Recent zbMATH articles in MSC 28https://zbmath.org/atom/cc/282022-11-17T18:59:28.764376ZWerkzeugTricky math, but trippy graphics: the quixotic search for the ``3D Mandelbrot''https://zbmath.org/1496.000432022-11-17T18:59:28.764376Z"Merow, Sophia D."https://zbmath.org/authors/?q=ai:merow.sophia-d(no abstract)The open dihypergraph dichotomy and the second level of the Borel hierarchyhttps://zbmath.org/1496.031832022-11-17T18:59:28.764376Z"Carroy, Raphaël"https://zbmath.org/authors/?q=ai:carroy.raphael"Miller, Benjamin D."https://zbmath.org/authors/?q=ai:miller.benjamin-david"Soukup, Dániel T."https://zbmath.org/authors/?q=ai:soukup.daniel-tamasSummary: We show that several dichotomy theorems concerning the second level of the Borel hierarchy are special cases of the \(\aleph_0\)-dimensional generalization of the open graph dichotomy, which itself follows from the usual proof(s) of the perfect set theorem. Under the axiom of determinacy, we obtain the generalizations of these results from analytic to separable metric spaces. We also consider connections between cardinal invariants and the chromatic numbers of the corresponding dihypergraphs.
For the entire collection see [Zbl 1454.03009].Measurable perfect matchings for acyclic locally countable Borel graphshttps://zbmath.org/1496.031862022-11-17T18:59:28.764376Z"Conley, Clinton T."https://zbmath.org/authors/?q=ai:conley.clinton-taylor"Miller, Benjamin D."https://zbmath.org/authors/?q=ai:miller.benjamin-davidSummary: We characterize the structural impediments to the existence of Borel perfect matchings for acyclic locally countable Borel graphs admitting a Borel selection of finitely many ends from their connected components. In particular, this yields the existence of Borel matchings for such graphs of degree at least three. As a corollary, it follows that acyclic locally countable Borel graphs of degree at least three generating \(\mu\)-hyperfinite equivalence relations admit \(\mu\)-measurable matchings. We establish the analogous result for Baire measurable matchings in the locally finite case, and provide a counterexample in the locally countable case.On the generalized nonmeasurability of some classical point setshttps://zbmath.org/1496.032022022-11-17T18:59:28.764376Z"Kharazishvili, A."https://zbmath.org/authors/?q=ai:kharazishvili.alexander-bSummary: The generalized nonmeasurability of certain classical point sets (such as Vitali sets, Bernstein sets, and Hamel bases) is considered in connection with \textbf{CH} and \textbf{MA}.Flett's theorem with infinite derivativeshttps://zbmath.org/1496.260022022-11-17T18:59:28.764376Z"Vîjîitu, Viorel"https://zbmath.org/authors/?q=ai:vijiitu.viorelIn this paper, the Flett mean value theorem is extended to continuous everywhere differentiable functions whose derivative may be infinite at some points.
The following is the main result obtained there.
Theorem. Let \(F:[a,b]\to R\) be a continuous function that has a (possibly infinite) derivative at any point of \([a,b]\), and satisfies \(F'(a)=F'(b)\). Then, there exists at least one point \(c\in (a,b)\), such that \(F'(c)=(F(c)-F(a))/(c-a)\).
Further aspects occasioned by these developments are also discussed.
Reviewer: Mihai Turinici (Iaşi)A new method on box dimension of Weyl-Marchaud fractional derivative of Weierstrass functionhttps://zbmath.org/1496.260072022-11-17T18:59:28.764376Z"Yao, Kui"https://zbmath.org/authors/?q=ai:yao.kui"Chen, Haotian"https://zbmath.org/authors/?q=ai:chen.haotian"Peng, W. L."https://zbmath.org/authors/?q=ai:peng.wenliang"Wang, Zekun"https://zbmath.org/authors/?q=ai:wang.zekun"Yao, Jia"https://zbmath.org/authors/?q=ai:yao.jia"Wu, Yipeng"https://zbmath.org/authors/?q=ai:wu.yipengSummary: A new method is applied to calculating fractal dimensions of fractional calculus of some fractal functions, by this method, we obtain Box dimension of Weyl-Marchaud fractional derivative of Weierstrass function.The asymptotic expansion for a class of convergent sequences defined by integralshttps://zbmath.org/1496.260082022-11-17T18:59:28.764376Z"Andrica, Dorin"https://zbmath.org/authors/?q=ai:andrica.dorin"Marinescu, Dan Ştefan"https://zbmath.org/authors/?q=ai:marinescu.dan-stefanSummary: We obtain the complete asymptotic expansion of the sequence defined by \(\int_0^1f(x)g(x^n)dx\), where the functions \(f\) and \(g\) satisfy various conditions. The main result is applied in Sect. 4 to find the complete asymptotic expansion of some classical sequences.
For the entire collection see [Zbl 1485.65002].When a convergence of filters is measure-theoretichttps://zbmath.org/1496.280012022-11-17T18:59:28.764376Z"Dolecki, Szymon"https://zbmath.org/authors/?q=ai:dolecki.szymonSummary: Convergence almost everywhere cannot be induced by a topology, and if measure is finite, it coincides with almost uniform convergence and is finer than convergence in measure, which is induced by a metrizable topology.
