Hausdorff Dimension Research Articles

Measuring the adaptive capacity of rangeland users under drought stress in North-easternIran: application of social network analysis

Drought events have significant impact on ecosystems and the livelihoods of rural communities in Iran. So, the purpose of this study was the analysis and evaluation of dimensions of adaptive capacity (AC) against drought in Bajestan county, Khorasan Razavi province, Iran. A questionnaire was used to collect data for social network analysis (SNA) and AC by the full network method. Then, a combined SNA-SEM model was developed to determine what was the social response of rangeland users in dealing with drought. In this study, the AC of rangeland users was compared in two groups of villages covered by the collaborative management or Carbon Sequestration Project (CSP) and uncovered by CSP. The results confirmed that the villages under CSP had higher levels of AC and social capital. Among five capitals, indicators of economic resources and information, skills and management were more effective on AC. Among the SNA indexes, effsize, constrain, indirects, density and betweenes centrality are more effective on AC of rangeland users. Results showed that identifying the key actors, who have the power and influence to determine the information flow and financial inflows in these components, can help increase the AC of rangeland users to mitigate effects of drought.

METRIC RESULTS FOR THE EVENTUALLY ALWAYS HITTING POINTS AND LEVEL SETS IN SUBSHIFT WITH SPECIFICATION

We study the set of eventually always hitting points for symbolic dynamics with specification. The measure and Hausdorff dimension of such fractal set are obtained. Moreover, we establish the stronger metric results by introducing a new quantity [Formula: see text] which describes the maximal length of string of zeros of the prefix among the first [Formula: see text] iterations of [Formula: see text] in symbolic space. The Hausdorff dimensions of the level sets for this quantity are also completely determined.

On the Dimension of Limit Sets on ℙ(ℝ3) via Stationary Measures: Variational Principles and Applications

Abstract This paper investigates the (semi)group action of $\textrm{SL}_{n}({\mathbb R})$ on ${\mathbb P}({\mathbb R}^{n})$, a primary example of non-conformal, non-linear, and non-strictly contracting action. We establish variational principles of the affinity exponent for two main examples: the Borel Anosov representations and the Rauzy gasket. In [ 32], they obtain a dimension formula for the stationary measures on ${\mathbb P}({\mathbb R}^{3})$. Combined with our result, it allows us to study the Hausdorff dimension of limit sets of Anosov representations in $\textrm{SL}_{3}({\mathbb R})$ and the Rauzy gasket. It yields the equality between the Hausdorff dimensions and the affinity exponents in both settings, generalizing the classical Patterson–Sullivan formula. In the appendix, we improve the numerical lower bound of the Hausdorff dimension of Rauzy gasket to $1.5$.

A New Insight on the Upwelling along the Atlantic Iberian Coasts and Warm Water Outflow in the Gulf of Cadiz from Multiscale Ultrahigh Resolution Sea Surface Temperature Imagery

The ATLAZUL project is an Interreg effort among 18 partners from Spain and Portugal along the Atlantic Iberian coasts. One of its objectives is the development of new methods and data processing for oceanic information to produce useful products for private and public stakeholders. This study proposes a new insight on the sea surface dynamic of the ATLAZUL area based on almost two years of multiscale high resolution sea surface temperature imagery. The use of techniques such as the Karhunen–Loève transform (Empirical Orthogonal Function) and the Maximum Entropy Spectral Analysis were applied to study long- and short-term features in the sea surface temperature imagery. Mathematical Morphology and the Geometrical Theory of Measure are utilized to compute the Medial Axis Transform and the Hausdorff dimension. The results can be summarized as follows: (i) the tow upwelling areas are identified along the Galician–Portugal coast as indicated in the second and third modes of KLT/EOF analysis, and they are partially affected by wind; (ii) the tow warm water outflows from the Bay of Cádiz to the Gulf of Cádiz are identified as the second and third modes of KLT/EOF analysis, which are also influenced by wind; (iii) the skeletons of the surface signature of the upwelling and of the warmer water outflow, along with their fractal dimensions, indicate a chaotic pattern of spatial distribution and (iv) the harmonic prediction model should be combined with the wind prediction.

