Recent zbMATH articles in MSC 28Ahttps://zbmath.org/atom/cc/28A2024-02-15T19:53:11.284213ZWerkzeugRandom walks, spectral gaps, and Khintchine's theorem on fractalshttps://zbmath.org/1526.110402024-02-15T19:53:11.284213Z"Khalil, Osama"https://zbmath.org/authors/?q=ai:khalil.osama"Luethi, Manuel"https://zbmath.org/authors/?q=ai:luethi.manuelThe authors prove a complete analogue of Khintchin's theorem for a certain type of fractal measures. These fractal measures arise from iterated function systems consisting of a finite number of similarities with rational entries. It is further assumed that the fractals have sufficiently small Hausdorff co-dimension. In order to establish their results, an effective equidistribution theorem for certain fractal measures on the homogeneous space \(\mathcal{L}_{d+1}\) of unimodular lattices is proven. This result is established employing a new technique involving the construction of \(S\)-arithmetic operators possessing a spectral gap and encoding the arithmetic structure of the maps generating the fractal. As a consequence of their methods, the authors also show that spherical averages of certain random walks naturally associated to the fractal measures effectively equidistribute on \(\mathcal{L}_{d+1}\).
Reviewer: Peter Massopust (München)On the differentiation of random measures with respect to homothecy invariant convex baseshttps://zbmath.org/1526.280012024-02-15T19:53:11.284213Z"Chubinidze, Kakha"https://zbmath.org/authors/?q=ai:chubinidze.kakha-a"Oniani, Giorgi"https://zbmath.org/authors/?q=ai:oniani.giorgi-giglaSummary: For every homothecy invariant convex density differentiation basis \(B\) in \(\mathbb{R}^d\), we characterize sequences of weights \(w=(w_j)_{j\in \mathbb{N}}\) for which the random measures \(\mu_{w,\theta}=\sum_{j=1}^\infty w_j \delta_{\theta_j}\) are differentiable with respect to the basis \(B\) for almost every selection of a sequence of points \(\theta_1,\theta_2,\ldots\) from the unit cube \([0,1]^d\).A fine property of sets at points of Lebesgue densityhttps://zbmath.org/1526.280022024-02-15T19:53:11.284213Z"Delladio, S."https://zbmath.org/authors/?q=ai:delladio.silvanoSummary: Consider an integer \(n\ge 2, m\in [n,+\infty)\) and put \(\overline{\alpha}:=\frac{m-1}{n-1} \). Moreover:
\begin{itemize}
\item[\(\bullet\)] Let \({\gamma_x}:(0,+\infty)\to{\mathbb{R}^n}\) be the line through \(x=({x_1},\dots ,{x_n})\in{\mathbb{R}^n}\) defined as \({\gamma_x}(t):=(t{x_1},{t^{\overline{\alpha}}}{x_2},\dots ,{t^{\overline{\alpha}}}{x_n})\);
\item[\(\bullet\)] If \(T\) is any motion of \({\mathbb{R}^n} \), then let \({\gamma_x^{(T)}}\) be the line through \(x=({x_1},\dots ,{x_n})\in{\mathbb{R}^n}\) defined as \({\gamma_x^{(T)}}:=T\circ{\gamma_{{T^{-1}}(x)}} \);
\item[\(\bullet\)] Let \({\mathcal{H}^1}\) denote the one-dimensional Hausdorff measure in \({\mathbb{R}^n} \).