Measures are assumed to be finite. It is proved that convergence in measure is the Urysohn modification of convergence almost everywhere, which is pseudotopological. Extensions of these convergences from sequences to arbitrary filters are discussed, and a concept of measure-theoretic convergence is introduced. A natural extension of convergence almost everywhere is neither measure-theoretic, nor finer than a natural extension of convergence in measure. A straightforward extension of almost uniform convergence is not pseudotopologically induced; it is finer than a natural extension of convergence in measure.Measures on effect algebrashttps://zbmath.org/1496.280022022-11-17T18:59:28.764376Z"Barbieri, Giuseppina"https://zbmath.org/authors/?q=ai:barbieri.giuseppina-gerarda"García-Pacheco, Francisco Javier"https://zbmath.org/authors/?q=ai:garcia-pacheco.francisco-javier"Moreno-Pulido, Soledad"https://zbmath.org/authors/?q=ai:moreno-pulido.soledadSummary: We study measures defined on effect algebras. We characterize real-valued measures on effect algebras and find a class of effect algebras, that include the natural effect algebras of sets, on which \(\sigma\)-additive measures with values in a finite dimensional Banach space are always bounded. We also prove that in effect algebras the Nikodym and the Grothendieck properties together imply the Vitali-Hahn-Saks property, and find an example of an effect algebra verifying the Vitali-Hahn-Saks property but failing to have the Nikodym property. Finally, we define the concept of variation for vector measures on effect algebras proving that in effect algebras verifying the Riesz Decomposition Property, the variation of a finitely additive vector measure is a finitely additive positive measure.On the Hausdorff and packing measures of typical compact metric spaceshttps://zbmath.org/1496.280032022-11-17T18:59:28.764376Z"Jurina, S."https://zbmath.org/authors/?q=ai:jurina.simon"MacGregor, N."https://zbmath.org/authors/?q=ai:macgregor.norman"Mitchell, A."https://zbmath.org/authors/?q=ai:mitchell.a-k|mitchell.aine|mitchell.alex|mitchell.andrew-j"Olsen, L."https://zbmath.org/authors/?q=ai:olsen.lars-ole-ronnow"Stylianou, A."https://zbmath.org/authors/?q=ai:stylianou.antonis-c|stylianou.anastasios|stylianou.athanasios-nSummary: We study the Hausdorff and packing measures of typical compact metric spaces belonging to the Gromov-Hausdorff space (of all compact metric spaces) equipped with the Gromov-Hausdorff metric.Absolute continuity of non-homogeneous self-similar measureshttps://zbmath.org/1496.280042022-11-17T18:59:28.764376Z"Saglietti, Santiago"https://zbmath.org/authors/?q=ai:saglietti.santiago"Shmerkin, Pablo"https://zbmath.org/authors/?q=ai:shmerkin.pablo-s"Solomyak, Boris"https://zbmath.org/authors/?q=ai:solomyak.borisSummary: We prove that self-similar measures on the real line are absolutely continuous for almost all parameters in the super-critical region, in particular confirming a conjecture of S.-M. Ngai and Y. Wang. While recently there has been much progress in understanding absolute continuity for homogeneous self-similar measures, this is the first improvement over the classical transversality method in the general (non-homogeneous) case. In the course of the proof, we establish new results on the dimension and Fourier decay of a class of random self-similar measures.Fractals and the method of iterated function systemshttps://zbmath.org/1496.280052022-11-17T18:59:28.764376Z"Anorova, Sh. A."https://zbmath.org/authors/?q=ai:anorova.sh-a"Adilova, G. P."https://zbmath.org/authors/?q=ai:adilova.g-p"Èrzhonov, M. O."https://zbmath.org/authors/?q=ai:erzhonov.m-o(no abstract)A quantification of a Besicovitch non-linear projection theorem via multiscale analysishttps://zbmath.org/1496.280062022-11-17T18:59:28.764376Z"Davey, Blair"https://zbmath.org/authors/?q=ai:davey.blair"Taylor, Krystal"https://zbmath.org/authors/?q=ai:taylor.krystalSummary: The Besicovitch projection theorem states that if a subset \(E\) of the plane has finite length in the sense of Hausdorff measure and is purely unrectifiable (so its intersection with any Lipschitz graph has zero length), then almost every orthogonal projection of \(E\) to a line will have zero measure. In other words, the Favard length of a purely unrectifiable 1-set vanishes. In this article, we show that when linear projections are replaced by certain non-linear projections called \textit{curve projections}, this result remains true. In fact, we go further and use multiscale analysis to prove a quantitative version of this Besicovitch non-linear projection theorem. Roughly speaking, we show that if a subset of the plane has finite length in the sense of Hausdorff and is nearly purely unrectifiable, then its \textit{Favard curve length} is very small. Our techniques build on those of \textit{T. Tao}, who in [Proc. Lond. Math. Soc. (3) 98, No. 3, 559--584 (2009; Zbl 1173.28001)] proves a quantification of the original Besicovitch projection theorem.Ledrappier-Young formulae for a family of nonlinear attractorshttps://zbmath.org/1496.280072022-11-17T18:59:28.764376Z"Jurga, Natalia"https://zbmath.org/authors/?q=ai:jurga.natalia"Lee, Lawrence D."https://zbmath.org/authors/?q=ai:lee.lawrence-dSummary: We study a natural class of invariant measures supported on the attractors of a family of nonlinear, non-conformal iterated function systems introduced by Falconer, Fraser and Lee. These are pushforward quasi-Bernoulli measures, a class which includes the well-known class of Gibbs measures for Hölder continuous potentials. We show that these measures are exact dimensional and that their exact dimensions satisfy a Ledrappier-Young formula.A novel permeability model in damaged tree-like bifurcating networks considering the influence of roughnesshttps://zbmath.org/1496.280082022-11-17T18:59:28.764376Z"Liu, Zhenjie"https://zbmath.org/authors/?q=ai:liu.zhenjie"Gao, Jun"https://zbmath.org/authors/?q=ai:gao.jun"Xiao, Boqi"https://zbmath.org/authors/?q=ai:xiao.boqi"Cao, Jiyin"https://zbmath.org/authors/?q=ai:cao.jiyin"Fang, Jing"https://zbmath.org/authors/?q=ai:fang.jing"Liang, Mingchao"https://zbmath.org/authors/?q=ai:liang.mingchao"Long, Gongbo"https://zbmath.org/authors/?q=ai:long.gongboContractive