Hausdorff dimension of the set of almost convergent sequences

2022 ◽  
pp. 1-7
Author(s):  
Alexandr Usachev
Keyword(s):  
Hausdorff Dimension ◽  
Summation Method ◽  
Binary Representation ◽  
Hausdorff Dimensions ◽  
Summation Methods ◽  
Wide Range ◽  
Matrix Summation ◽  
Convergent Sequences

Abstract The paper deals with the sets of numbers from [0,1] such that their binary representation is almost convergent. The aim of the study is to compute the Hausdorff dimensions of such sets. Previously, the results of this type were proved for a single summation method (e.g. Cesàro, Abel, Toeplitz). This study extends the results to a wide range of matrix summation methods.

scholarly journals The Dimension of the Boundary of a Liouville Quantum Gravity Metric Ball

2020 ◽  
Vol 378 (1) ◽  
pp. 625-689 ◽  
Author(s):  
Ewain Gwynne
Keyword(s):  
Hausdorff Dimension ◽  
Quantum Gravity ◽  
Free Field ◽  
Gaussian Free Field ◽  
Hausdorff Dimensions ◽  
Thick Points

Abstract Let $$\gamma \in (0,2)$$ γ ∈ ( 0 , 2 ) , let h be the planar Gaussian free field, and consider the $$\gamma $$ γ -Liouville quantum gravity (LQG) metric associated with h. We show that the essential supremum of the Hausdorff dimension of the boundary of a $$\gamma $$ γ -LQG metric ball with respect to the Euclidean (resp. $$\gamma $$ γ -LQG) metric is $$2 - \frac{\gamma }{d_\gamma }\left( \frac{2}{\gamma } + \frac{\gamma }{2} \right) + \frac{\gamma ^2}{2d_\gamma ^2}$$ 2 - γ d γ 2 γ + γ 2 + γ 2 2 d γ 2 (resp. $$d_\gamma -1$$ d γ - 1 ), where $$d_\gamma $$ d γ is the Hausdorff dimension of the whole plane with respect to the $$\gamma $$ γ -LQG metric. For $$\gamma = \sqrt{8/3}$$ γ = 8 / 3 , in which case $$d_{\sqrt{8/3}}=4$$ d 8 / 3 = 4 , we get that the essential supremum of Euclidean (resp. $$\sqrt{8/3}$$ 8 / 3 -LQG) dimension of a $$\sqrt{8/3}$$ 8 / 3 -LQG ball boundary is 5/4 (resp. 3). We also compute the essential suprema of the Euclidean and $$\gamma $$ γ -LQG Hausdorff dimensions of the intersection of a $$\gamma $$ γ -LQG ball boundary with the set of metric $$\alpha $$ α -thick points of the field h for each $$\alpha \in \mathbb R$$ α ∈ R . Our results show that the set of $$\gamma /d_\gamma $$ γ / d γ -thick points on the ball boundary has full Euclidean dimension and the set of $$\gamma $$ γ -thick points on the ball boundary has full $$\gamma $$ γ -LQG dimension.

scholarly journals NEI Modelling of the ISM - Turbulent Dissipation and Hausdorff Dimension

2009 ◽  
Vol 5 (H15) ◽  
pp. 468-469 ◽  
Author(s):  
Miguel A. de Avillez ◽  
Dieter Breitschwerdt
Keyword(s):  
High Resolution ◽  
Hausdorff Dimension ◽  
Dissipative Structures ◽  
Self Similarity ◽  
Extended Self ◽  
Turbulent Dissipation ◽  
Independent Information ◽  
Ionization Structure ◽  
Wide Range ◽  
Self Gravity

AbstractHigh-resolution non-ideal magnetohydrodynamical simulations of the turbulent magnetized ISM, powered by supernovae types Ia and II at Galactic rate, including self-gravity and non-equilibriuim ionization (NEI), taking into account the time evolution of the ionization structure of H, He, C, N, O, Ne, Mg, Si, S and Fe, were carried out. These runs cover a wide range (from kpc to sub-parsec) of scales, providing resolution independent information on the injection scale, extended self-similarity and the fractal dmension of the most dissipative structures.

Dimension spectra of lines1

Computability ◽  
2021 ◽  
pp. 1-28
Author(s):  
Neil Lutz ◽  
D.M. Stull
Keyword(s):  
Hausdorff Dimension ◽  
Kolmogorov Complexity ◽  
Euclidean Plane ◽  
Unit Interval ◽  
Packing Dimension ◽  
Hausdorff Dimensions ◽  
Dimension Spectrum ◽  
Algorithmic Dimension

This paper investigates the algorithmic dimension spectra of lines in the Euclidean plane. Given any line L with slope a and vertical intercept b, the dimension spectrum sp ( L ) is the set of all effective Hausdorff dimensions of individual points on L. We draw on Kolmogorov complexity and geometrical arguments to show that if the effective Hausdorff dimension dim ( a , b ) is equal to the effective packing dimension Dim ( a , b ), then sp ( L ) contains a unit interval. We also show that, if the dimension dim ( a , b ) is at least one, then sp ( L ) is infinite. Together with previous work, this implies that the dimension spectrum of any line is infinite.

