Recent zbMATH articles in MSC 28Ahttps://www.zbmath.org/atom/cc/28A2021-04-16T16:22:00+00:00WerkzeugMorse spectra, homology measures, spaces of cycles and parametric packing problems.https://www.zbmath.org/1456.530352021-04-16T16:22:00+00:00"Gromov, Misha"https://www.zbmath.org/authors/?q=ai:gromov.mikhaelA motivating question for this long and densely-written paper is the following: For an ensemble of moving particles in a space, what happens if the effectively observable number of states is replaced by the number of effective/persistent degrees of freedom? ``We suggest in this paper several mathematical counterparts to the idea of persistent degrees of freedom and formulate specific questions, many of which are inspired by Larry Guth's results and ideas on the Hermann Weyl kind of asymptotics of the Morse (co)homology spectra of the volume energy function on the spaces of cycles in a ball. And often we present variable aspects of the same ideas in different sections of the paper.''
For the entire collection see [Zbl 1437.55002].
Reviewer: Bruno Zimmermann (Trieste)On volume and surface densities of dynamical germ-grain models with ellipsoidal growth: a rigorous approach with applications to materials science.https://www.zbmath.org/1456.600452021-04-16T16:22:00+00:00"Villa, Elena"https://www.zbmath.org/authors/?q=ai:villa.elena"Rios, Paulo R."https://www.zbmath.org/authors/?q=ai:rios.paulo-rSummary: Many engineering materials of interest are polycrystals: an aggregate of many crystals with size usually below \(100\mu\)m. Those small crystals are called the grains of the polycrystal, and are often equiaxed. However, because of processing, the grain shape may become anisotropic; for instance, during recrystallization or phase transformations, the new grains may grow in the form of ellipsoids. Heavily anisotropic grains may result from a process, such as rolling, and they may have most of their interfacial area parallel to the rolling plane. Therefore, to a first approximation, these heavily deformed grains may be approximated by random parallel planes; as a consequence, the nucleation process may be assumed to take place on random parallel planes. The case of nucleation on random parallel planes and subsequent ellipsoidal growth is also possible. In this paper we model such situations employing time dependent germ grain processes with ellipsoidal growth. We provide explicit formulas for the mean volume and surface densities and related quantities. The known results for the spherical growth follow here as a particular case. Although this work has been done bearing applications to Materials Science in mind, the results obtained here may be applied to nucleation and growth reactions in general. Moreover, a generalization of the so called mean value property, crucial in finding explicit analytical formulas in the paper, is also provided as a further result in the Appendix A.Lebesgue integral.https://www.zbmath.org/1456.260022021-04-16T16:22:00+00:00"Florescu, Liviu C."https://www.zbmath.org/authors/?q=ai:florescu.liviu-cPublisher's description: This book presents a compact and self-contained introduction to the theory of measure and integration. The introduction into this theory is as necessary (because of its multiple applications) as difficult for the uninitiated. Most measure theory treaties involve a large amount of prerequisites and present crucial theoretical challenges. By taking on another approach, this textbook provides less experienced readers with material that allows an easy access to the definition and main properties of the Lebesgue integral.
The book will be welcomed by upper undergraduate/early graduate students who wish to better understand certain concepts and results of probability theory, statistics, economic equilibrium theory, game theory, etc., where the Lebesgue integral makes its presence felt throughout. The book can also be useful to students in the faculties of mathematics, physics, computer science, engineering, life sciences, as an introduction to a more in-depth study of measure theory.Box dimension of fractal attractors and their numerical computation.https://www.zbmath.org/1456.280062021-04-16T16:22:00+00:00"Freiberg, Uta"https://www.zbmath.org/authors/?q=ai:freiberg.uta-renata"Kohl, Stefan"https://www.zbmath.org/authors/?q=ai:kohl.stefanThe authors provide a general algorithmic approach to calculate the box dimension of Lorenz attractors. The algorithm behaves very well on self-similar fractals and fractal graphs of deterministic as well as random functions; however, in the case of the Lorenz attractor the algorithm seems to fail. The reason for this phenomenon is discussed, the notion of local box dimension is introduced, and the algorithm reveals a hidden multifractal structure of the Lorenz attractor.