\end{itemize}
The main goal of this paper is to prove the following property: If \({x_0}\) is an \(m\)-density point of a Lebesgue measurable set \(E\) and \(T\) is an arbitrary motion of \({\mathbb{R}^n}\) mapping the origin to \({x_0} \), then we have
\[
\underset{t\to 0+}{\limsup}\frac{{\mathcal{H}^1}(E\cap{\gamma_x^{(T)}}((0,t]))}{{\mathcal{H}^1}({\gamma_x^{(T)}}((0,t]))}=1
\] for almost every \(x\in T(\{1\}\times{\mathbb{R}^{n-1}})\). An application of this result to locally finite perimeter sets is provided.Dimensions of fractional Brownian imageshttps://zbmath.org/1526.280032024-02-15T19:53:11.284213Z"Burrell, Stuart A."https://zbmath.org/authors/?q=ai:burrell.stuart-aSummary: This paper concerns the intermediate dimensions, a spectrum of dimensions that interpolate between the Hausdorff and box dimensions. Potential-theoretic methods are used to produce dimension bounds for images of sets under Hölder maps and certain stochastic processes. We apply this to compute the almost-sure value of the dimension of Borel sets under index-\( \alpha\) fractional Brownian motion in terms of dimension profiles defined using capacities. As a corollary, this establishes continuity of the profiles for Borel sets and allows us to obtain an explicit condition showing how the Hausdorff dimension of a set may influence the typical box dimension of Hölder images such as projections. The methods used propose a general strategy for related problems; dimensional information about a set may be learned from analysing particular fractional Brownian images of that set. To conclude, we obtain bounds on the Hausdorff dimension of exceptional sets, with respect to intermediate dimensions, in the setting of projections.Spectral decimation for a graph-directed fractal pairhttps://zbmath.org/1526.280042024-02-15T19:53:11.284213Z"Cao, Shiping"https://zbmath.org/authors/?q=ai:cao.shiping"Qiu, Hua"https://zbmath.org/authors/?q=ai:qiu.hua"Tian, Haoran"https://zbmath.org/authors/?q=ai:tian.haoran"Yang, Lijian"https://zbmath.org/authors/?q=ai:yang.lijianSummary: We introduce a graph-directed pair of planar self-similar sets that possess fully symmetric Laplacians. For these two fractals, due to Shima's celebrated criterion, we point out that one admits the spectral decimation by the canonic graph approximation and the other does not. For the second fractal, we adjust to choosing a new graph approximation guided by the directed graph, which still admits spectral decimation. Then we make a full description of the Dirichlet and Neumann eigenvalues and eigenfunctions of both of these two fractals.Regularity of Kleinian limit sets and Patterson-Sullivan measureshttps://zbmath.org/1526.300552024-02-15T19:53:11.284213Z"Fraser, Jonathan M."https://zbmath.org/authors/?q=ai:fraser.jonathan-mSummary: We consider several (related) notions of geometric regularity in the context of limit sets of geometrically finite Kleinian groups and associated Patterson-Sullivan measures. We begin by computing the upper and lower regularity dimensions of the Patterson-Sullivan measure, which involves controlling the relative measure of concentric balls. We then compute the Assouad and lower dimensions of the limit set, which involves controlling local doubling properties. Unlike the Hausdorff, packing, and box-counting dimensions, we show that the Assouad and lower dimensions are not necessarily given by the Poincaré exponent.From weak to strong convergence of the gradients for Finsler \(p\)-Laplacian problems as \(p \to \infty\)https://zbmath.org/1526.351992024-02-15T19:53:11.284213Z"Van Thanh Nguyen"https://zbmath.org/authors/?q=ai:van-thanh-nguyen.Summary: In this paper we investigate limits as \(p \to \infty\) of solutions \(u_p\) to Finsler \(p\)-Laplacian problems \(- \operatorname{div} \big(F^\ast (x, \nabla u_p)^{p - 1} \partial_\xi F^\ast (x, \nabla u_p)\big) = f\) with \(f > 0\), coupled with a Dirichlet boundary condition \(u_p = g\) on \(\partial \varOmega\). We prove that the whole sequence of solutions \(\{u_p\}\) converges to the limit function \(u_\infty\) \textit{strongly} in \(W^{1, m} (\varOmega)\) for any \(1 \leq m < \infty\), provided that \(F^\ast (x, .)