Metric theory for continued β-fractions

For a fixed real number β>1, let ϵ⁎(1,β)=(ϵ1⁎(1,β),ϵ2⁎(1,β),⋯) be the infinite β-expansion of 1. For any n≥1, we define ℓn(β)=sup⁡{j≥0:ϵn⁎(1,β)=⋯=ϵn+j⁎(1,β)=0} be the maximal length of consecutive 0's in ϵ⁎(1,β) starting from ϵn⁎(1,β). Write A0={β>1:{ℓn(β)}is bounded} and let φ:N→R+ be a positive function. For any fixed β∈A0, we show that the Lebesgue measure of the following setE(φ)={x∈[0,1):an(x)≥φ(n)for infinitely manyn∈N} is null or full according to the convergence or divergence of the series ∑n=1∞1φ(n) respectively, where an(x) denotes the n-th partial quotient of the continued β-fraction expansion of x. We further determine the Hausdorff dimension of the set E(φ). Such measure and dimension results generalize a classical theorem of Borel–Bernstein and a well-known result of Wang–Wu (2008) [28] on regular continued fractions.

A Mañé-Manning formula for expanding measures for endomorphisms of ℙ^{𝕜}

Let k ≥ 1 k \ge 1 be an integer and f f a holomorphic endomorphism of P k ( C ) \mathbb {P}^k (\mathbb {C}) of algebraic degree d ≥ 2 d\geq 2 . We introduce a dynamical volume dimension for ergodic f f -invariant probability measures with strictly positive Lyapunov exponents. In particular, this class of measures includes all ergodic measures whose measure-theoretic entropy is strictly larger than ( k − 1 ) log ⁡ d (k-1)\log d , a natural generalization of the class of measures of positive measure-theoretic entropy in dimension 1. The volume dimension is equivalent to the Hausdorff dimension when k = 1 k=1 , but depends on the dynamics of f f to incorporate the absence of an analogue of Koebe’s theorem and the non-conformality of holomorphic endomorphisms for k ≥ 2 k\geq 2 . If ν \nu is an ergodic f f -invariant probability measure with strictly positive Lyapunov exponents, we prove a generalization of the Mañé-Manning formula relating the volume dimension, the measure-theoretic entropy, and the sum of the Lyapunov exponents of ν \nu . As a consequence, we give a characterization of the first zero of a natural pressure function for such expanding measures in terms of their volume dimensions. For hyperbolic maps, such zero also coincides with the volume dimension of the Julia set, and with the exponent of a natural (volume-)conformal measure. This generalizes results by Denker-Urbański [Nonlinearity 4 (1991), pp. 365–384; Trans. Amer. Math. Soc. 328 (1991), pp. 563–587] and McMullen [Comment. Math. Helv. 75 (2000), pp. 535–593] in dimension 1 to any dimension k ≥ 1 k\geq 1 . Our methods mainly rely on a theorem by Berteloot-Dupont-Molino [Ann. Inst. Fourier (Grenoble) 58 (2008), pp. 2137–2168], which gives a precise control on the distortion of inverse branches of endomorphisms along generic inverse orbits with respect to measures with strictly positive Lyapunov exponents.

Quasiperiodic disorder induced critical phases in a periodically driven dimerized p-wave Kitaev chain