Reviewer: George Stoica (Saint John)Cauchy transforms of self-similar measures: starlikeness and univalence.https://www.zbmath.org/1456.280052021-04-16T16:22:00+00:00"Dong, Xin-Han"https://www.zbmath.org/authors/?q=ai:dong.xinhan"Lau, Ka-Sing"https://www.zbmath.org/authors/?q=ai:lau.kasing"Wu, Hai-Hua"https://www.zbmath.org/authors/?q=ai:wu.haihuaSummary: For the contractive iterated function system \( S_kz=e^{2\pi ik/m}+{\rho (z-e^{2\pi ik/m})}\) with \( 0<\rho <1, k=0,\cdots , m-1\), we let \( K\subset \mathbb{C}\) be the attractor, and let \( \mu \) be a self-similar measure defined by \( \mu =\frac 1m\sum _{k=0}^{m-1}\mu \circ S_k^{-1}\). We consider the Cauchy transform \( F\) of \( \mu \). It is known that the image of \( F\) at a small neighborhood of the boundary of \( K\) has very rich fractal structure, which is coined the Cantor boundary behavior. In this paper, we investigate the behavior of \( F\) away from \( K\); it has nice geometric and analytic properties, such as univalence, starlikeness and convexity. We give a detailed investigation for those properties in the general situation as well as certain classical cases of self-similar measures.Multifractal detrended moving average analysis of global temperature records.https://www.zbmath.org/1456.860092021-04-16T16:22:00+00:00"Mali, Provash"https://www.zbmath.org/authors/?q=ai:mali.provashOn the quantitative isoperimetric inequality in the plane.https://www.zbmath.org/1456.490342021-04-16T16:22:00+00:00"Bianchini, Chiara"https://www.zbmath.org/authors/?q=ai:bianchini.chiara"Croce, Gisella"https://www.zbmath.org/authors/?q=ai:croce.gisella"Henrot, Antoine"https://www.zbmath.org/authors/?q=ai:henrot.antoineSummary: In this paper we study the quantitative isoperimetric inequality in the plane. We prove the existence of a set \(\Omega\), different from a ball, which minimizes the ratio \(\delta(\Omega)/\lambda^{2}(\Omega)\), where \(\delta\) is the isoperimetric deficit and \(\lambda\) the Fraenkel asymmetry, giving a new proof of the quantitative isoperimetric inequality. Some new properties of the optimal set are also shown.The strong uniqueness property of invariant measures in infinite dimensional topological vector spaces.https://www.zbmath.org/1456.280012021-04-16T16:22:00+00:00"Khachidze, Marika"https://www.zbmath.org/authors/?q=ai:khachidze.marika"Kirtadze, Aleks"https://www.zbmath.org/authors/?q=ai:kirtadze.aleks-pIn [\textit{A. B. Kharazishvili}, Bull. Acad. Sci. GSSR 114, No. 1, 41--48 (1984)] a normalized \(\sigma\)-finite metrically transitive Borel measure \(\chi\) in \(\mathbb{R}^\omega\) is constructed which is invariant with respect to
\[G=c_{00}:=\{x \in \mathbb{R}^\omega : \ \hbox{supp}~ x < \omega\}\]
Let \(s_0\) be the central symmetry of \(\mathbb{R}^\omega\) with respect to the origin and let \(S_\omega\) be the group generated by \(s_0\) and \(G\). In the present paper a \(\sigma\)-finite Borel measure on \(\mathbb{R}^\omega\) is costructed which is invariant with respect to the group \(S_\omega\).