affine generalized iterated function systems which are topologically contractinghttps://zbmath.org/1496.280092022-11-17T18:59:28.764376Z"Miculescu, Radu"https://zbmath.org/authors/?q=ai:miculescu.radu"Mihail, Alexandru"https://zbmath.org/authors/?q=ai:mihail.alexandru"Urziceanu, Silviu-Aurelian"https://zbmath.org/authors/?q=ai:urziceanu.silviu-aurelianSummary: In this paper we provide alternative descriptions of those generalized possibly infinite iterated function systems whose constitutive functions are affine contractions which are topologically contracting generalized iterated function systems. As a by-product, we establish conditions under which the attractor of a contractive affine generalized possibly infinite iterated function system is compact.Methods for constructing equations of objects of fractal geometryhttps://zbmath.org/1496.280102022-11-17T18:59:28.764376Z"Nazirov, Sh. A."https://zbmath.org/authors/?q=ai:nazirov.shodmankula-abdirozikovich"Èrzhanov, M. O."https://zbmath.org/authors/?q=ai:erzhanov.m-o"Tashmukhamedova, G. Kh."https://zbmath.org/authors/?q=ai:tashmukhamedova.g-kh(no abstract)Application of the \(R\)-functions method in fractal geometryhttps://zbmath.org/1496.280112022-11-17T18:59:28.764376Z"Nazirov, Sh. A."https://zbmath.org/authors/?q=ai:nazirov.shodmankula-abdirozikovich"Èrzhonov, M. O."https://zbmath.org/authors/?q=ai:erzhonov.m-o(no abstract)Several special functions in fractals and applications of the fractal in machine learninghttps://zbmath.org/1496.280122022-11-17T18:59:28.764376Z"Wang, Jun"https://zbmath.org/authors/?q=ai:wang.jun.24"Cao, Lei"https://zbmath.org/authors/?q=ai:cao.lei.1|cao.lei"Chen, Xiliang"https://zbmath.org/authors/?q=ai:chen.xiliang"Tang, Wei"https://zbmath.org/authors/?q=ai:tang.wei"Xu, Zhixiong"https://zbmath.org/authors/?q=ai:xu.zhixiongOn a Lusin theorem for capacitieshttps://zbmath.org/1496.280132022-11-17T18:59:28.764376Z"Wiesel, Johannes"https://zbmath.org/authors/?q=ai:wiesel.johannes-c-wIn this paper, the author proves a Lusin type theorem for a subadditive capacity defined on a compact metric space. Lusin's theorem is valid if and only if the capacity is continuous from above.
Reviewer: Alina Gavrilut (Iasi)Repeated quasi-integration on locally compact spaceshttps://zbmath.org/1496.280142022-11-17T18:59:28.764376Z"Butler, Svetlana V."https://zbmath.org/authors/?q=ai:butler.svetlana-vLet \(C_c(X)\) be the space of continuous functions with compact support in \(X\), a locally compact Hausdorff space. A quasi-integral \(\rho\) (also we say quasi-linear functional) is a real map on \(C_c(X)\), homogeneous, additive, and positive. A topological measure on \(X\) is a set of positive functions on the set of open sets union of the set of closed sets in \(X\) and it is simple when it takes only two values zero and one.
In the third section, the author studies (almost) simple quasi-integrals, i.e., the corresponding topological measures are simple. Also, she presents equivalent conditions for a quasi-integral to be (almost) simple. In the fourth section, she studies repeated quasi-integrals and particularly she provides criteria for repeated quasi-integration to produce a quasi-linear functional and for a double quasi-integral to be (almost) simple. In the fifth section, the author furnishes formulas describing how a product of compact-finite topological measures acts on open and compact sets.
Reviewer: Mohammed El Aïdi (Bogotá)Semisolid sets and topological measureshttps://zbmath.org/1496.280152022-11-17T18:59:28.764376Z"Butler, Svetlana V."https://zbmath.org/authors/?q=ai:butler.svetlana-vThe paper investigates topological measures, that is, measures that are defined on open and closed sets of a locally compact space and which are finitely additive on the collection of open and compact sets, inner regular on open sets and outer regular on closed sets. In this context, semisolid sets and solid-set functions are studied and several examples of finite and infinite topological measures are also provided.
Reviewer: Alina Gavrilut (Iasi)A new approach for an intuitionistic fuzzy Sugeno integral using morphological gradient edge detectorhttps://zbmath.org/1496.280162022-11-17T18:59:28.764376Z"Martínez, Gabriela E."https://zbmath.org/authors/?q=ai:martinez.gabriela-e"Melin, Patricia"https://zbmath.org/authors/?q=ai:melin.patricia"Castillo, Oscar"https://zbmath.org/authors/?q=ai:castillo.oscarSummary: In this paper, an extension to the Sugeno integral using intuitionistic fuzzy sets for morphological gradient edge detection is presented. The proposed method is used as an aggregation operator to combine the four gradients of the morphological gradient edge detector and enables the calculation of the Sugeno integral for combining multiple sources of information with a membership degree and non-membership using intuitionistic fuzzy sets. In this paper, the focus is on aggregation operators that use measures with intuitionistic fuzzy sets, in particular, the Sugeno integral. The performance of the proposed method is compared with the traditional Sugeno integral and other aggregation operators, using images of the Berkeley database (BSDS) and with synthetic images.
For the entire collection see [Zbl 1478.03003].Hadamard integral inequality for the class of semi-harmonically convex functionshttps://zbmath.org/1496.280172022-11-17T18:59:28.764376Z"Vosoughian, Hamid"https://zbmath.org/authors/?q=ai:vosoughian.hamidThe Poincaré exponent and the dimensions of Kleinian limit setshttps://zbmath.org/1496.300222022-11-17T18:59:28.764376Z"Fraser, Jonathan M."https://zbmath.org/authors/?q=ai:fraser.jonathan-mA Kleinian group \(\Gamma\) is a discrete group of orientation-preserving isometries of the hyperbolic space \( \mathbb H^n\). In the Poincaré disk model, the boundary at infinity of \( \mathbb H^n\) can be identified with the unit sphere \( S^{n-1}\) in \( \mathbb R^n\), and hyperbolic isometries act by conformal diffeomorphisms of the unit disk \( \mathbb D^n\).