The intricate relationship between topology and disorder in non-equilibrium quantum systems presents a captivating avenue for exploring localization phenomenon. Here, we look for a suitable platform that enables an in-depth investigation of the topic. To this end, we delve into the nuanced analysis of the topological and localization characteristics exhibited by a one-dimensional dimerized Kitaev chain under periodic driving and perform detailed analyses of the Floquet Majorana modes. Such a non-equilibrium scenario is made further interesting by including a spatially varying quasiperiodic potential with a temporally modulated amplitude. Apriori, the motivation is to explore an interplay between dimerization and a quasiperiodic disorder in a topological setting which is also known to demonstrate unique (re-entrant) localization properties. While the topological properties of the driven system confirm the presence of zero and π\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\pi $$\\end{document} Majorana modes, the phase diagram obtained by constructing a pair of topological invariants (Z×Z\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Z} \ imes \\mathbb {Z} $$\\end{document}), also referred to as the real space winding numbers, at different driving frequencies reveal intriguing features that are distinct from the static scenario. In particular, at either low or intermediate frequency regimes, the phase diagram concerning the zero mode involves two distinct phase transitions, one from a topologically trivial to a non-trivial phase, and another from a topological phase to an Anderson localized phase. On the other hand, the study of the Majorana π\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\pi $$\\end{document} mode unveils the emergence of a unique topological phase, characterized by complete localization of both the bulk and the edge modes, which may be called as the Floquet topological Anderson phase. Moreover, different frequency regimes showcase distinct localization features which can be examined via the localization toolbox, namely, the inverse and the normalized participation ratios. Specifically, the low and high-frequency regimes demonstrate the existence of completely extended and localized phases, respectively. While at intermediate frequencies, we observe the critical (multifractal) phase of the model which is further investigated via a finite-size scaling analysis of the fractal dimension. Finally, to add depth into our study, we have performed a mean level spacing analyses and computed the Hausdorff dimension which yields specific characteristics inherent to the critical phase, offering profound insights into its underlying properties.

Hausdorff dimension of collision times in one-dimensional log-gases

We consider systems of multiple Brownian particles in one dimension that repel mutually via a logarithmic potential on the real line, more specifically the Dyson model. These systems are characterized by a parameter that controls the strength of the interaction, k > 0. Despite being a one-dimensional system, this system is interesting due to the properties that arise from the long-range interaction between particles. It is a well-known fact that when k is small enough, particle collisions occur almost surely, while when k is large, collisions never occur. However, aside from this fact, there was no characterization of the collision times until now. In this paper, we derive the fractal (Hausdorff) dimension of the set of collision times by generalizing techniques introduced by L. Liu and Y. Xiao [Probab. Math. Stat. 18(2), 369–383 (1998)] to study the return times to the origin of self-similar Markov processes. In our case, we consider the return times to configurations where at least one collision occurs, which is a condition that defines unbounded sets as opposed to a single point, namely, the origin. We find that the fractal dimension characterizes the collision behavior of these systems and establishes a clear delimitation between the colliding and non-colliding regions in a way similar to that of a thermodynamic function.

Projections of four corner Cantor set: Total self-similarity, spectrum and unique codings

Given ρ∈(0,1/4], the four corner Cantor set E⊂R2 is a self-similar set generated by the iterated function system {(ρx,ρy),(ρx,ρy+1−ρ),(ρx+1−ρ,ρy),(ρx+1−ρ,ρy+1−ρ)}. For θ∈[0,π) let Eθ be the orthogonal projection of E onto a line with an angle θ to the x-axis. In principle, Eθ is a self-similar set having overlaps. In this paper we give a complete characterization on which the projection Eθ is totally self-similar. We also study the spectrum of Eθ, which turns out that the spectrum achieves its maximum value if and only if Eθ is totally self-similar. Furthermore, when Eθ is totally self-similar, we calculate its Hausdorff dimension and study the subset Uθ which consists of all x∈Eθ having a unique coding. In particular, we show that dimHUθ=dimHEθ for Lebesgue almost every θ∈[0,π). Finally, for ρ=1/4 we prove that the possibility for Eθ to contain an interval is strictly smaller than that for Eθ to have an exact overlap.

Assessing the Sustainability of County-level Rural E-commerce Ecosystems in China: A Multi-dimensional Framework

2. Enhancing the sustainability of the rural e-commerce industry chain and achieving sustainable development within County-level Rural E-commerce Ecosystems (CREE) is a crucial endeavor in China's rural revitalization phase. This study aims to design an assessment tool named SREEM to evaluate the sustainability of China County-level Rural E-commerce Ecosystem. it provides a comprehensive exposition of SREEM, elucidating its foundational principles and indicator origins. From the dimensions of Robustness, Niche Creation, and Endogenous Development Capacity, the study explicates the various levels of indicators within the preliminary SREEM framework. Employing a combined approach of Modified Delphi and AHP, the study screens SREEM assessment indicators and determines indicator weight. The final iteration of SREEM includes 3 first-level indicators, 8 second-level indicators, 23 third-level indicators, and 35 fourth-level indicators.