Reviewer: Daniele Puglisi (Catania)Approximating the Ising model on fractal lattices of dimension less than two.https://www.zbmath.org/1456.821042021-04-16T16:22:00+00:00"Codello, Alessandro"https://www.zbmath.org/authors/?q=ai:codello.alessandro"Drach, Vincent"https://www.zbmath.org/authors/?q=ai:drach.vincent"Hietanen, Ari"https://www.zbmath.org/authors/?q=ai:hietanen.ariCorrigendum to: ``Local dimensions of measures of finite type III -- measures that are not equicontractive''.https://www.zbmath.org/1456.280072021-04-16T16:22:00+00:00"Hare, Kathryn E."https://www.zbmath.org/authors/?q=ai:hare.kathryn-e"Hare, Kevin G."https://www.zbmath.org/authors/?q=ai:hare.kevin-g"Simms, Grant"https://www.zbmath.org/authors/?q=ai:simms.grantFrom the text: In the authors's paper [ibid. 458, No. 2, 1653--1677 (2018; Zbl 1376.28008)], Definition 3.5 was incorrectly stated. The correct definition was used, or bypassed for all proofs and calculations, and all results of the paper remain valid. We correct it here only to alleviate possible confusion.Vitali-Hahn-Saks property in coverings of sets algebras.https://www.zbmath.org/1456.280032021-04-16T16:22:00+00:00"López-Alfonso, S."https://www.zbmath.org/authors/?q=ai:lopez-alfonso.salvadorLet \(ba(\mathcal{A})\) denote the Banach space of the real (or complex) finitely additive measures of bounded variation defined on an algebra \(\mathcal{A}\) of subsets of \(\Omega\). A subset \(\mathcal{B}\) of \(\mathcal{A}\) is a Nikodým set for \(ba(\mathcal{A})\) if each \(\mathcal{B}\)-pointwise bounded subset \(M\) of \(ba(\mathcal{A})\) is uniformly bounded on \(\mathcal{A}\); while \(\mathcal{B}\) is a strong Nikodým set for \(ba(\mathcal{A})\) if each increasing covering \((\mathcal{B}_m )_{m=1}^\infty\) of \(\mathcal{B}\) contains a \(\mathcal{B}_n\) which is a Nikodým set for \(ba(\mathcal{A})\). The subset \(\mathcal{B}\) has (VHS) property if \(\mathcal{B}\) is a Nikodým set for \(ba(\mathcal{A})\) and for every \(\mu\in ba(\mathcal{A})\) and sequence \((\mu_m)_{m=1}^\infty \subseteq ba(\mathcal{A})\) such that \(\lim_{m\rightarrow \infty} \mu_m(B) = \mu(B)\), for each \(B \in \mathcal{B}\), then \((\mu_m)_{m=1}^\infty\) weakly converges to \(\mu\). It is proved that if \((\mathcal{B}_m)_{m=1}^\infty\) is an increasing covering of \(\mathcal{A}\) that has (VHS) property and there exist a \(\mathcal{B}_p\) which is a Nikodým set for \(ba(\mathcal{A})\), then there exists \(q \geq p\) such that \(\mathcal{B}_q\) has (VHS) property. In particular, if \((\mathcal{B}_m )_{m=1}^\infty\) is an increasing covering of a \(\sigma\)-algebra then some \(\mathcal{B}_q\) has (VHS) property. In 2013 Valdivia proved that every \(\sigma\)-algebra has strong Nikodým property. Another proof of this result is presented, here. Some related questions with respect to the problem if Nikodým property in an algebra implies strong Nikodým property are also considered.
Reviewer: Daniele Puglisi (Catania)Regularity of area minimizing currents mod \(p\).https://www.zbmath.org/1456.490322021-04-16T16:22:00+00:00"De Lellis, Camillo"https://www.zbmath.org/authors/?q=ai:de-lellis.camillo"Hirsch, Jonas"https://www.zbmath.org/authors/?q=ai:hirsch.jonas"Marchese, Andrea"https://www.zbmath.org/authors/?q=ai:marchese.andrea"Stuvard, Salvatore"https://www.zbmath.org/authors/?q=ai:stuvard.salvatoreRegularity of area minimizing currents mod(p) have had an important role in several classical problems of geometric measure theory and mathematical physics. Several authors studied the regularity of area [\textit{C. De Lellis} and \textit{E.N. Spadaro}, Ann. of Math. (2), 183, No. 2, 499--575 (2016; Zbl 1345.49052); \textit{C. De Lellis} and \textit{E.N. Spadaro}, Geom. Funct. Anal., 24, No. 6, 1831--1884 (2014; Zbl 1307.49043); \textit{C. De Lellis} and \textit{E. N. Spadaro}, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 14, No. 4, 1239--1269 (2015; Zbl 1343.49073); \textit{T. De Pauw} and \textit{R. Hardt}, Am. J. Math., 134, No. 1, 1--69 (2012; Zbl 1252.49070); \textit{T. De Pauw} and \textit{R. Hardt}, J. Math. Anal. Appl., 418, No. 2, 1047--1061 (2014; Zbl 1347.49073); \textit{L. Simon}, J. Diff. Geom. 38, No. 3, 585--652 (1993; Zbl 0819.53029); \textit{L. Spolaor}, Adv. Math., 350, 747--815 (2019; Zbl 1440.49048)].