An interesting phenomenon is that the orbit of a point under the action of a Kleinian group in \( \Gamma \) can accumulate to the sphere at infinity. The rate at which this happens is measured by the so-called Poincaré exponent \( \delta (\Gamma) \) of \( \Gamma \). The subspace of the sphere at infinity consisting of all the accumulation points of an orbit is called the limit set \( L (\Gamma) \) of \( \Gamma \), and often displays intricate fractal geometry. A celebrated result in this area is that for a geometrically finite nonelementary Kleinian group \( \Gamma \) the Poincaré exponent is equal to the Hausdorff dimension and the upper box dimension of the limit set.
In this paper the author proposes an elementary proof of the fact that the Poincaré exponent \( \delta (\Gamma) \) of a nonelementary Kleinian group is a lower bound for the upper box dimension of the limit set \( L (\Gamma) \). The proof is based on simple estimates for the Euclidean volume of hyperbolic balls, and only involves elementary methods from hyperbolic and fractal geometry.
Reviewer: Lorenzo Ruffoni (Medford)The mutual singularity of harmonic measure and Hausdorff measure of codimension smaller than onehttps://zbmath.org/1496.310052022-11-17T18:59:28.764376Z"Tolsa, Xavier"https://zbmath.org/authors/?q=ai:tolsa.xavierThis article discusses the mutual singularity of the harmonic and Hausdorff measure of codimension smaller than one. Precisely, let \(\Omega\subset {\mathbb R}^{n+1}\), \(n\geq 2\), be an open set, \(E\subset \partial \Omega\) and \(s\in (n, n+1)\). Suppose that:
(i) There exists \(r_E>0\) and \(c_E>0\) such that \(\mathrm{Cap}(B(x,r)\cap \Omega^c)\geq c_E r^{n-1}\) for all \(0<r\leq r_E\) and all \(x\in E\);
(ii) The harmonic measure and the \(s\)-Hausdorff measure on \(E\) are mutually absolute continuous.
The main result of the article establishes that under the above assumptions, both the harmonic measure and the \(s\)-Hausdorff measure of \(E\) are zero.
Reviewer: Marius Ghergu (Dublin)Generalized fractal dimensions of invariant measures of full-shift systems over compact and perfect spaces: generic behaviorhttps://zbmath.org/1496.370092022-11-17T18:59:28.764376Z"Carvalho, Silas L."https://zbmath.org/authors/?q=ai:carvalho.silas-l"Condori, Alexander"https://zbmath.org/authors/?q=ai:condori.alexanderSummary: In this paper, we show that, for topological dynamical systems with a dense set (in the weak topology) of periodic measures, a typical (in Baire's sense) invariant measure has, for each \(q>0\), zero lower \(q\)-generalized fractal dimension. This implies, in particular, that a typical invariant measure has zero upper Hausdorff dimension and zero lower rate of recurrence. Of special interest is the full-shift system \((X,T)\) (where \(X=M^{\mathbb{Z}}\) is endowed with a sub-exponential metric and the alphabet \(M\) is a compact and perfect metric space), for which we show that a typical invariant measure has, for each \(q>1\), infinite upper \(q\)-correlation dimension. Under the same conditions, we show that a typical invariant measure has, for each \(s\in(0,1)\) and each \(q>1\), zero lower \(s\)-generalized and infinite upper \(q\)-generalized dimensions.Analogues of Khintchine's theorem for random attractorshttps://zbmath.org/1496.370532022-11-17T18:59:28.764376Z"Baker, Simon"https://zbmath.org/authors/?q=ai:baker.simon|baker.simon.1"Troscheit, Sascha"https://zbmath.org/authors/?q=ai:troscheit.saschaSummary: In this paper we study random iterated function systems. Our main result gives sufficient conditions for an analogue of a well known theorem due to \textit{A. Khintchine} [Math. Ann. 92, 115--125 (1924; JFM 50.0125.01)] from Diophantine approximation to hold almost surely for stochastically self-similar and self-affine random iterated function systems.Approximation by mixed operators of max-product-Choquet typehttps://zbmath.org/1496.410112022-11-17T18:59:28.764376Z"Gal, Sorin G."https://zbmath.org/authors/?q=ai:gal.sorin-gheorghe"Iancu, Ionut T."https://zbmath.org/authors/?q=ai:iancu.ionut-tSummary: The main aim of this chapter is to introduce several mixed operators between Choquet integral operators and max-product operators and to study their approximation, shape preserving, and localization properties. Section 2 contains some preliminaries on the Choquet integral. In Sect. 3, we obtain quantitative estimates in uniform and pointwise approximation for the following mixed type operators: max-product Bernstein-Kantorovich-Choquet operator, max-product Szász-Mirakjan-Kantorovich-Choquet operators, nontruncated and truncated cases, and max-product Baskakov-Kantorovich-Choquet operators, nontruncated and truncated cases. We show that for large classes of functions, the max-product Bernstein-Kantorovich-Choquet operators approximate better than their classical correspondents, and we construct new max-product Szász-Mirakjan-Kantorovich-Choquet and max-product Baskakov-Kantorovich-Choquet operators, which approximate uniformly \(f\) in each compact subinterval of \([0, +\infty)\) with the order \(\omega_1(f; \sqrt{\lambda_n})\), where \(\lambda_n \searrow 0\) arbitrary fast. Also, shape preserving and localization results for max-product Bernstein-Kantorovich-Choquet operators are obtained. Section 4 contains quantitative approximation results for discrete max-product Picard-Kantorovich-Choquet, discrete max-product Gauss-Weierstrass-Kantorovich-Choquet operators, and discrete max-product Poisson-Cauchy-Kantorovich-Choquet operators. Section 5 deals with the approximation properties of the max-product Kantorovich-Choquet operators based on \((\varphi, \psi)\)-kernels. It is worth to mention that with respect to their max-product correspondents, while they keep their good properties, the mixed max-product Choquet operators present, in addition, the advantage of a great flexibility by the many possible choices for the families of set functions used in their definitions. The results obtained present potential applications in sampling theory, neural networks, and learning theory.
