scholarly journals The horizon problem for prevalent surfaces

2011 ◽  
Vol 151 (2) ◽  
pp. 355-372 ◽  
Author(s):  
K. J. FALCONER ◽  
J. M. FRASER
Keyword(s):  
Box Dimension ◽  
Fractal Surface ◽  
Lipschitz Spaces ◽  
Horizon Problem ◽  
Box Dimensions ◽  
The Way

We investigate the box dimensions of the horizon of a fractal surface defined by a functionf∈C[0,1]2. In particular we show that a prevalent surface satisfies the ‘horizon property’, namely that the box dimension of the horizon is one less than that of the surface. Since a prevalent surface has box dimension 3, this does not give us any information about the horizon of surfaces of dimension strictly less than 3. To examine this situation we introduce spaces of functions with surfaces of upper box dimension at most α, for α ∈ [2,3). In this setting the behaviour of the horizon is more subtle. We construct a prevalent subset of these spaces where the lower box dimension of the horizon lies between the dimension of the surface minus one and 2. We show that in the sense of prevalence these bounds are as tight as possible if the spaces are defined purely in terms of dimension. However, if we work in Lipschitz spaces, the horizon property does indeed hold for prevalent functions. Along the way, we obtain a range of properties of box dimensions of sums of functions.

scholarly journals Dimensions of Fractional Brownian Images

2021 ◽  
Author(s):  
Stuart A. Burrell
Keyword(s):  
Brownian Motion ◽  
Hausdorff Dimension ◽  
Stochastic Processes ◽  
Fractional Brownian Motion ◽  
Box Dimension ◽  
General Strategy ◽  
Borel Sets ◽  
Exceptional Sets ◽  
Explicit Condition ◽  
Box Dimensions

AbstractThis 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-$$\alpha $$ α 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.

scholarly journals A Method for Identification of Transformer Inrush Current Based on Box Dimension

2017 ◽  
Vol 2017 ◽  
pp. 1-6
Author(s):  
Maofa Gong ◽  
Ran Zheng ◽  
Linyuan Hou ◽  
Jingyu Wei ◽  
Na Wu
Keyword(s):  
Fundamental Difference ◽  
Box Dimension ◽  
Inrush Current ◽  
Fault Current ◽  
Differential Protection ◽  
Phase Current ◽  
Magnetizing Inrush Current ◽  
Three Phase ◽  
Transformer Differential Protection ◽  
Box Dimensions

Magnetizing inrush current can lead to the maloperation of transformer differential protection. To overcome such an issue, a method is proposed to distinguish inrush current from inner fault current based on box dimension. According to the fundamental difference in waveform between the two, the algorithm can extract the three-phase current and calculate its box dimensions. If the box dimension value is smaller than the setting value, it is the inrush current; otherwise, it is inner fault current. Using PSACD and MATLAB, the simulation has been performed to prove the efficiency reliability of the presented algorithm in distinguishing inrush current and fault current.

scholarly journals Box dimension, oscillation and smoothness in function spaces

2005 ◽  
Vol 3 (3) ◽  
pp. 287-320 ◽  
Author(s):  
Abel Carvalho
Keyword(s):  
Compact Support ◽  
Besov Spaces ◽  
Function Spaces ◽  
Box Dimension ◽  
Real Functions ◽  
Box Dimensions

The aim of this paper is twofold. First we relate upper and lower box dimensions with oscillation spaces, and we develop embeddings or inclusions between oscillation spaces and Besov spaces. Secondly, given a point in the (1p,s)-plane we determine maximal and minimal values for the upper box dimension (also the maximal value for lower box dimension) for the graphs of continuous real functions with a compact support, represented by this point.

BOX DIMENSIONS OF α-FRACTAL FUNCTIONS

Fractals ◽  
2016 ◽  
Vol 24 (03) ◽  
pp. 1650037 ◽  
Author(s):  
MD. NASIM AKHTAR ◽  
M. GURU PREM PRASAD ◽  
M. A. NAVASCUÉS
Keyword(s):  
Continuous Function ◽  
Differentiable Function ◽  
Iterated Function System ◽  
Box Dimension ◽  
Data Sets ◽  
Fractal Function ◽  
Iterated Function ◽  
Nowhere Differentiable Function ◽  
Fractal Functions ◽  
Box Dimensions

The box dimension of the graph of non-affine, continuous, nowhere differentiable function [Formula: see text] which is a fractal analogue of a continuous function [Formula: see text] corresponding to a certain iterated function system (IFS), is investigated in the present paper. The estimates for box dimension of the graph of [Formula: see text]-fractal function [Formula: see text] for equally spaced as well as arbitrary data sets are found.