On Hausdorff dimension monotonicity of a family of dynamical subsets of Rauzy fractals

2015 ◽  
Vol 11 (04) ◽  
pp. 1089-1098 ◽  
Author(s):  
W. Georg Nowak ◽  
Klaus Scheicher ◽  
Víctor F. Sirvent
Keyword(s):  
Hausdorff Dimension ◽  
Hausdorff Dimensions ◽  
The Third

We consider a family of dynamically defined subsets of Rauzy fractals in the plane. These sets were introduced in the context of the study of symmetries of Rauzy fractals. We prove that their Hausdorff dimensions form an ultimately increasing sequence of numbers converging to 2. These results answer a question stated by the third author in 2012.

scholarly journals Directional Multifractal Analysis in the Lp Setting

2019 ◽  
Vol 2019 ◽  
pp. 1-12
Author(s):  
Mourad Ben Slimane ◽  
Ines Ben Omrane ◽  
Moez Ben Abid ◽  
Borhen Halouani ◽  
Farouq Alshormani
Keyword(s):  
Hausdorff Dimension ◽  
Multifractal Analysis ◽  
Hölder Regularity ◽  
Wavelet Coefficients ◽  
Wide Range ◽  
Bounded Functions ◽  
Set Of Points ◽  
Holder Regularity

The classical Hölder regularity is restricted to locally bounded functions and takes only positive values. The local Lp regularity covers unbounded functions and negative values. Nevertheless, it has the same apparent regularity in all directions. In the present work, we study a recent notion of directional local Lp regularity introduced by Jaffard. We provide its characterization by a supremum of a wide range oriented anisotropic Triebel wavelet coefficients and leaders. In addition, we deduce estimates on the Hausdorff dimension of the set of points where the directional local Lp regularity does not exceed a given value. The obtained results are illustrated by some examples of self-affine cascade functions.

scholarly journals Using matrix summation method for three dimensional dose calculation in brachytherapy

2012 ◽  
Vol 17 (2) ◽  
pp. 110-114
Author(s):  
Mahmoud Zibandeh-Gorji ◽  
Ali Asghar Mowlavi ◽  
Saeed Mohammadi
Keyword(s):  
Three Dimensional ◽  
Dose Calculation ◽  
Summation Method ◽  
Matrix Summation

scholarly journals Hausdorff dimension, heavy tails, and generalization in neural networks*

2021 ◽  
Vol 2021 (12) ◽  
pp. 124014
Author(s):  
Umut Şimşekli ◽  
Ozan Sener ◽  
George Deligiannidis ◽  
Murat A Erdogdu
Keyword(s):  
Neural Networks ◽  
Hausdorff Dimension ◽  
Heavy Tails ◽  
Stochastic Gradient Descent ◽  
Generalization Error ◽  
Generalization Bounds ◽  
Wide Range ◽  
Important Challenge ◽  
Rigorous Treatment ◽  
Heavy Tailed

Abstract Despite its success in a wide range of applications, characterizing the generalization properties of stochastic gradient descent (SGD) in non-convex deep learning problems is still an important challenge. While modeling the trajectories of SGD via stochastic differential equations (SDE) under heavy-tailed gradient noise has recently shed light over several peculiar characteristics of SGD, a rigorous treatment of the generalization properties of such SDEs in a learning theoretical framework is still missing. Aiming to bridge this gap, in this paper, we prove generalization bounds for SGD under the assumption that its trajectories can be well-approximated by a Feller process, which defines a rich class of Markov processes that include several recent SDE representations (both Brownian or heavy-tailed) as its special case. We show that the generalization error can be controlled by the Hausdorff dimension of the trajectories, which is intimately linked to the tail behavior of the driving process. Our results imply that heavier-tailed processes should achieve better generalization; hence, the tail-index of the process can be used as a notion of ‘capacity metric’. We support our theory with experiments on deep neural networks illustrating that the proposed capacity metric accurately estimates the generalization error, and it does not necessarily grow with the number of parameters unlike the existing capacity metrics in the literature.

scholarly journals Hausdorff Dimension of Escaping Sets of Nevanlinna Functions

2019 ◽  
Author(s):  
Weiwei Cui
Keyword(s):  
Hausdorff Dimension ◽  
Differential Equations ◽  
Complex Plane ◽  
Meromorphic Functions ◽  
Second Order ◽  
Hausdorff Dimensions ◽  
Second Order Differential Equations ◽  
Schwarzian Derivatives ◽  
Nevanlinna Functions

Abstract We determine the exact values of Hausdorff dimensions of escaping sets of meromorphic functions with polynomial Schwarzian derivatives. This will follow from the relation between these functions and the second-order differential equations in the complex plane.

On Weierstrass-like functions and random recurrent sets

1989 ◽  
Vol 106 (2) ◽  
pp. 325-342 ◽  
Author(s):  
Tim Bedford
Keyword(s):  
Hausdorff Dimension ◽  
Random Matrix ◽  
Hausdorff Dimensions ◽  
Random Matrix Products ◽  
Matrix Products ◽  
Recurrent Set

AbstractA construction of Weierstrass-like functions using recurrent sets is described, and the Hausdorff dimensions of the graphs computed. An important part of the proof is the notion of a globally random recurrent set. The Hausdorff dimension of a class of such sets is calculated using techniques of random matrix products.

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