For the spherical mean operators $\scr{A}_t$ in $\Bbb{R}^d$, $d\ge 2$, we consider the maximal functions $M_Ef=\sup_{t\in E}|\scr{A}_t f|$, with dilation sets $E\subset 1,2$. In this paper we give a ...surprising characterization of the closed convex sets which can occur as closure of the sharp $L^p$ improving region of $M_E$ for some $E$. This region depends on the Minkowski dimension of $E$, but also other properties of the fractal geometry such as the Assouad spectrum of $E$ and subsets of $E$. A key ingredient is an essentially sharp result on $M_E$ for a class of sets called (quasi-)Assouad regular which is new in two dimensions.
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.
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The mobility edges (MEs) in energy that separate extended and localized states are a central concept in understanding the localization physics. In one-dimensional (1D) quasiperiodic systems, while ...MEs may exist for certain cases, the analytic results that allow for an exact understanding are rare. Here we uncover a class of exactly solvable 1D models with MEs in the spectra, where quasiperiodic on-site potentials are inlaid in the lattice with equally spaced sites. The analytical solutions provide the exact results not only for the MEs, but also for the localization and extended features of all states in the spectra, as derived through computing the Lyapunov exponents from Avila's global theory and also numerically verified by calculating the fractal dimension. We further propose a novel scheme with experimental feasibility to realize our model based on an optical Raman lattice, which paves the way for experimental exploration of the predicted exact ME physics.
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Analytical calculation of τ(q) for multifractals del Río Correa, J L; López García, J; Álvarez Ballesteros, Y A
Journal of physics. Conference series,
09/2022, Volume:
2307, Issue:
1
Journal Article
Peer reviewed
Open access
Abstract
To find the Hausdorff dimensional spectrum that characterizes a multifractal, it is required to find the mass function
τ
(
q
), which is implicitly defined by the Halsey equation, which in ...general can only be solved numerically, however there are particular cases in which it is possible to find analytically
τ
(
q
) as a parametrically defined function. In this paper we present and discuss the Piña’s method to find analytically
τ
(
q
), the Hausdorff dimension spectrum of the multifractal is found.
We demonstrate many-body multifractality of the Bose-Hubbard Hamiltonian's ground state in Fock space, for arbitrary values of the interparticle interaction. Generalized fractal dimensions ...unambiguously signal, even for small system sizes, the emergence of a Mott insulator that cannot, however, be naively identified with a localized phase in Fock space. We show that the scaling of the derivative of any generalized fractal dimension with respect to the interaction strength encodes the critical point of the superfluid to the Mott insulator transition, and provides an efficient way to accurately estimate its position. We further establish that the transition can be quantitatively characterized by one single wave function amplitude from the exponentially large Fock space.
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In contrast with Anderson localization where a genuine localization is observed in real space, the many-body localization (MBL) problem is much less understood in Hilbert space, the support of the ...eigenstates. In this Letter, using exact diagonalization techniques we address the ergodicity properties in the underlying N-dimensional complex networks spanned by various computational bases for up to L=24 spin-1/2 particles (i.e., Hilbert space of size N≃2.7×106). We report fully ergodic eigenstates in the delocalized phase (irrespective of the computational basis), while the MBL regime features a generically (basis-dependent) multifractal behavior, delocalized but nonergodic. The MBL transition is signaled by a nonuniversal jump of the multifractal dimensions.
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Abstract
It is focused on the point that, factually, the material’s structure is a primary cause of its properties. The elements of hardened cement stone (inner interfaces), which are the source of ...its properties, are revealed. It is supposed that the inner interfaces branched chain (patterns) are carriers of certain information. It is proposed to describe this information in quantitative values by fractal dimensions. With such approach the research process is described by sole triad «structure-information-fractality». This article states experimental results proving the effectiveness of the approach proposed.
The heritage of Central Sulawesi batik have to be prevented in order to maintain Indonesian culture. Nowadays, available batik motif of Central Sulawesi needs to be explored to promote its cultural ...wealth. Using the main concept of fractal geometry, repetition and similarity, some motifs are designed in this research. The concepts make the motifs are displayed regularly. The base of the design is the diversity of the Central Sulawesi that located in Wallacea zone. This principal base makes the uniqueness of the motifs design. The generated motifs are varied by some parameters values that represent some mathematical operation and simulated by jbatik software. The operations are translation, rotation, dilatation, and reflection. This research results some motifes that are namely the waving cactus bloom, cesara, moringa chem, eboni herb, sigi fiesta, and harmony nature. The other advantage of the result is some motifs are also could be applicated as woven motif, the other Indonesian heritage clothing.
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