For the entire collection see [Zbl 1485.65002].Hardy-Littlewood maximal operator on variable Lebesgue spaces with respect to a probability measurehttps://zbmath.org/1496.420252022-11-17T18:59:28.764376Z"Moreno, Jorge"https://zbmath.org/authors/?q=ai:moreno.jorge"Pineda, Ebner"https://zbmath.org/authors/?q=ai:pineda.ebner"Rodriguez, Luz"https://zbmath.org/authors/?q=ai:rodriguez.luz"Urbina, Wilfredo O."https://zbmath.org/authors/?q=ai:urbina-romero.wilfredo-oIn this paper, the authors established the strong and weak boundedness of Hardy-Littlewood maximal operators on variable Lebesgue spaces \(L^{p(\cdot)}(\mu)\) with respect to a probability Borel measure \(\mu\) for two conditions of regularity on the exponent function \(p(\cdot)\).
To be more precise, let \(\mu\) be a Radon measure and \(f\in L^1_{\mathrm{loc}}(\mathbb{R}^d,\mu)\), The Hardy-Littlewood non-centered maximal function of \(f\), with respect to \(\mu\), is defined by
\[
M_\mu f(x)=\sup_{B\ni x}\fint_{B}|f(x)|\mu(dy),
\]
and the Hardy-Littlewood centered maximal function of \(f\), with respect to \(\mu\), is defined by
\[
M_\mu^c f(x)=\sup_{r>0}\fint_{B(x,r)}|f(y)|\mu(dy),
\]
The authors first prove the boundedness with the condition \(\mathcal{P}^0_\mu(\mathbb{R}^d)\).
Theorem 1. Let \(p(\cdot)\in\mathcal{P}^0_\mu(\mathbb{R}^d)\) be continuous with \(p_- > 1\).
(i) There exists \(c>0\) depending on \(p\) such that
\[
\|M_\mu^c f\|_{p(\cdot),\mu}\leq c\|f\|_{p(\cdot),\mu}.
\]
(ii) If \(M_\mu\) is bounded on \(L^p-(\mathbb{R}^d,\mu)\) then there exists \(c>0\) depending on \(p\) such that
\[
\|M_\mu f\|_{p(\cdot),\mu}\leq c\|f\|_{p(\cdot),\mu}.
\]
Then, they gave the boundedness with the condition \(\mathcal{P}_\mu(\mathbb{R}^d)\).
Theorem 2. Let \(p(\cdot)\in\mathcal{P}_\mu(\mathbb{R}^d)\) with \(p_- > 1\) be such that \(1/{p(\cdot)}\) is continuous. If \(M_\mu\) is bounded on \(L^p-(\mathbb{R}^d,\mu)\) then there exists \(c>0\) depending on \(p\) such that
\[
\|M_\mu^c f\|_{p(\cdot),\mu}\leq c\|f\|_{p(\cdot),\mu}.
\]
Theorem 3. Let \(p(\cdot)\in\mathcal{P}_\mu(\mathbb{R}^d)\) be such that \(1/{p(\cdot)}\) is continuous, then there exists \(C>0\) depending on \(p\) such that
\[
\|t\chi_{\{x\in\mathbb{R}^d:M_\mu^c f(x)>t\}}\|_{p(\cdot),\mu}\leq C\|f\|_{p(\cdot),\mu}
\]
for all \(f\in L^1_{\mathrm{loc}}(\mathbb{R}^d,\mu)\), \(t>0\).
They also extended some properties of this operator to a probability Borel measure \(\mu\), the key to extending these results is using the Besicovitch covering lemma instead of the Calderón Zygmund decomposition.
Reviewer: Qingying Xue (Beijing)A generalized polar-coordinate integration formula with applications to the study of convolution powers of complex-valued functions on \(\mathbb{Z}^d\)https://zbmath.org/1496.430062022-11-17T18:59:28.764376Z"Bui, Huan Q."https://zbmath.org/authors/?q=ai:bui.huan-q"Randles, Evan"https://zbmath.org/authors/?q=ai:randles.evanSummary: In this article, we consider a class of functions on \(\mathbb{R}^d\), called positive homogeneous functions, which interact well with certain continuous one-parameter groups of (generally anisotropic) dilations. Generalizing the Euclidean norm, positive homogeneous functions appear naturally in the study of convolution powers of complex-valued functions on \(\mathbb{Z}^d\). As the spherical measure is a Radon measure on the unit sphere which is invariant under the symmetry group of the Euclidean norm, to each positive homogeneous function \(P\), we construct a Radon measure \(\sigma_P\) on \(S=\{\eta \in\mathbb{R}^d:P(\eta)=1\}\) which is invariant under the symmetry group of \(P\). With this measure, we prove a generalization of the classical polar-coordinate integration formula and deduce a number of corollaries in this setting. We then turn to the study of convolution powers of complex functions on \(\mathbb{Z}^d\) and certain oscillatory integrals which arise naturally in that context. Armed with our integration formula and the Van der Corput lemma, we establish sup-norm-type estimates for convolution powers; this result is new and partially extends results of Randles and Saloff-Coste
[\textit{E.~Randles} and \textit{L.~Saloff-Coste}, Rev. Mat. Iberoam. 33, No.~3, 1045--1121 (2017; Zbl 1377.42012)].Rigidity of the Pu inequality and quadratic isoperimetric constants of normed spaceshttps://zbmath.org/1496.460112022-11-17T18:59:28.764376Z"Creutz, Paul"https://zbmath.org/authors/?q=ai:creutz.paulThe author furnishes an enhanced bound on the filling areas curves (not closed geodesics) in Banach spaces. He shows rigidity of \textit{P. M. Pu}'s classical systolic inequality [Pac. J. Math. 2, 55--71 (1952; Zbl 0046.39902)] and examines the isoperimetric constants of normed spaces.