\) has some strict convexity on its unit sphere. We also characterize an explicit expression of the limit function.Accessibility and porosity of harmonic measure at bifurcation locushttps://zbmath.org/1526.370542024-02-15T19:53:11.284213Z"Graczyk, Jacek"https://zbmath.org/authors/?q=ai:graczyk.jacek"Świątek, Grzegorz"https://zbmath.org/authors/?q=ai:swiatek.grzegorzSummary: We study hyperbolic geodesics running from \(\infty\) to a generic point, by the harmonic measure with the pole at \(\infty \), on the boundary of the connectedness locus \(\mathcal{M}_d\) for unicritical polynomials \(f_c(z)=z^d+c\). It is known that a generic parameter \(c\in \partial\mathcal{M}_d\) is not accessible within a John angle and \(\partial\mathcal{M}_d\) spirals round them infinitely many times in both directions. We prove that almost every point from \(\partial\mathcal{M}_d\) is asymptotically accessible by a flat angle with apperture decreasing slower than \((\log \circ \dots \circ \log\operatorname{dist}(c,\partial\mathcal{M}_d))^{-1}\) for any iterate of \(\log \). This is a consequence of an iterated large deviation estimate for exponential distribution. Additionally, for an arbitrary \(\beta >0\), the bifurcation locus is not \(\beta \)-porous on a set of scales of positive density along almost every external ray with respect to the harmonic measure.Mandelbrot and Julia sets of complex polynomials involving sine and cosine functions via Picard-Mann orbithttps://zbmath.org/1526.370552024-02-15T19:53:11.284213Z"Hamada, Nuha"https://zbmath.org/authors/?q=ai:hamada.nuha-h"Kharbat, Faten"https://zbmath.org/authors/?q=ai:kharbat.fatenSummary: The purpose of this paper is to generate fractals of sine and cosine functions for a complex polynomial \(z^k + c\) via Picard-Mann iterations. To generate Mandelbrot and Julia sets, escape criteria are established and proven for \(\sin z^k + c\) and \(\cos z^k + c\) for Picard-Mann iterations. Various input parameters for different \(k\) are used to compare images between sine and cosine. generation time and average number of iterations of the generated Mandelbrot and Julia sets are presented.Dimensional estimates for measures on quaternionic sphereshttps://zbmath.org/1526.420182024-02-15T19:53:11.284213Z"Ayoush, Rami"https://zbmath.org/authors/?q=ai:ayoush.rami"Wojciechowski, Michał"https://zbmath.org/authors/?q=ai:wojciechowski.michalSummary: In this article we provide lower bounds for the lower Hausdorff dimension of finite measures assuming certain restrictions on their quaternionic spherical harmonics expansions. This estimate is an analog of a result previously obtained by the authors for the complex spheres.Sharp weak-type estimates for maximal operators associated to rare baseshttps://zbmath.org/1526.420292024-02-15T19:53:11.284213Z"Hagelstein, Paul"https://zbmath.org/authors/?q=ai:hagelstein.paul-alton"Oniani, Giorgi"https://zbmath.org/authors/?q=ai:oniani.giorgi-gigla|oniani.giorgi"Stokolos, Alex"https://zbmath.org/authors/?q=ai:stokolos.alexThe authors consider a geometric maximal operator \(M_{\mathcal{B}}\) associated with a translation invariant collection \(\mathcal{B}\) of intervals in \(\mathbb{R}^n\), i.e.
\[
M_{\mathcal{B}} f(x) = \sup_{ R \in \mathcal{B} : x \in R } \frac{1}{|R|} \int_R |f| .
\]
Such a collection \(\mathcal{B}\) is called a rare basis. An interval in \(\mathbb{R}^n\) is a rectangular parallelepiped whose sides are parallel to the coordinate axes. Then, sufficient conditions on a rare basis \(\mathcal{B}\) are given so that the weak-type \(L(1+ \log^+ L)^{n-1}\) estimate
\[
\tag{1} | \{ x \in \mathbb{R}^n : M_{\mathcal{B}} f(x) > \alpha \} | \leq C_n \int_{\mathbb{R}^n} \frac{|f|}{\alpha} \bigg( 1 + \log^+ \frac{|f|}{\alpha} \bigg)^{n-1}
\]
is sharp for \(M_{\mathcal{B}}\). Subsequently, several applications on sharp weak-type \(L(1+ \log^+ L)^{n-1}\) estimates are provided for maximal operators associated to several classes of rare bases.
Specifically, the spectrum of \(\mathcal{B}\), denoted by \(W_{\mathcal{B}}\), is defined as the set of all \(n\)-tuples of the type \( ( \lceil \log_2 |R_1| \rceil , \ldots , \lceil \log_2 |R_n| \rceil ) \), where \(R_1 \times \ldots \times R_n \in \mathcal{B}\) and \(\lceil x \rceil\) denotes the least integer greater than or equal to \(x\).
A set \(W \subset \mathbb{Z}\) is called a net for a set \(S \subset \mathbb{Z}\) if there exists a natural number \(N\) such that for every \(s \in S\) there exists \(w \in W\) with \( |s-w| \leq N \).
Given a set \(W \subset \mathbb{Z}^n\) and \(t \in \mathbb{Z}^{n-1}\), denote \(W_t = \{ \tau \in \mathbb{Z} : (t, \tau) \in W \}\). Then, a set \(W \subset \mathbb{Z}^n\) is called a net for a set \(S \subset \mathbb{Z}^n\) if \(W_t\) is a net for \(S_t\) for every \(t \in \mathbb{Z}^{n-1}\).
