Hausdorff Dimension Research Articles

On the exact rate of convergence of digits in Engel expansions

Let ϕ:N→R+ be a function satisfying ϕ(n)→∞ as n→∞. Denote by 〈d1(x),d2(x),⋯,dn(x),⋯〉 the Engel expansion of an irrational number x∈(0,1). We study the Baire classification and Hausdorff dimension ofDϕ(α,β):={x∈(0,1)﹨Q:liminfn→∞log⁡dn(x)−nϕ(n)=α,limsupn→∞log⁡dn(x)−nϕ(n)=β} for α,β∈[−∞,∞] with α≤β. Under the condition that ϕ(n+1)−ϕ(n)→0 as n→∞, we show that the set Dϕ(α,β) is residual if and only if α=−∞ and β=∞. Under the conditions that ϕ is increasing and ϕ(n+1)−ϕ(n)→0 as n→∞, we prove that Dϕ(α,β) has full Hausdorff dimension for any −∞≤α≤β≤∞, which improves the result of Liu and Wu (2003). We also do some similar analyses for the gap of consecutive digits.

Two-Sided Boundary Points of Sobolev Extension Domains on Euclidean Spaces

We prove an estimate on the Hausdorff dimension of the set of two-sided boundary points of general Sobolev extension domains on Euclidean spaces. We also present examples showing lower bounds on possible dimension estimates of this type.

How external monitoring can mitigate cyberloafing: understanding the mediating and moderating roles of employees’ self-control

ABSTRACT Organisations widely adopt external monitoring to regulate employees’ cyberloafing, which refers to employees’ non-work-related internet usage during work time. Although prior studies have examined the effect of external monitoring on cyberloafing, the underlying mechanisms through which external monitoring affects cyberloafing remain unclear. Drawing on counteractive control theory and the literature on the relationship between external controls and self-control, we investigate how employees’ self-control can play an important role in understanding the effect of external monitoring on cyberloafing. Based on a survey of 227 employees, we find that external monitoring promotes employees’ self-control motivation, which in turn, decreases their cyberloafing. We further find that the above relationships were contingent upon employees’ dispositional self-control capacity. Our study makes contributions by advancing researchers’ understanding of how external monitoring affects employees’ cyberloafing. Our study also extends the counteractive control theory by introducing the concept of self-control motivation, which is a complementary dimension of self-control capacity. In addition, the findings have important implications for alleviating the potential side effect of external monitoring used in organisations.

Beurling densities of regular maximal orthogonal sets of self-similar spectral measure with consecutive digit sets

Abstract Beurling density plays a key role in the study of frame-spectrality of normalized Lebesgue measure restricted to a set. Accordingly, in this paper, the authors study the s-Beurling densities of regular maximal orthogonal sets of a class of self-similar spectral measures, where s is the Hausdorff dimension of its support and obtain their exact upper bound of the densities.

Uniqueness for volume-constraint local energy-minimizing sets in a half-space or a ball

Abstract In this paper, we prove a Poincaré-type inequality for any set of finite perimeter which is stable with respect to the free energy among volume-preserving perturbation, provided that the Hausdorff dimension of its singular set is at most n - 3 {n-3} . With this inequality, we classify all volume-constraint local energy-minimizing sets in a unit ball, a half-space or a wedge-shaped domain. In particular, we prove that the relative boundary of any energy-minimizing set is smooth.

Quantitative Analyses of the Role of Relational Capital on Absorptive Capacity in Knowledge-intensive SME's

The research aimed to measure the relational capital influence (RC) on absorption capacity in knowledge-intensive companies. The direct relationship between the different components of relational capital and absorption capacity was analysed through a model of structural equations of partial least squares (PLS-SEM), tested in SmartPLS. This method was chosen for the following reasons: the use of SEM-PLS allows testing causal paths between second-order latent variables, and in addition to offering extensive and flexible causal modelling resources, the technique is recommended for more complex models, with constructs composed of a greater number of variables and with a smaller number of data, as observed in this research. The method is also justified because this research is based on a composite measurement model with a reflective design approach, which means there are correlations between indicators and dimensions. The SmartPLS 3.0 Software was used to carry out the global model evaluation and measurement and structural model evaluation steps. These analyses were conducted in a sample of 174 small and medium-sized technology enterprises (SMEs) that are part of different innovation networks in Brazil. This study highlights that the development of relational capital is supported by collaborative relations of cooperation, trust, communication, and resources invested by enterprises established in different networks. The proposed statistical model allowed proving the relationship factors that help strengthen the relations between the network actors that facilitate the transfer of knowledge. This relationship still needs to be investigated, especially for small and medium-sized knowledge-intensive companies in emerging countries like Brazil. The research conclusion supported the research hypothesis and proved that relational capital is an independent variable that directly and positively influences the absorption capacity process. This study, quantitatively combining the external perspective of relational capital and the internal organisational dimension of absorption capacity, provides valuable information about using quantitative methodologies to explore intangible organisational resources to promote innovation.