Reviewer: Mohammed El Aïdi (Bogotá)On Nikodým and Rainwater sets for \(ba(\mathcal{R})\) and a problem of {M. Valdivia}https://zbmath.org/1496.460212022-11-17T18:59:28.764376Z"Ferrando, J. C."https://zbmath.org/authors/?q=ai:ferrando.juan-carlos"López-Alfonso, S."https://zbmath.org/authors/?q=ai:lopez-alfonso.salvador"López-Pellicer, M."https://zbmath.org/authors/?q=ai:lopez-pellicer.manuelSummary: If \(\mathcal{R}\) is a ring of subsets of a set \(\Omega\) and \(ba(\mathcal{R})\) is the Banach space of bounded finitely additive measures defined on \(\mathcal{R}\) equipped with the supremum norm, a subfamily \(\Delta\) of \(\mathcal{R}\) is called a \textit{Nikodým set} for \(ba(\mathcal{R})\) if each set \(\{\mu_\alpha:\alpha\in\Lambda\}\) in \(ba(\mathcal{R})\) which is pointwise bounded on \(\Delta\) is norm-bounded in \(ba(\mathcal{R})\). If the whole ring \(\mathcal{R}\) is a Nikodým set, \(\mathcal{R}\) is said to have property \((N)\), which means that \(\mathcal{R}\) satisfies the Nikodým-Grothendieck boundedness theorem. In this paper we find a class of rings with property \((N)\) that fail Grothendieck's property \((G)\) and prove that a ring \(\mathcal{R}\) has property \((G)\) if and only if the set of the evaluations on the sets of \(\mathcal{R}\) is a so-called \textit{Rainwater set} for \(ba(\mathcal{R})\). Recalling that \(\mathcal{R}\) is called a \((wN)\)-ring if each increasing web in \(\mathcal{R}\) contains a strand consisting of Nikodým sets, we also give a partial solution to a question raised by \textit{M. Valdivia} [Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., RACSAM 107, No. 2, 355--372 (2013; Zbl 1290.46019)] by providing a class of rings without property \((G)\) for which the relation \((N)\Leftrightarrow(wN)\) holds.On entropy for general quantum systemshttps://zbmath.org/1496.460632022-11-17T18:59:28.764376Z"Majewski, W. A."https://zbmath.org/authors/?q=ai:majewski.wladyslaw-adam"Labuschagne, L. E."https://zbmath.org/authors/?q=ai:labuschagne.louis-eSummary: In these notes we will give an overview and road map for a definition and characterization of (relative) entropy for both classical and quantum systems. In other words, we will provide a consistent treatment of entropy which can be applied within the recently developed Orlicz space based approach to large systems. This means that the proposed approach successfully provides a refined framework for the treatment of entropy in each of classical statistical physics, Dirac's formalism of Quantum Mechanics, large systems of quantum statistical physics, and finally also for Quantum Field Theory.Area of intrinsic graphs and coarea formula in Carnot groupshttps://zbmath.org/1496.530432022-11-17T18:59:28.764376Z"Julia, Antoine"https://zbmath.org/authors/?q=ai:julia.antoine"Nicolussi Golo, Sebastiano"https://zbmath.org/authors/?q=ai:nicolussi-golo.sebastiano"Vittone, Davide"https://zbmath.org/authors/?q=ai:vittone.davideThe authors consider submanifolds of sub-Riemannian Carnot groups with intrinsic \(C^1\) regularity \((C^1_H )\). The first main result in the present paper is an area formula for \(C^1_H\) intrinsic graphs; as an application, the authors deduce density properties for Hausdorff measures on rectifiable sets. The second main result is a coarea formula for slicing \(C^1_H\) submanifolds into level sets of a \(C^1_H\) function
Reviewer: Peibiao Zhao (Nanjing)Correction to: ``On the equivalence of conglomerability and disintegrability for unbounded random variables''https://zbmath.org/1496.600032022-11-17T18:59:28.764376Z"Schervish, Mark J."https://zbmath.org/authors/?q=ai:schervish.mark-j"Seidenfeld, Teddy"https://zbmath.org/authors/?q=ai:seidenfeld.teddy"Kadane, Joseph B."https://zbmath.org/authors/?q=ai:kadane.joseph-bornCorrection to the authors' paper [ibid. 23, No. 4, 501--518 (2014; Zbl 1477.60011)].Corrigendum to: ``On iterated function systems with place-dependent probabilities''https://zbmath.org/1496.600332022-11-17T18:59:28.764376Z"Bárány, Balázs"https://zbmath.org/authors/?q=ai:barany.balazsFrom the text: In my paper [ibid. 143, No. 1, 419--432 (2015; Zbl 1308.60044)], the proof of the main theorem contains a crucial error. The estimates in page 427 line 2--5 and in page 429 line 6--7 are incorrect. The solution of this problem was far from being trivial and so the proof of this paper remains incomplete.