The principal objective in this paper is to establish a first general partial regularity theorem for area minimizing currents mod(p), for every p, in any dimension and codimension. More precisely, the authors prove that the Hausdorff dimension of the interior singular set of an m-dimensional area minimizing current mod(p) cannot be larger than \(m-1\).
Reviewer: Lakehal Belarbi (Mostaganem)Hausdorff dimension of limsup sets of rectangles in the Heisenberg group.https://www.zbmath.org/1456.600382021-04-16T16:22:00+00:00"Ekström, Fredrik"https://www.zbmath.org/authors/?q=ai:ekstrom.fredrik"Järvenpää, Esa"https://www.zbmath.org/authors/?q=ai:jarvenpaa.esa"Järvenpää, Maarit"https://www.zbmath.org/authors/?q=ai:jarvenpaa.maaritThe main findings of the paper refer to computing the almost sure value of the Hausdorff dimension of limsup sets generated by randomly distributed rectangles in the Heisenberg group, in terms of directed singular value functions.
Reviewer: George Stoica (Saint John)Existence of similar point configurations in thin subsets of \(\mathbb{R}^d\).https://www.zbmath.org/1456.520202021-04-16T16:22:00+00:00"Greenleaf, Allan"https://www.zbmath.org/authors/?q=ai:greenleaf.allan"Iosevich, Alex"https://www.zbmath.org/authors/?q=ai:iosevich.alex"Mkrtchyan, Sevak"https://www.zbmath.org/authors/?q=ai:mkrtchyan.sevakSummary: We prove the existence of similar and multi-similar point configurations (or simplexes) in sets of fractional Hausdorff dimension in Euclidean space. Let \(d \ge 2\) and \(E \subset \mathbb{R}^d\) be a compact set. For \(k \ge 1\), define
\[
\Delta_k(E) = \left\{\left(|x^1 - x^2|, \ldots, |x^i - x^j|, \ldots, |x^k - x^{k+1}|\right) : \left\{x^i\right\}_{i = 1}^{k + 1} \subset E \right\} \subset \mathbb{R}^{k(k+1)/2},
\]
the \((k+1)\)-point configuration set of \(E\). For \(k \le d\), this is (up to permutations) the set of congruences of \((k+1)\)-point configurations in \(E\); for \(k > d\), it is the edge-length set of \((k + 1)\)-graphs whose vertices are in \(E\). Previous works by a number of authors have found values \(s_{k, d} < d\) so that if the Hausdorff dimension of \(E\) satisfies \(\dim_{\mathcal H}(E) > s_{k, d} \), then \(\Delta_k(E)\) has positive Lebesgue measure. In this paper we study more refined properties of \(\Delta_k(E)\), namely the existence of similar or multi-similar configurations. For \(r \in \mathbb{R}, r > 0\), let
\[
\Delta_k^r(E) := \{\mathbf{t}\in \Delta_k (E) : r \mathbf{t} \in \Delta_k (E)\} \subset \Delta_k (E).
\]
We show that if \(\dim_{\mathcal H}(E) > s_{k, d}\), for a natural measure \(\nu_k\) on \(\Delta_k(E)\), one has \(\nu_k \left(\delta^r_k(E)\right)\) all \(r \in \mathbb{R}_+\). Thus, in \(E\) there exist many pairs of \((k + 1)\)-point configurations which are similar by the scaling factor \(r\). We extend this to show the existence of multi-similar configurations of any multiplicity. These results can be viewed as variants and extensions, for compact thin sets, of theorems of Furstenberg, Katznelson and Weiss [7], Bourgain [2] and Ziegler [11] for sets of positive density in \(\mathbb{R}^d\).On the computational complexity of the Dirichlet problem for Poisson's equation.https://www.zbmath.org/1456.030692021-04-16T16:22:00+00:00"Kawamura, Akitoshi"https://www.zbmath.org/authors/?q=ai:kawamura.akitoshi"Steinberg, Florian"https://www.zbmath.org/authors/?q=ai:steinberg.florian"Ziegler, Martin"https://www.zbmath.org/authors/?q=ai:ziegler.martinSummary: The last years have seen an increasing interest in classifying (existence claims in) classical mathematical theorems according to their strength. We pursue this goal from the refined perspective of computational complexity. Specifically, we establish that rigorously solving the Dirichlet Problem for Poisson's Equation is in a precise sense `complete' for the complexity class \(\#\mathcal{P}\) and thus as hard or easy as parametric Riemann integration [\textit{H. Friedman}, Adv. Math. 53, 80--98 (1984; Zbl 0563.03023); \textit{K.