A Mechanical Picture of Fractal Darcy’s Law

The main goal of this manuscript is to generalize Darcy’s law from conventional calculus to fractal calculus in order to quantify the fluid flow in subterranean heterogeneous reservoirs. For this purpose, the inherent features of fractal sets are scrutinized. A set of fractal dimensions is incorporated to describe the geometry, morphology, and fractal topology of the domain under study. These characteristics are known through their Hausdorff, chemical, shortest path, and elastic backbone dimensions. Afterward, fractal continuum Darcy’s law is suggested based on the mapping of the fractal reservoir domain given in Cartesian coordinates xi into the corresponding fractal continuum domain expressed in fractal coordinates ξi by applying the relationship ξi=ϵ0(xi/ϵ0)αi−1, which possesses local fractional differential operators used in the fractal continuum calculus framework. This generalized version of Darcy’s law describes the relationship between the hydraulic gradient and flow velocity in fractal porous media at any scale including their geometry and fractal topology using the αi-parameter as the Hausdorff dimension in the fractal directions ξi, so the model captures the fractal heterogeneity and anisotropy. The equation can easily collapse to the classical Darcy’s law once we select the value of 1 for the alpha parameter. Several flow velocities are plotted to show the nonlinearity of the flow when the generalized Darcy’s law is used. These results are compared with the experimental data documented in the literature that show a good agreement in both high-velocity and low-velocity fractal Darcian flow with values of alpha equal to 0<α1<1 and 1<α1<2, respectively, whereas α1=1 represents the standard Darcy’s law. In that way, the alpha parameter describes the expected flow behavior which depends on two fractal dimensions: the Hausdorff dimension of a porous matrix and the fractal dimension of a cross-section area given by the intersection between the fractal matrix and a two-dimensional Cartesian plane. Also, some physical implications are discussed.

Hausdorff dimension of limit sets for projective Anosov representations

Hausdorff dimension of limit sets for projective Anosov representations

The scale of homogeneity in the Rh = ct universe

ABSTRACT Studies of the Universe’s transition to smoothness in the context of Lambda cold dark matter (ΛCDM) have all pointed to a transition radius no larger than ∼300 Mpc. These are based on a broad array of tracers for the matter power spectrum, including galaxies, clusters, quasars, the Ly-α forest, and anisotropies in the cosmic microwave background. It is therefore surprising, if not anomalous, to find many structures extending out over scales as large as ∼2 Gpc, roughly an order of magnitude greater than expected. Such a disparity suggests that new physics may be contributing to the formation of large-scale structure, warranting a consideration of the alternative Friedmann–Lemaître–Robertson–Walker cosmology known as the Rh = ct universe. This model has successfully eliminated many other problems in ΛCDM. In this paper, we calculate the fractal (or Hausdorff) dimension in this cosmology as a function of distance, showing a transition to smoothness at ∼2.2 Gpc, fully accommodating all of the giant structures seen thus far. This outcome adds further observational support for Rh = ct over the standard model.

A Peak Set with a Maximal Hausdorff Dimension on Each Slice

We consider a bounded balanced strictly convex domain Omega subset {mathbb {C}}^{d} with C^{2} boundary. Then there exists a peak set E with Hausdorff dimension equal to 1 on each slice. In particular E has maximal possible Hausdorff dimension equal to 2d-1.