Let \(\pi_k\) denote the usual projection \( \pi_k (x_1, \ldots , x_n) = (x_1, \ldots ,x_k) \). So, a set \(W \subset \mathbb{Z}^n\) is said to be dense in a set \(S_1 \times \ldots \times S_n \subset \mathbb{Z}^n\) if the sets \(\pi_1 (W), \ldots, \pi_n (W)\) are nets for the sets \( S_1, \pi_1 (W) \times S_2, \ldots , \pi_{n-1} (W) \times S_n \) respectively.
The main theorem of this article is the following.
Theorem. If for a rare basis \(\mathcal{B}\), there exist infinite sets \(S_1, \ldots , S_n \subset \mathbb{Z}\) for which the spectrum of \(\mathcal{B}\) is dense in \(S_1 \times \ldots \times S_n\), then the maximal operator \(M_{\mathcal{B}}\) satisfies a sharp weak-type \(L(1+ \log^+ L)^{n-1}\) estimate as in (1). Moreover, for every \(\alpha \in (0,1)\), there exists a bounded set \(E_\alpha \subset \mathbb{R}^n\) with positive measure such that
\[
| \{ x \in \mathbb{R}^n : M_{\mathcal{B}} (\chi_{E_\alpha}) (x) > \alpha \} | \geq c_n \frac{1}{\alpha} \bigg( 1 + \log \frac{1}{\alpha} \bigg)^{n-1} |E_\alpha| .
\]
Geometric maximal operators associated with rare bases occupy a fascinating middle ground between the Hardy-Littlewood maximal operator (the basis \(\mathcal{B}\) consists of all cubic intervals) and the strong maximal operator (the basis \(\mathcal{B}\) consists of all intervals). Significant mathematical work on the topic on rare bases has been done by, among others, Zygmund, Córdoba, Soria, and Rey.
Reviewer: Guillermo Flores (Córdoba)Nonlocal cross-interaction systems on graphs: nonquadratic Finslerian structure and nonlinear mobilitieshttps://zbmath.org/1526.490052024-02-15T19:53:11.284213Z"Heinze, Georg"https://zbmath.org/authors/?q=ai:heinze.georg"Pietschmann, Jan-Frederik"https://zbmath.org/authors/?q=ai:pietschmann.jan-frederik"Schmidtchen, Markus"https://zbmath.org/authors/?q=ai:schmidtchen.markusThis research at hand delves into the dynamic evolution of a complex system involving two species, characterized by nonlinear mobility and nonlocal interactions on a graph. Notably, the vertices of this graph are defined by an arbitrary yet positive measure, introducing a nuanced dimension to the analysis. The study extends a recently introduced 2-Wasserstein-type quasi-metric, originally designed for generalized graphs, to accommodate the intricacies of a two-species system. This extension is particularly tailored to account for concave, nonlinear mobilities and situations where the parameter \( p \) is not equal to 2.
A key innovation of the research lies in the development of a rigorous interpretation of the interaction system within the framework of a gradient flow in the Finslerian setting. This conceptualization stems from the introduction of the novel quasi-metric, showcasing the adaptability and versatility of the proposed metric in capturing the dynamics of such complex biological systems.
The utilization of an upwind-interpolation technique further enhances the applicability of the quasi-metric, demonstrating its robustness in dealing with the complexities introduced by nonlinear mobilities and diverse values of \( p \). This methodological choice underlines the researcher's commitment to precision and depth in addressing the challenges posed by the system under consideration.
In conclusion, this research not only contributes to the expanding field of dynamic systems but also introduces a sophisticated framework for analyzing two-species systems with nonlinear mobility and nonlocal interactions on graphs defined by arbitrary positive measures. The integration of a tailored quasi-metric and the establishment of a gradient flow interpretation underscore the scholarly depth and practical relevance of the findings, positioning this work as a valuable addition to the literature on complex systems in mathematical biology.
Reviewer: Ankit Gupta (Delhi)\(L_p\)-Steiner quermassintegralshttps://zbmath.org/1526.520082024-02-15T19:53:11.284213Z"Tatarko, Kateryna"https://zbmath.org/authors/?q=ai:tatarko.kateryna"Werner, Elisabeth M."https://zbmath.org/authors/?q=ai:werner.elisabeth-mThe authors introduce a new concept of the \(L_p\)-Steiner quermassintegrals, which include the classical mixed volumes, the dual mixed volumes, the \(L_p\) affine surface areas and the mixed \(L_p\) affine surface areas. Properties of the \(L_p\)-Steiner quermassintegrals are investigated. Connection with rotation and reflection invariant valuations in a special class of convex bodies is studied.