-I Ko}, Complexity theory of real functions. Boston, MA etc.: Birkhäuser (1991; Zbl 0791.03019)].Pulling self-interacting linear polymers on a family of fractal lattices embedded in three-dimensional space.https://www.zbmath.org/1456.829502021-04-16T16:22:00+00:00"Elezović-Hadžić, S."https://www.zbmath.org/authors/?q=ai:elezovic-hadzic.suncica"Živić, I."https://www.zbmath.org/authors/?q=ai:zivic.ivanDynamic scaling behaviors of the discrete growth models on fractal substrates.https://www.zbmath.org/1456.829822021-04-16T16:22:00+00:00"Xun, Zhipeng"https://www.zbmath.org/authors/?q=ai:xun.zhipeng"Zhang, Yongwei"https://www.zbmath.org/authors/?q=ai:zhang.yongwei"Li, Yan"https://www.zbmath.org/authors/?q=ai:li.yan.6|li.yan.1|li.yan.8|li.yan.4|li.yan.5|li.yan|li.yan.3|li.yan.2|li.yan.10|li.yan.7|li.yan.9"Xia, Hui"https://www.zbmath.org/authors/?q=ai:xia.hui"Hao, Dapeng"https://www.zbmath.org/authors/?q=ai:hao.da-peng"Tang, Gang"https://www.zbmath.org/authors/?q=ai:tang.gangA. Kharazishvili's some results of on the structure of pathological functions.https://www.zbmath.org/1456.260032021-04-16T16:22:00+00:00"Kirtadze, Aleks"https://www.zbmath.org/authors/?q=ai:kirtadze.aleks-p"Pantsulaia, Gogi"https://www.zbmath.org/authors/?q=ai:pantsulaia.gogi-rauliThe authors present a brief survey of A. Kharazishvili's works devoted to real-valued functions with strange, pathological and paradoxical structural properties, e.g. absolutely non-measurable functions, Sierpiński-Zygmund functions, sup-measurable and weakly sup-measurable functions of two real variables, and non-measurable functions of two real variables for which both iterated integrals exist.
Reviewer: George Stoica (Saint John)Characterization of sets of singular rotations for a class of differentiation bases.https://www.zbmath.org/1456.280022021-04-16T16:22:00+00:00"Oniani, G."https://www.zbmath.org/authors/?q=ai:oniani.giorgi-gigla"Chubinidze, K."https://www.zbmath.org/authors/?q=ai:chubinidze.kakha-aSummary: We study the dependence of differential properties of an indefinite integral on rotations of the coordinate system. Namely, the following problem is studied: For a summable function \(f\) of what kind can be the set of rotations \(\gamma\) for which \(\int f\) is not differentiable with respect to the \(\gamma \)-rotation of a given basis \(B\)? The result obtained in the paper implies a solution of the problem for any homothecy invariant differentiation basis \(B\) of two-dimensional intervals which has symmetric structure.Rough basin boundaries in high dimension: can we classify them experimentally?https://www.zbmath.org/1456.370962021-04-16T16:22:00+00:00"Bódai, Tamás"https://www.zbmath.org/authors/?q=ai:bodai.tamas"Lucarini, Valerio"https://www.zbmath.org/authors/?q=ai:lucarini.valerioSummary: We show that a known condition for having rough basin boundaries in bistable 2D maps holds for high-dimensional bistable systems that possess a unique nonattracting chaotic set embedded in their basin boundaries. The condition for roughness is that the cross-boundary Lyapunov exponent \(\lambda_x\) on the nonattracting set is not the maximal one. Furthermore, we provide a formula for the generally noninteger co-dimension of the rough basin boundary, which can be viewed as a generalization of the Kantz-Grassberger formula. This co-dimension that can be at most unity can be thought of as a partial co-dimension, and, so, it can be matched with a Lyapunov exponent. We show in 2D noninvertible- and 3D invertible-minimal models, that, formally, it cannot be matched with \(\lambda_x\). Rather, the partial dimension \(D_0^{(x)}\) that \(\lambda_x\) is associated with in the case of rough boundaries is trivially unity. Further results hint that the latter holds also in higher dimensions. This is a peculiar feature of rough fractals. Yet, \(D_0^{(x)}\) cannot be measured via the uncertainty exponent along a line that traverses the boundary. Consequently, one cannot determine whether the boundary is a rough or a filamentary fractal by measuring fractal dimensions. Instead, one needs to measure both the maximal and cross-boundary Lyapunov exponents numerically or experimentally.
