and the loop-erased self-repelling walk. We reproduce in a unified manner the proof of the existence of the continuum limit and the self-avoiding property of the limit processes. Our limit processes include not only all the processes obtained from the previously studied self-avoiding walk models, but also the ones that have not been constructed before. While the paths of limit processes appearing in the previous works were self-avoiding or filled the whole space, our family includes continuous processes whose path is self-intersecting but does not fill the whole space.Multiscale functional inequalities in probability: constructive approachhttps://zbmath.org/1472.600862021-11-25T18:46:10.358925Z"Duerinckx, Mitia"https://zbmath.org/authors/?q=ai:duerinckx.mitia"Gloria, Antoine"https://zbmath.org/authors/?q=ai:gloria.antoineSummary: Consider an ergodic stationary random field \(A\) on the ambient space \(\mathbb{R}^d\). In order to establish concentration properties for nonlinear functions \(Z(A)\), it is standard to appeal to functional inequalities like Poincaré or logarithmic Sobolev inequalities in the probability space. These inequalities are however only known to hold for a restricted class of laws (product measures, Gaussian measures with integrable covariance, or more general Gibbs measures with nicely behaved Hamiltonians). In this contribution, we introduce variants of these inequalities, which we refer to as \textit{multiscale functional inequalities} and which still imply fine concentration properties, and we develop a constructive approach to such inequalities. We consider random fields that can be viewed as transformations of a product structure, for which the question is reduced to devising approximate chain rules for nonlinear random changes of variables. This approach allows us to cover most examples of random fields arising in the modelling of heterogeneous materials in the applied sciences, including Gaussian fields with arbitrary covariance function, Poisson random inclusions with (unbounded) random radii, random parking and Matérn-type processes, as well as Poisson random tessellations. The obtained multiscale functional inequalities, which we primarily develop here in view of their application to concentration and to quantitative stochastic homogenization, are of independent interest.Multifractal formalisms for multivariate analysishttps://zbmath.org/1472.620732021-11-25T18:46:10.358925Z"Jaffard, Stéphane"https://zbmath.org/authors/?q=ai:jaffard.stephane"Seuret, Stéphane"https://zbmath.org/authors/?q=ai:seuret.stephane"Wendt, Herwig"https://zbmath.org/authors/?q=ai:wendt.herwig"Leonarduzzi, Roberto"https://zbmath.org/authors/?q=ai:leonarduzzi.roberto"Abry, Patrice"https://zbmath.org/authors/?q=ai:abry.patriceSummary: Multifractal analysis, that quantifies the fluctuations of regularities in time series or textures, has become a standard signal/image processing tool. It has been successfully used in a large variety of applicative contexts. Yet, successes are confined to the analysis of one signal or image at a time (univariate analysis). This is because multivariate (or joint) multifractal analysis remains so far rarely used in practice and has barely been studied theoretically. In view of the myriad of modern real-world applications that rely on the joint (multivariate) analysis of collections of signals or images, univariate analysis constitutes a major limitation. The goal of the present work is to theoretically ground multivariate multifractal analysis by studying the properties and limitations of the most natural extension of the univariate formalism to a multivariate formulation. It is notably shown that while performing well for a class of model processes, this natural extension is not valid in general. Based on the theoretical study of the mechanisms leading to failure, we propose alternative formulations and examine their mathematical properties.Scaling property for fragmentation processes related to avalancheshttps://zbmath.org/1472.741552021-11-25T18:46:10.358925Z"Beznea, Lucian"https://zbmath.org/authors/?q=ai:beznea.lucian"Deaconu, Madalina"https://zbmath.org/authors/?q=ai:deaconu.madalina"Lupaşcu-Stamate, Oana"https://zbmath.org/authors/?q=ai:lupascu-stamate.oanaSummary: We emphasize a scaling property for the continuous time fragmentation processes related to a stochastic model for the fragmentation phase of an avalanche. We present numerical results that confirm the validity of the scaling property for our model, based on the appropriate stochastic differential equation of fragmentation and on a fractal property of the solution.
For the entire collection see [Zbl 1468.74003].