Measuring the farmers’ vulnerability to climate change in Tashk and Bakhtegan Lakes in Iran

ABSTRACT This study used a descriptive-analytical method to assess how vulnerable farmers in Fars Province's rural regions near the Tashk and Bakhtegan lakes are to the effects of climate change. For this purpose, 17 villages around Tashk and Bakhtegan lakes with 2511 households, who are engaged in agricultural activities, were investigated. To achieve the research goals, a broad range of indexes were determined with the dimensions of exposure, sensitivity, and adaptive capacity to climate change according to the Climate Vulnerability Index (CVI) and were evaluated in the field studies. According to the results, 52.93% of communities were highly vulnerable to climate change. On the other hand, only 23.52% of farmers had very low vulnerability, and 23.52% had moderate vulnerability. The local farmers were aware of the climate change in the studied region with an average of 4.56. Given that more than half of the villagers are highly vulnerable to climate change, the government and relevant institutions need to increase their resilience to vulnerability. Increasing vulnerability requires careful measurement and monitoring of all related variables, increasing adaptation to the generalization of insurance, access to facilities, and reducing climatic sensitivities by increasing livelihood diversification, reducing unemployment, and using drought-resistant plants.

And empirical measures, the irregular set and entropy

AbstractFor integers a and $b\geq 2$ , let $T_a$ and $T_b$ be multiplication by a and b on $\mathbb {T}=\mathbb {R}/\mathbb {Z}$ . The action on $\mathbb {T}$ by $T_a$ and $T_b$ is called $\times a,\times b$ action and it is known that, if a and b are multiplicatively independent, then the only $\times a,\times b$ invariant and ergodic measure with positive entropy of $T_a$ or $T_b$ is the Lebesgue measure. However, it is not known whether there exists a non-trivial $\times a,\times b$ invariant and ergodic measure. In this paper, we study the empirical measures of $x\in \mathbb {T}$ with respect to the $\times a,\times b$ action and show that the set of x such that the empirical measures of x do not converge to any measure has Hausdorff dimension one and the set of x such that the empirical measures can approach a non-trivial $\times a,\times b$ invariant measure has Hausdorff dimension zero. Furthermore, we obtain some equidistribution result about the $\times a,\times b$ orbit of x in the complement of a set of Hausdorff dimension zero.

A dichotomy phenomenon for bad minus normed Dirichlet

AbstractGiven a norm ν on , the set of ν‐Dirichlet improvable numbers was defined and studied in the papers (Andersen and Duke, Acta Arith. 198 (2021) 37–75 and Kleinbock and Rao, Internat. Math. Res. Notices 2022 (2022) 5617–5657). When ν is the supremum norm, , where is the set of badly approximable numbers. Each of the sets , like , is of measure zero and satisfies the winning property of Schmidt. Hence for every norm ν, is winning and thus has full Hausdorff dimension. In this article, we prove the following dichotomy phenomenon: either or else has full Hausdorff dimension. We give several examples for each of the two cases. The dichotomy is based on whether the critical locus of ν intersects a precompact ‐orbit, where is the one‐parameter diagonal subgroup of acting on the space X of unimodular lattices in . Thus, the aforementioned dichotomy follows from the following dynamical statement: for a lattice , either is unbounded (and then any precompact ‐orbit must eventually avoid a neighborhood of Λ), or not, in which case the set of lattices in X whose ‐trajectories are precompact and contain Λ in their closure has full Hausdorff dimension.

Localized growth speed of the digits in Engel expansions

We introduce a localized version of a Galambos's question about the growth speed of the digits in Engel expansion, namely the set{x∈(0,1]:limn→∞⁡1nlog⁡dn(x)=α(x)}, where α:[0,1]→[0,∞] is a nonnegative continuous function and dn(x) denotes the nth digits in the Engel expansion of x. The Hausdorff dimension is shown to be irrelevant of the function α(x). As applications, this answers Galambos's question and strengthens some results by Liu & Wu.

Distribution of pinned distance trees in the plane [formula omitted

The recent result of Guth, Iosevich, Ou, and Wang (2019) on the Falconer distance problem states that for a compact set A⊂R2, if the Hausdorff dimension of A is greater than 54, then the distance set Δ(A) has positive Lebesgue measure. With a completely different approach, Murphy, Petridis, Pham, Rudnev, and Stevens (2022) proved the prime field version of this result, namely, for E⊂Fp2 with |E|≫p5/4, there exist many points x∈E such that the number of distinct distances from x is at least cp. The main purpose of this paper is to prove extensions in the more general structure of pinned trees. The case of distances on small sets will also be addressed in this paper.