{\copyright 2020 American Institute of Physics}Number theory problems related to the spectrum of Cantor-type measures with consecutive digits.https://www.zbmath.org/1456.280042021-04-16T16:22:00+00:00"Ai, Wen-Hui"https://www.zbmath.org/authors/?q=ai:ai.wenhuiThe present paper deals with the scaling of spectra of a class of self-similar measures with consecutive digits and extends results obtained in some researches of other authors.
A brief survey of this paper is devoted to introduce a new area of research for a harmonic analysis on fractals, to construct the
first singular, nonatomic, spectral measure by Jorgensen and Pedersen, and to classes of fractal measures (defined by self-similarity) which have spectra, i.e., admit \(L^2\)-orthogonal Fourier expansions. Also, some related investigations are considered.
The author gives the following description of the present research:
``For integers \(p,b \ge 2\), let \(D=\{0,1,\dots , b-1\}\) be a set of consecutive digits. It is known that the Cantor measure \(\mu_{pb, D}\) generated by the iterated function system \(\{(pb)^{-1}(x+d)\}_{x\in\mathbb R, d\in D}\) is a spectral measure with spectrum
\[
\Lambda (pb, S)=\left\{\sum^{\text{finite}} _{j=0}{(pb)^{j}s_j}: s_j \in S \right\},
\]
where \(S=pD\). We give conditions on \(\tau \in \mathbb Z\) under which the scaling set \(\tau \Lambda (pb, S)\) is also a spectrum of
\(\mu_{pb, D}\). These investigations link number theory and spectral measures.''
One can note that necessary or sufficient conditions under which a composite number is complete or incomplete, are studied. Connections between obtained results and known ones are described.
Reviewer: Symon Serbenyuk (Kyïv)Intrinsic differentiability and intrinsic regular surfaces in Carnot groups.https://www.zbmath.org/1456.352062021-04-16T16:22:00+00:00"Di Donato, Daniela"https://www.zbmath.org/authors/?q=ai:di-donato.danielaSummary: A Carnot group \(\mathbb{G}\) is a connected, simply connected, nilpotent Lie group with stratified Lie algebra. Intrinsic regular surfaces in Carnot groups play the same role as \(\mathbb{C}^1\) surfaces in Euclidean spaces. As in Euclidean spaces, intrinsic regular surfaces can be locally defined in different ways: e.g. as non critical level sets or as continuously intrinsic differentiable graphs. The equivalence of these natural definitions is the problem that we are studying. Precisely our aim is to generalize the results on \textit{L. Ambrosio} et al. [J. Geom. Anal. 16, No. 2, 187--232 (2006; Zbl 1085.49045)] valid in Heisenberg groups to the more general setting of Carnot groups.On fractional and fractal Einstein's field equations.https://www.zbmath.org/1456.830062021-04-16T16:22:00+00:00"El-Nabulsi, Rami Ahmad"https://www.zbmath.org/authors/?q=ai:el-nabulsi.rami-ahmad"Golmankhaneh, Alireza Khalili"https://www.zbmath.org/authors/?q=ai:golmankhaneh.alireza-khaliliA domain-theoretic investigation of posets of sub-\(\sigma\)-algebras (extended abstract).https://www.zbmath.org/1456.060112021-04-16T16:22:00+00:00"Battenfeld, Ingo"https://www.zbmath.org/authors/?q=ai:battenfeld.ingoSummary: Given a measurable space \((X,\mathcal{M})\) there is a (Galois) connection between sub-\(\sigma\)-algebras of \(\mathcal{M}\) and equivalence relations on \(X\). On the other hand equivalence relations on \(X\) are closely related to congruences on stochastic relations. In recent work, Doberkat has examined lattice properties of posets of congruences on a stochastic relation and motivated a domain-theoretic investigation of these ordered sets. Here we show that the posets of sub-\(\sigma\)-algebras of a measurable space do not enjoy desired domain-theoretic properties and that our counterexamples can be applied to the set of smooth equivalence relations on an analytic space, thus giving a rather unsatisfactory answer to Doberkat's question.