Dimensions associated with recurrent self-similar sets

1991 ◽  
Vol 110 (2) ◽  
pp. 327-336 ◽  
Author(s):  
Anca Deliu ◽  
J. S. Geronimo ◽  
R. Shonkwiler ◽  
D. Hardin
Keyword(s):  
Box Dimension ◽  
Two Dimensional ◽  
Affine Maps ◽  
Self Similar ◽  
Box Dimensions

AbstractThe Hausdorff and box dimensions for measures associated with recurrent self-similar sets generated by similitudes is explicitly given. The box dimension of the attractor associated with a class of two-dimensional affine maps is also computed.

BOX DIMENSIONS OF THE RIEMANN–LIOUVILLE FRACTIONAL INTEGRAL OF HÖLDER CONTINUOUS MULTIVARIATE FUNCTIONS

Fractals ◽  
2020 ◽  
Vol 28 (06) ◽  
pp. 2050113
Author(s):  
JING LEI ◽  
KANGJIE LIU ◽  
YINGZI DAI
Keyword(s):  
Lipschitz Condition ◽  
Fractional Integral ◽  
Box Dimension ◽  
Multivariate Functions ◽  
Hölder Continuous ◽  
Multivariate Function ◽  
Box Dimensions

If a continuous multivariate function satisfies a Lipschitz condition on its domain, Box dimension of its graph equals to the number of its arguments. Furthermore, Box dimension of the graph of its Riemann–Liouville fractional integral also equals to the number of its arguments.

scholarly journals Accumulation of capital and divisible reserves in labour managed firms

2005 ◽  
Vol 2 (3) ◽  
pp. 19-27
Author(s):  
Ermanno C. Tortia
Keyword(s):  
Past Research ◽  
Specific Mechanism ◽  
Horizon Problem ◽  
Worker Cooperatives ◽  
The Way

The problem of the accumulation of capital in labour managed firms and worker cooperatives has been attracting considerable attention by past research. The Furubotn-Pejovich effect is considered to be the source undercapitalisation. The paper seeks to show that the presence of undercapitalisation is due to a specific mechanism of reinvestment, i.e. reinvestment of self-financed capital funds in indivisible reserves. The introduction of divisible reserves appropriable by worker members at some point in time would solve the horizon problem. However, it is likely to engender new and unexplored problems connected with the way in which net surpluses are distributed, the reinvestment of individual shares of net surpluses and the reimbursement of individual capital quotas.

BOX DIMENSION OF A NONLINEAR FRACTAL INTERPOLATION CURVE

Fractals ◽  
2019 ◽  
Vol 27 (03) ◽  
pp. 1950023 ◽  
Author(s):  
SONG-IL RI
Keyword(s):  
Mean Value ◽  
Mean Value Theorem ◽  
Box Dimension ◽  
Fractal Interpolation ◽  
Monotone Property ◽  
Interpolation Curve ◽  
Function Convex ◽  
Differential Mean Value Theorem ◽  
Box Dimensions

In this paper, we present a delightful method to estimate the lower and upper box dimensions of a special nonlinear fractal interpolation curve. We use Rakotch contractibility and monotone property of function in the estimation of upper box dimension, and we use Rakotch contractibility, noncollinearity of interpolation points, nondecreasing property of function, convex (or concave) property of function and differential mean value theorem in the estimation of lower box dimension. In particular, we propose a well-founded conjecture motivated by our results.

Fault Diagnosis of the Relief Valve Based on Fractal Box Dimension of Autoregressive Bispectrum Slices

2013 ◽  
Vol 274 ◽  
pp. 70-73
Author(s):  
Ying Xiao ◽  
Xiao Mei Liu ◽  
Jian Can Chen
Keyword(s):  
Fault Diagnosis ◽  
Data Preprocessing ◽  
Accurate Method ◽  
Box Dimension ◽  
Relief Valve ◽  
Fault Pattern ◽  
Box Dimensions

After data preprocessing,the autoregressive(AR) bispectrum diagonal slices were figured and fractal box dimensions were calculated. The results show that the box dimensions are different in evidence under different conditions. So the fault pattern recognitions of the relief valve are effective by the method of using fractal box dimension of AR bispectrum diagonal slices, it provides a simple and accurate method for fault diagnosis of the relief valve.

Box and packing dimensions of projections and dimension profiles

2001 ◽  
Vol 130 (1) ◽  
pp. 135-160 ◽  
Author(s):  
J. D. HOWROYD
Keyword(s):  
Packing Dimension ◽  
Dimension Theory ◽  
Box Dimension ◽  
Orthogonal Projections ◽  
Packing Dimensions ◽  
Almost All ◽  
Box Dimensions

For E a subset of ℝn and s ∈ [0, n] we define upper and lower box dimension profiles, B-dimsE and B-dimsE respectively, that are closely related to the box dimensions of the orthogonal projections of E onto subspaces of ℝn. In particular, the projection of E onto almost all m-dimensional subspaces has upper box dimension B-dimmE and lower box dimension B-dimmE. By defining a packing type measure with respect to s-dimensional kernels we are able to establish the connection to an analogous packing dimension theory.

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