Reviewer: Boris Rubin (Baton Rouge)Every component of a fractal square is a Peano continuumhttps://zbmath.org/1526.540092024-02-15T19:53:11.284213Z"Luo, Jun"https://zbmath.org/authors/?q=ai:luo.jun.1"Rao, Hui"https://zbmath.org/authors/?q=ai:rao.hui"Xiong, Ying"https://zbmath.org/authors/?q=ai:xiong.yingIn this article, the authors investigate the question of local connectedness. The first result of the article states that every self-similar set is locally connected if and only if it has finitely many components. The proof is straightforward and motivated by several examples, the authors ask whether these components are locally connected even if there are infinitely many components.
To do so, the authors consider fractal squares: Let \(n\in\mathbb{N}\) be such that \(n\geq 2\). A \textit{fractal square} \(F\subset \mathbb{R}^2\) is a self-similar carpet satisfying
\[
F = \bigcup_{d\in D}(F+d)/n,
\]
where \(D \subset \left\{ 0,1,\dots,n-1 \right\}^2\). This is a generalisation of the Siepinski carpet that still retains much self-similar structure. These sets coincide with the special class of self-similar (rather than self-affine) Bedford-McMullen carpets.
The main achievement of the article is Theorem 2, which states that every component of a fractal square is locally connected.
The article is written in a clear manner and the authors provide several illustrative examples, including that an analogue of Theorem 2 does not hold in higher dimensions.
Reviewer: Sascha Troscheit (Oulu)Geometric inequalities on Riemannian and sub-Riemannian manifolds by heat semigroups techniqueshttps://zbmath.org/1526.580112024-02-15T19:53:11.284213Z"Baudoin, Fabrice"https://zbmath.org/authors/?q=ai:baudoin.fabriceIn this survey paper, some functional inequalities are introduced for elliptic diffusion operators using curvature-dimension conditions, which include the Soblove/isoperimetric inequalities and Li-Yau type parabolic Harnack inequality. These inequalties are also extended to the sub-Riemannian setting by using generalized curvature-dimension conditions.
For the entire collection see [Zbl 1481.28001].
Reviewer: Feng-Yu Wang (Tianjin)Measure theory, probability, and stochastic processeshttps://zbmath.org/1526.600012024-02-15T19:53:11.284213Z"Le Gall, Jean-François"https://zbmath.org/authors/?q=ai:le-gall.jean-francoisThis book covers probability theory and stochastic processes at a graduate level. It begins with measure theory, allowing readers with a background in measure theory to skip ahead and start from Part II, which focuses on probability theory. Part II covers most topics commonly found in other graduate-level probability books. Part III, stochastic processes, delves into martingale theory, Markov chains, and Brownian motions. However, it's worth noting that while this book covers Brownian motion, it does not delve into the general theory of Markov processes with uncountable state spaces and continuous-time martingales, which is based on more advanced functional analysis. For a comprehensive treatment of these advanced topics, one may refer to [Markov processes. Characterization and convergence. Hoboken, NJ: John Wiley \& Sons (2005; Zbl 1089.60005)] by \textit{S. N. Ethier} and \textit{T. G. Kurtz}. Additionally, this book provides a set of good exercise problems at the end of each chapter, although the number of problems is relatively small. Many topics covered in Part III have applications in finance, but it's essential to understand that this book focuses solely on theoretical aspects and does not address these practical applications. In conclusion, this book is exceptionally well written in a concise manner and is suitable for individuals with a strong background in undergraduate real analysis and undergraduate probability.
Reviewer: Eunghyun Lee (Astana)Investigations on stress-dependent thermal conductivity of fractured rock by fractal theoryhttps://zbmath.org/1526.740482024-02-15T19:53:11.284213Z"Miao, Tongjun"https://zbmath.org/authors/?q=ai:miao.tongjun"Chen, Aimin"https://zbmath.org/authors/?q=ai:chen.aimin"Yang, Xiaoya"https://zbmath.org/authors/?q=ai:yang.xiaoya"Li, Zun"https://zbmath.org/authors/?q=ai:li.zun"Liu, Hao"https://zbmath.org/authors/?q=ai:liu.hao|liu.hao.4|liu.hao.6|liu.hao.1|liu.hao.2"Yu, Boming"https://zbmath.org/authors/?q=ai:yu.boming(no abstract)