In our recent paper with my coauthors, \textit{K. Simon}, \textit{B. Solomyak} and \textit{A. Śpiewak}, we were able to circumvent the problem. Under more restrictive assumptions on the smoothness of a one-dimensional parameter family of iterated function systems, we showed that the claim of the main theorem remains valid even for a wider class of parameter dependent family of invariant measures. For details, we refer the reader to [Adv. Math. 399, Article ID 108258, 73 p. (2022; Zbl 07496420)].Hausdorff dimensions for shared endpoints of disjoint geodesics in the directed landscapehttps://zbmath.org/1496.601152022-11-17T18:59:28.764376Z"Bates, Erik"https://zbmath.org/authors/?q=ai:bates.erik"Ganguly, Shirshendu"https://zbmath.org/authors/?q=ai:ganguly.shirshendu"Hammond, Alan"https://zbmath.org/authors/?q=ai:hammond.alanSummary: Within the Kardar-Parisi-Zhang universality class, the space-time Airy sheet is conjectured to be the canonical scaling limit for last passage percolation models. In recent work [``The directed landscape'', Preprint, \url{arXiv:1812.00309}] of \textit{D. Dauvergne} et al., this object was constructed and, upon a parabolic correction, shown to be the limit of one such model: Brownian last passage percolation. The limit object without parabolic correction, called the directed landscape, admits geodesic paths between any two space-time points \((x,s)\) and \((y,t)\) with \(s< t\). In this article, we examine fractal properties of the set of these paths. Our main results concern exceptional endpoints admitting disjoint geodesics. First, we fix two distinct starting locations \(x_1\) and \(x_2\), and consider geodesics traveling \((x_1, 0)\to (y,1)\) and \((x_2, 0)\to (y,1)\). We prove that the set of \(y\in \mathbb{R}\) for which these geodesics coalesce only at time 1 has Hausdorff dimension one-half. Second, we consider endpoints \((x,0)\) and \((y,1)\) between which there exist two geodesics intersecting only at times 0 and 1. We prove that the set of such \((x,y)\in\mathbb{R}^2\) also has Hausdorff dimension one-half. The proofs require several inputs of independent interest, including (i) connections to the so-called \textit{difference weight profile} studied in
[\textit{R. Basu} et al., Ann. Probab. 49, No. 1, 485--505 (2021; Zbl 1457.82165)]; and (ii) a tail estimate on the number of disjoint geodesics starting and ending in small intervals. The latter result extends the analogous estimate proved for the prelimiting model in [\textit{A. Hammond}, Proc. Lond. Math. Soc. (3) 120, No. 3, 370--433 (2020; Zbl 1453.82078)].In reference to a self-referential approach towards smooth multivariate approximationhttps://zbmath.org/1496.650222022-11-17T18:59:28.764376Z"Pandey, K. K."https://zbmath.org/authors/?q=ai:pandey.kshitij-kumar"Viswanathan, P."https://zbmath.org/authors/?q=ai:viswanathan.puthan-veeduSummary: Approximation of a multivariate function is an important theme in the field of numerical analysis and its applications, which continues to receive a constant attention. In this paper, we provide a parameterized family of self-referential (fractal) approximants for a given multivariate smooth function defined on an axis-aligned hyper-rectangle. Each element of this class preserves the smoothness of the original function and interpolates the original function at a prefixed gridded data set. As an application of this construction, we deduce a fractal methodology to approach a multivariate Hermite interpolation problem. This part of our paper extends the classical bivariate Hermite's interpolation formula by \textit{A. C. Ahlin} [Math. Comput. 18, 264--273 (1964; Zbl 0122.12501)] in a twofold sense: (i) records, in particular, a multivariate generalization of this bivariate interpolation theory; (ii) replaces the unicity of the Hermite interpolant with a parameterized family of self-referential Hermite interpolants which contains the multivariate analogue of Ahlin's interpolant as a particular case. Some related aspects including the approximation by multivariate self-referential functions preserving Popoviciu convexity are given, too.A novel variational approach for fractal Ginzburg-Landau equationhttps://zbmath.org/1496.651952022-11-17T18:59:28.764376Z"Wang, Kang-Le"https://zbmath.org/authors/?q=ai:wang.kangle"Wang, Hao"https://zbmath.org/authors/?q=ai:wang.hao.13|wang.hao.5|wang.hao.9|wang.hao.3|wang.hao.1|wang.hao.2|wang.hao.11|wang.hao.12|wang.hao.4|wang.hao.7|wang.hao.10|wang.hao.6Bounding the dimension of points on a linehttps://zbmath.org/1496.681602022-11-17T18:59:28.764376Z"Lutz, Neil"https://zbmath.org/authors/?q=ai:lutz.neil"Stull, D. M."https://zbmath.org/authors/?q=ai:stull.donald-mSummary: We use Kolmogorov complexity methods to give a lower bound on the effective Hausdorff dimension of the point \((x, ax + b)\), given real numbers \(a, b\), and \(x\). We apply our main theorem to a problem in fractal geometry, giving an improved lower bound on the (classical) Hausdorff dimension of generalized sets of Furstenberg type.Fractal-like actuator disc theory for optimal energy extractionhttps://zbmath.org/1496.760372022-11-17T18:59:28.764376Z"Dehtyriov, D."https://zbmath.org/authors/?q=ai:dehtyriov.daniel"Schnabl, A. M."https://zbmath.org/authors/?q=ai:schnabl.a-m"Vogel, C. R."https://zbmath.org/authors/?q=ai:vogel.christopher-r"Draper, S."https://zbmath.org/authors/?q=ai:draper.scott"Adcock, T. A. A."https://zbmath.org/authors/?q=ai:adcock.thomas-a-a"Willden, R. H. J."https://zbmath.org/authors/?q=ai:willden.richard-h-jSummary: The limit of power extraction by a device which makes use of constructive interference, i.e. local blockage, is investigated theoretically. The device is modelled using actuator disc theory in which we allow the device to be split into arrays and these then into sub-arrays an arbitrary number of times so as to construct an \(n\)-level multi-scale device in which the original device undergoes \(n-1\) sub-divisions. The alternative physical interpretation of the problem is a planar system of arrayed turbines in which groups of turbines are homogeneously arrayed at the smallest \(n\text{th}\) scale, and then these groups are homogeneously spaced relative to each other at the next smallest \(n-1\)th scale, with this pattern repeating at all subsequent larger scales. The scale-separation idea of \textit{T. Nishino} and \textit{R. H. J. Willden} [J. Fluid Mech. 708, 596--606 (2012; Zbl 1275.76045)] is employed, which assumes mixing within a sub-array occurs faster than mixing of the by-pass flow around that sub-array, so that in the \(n\)-scale device mixing occurs from the inner scale to the outermost scale in that order. We investigate the behaviour of an arbitrary level multi-scale device, and determine the arrangement of actuator discs (\(n\)th level devices) which maximises the power coefficient (ratio of power extracted to undisturbed kinetic energy flux through the net disc frontal area). We find that this optimal arrangement is close to fractal, and fractal arrangements give similar results. With the device placed in an infinitely wide channel, i.e. zero