Regularity for one-phase Bernoulli problems with discontinuous weights and applications

We study a one-phase Bernoulli free boundary problem with weight function admitting a discontinuity along a smooth jump interface. In any dimension N ≥ 2 N\ge 2 , we show the C 1 , α C^{1, \alpha } regularity of the free boundary outside of a singular set of Hausdorff dimension at most N − 3 N-3 . In particular, we prove that the free boundaries are C 1 , α C^{1, \alpha } regular in dimension N = 2 N=2 , while in dimension N = 3 N=3 the singular set can contain at most a finite number of points. We use this result to construct singular free boundaries in dimension N = 2 N=2 , which are minimizing for one-phase functionals with weight functions in L ∞ L^\infty that are arbitrarily close to a positive constant.

On the Support of Anomalous Dissipation Measures

By means of a unifying measure-theoretic approach, we establish lower bounds on the Hausdorff dimension of the space-time set which can support anomalous dissipation for weak solutions of fluid equations, both in the presence or absence of a physical boundary. Boundary dissipation, which can occur at both the time and the spatial boundary, is analyzed by suitably modifying the Duchon & Robert interior distributional approach. One implication of our results is that any bounded Euler solution (compressible or incompressible) arising as a zero viscosity limit of Navier–Stokes solutions cannot have anomalous dissipation supported on a set of dimension smaller than that of the space. This result is sharp, as demonstrated by entropy-producing shock solutions of compressible Euler (Drivas and Eyink in Commun Math Phys 359(2):733–763, 2018. https://doi.org/10.1007/s00220-017-3078-4; Majda in Am Math Soc 43(281):93, 1983. https://doi.org/10.1090/memo/0281) and by recent constructions of dissipative incompressible Euler solutions (Brue and De Lellis in Commun Math Phys 400(3):1507–1533, 2023. https://doi.org/10.1007/s00220-022-04626-0 624; Brue et al. in Commun Pure App Anal, 2023), as well as passive scalars (Colombo et al. in Ann PDE 9(2):21–48, 2023. https://doi.org/10.1007/s40818-023-00162-9; Drivas et al. in Arch Ration Mech Anal 243(3):1151–1180, 2022. https://doi.org/10.1007/s00205-021-01736-2). For LtqLxr\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^q_tL^r_x$$\\end{document} suitable Leray–Hopf solutions of the d-\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d-$$\\end{document}dimensional Navier–Stokes equation we prove a bound of the dissipation in terms of the Parabolic Hausdorff measure Ps\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {P}^{s}$$\\end{document}, which gives s=d-2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$s=d-2$$\\end{document} as soon as the solution lies in the Prodi–Serrin class. In the three-dimensional case, this matches with the Caffarelli–Kohn–Nirenberg partial regularity.