For the entire collection see [Zbl 1391.03010].Fourier quasicrystals with unit masses.https://www.zbmath.org/1456.420322021-04-16T16:22:00+00:00"Olevskii, Alexander"https://www.zbmath.org/authors/?q=ai:olevskii.alexander-moiseevich"Ulanovskii, Alexander"https://www.zbmath.org/authors/?q=ai:ulanovskii.alexander-mSummary: The sum of \(\delta\)-measures sitting at the points of a discrete set \(\Lambda\subset\mathbb{R}\) forms a Fourier quasicrystal if and only if \(\Lambda\) is the zero set of an exponential polynomial with imaginary frequencies.On fractional differentiation.https://www.zbmath.org/1456.260082021-04-16T16:22:00+00:00"Gladkov, S. O."https://www.zbmath.org/authors/?q=ai:gladkov.sergei-oktyabrinovich"Bogdanova, S. B."https://www.zbmath.org/authors/?q=ai:bogdanova.sofya-borisovnaSummary: Due to the operation of fractional differentiation introduced with the help of Fourier integral, the results of calculating fractional derivatives for certain types of functions are given. Using the numerical method of integration, the values of fractional derivatives for arbitrary dimensionality \(\varepsilon \), (where \(\varepsilon\) is any number greater than zero) are calculated. It is proved that for integer values of \(\varepsilon\) we obtain ordinary derivatives of the first, second and more high orders. As an example it was considered heat conduction equation of Fourier, where spatial derivation was realized with the use of fractional derivatives. Its solution is given by Fourier integral. Mmoreover, it was shown that integral went into the required results in special case of the whole \(\varepsilon\) obtained in \(n\)-dimensional case, where \(n = 1, 2\dots \), etc.Multifractal analysis for core-periphery structure of complex networks.https://www.zbmath.org/1456.051552021-04-16T16:22:00+00:00"Liu, Jin-Long"https://www.zbmath.org/authors/?q=ai:liu.jinlong"Yu, Zu-Guo"https://www.zbmath.org/authors/?q=ai:yu.zuguo"Anh, Vo"https://www.zbmath.org/authors/?q=ai:anh.vo-v|anh.vo-ngocEstimates for Taylor coefficients of Cauchy transforms of some Hausdorff measures. II.https://www.zbmath.org/1456.300672021-04-16T16:22:00+00:00"Li, Hong-Guang"https://www.zbmath.org/authors/?q=ai:li.hongguang"Dong, Xin-Han"https://www.zbmath.org/authors/?q=ai:dong.xinhan"Zhang, Peng-Fei"https://www.zbmath.org/authors/?q=ai:zhang.pengfeiSummary: Let \(q\) be even, \(F\) be the Cauchy transform of the self-similar measure \(\mu = \frac{1}{q} \sum_{j = 0}^{q - 1} \mu \circ S_j\) where \(S_j(z) = e^{2j \pi i / q} + \rho(z - e^{2 j \pi i / q})\) with \(\rho \in (0, 1)\), and \(K\) be the attractor of \(\{S_j\}_{j = 0}^{q - 1}\) and \(R_q = \text{dist}(0, K)\). As a follow-up of [the authors, ibid., Article ID 108653, (2021; Zbl 07306981)], we not only study the asymptotic formula for the Taylor coefficients \(\{ b_{qk - 1} \}_{k = 1}^\infty\) of \(F\) in \(|z| < R_q\), but also the chaotic behavior of \(F^{(n)}\) near the attractor \(K\).
For part I, see [Zbl 1456.30068].Estimates for Taylor coefficients of Cauchy transforms of some Hausdorff measures. I.https://www.zbmath.org/1456.300682021-04-16T16:22:00+00:00"Li, Hong-Ping"https://www.zbmath.org/authors/?q=ai:li.hongping"Dong, Xin-Han"https://www.zbmath.org/authors/?q=ai:dong.xinhan"Zhang, Peng-Fei"https://www.zbmath.org/authors/?q=ai:zhang.pengfei"Wu, Hai-Hua"https://www.zbmath.org/authors/?q=ai:wu.haihuaSummary: Suppose that \(q = 2m + 1 \geq 5\). Let \(F\) be the Cauchy transform of the self-similar measure \(\mu = \frac{1}{q} \sum_{j = 0}^{q - 1} \mu \circ S_j\) where \(S_j(z) = e^{2j \pi i / q} + \rho (z - e^{2j \pi i / q})\) with \(\rho \in (0, 1)\). Let \(K\) be the attractor of \(\{S_j\}_{j = 0}^{q - 1}\) and \(R_q = \text{dist}(0, K)\). The Laurent coefficients of \(F\) in \(| z | > 1\) was studied in [\textit{J.-P. Lund} et al., Exp. Math. 7, No. 3, 177--190 (1998; Zbl 0959.28006)] and [\textit{X.-H. Dong} and \textit{K.-S. Lau}, J. Funct. Anal. 202, No. 1, 67--97 (2003; Zbl 1032.28005)].