global blockage, we find that the optimum power coefficient tends to unity as the number of device scales tends to infinity, a 27/16 increase over the Lanchester-Betz limit of 0.593. For devices in finite width channels, i.e. non-zero global blockage, similar observations can be made with further uplift in the maximum power coefficient. We discuss the fluid mechanics of this energy extraction process and examine the scale distribution of thrust and wake velocity coefficients. Numerical demonstration of performance uplift due to multi-scale dynamics is also provided. We demonstrate that bypass flow remixing and ensuing energy losses increase the device power coefficient above the limits for single devices, so that although the power coefficient can be made to increase, this is at the expense of the overall efficiency of energy extraction which decreases as wake-scale remixing losses necessarily rise. For multi-scale devices in finite overall blockage two effects act to increase extractable power; an overall streamwise pressure gradient associated with finite blockage, and wake pressure recoveries associated with bypass-scale remixing.The Van Vleck formula on Ehrenfest time scales and stationary phase asymptotics for frequency-dependent phaseshttps://zbmath.org/1496.810592022-11-17T18:59:28.764376Z"Blair, Matthew D."https://zbmath.org/authors/?q=ai:blair.matthew-dSummary: The Van Vleck formula is a semiclassical approximation to the integral kernel of the propagator associated to a time-dependent Schrödinger equation. Under suitable hypotheses, we present a rigorous treatment of this approximation which is valid on \textit{Ehrenfest time scales}, i.e. \(\hbar\)-dependent time intervals which most commonly take the form \(|t| \le c|\log \hbar|\). Our derivation is based on an approximation to the integral kernel often called the \textit{Herman-Kluk approximation}, which realizes the kernel as an integral superposition of Gaussians parameterized by points in phase space. As was shown by \textit{D. Robert} [Rev. Math. Phys. 22, No. 10, 1123--1145 (2010; Zbl 1206.35212)], this yields effective approximations over Ehrenfest time intervals. In order to derive the Van Vleck approximation from the Herman-Kluk approximation, we are led to develop stationary phase asymptotics where the phase functions depend on the frequency parameter in a nontrivial way, a result which may be of independent interest.On a generalized central limit theorem and large deviations for homogeneous open quantum walkshttps://zbmath.org/1496.810602022-11-17T18:59:28.764376Z"Carbone, Raffaella"https://zbmath.org/authors/?q=ai:carbone.raffaella"Girotti, Federico"https://zbmath.org/authors/?q=ai:girotti.federico"Hernandez, Anderson Melchor"https://zbmath.org/authors/?q=ai:hernandez.anderson-melchorSummary: We consider homogeneous open quantum walks on a lattice with finite dimensional local Hilbert space and we study in particular the position process of the quantum trajectories of the walk. We prove that the properly rescaled position process asymptotically approaches a mixture of Gaussian measures. We can generalize the existing central limit type results and give more explicit expressions for the involved asymptotic quantities, dropping any additional condition on the walk. We use deformation and spectral techniques, together with reducibility properties of the local channel associated with the open quantum walk. Further, we can provide a large deviation principle in the case of a fast recurrent local channel and at least lower and upper bounds in the general case.Instanton effects vs resurgence in the \(O(3)\) sigma modelhttps://zbmath.org/1496.810792022-11-17T18:59:28.764376Z"Bajnok, Zoltán"https://zbmath.org/authors/?q=ai:bajnok.zoltan"Balog, János"https://zbmath.org/authors/?q=ai:balog.janos"Hegedűs, Árpád"https://zbmath.org/authors/?q=ai:hegedus.arpad"Vona, István"https://zbmath.org/authors/?q=ai:vona.istvanSummary: We investigate the ground-state energy of the integrable two dimensional \(O(3)\) sigma model in a magnetic field. By determining a large number of perturbative coefficients we explore the closest singularities of the corresponding Borel function. We then confront its median resummation to the high precision numerical solution of the exact integral equation and observe that the leading exponentially suppressed contribution is not related to the asymptotics of the perturbative coefficients. By analytically expanding the integral equation we calculate the leading non-perturbative contributions up to fourth order and find complete agreement. These anomalous terms could be attributed to instantons, while the asymptotics of the perturbative coefficients seems to be related to renormalons.Four-dimensional factorization of the fermion determinant in lattice QCDhttps://zbmath.org/1496.810942022-11-17T18:59:28.764376Z"Giusti, Leonardo"https://zbmath.org/authors/?q=ai:giusti.leonardo"Saccardi, Matteo"https://zbmath.org/authors/?q=ai:saccardi.matteoSummary: In the last few years it has been proposed a one-dimensional factorization of the fermion determinant in lattice QCD with Wilson-type fermions that leads to a block-local action of the auxiliary bosonic fields. Here we propose a four-dimensional generalization of this factorization. Possible applications are more efficient parallelizations of Monte Carlo algorithms and codes, master field simulations, and multi-level integration.Multifractal characteristics of singular signalshttps://zbmath.org/1496.940172022-11-17T18:59:28.764376Z"Oświęcimka, Paweł"https://zbmath.org/authors/?q=ai:oswiecimka.pawel"Minati, Ludovico"https://zbmath.org/authors/?q=ai:minati.ludovicoStarting with locally Hölder continuous functions satisfying \[g(x+h)-g(x)\propto C\cdot h^{\alpha(x)},\] multifractal processes are characterized by a typically concave form of the ``singularity spectrum'' \(f(\alpha)\). The spectrum is defined as \(f(\alpha)=d(\{x\mid \alpha(x)\stackrel{!}{=}\alpha\})\), where \(d(\cdot)\) denotes the Hausdorff dimension of the respective set. Such processes are applied in such diverse areas as financial time series or medical applications [\textit{A. L. Karperien} et al., Banach Cent. Publ. 109, 23--45 (2016; Zbl 1355.28014)].
The present paper compares two spectrum estimation techniques, namely the ``Multifractal Detrended Fluctuation Analysis'' (MDFA) [\textit{J. W. Kantelhardt} et al., Physica A 316, No. 1--4, 87--114 (2002; Zbl 1001.62029)] and the ``Wavelet Leader'' (WL)-method [\textit{S. Jaffard}, Proc. Symp. Pure Math. 72, 91--151 (2004; Zbl 1093.28005)] on three data sets. Two of them exhibit typical recursive multiscale structures like Binomial cascades or similar Cantor-like constructions.
The third signal, however, exhibits clearly visible isolated singularities, and a comparision of MDFA/WL results with a standard wavelet analysis diagram -- clearly locating these isolated singularities -- indicates, that a multifractal interpretation of the signal, based on MDFA/WL might be misleading in this case.
For the entire collection see [Zbl 1491.93003].
Reviewer: Hans-Georg Stark (Aschaffenburg)