Metric Mean Dimension and Mean Hausdorff Dimension Varying the Metric

Let f:M→M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f:\\mathbb {M}\\rightarrow \\mathbb {M}$$\\end{document} be a continuous map on a compact metric space M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {M}$$\\end{document} equipped with a fixed metric d, and let τ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au $$\\end{document} be the topology on M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {M}$$\\end{document} induced by d. We denote by M(τ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {M}(\ au )$$\\end{document} the set consisting of all metrics on M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {M}$$\\end{document} that are equivalent to d. Let mdimM(M,d,f)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \ ext {mdim}_{\ ext {M}}(\\mathbb {M},d, f)$$\\end{document} and mdimH(M,d,f)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \ ext {mdim}_{\ ext {H}} (\\mathbb {M},d, f)$$\\end{document} be, respectively, the metric mean dimension and mean Hausdorff dimension of f. First, we will establish some fundamental properties of the mean Hausdorff dimension. Furthermore, it is important to note that mdimM(M,d,f)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \ ext {mdim}_{\ ext {M}}(\\mathbb {M},d, f)$$\\end{document} and mdimH(M,d,f)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \ ext {mdim}_{\ ext {H}} (\\mathbb {M},d, f)$$\\end{document} depend on the metric d chosen for M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {M}$$\\end{document}. In this work, we will prove that, for a fixed dynamical system f:M→M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f:\\mathbb {M}\\rightarrow \\mathbb {M}$$\\end{document}, the functions mdimM(M,f):M(τ)→R∪{∞}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {mdim}_{\ ext {M}} (\\mathbb {M}, f):\\mathbb {M}(\ au )\\rightarrow \\mathbb {R}\\cup \\{\\infty \\}$$\\end{document} and mdimH(M,f):M(τ)→R∪{∞}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \ ext {mdim}_{\ ext {H}}(\\mathbb {M}, f): \\mathbb {M}(\ au )\\rightarrow \\mathbb {R}\\cup \\{\\infty \\}$$\\end{document} are not continuous, where mdimM(M,f)(ρ)=mdimM(M,ρ,f)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \ ext {mdim}_{\ ext {M}}(\\mathbb {M}, f) (\\rho )= \ ext {mdim}_{\ ext {M}} (\\mathbb {M},\\rho , f)$$\\end{document} and mdimH(M,f)(ρ)=mdimH(M,ρ,f)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \ ext {mdim}_{\ ext {H}}(\\mathbb {M}, f) (\\rho )= \ ext {mdim}_{\ ext {H}} (\\mathbb {M},\\rho , f)$$\\end{document} for any ρ∈M(τ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho \\in \\mathbb {M}(\ au )$$\\end{document}. Furthermore, we will present examples of certain classes of metrics for which the metric mean dimension is a continuous function.

Packing measure and dimension of the limit sets of IFSs of generalized complex continued fractions

We consider a family of conformal iterated function systems (for short, CIFSs) of generalized complex continued fractions. Note that in our previous paper we showed that the proper-dimensional Hausdorff measure of the limit set of each CIFS is zero and the packing measure of the limit set with respect to the Hausdorff dimension is positive. In this paper, we show that the packing dimension and the Hausdorff dimension of the limit set of each CIFS in the family are equal, and the proper-dimensional packing measure of the limit set is finite.

The Pore Structure Multifractal Evolution of Vibration-Affected Tectonic Coal and the Gas Diffusion Response Characteristics

Vibrations caused by downhole operations often induce coal and gas outburst accidents in tectonic zone coal seams. To clarify how vibration affects the pore structure, gas desorption, and diffusion capacity of tectonic coal, isothermal adsorption-desorption experiments under different vibration frequencies were carried out. In this study, high-pressure mercury intrusion experiments and low-pressure liquid nitrogen adsorption experiments were conducted to determine the pore structures of tectonic coal before and after vibration. The pore distribution of vibration-affected tectonic coal, including local concentration, heterogeneity, and connectivity, was analyzed using multifractal theory. Further, a correlation analysis was performed between the desorption diffusion characteristic parameters and the pore fractal characteristic parameters to derive the intrinsic relationship between the pore fractal evolution characteristics and the desorption diffusion characteristics. The results showed that the vibration increased the pore volume of the tectonic coal, and the pore volume increased as the vibration frequency increased in the 50 Hz range. The pore structure of the vibration-affected tectonic coal showed multifractal characteristics, and the multifractal parameters affected the gas desorption and diffusion capacity by reflecting the density, uniformity, and connectivity of the pore distribution in the coal. The increases in the desorption amount (Q), initial desorption velocity (V0), initial diffusion coefficient (D0), and initial effective diffusion coefficient (De) of the tectonic coal due to vibration indicated that the gas desorption and diffusion capacity of the tectonic coal were improved at the initial desorption stage. Q, V0, D0, and De had significant positive correlations with pore volume and the Hurst index, and V0, D0, and De had negative correlations with the Hausdorff dimension. To a certain extent, vibration reduced the local density regarding the pore distribution in the coal. As a result, the pore size distribution was more uniform, and the pore connectivity was improved, thereby enhancing the gas desorption and diffusion capacity of the coal.