In this paper, we study the asymptotic formula of the Taylor coefficients \(\{ b_{qk - 1} \}_{k = 1}^\infty\) of \(F\) in \(|z| < R_q\). We prove that the supremum of \(\beta\) satisfying \(\{ ( q k )^\beta R_q^{qk} b_{qk - 1} \}_{k = 1}^\infty \in l^\infty\) is the Hausdorff dimension \(\alpha\) of \(K\), and that the accumulation points set of \(\{(qk)^\alpha R_q^{qk} b_{qk - 1}\}_{k = 1}^\infty\) is a union of some non-degenerate line segments.On spectral eigenvalue problem of a class of self-similar spectral measures with consecutive digits.https://www.zbmath.org/1456.420332021-04-16T16:22:00+00:00"Wang, Cong"https://www.zbmath.org/authors/?q=ai:wang.cong"Wu, Zhi-Yi"https://www.zbmath.org/authors/?q=ai:wu.zhi-yiSummary: Let \(\mu_{p,q}\) be a self-similar spectral measure with consecutive digits generated by an iterated function system \(\{f_i(x)=\frac{x}{p}+\frac{i}{q}\}_{i=0}^{q-1} \), where \(2\le q\in\mathbb{Z}\) and \(q|p\). It is known that for each \(w=w_1w_2\cdots \in \{-1,1\}^\infty :=\{i_1i_2\cdots :\text{ all }i_k \in \{-1,1\}\} \), the set
\[
\Lambda_w=\left\{\sum_{j=1}^na_j w_j p^{j-1}:a_j\in \{0,1,\ldots ,q-1\},n\geq 1\right\}
\]
is a spectrum of \(\mu_{p,q}\). In this paper, we study the possible real number \(t\) such that the set \(t\Lambda_w\) are also spectra of \(\mu_{p,q}\) for all \(w\in \{-1,1\}^\infty\).Scalings and fractals in information geometry: Ornstein-Uhlenbeck processes.https://www.zbmath.org/1456.601282021-04-16T16:22:00+00:00"Oxley, William"https://www.zbmath.org/authors/?q=ai:oxley.william"Kim, Eun-Jin"https://www.zbmath.org/authors/?q=ai:kim.eunjinHigher dimensional divergence for mapping class groups.https://www.zbmath.org/1456.200412021-04-16T16:22:00+00:00"Behrstock, Jason"https://www.zbmath.org/authors/?q=ai:behrstock.jason-a"Druţu, Cornelia"https://www.zbmath.org/authors/?q=ai:drutu.corneliaSummary: In this paper we investigate the higher dimensional divergence functions of mapping class groups of surfaces. We show that these functions exhibit phase transitions at the quasi-flat rank (as measured by \(3 \cdot \text{ genus }+ \text{ number of punctures }- 3)\).Appendix to the paper ``On the Billingsley dimension of Birkhoff average in the countable symbolic space''.https://www.zbmath.org/1456.370052021-04-16T16:22:00+00:00"Selmi, Bilel"https://www.zbmath.org/authors/?q=ai:selmi.bilelSummary: This appendix gives a lower bound of the Billingsley-Hausdorff dimension of a level set related to Birkhoff average in the \textit{``non-compact''} symbolic space \(\mathbb{N}^{\mathbb{N}}\), defined by Gibbs measure.
This is an appendix to [\textit{N. Attia} and the author, C. R., Math., Acad. Sci. Paris 358, No. 3, 255--265 (2020; Zbl 1445.28005)].Fractal transformed doubly reflected Brownian motions.https://www.zbmath.org/1456.602092021-04-16T16:22:00+00:00"Ehnes, Tim"https://www.zbmath.org/authors/?q=ai:ehnes.tim"Freiberg, Uta"https://www.zbmath.org/authors/?q=ai:freiberg.uta-renata