Vertical projections in the Heisenberg group via cinematic functions and point-plate incidences

Let {πe:H→We:e∈S1} be the family of vertical projections in the first Heisenberg group H. We prove that if K⊂H is a Borel set with Hausdorff dimension dimH⁡K∈[0,2]∪{3}, thendimH⁡πe(K)≥dimH⁡K for H1 almost every e∈S1. This was known earlier if dimH⁡K∈[0,1].The proofs for dimH⁡K∈[0,2] and dimH⁡K=3 are based on different techniques. For dimH⁡K∈[0,2], we reduce matters to a Euclidean problem, and apply the method of cinematic functions due to Pramanik, Yang, and Zahl.To handle the case dimH⁡K=3, we introduce a point-line duality between horizontal lines and conical lines in R3. This allows us to transform the Heisenberg problem into a point-plate incidence question in R3. To solve the latter, we apply a Kakeya inequality for plates in R3, due to Guth, Wang, and Zhang. This method also yields partial results for Borel sets K⊂H with dimH⁡K∈(5/2,3).

Assessing the crisis management of the COVID-19 pandemic: a study of inquiry commission reports in Norway and Sweden

Abstract This article examines the inquiry reports from the commissions charged with investigating government crisis management of the COVID-19 pandemic in Norway and Sweden. Such postcrises commissions have been a common feature in many countries as they seek to systematize their experiences and learn from the crisis. In this article, we used various dimensions of governance capacity and governance legitimacy as assessment criteria. It reveals that the commissions’ assessment criteria were not very specific in their reports, but a reanalysis of their findings shows that governance capacity and governance legitimacy dimensions are useful to assess the reports themselves. The two reports reveal a lack of preparedness in both countries, but they differ in their conclusions about governance regulation and output legitimacy.

Absolute continuity in families of parametrised non-homogeneous self-similar measures

Let \mu be a planar self-similar measure with similarity dimension exceeding 1 , satisfying a mild separation condition, and such that the fixed points of the associated similitudes do not share a common line. Then, we prove that the orthogonal projections \pi_{e\sharp}(\mu) are absolutely continuous for all e \in S^{1} \setminus E , where the exceptional set E has zero Hausdorff dimension. The result is obtained from a more general framework which applies to certain parametrised families of self-similar measures on the real line. Our results extend the previous work of Shmerkin and Solomyak from 2016, where it was assumed that the similitudes associated with \mu have a common contraction ratio.

Dimensions of projected sets and measures on typical self-affine sets

Let T1,…,Tm be a family of d×d invertible real matrices with ‖Ti‖<1/2 for 1≤i≤m. For a=(a1,…,am)∈Rmd, let πa:Σ={1,…,m}N→Rd denote the coding map associated with the affine IFS {Tix+ai}i=1m. We show that for every Borel probability measure μ on Σ, each of the following dimensions (lower and upper Hausdorff dimensions, lower and upper packing dimensions) of π⁎aμ is constant for Lmd-a.e. a∈Rmd, where π⁎aμ stands for the push-forward of μ by πa. In particular, we give a necessary and sufficient condition on μ so that π⁎aμ is exact dimensional for Lmd-a.e. a∈Rmd. Moreover, for every analytic set E⊂Σ, each of the Hausdorff, packing, lower and upper box-counting dimensions of πa(E) is constant for Lmd-a.e. a∈Rmd. Formal dimension formulas of these projected measures and sets are given. The Hausdorff dimensions of exceptional sets are estimated.

On the dimension of stationary measures for random piecewise affine interval homeomorphisms

AbstractWe study stationary measures for iterated function systems (considered as random dynamical systems) consisting of two piecewise affine interval homeomorphisms, called Alsedà–Misiurewicz (AM) systems. We prove that for an open set of parameters, the unique non-atomic stationary measure for an AM system has Hausdorff dimension strictly smaller than $1$ . In particular, we obtain singularity of these measures, answering partially a question of Alsedà and Misiurewicz [Random interval homeomorphisms. Publ. Mat.58(suppl.) (2014), 15–36].