Exact partition function of the Potts model on the Sierpinski gasket and the Hanoi lattice

We present an analytic study of the Potts model partition function on the Sierpinski and Hanoi lattices, which are self-similar lattices of triangular shape with non integer Hausdorff dimension. Both lattices are examples of non-trivial thermodynamics in less than two dimensions, where mean field theory does not apply. We used and explain a method based on ideas of graph theory and renormalization group theory to derive exact equations for appropriate variables that are similar to the restricted partition functions. We benchmark our method with Metropolis Monte Carlo simulations. The analysis of fixed points reveals information of location of the Fisher zeros and we provide a conjecture about the location of zeros in terms of the boundary of the basins of attraction.

Optimizers of three-point energies and nearly orthogonal sets

This paper is devoted to spherical measures and point configurations optimizing three-point energies. Our main goal is to extend the classic optimization problems based on pairs of distances between points to the context of three-point potentials. In particular, we study three-point analogues of the sphere packing problem and the optimization problem for p p -frame energies based on three points. It turns out that both problems are inherently connected to the problem of nearly orthogonal sets by Erdős. As the outcome, we provide a new solution of the Erdős problem from the three-point packing perspective. We also show that the orthogonal basis uniquely minimizes the p p -frame three-point energy when 0 > p > 1 0>p>1 in all dimensions. The arguments make use of multivariate polynomials employed in semidefinite programming and based on the classical Gegenbauer polynomials. For p = 1 p=1 , we completely solve the analogous problem on the circle. As for higher dimensions, we show that the Hausdorff dimension of minimizers is not greater than d − 2 d-2 for measures on S d − 1 \mathbb {S}^{d-1} .

Semi-Regular Continued Fractions with Fast-Growing Partial Quotients

In number theory, continued fractions are essential tools because they provide distinct representations of real numbers and provide information about their characteristics. Regular continued fractions have been examined in great detail, but less research has been carried out on their semi-regular counterparts, which are produced from the sequences of alternating plus and minus ones. In this study, we investigate the structure and features of semi-regular continuous fractions through the lens of dimension theory. We prove a primary result about the Hausdorff dimension of number sets whose partial quotients increase more quickly than a given pace. Furthermore, we conduct numerical analyses to illustrate the differences between regular and semi-regular continued fractions, shedding light on potential future directions in this field.

Shrinking target problems in d-decaying Gauss-like iterated function systems

Let Φ={ϕn}n≥1 be an infinite iterated function system (IFS) on [0,1] satisfying the open set condition with the open set (0,1) and let J be its attractor. For any x∈J, there is a unique integer sequence {ωn(x)}n≥1, called the digit sequence of x, such thatx=limn→∞⁡ϕω1(x)∘⋯∘ϕωn(x)(1) except at countably many points. In this paper, we investigate the shrinking target problem in the infinite iterated function system with a polynomial decay of the derivative. More precisely, we consider the size of the setW(T,x0)={x∈J:Tn(x)∈Itn(x0) for infinitely many n∈N}, where T:J⟼J is the expanding map induced by the left shift and Itn(x0) denotes the tn-th cylinder of x0∈J. As a continuation of one result of Li et al. (2014) [11], we obtain the exact Hausdorff dimension of the set W(T,x0) in some general infinite function systems.

Badly approximable grids and k$k$‐divergent lattices

AbstractLet be a matrix. In this paper, we investigate the set of badly approximable targets for , where is the ‐torus. It is well known that is a winning set for Schmidt's game and hence is a dense subset of full Hausdorff dimension. We investigate the relationship between the measure of and Diophantine properties of . On the one hand, we give the first examples of a nonsingular such that has full measure with respect to some nontrivial algebraic measure on the torus. For this, we use transference theorems due to Jarnik and Khintchine, and the parametric geometry of numbers in the sense of Roy. On the other hand, we give a novel Diophantine condition on that slightly strengthens nonsingularity, and show that under the assumption that satisfies this condition, is a null‐set with respect to any nontrivial algebraic measure on the torus. For this, we use naive homogeneous dynamics, harmonic analysis, and a novel concept that we refer to as mixing convergence of measures.