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

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.

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HÖLDER CONTINUITY AND BOX DIMENSION FOR THE WEYL FRACTIONAL INTEGRAL

Fractals ◽

10.1142/s0218348x20500322 ◽

2020 ◽

Vol 28 (02) ◽

pp. 2050032 ◽

Cited By ~ 1

Author(s):

LONG TIAN

Keyword(s):

Continuous Function ◽

Fractional Integral ◽

Hölder Continuity ◽

Box Dimension ◽

Holder Continuity ◽

Hölder Continuous ◽

Weyl Fractional Integral

In this paper, we investigate the Hölder continuity and the estimate for the box dimension of the Weyl fractional integral of some continuous function [Formula: see text], denoted by [Formula: see text]. We obtain that if [Formula: see text] is [Formula: see text]-order Hölder continuous, then [Formula: see text] is [Formula: see text]-order Hölder continuous. Moreover, if [Formula: see text] belongs to [Formula: see text], then [Formula: see text] is [Formula: see text]-order Hölder continuous with [Formula: see text].

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THE EFFECTS OF THE RIEMANN–LIOUVILLE FRACTIONAL INTEGRAL ON THE BOX DIMENSION OF FRACTAL GRAPHS OF HÖLDER CONTINUOUS FUNCTIONS

Fractals ◽

10.1142/s0218348x20500528 ◽

2020 ◽

Vol 28 (03) ◽

pp. 2050052

Author(s):

JUNRU WU

Keyword(s):

Continuous Function ◽

Fractional Integral ◽

Closed Interval ◽

Open Interval ◽

Continuous Functions ◽

Box Dimension ◽

Hölder Continuous ◽

Positive Order ◽

Lipschitz Continuous ◽

Hölder Continuous Functions

In this paper, the linearity of the dimensional-decrease effect of the Riemann–Liouville fractional integral is mainly explored. It is proved that if the Box dimension of the graph of an [Formula: see text]-Hölder continuous function is greater than one and the positive order [Formula: see text] of the Riemann–Liouville fractional integral satisfies [Formula: see text], the upper Box dimension of the Riemann–Liouville fractional integral of the graph of this function will not be greater than [Formula: see text]. Furthermore, it is proved that the Riemann–Liouville fractional integral of a Lipschitz continuous function defined on a closed interval is continuously differentiable on the corresponding open interval.

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ESTIMATION OF FRACTAL DIMENSION OF FRACTIONAL CALCULUS OF THE HÖLDER CONTINUOUS FUNCTIONS

Fractals ◽

10.1142/s0218348x20501236 ◽

2020 ◽

Vol 28 (07) ◽

pp. 2050123

Author(s):

YONG-SHUN LIANG

Keyword(s):

Fractal Dimension ◽

Fractional Calculus ◽

Fractional Derivative ◽

Fractional Integral ◽

Continuous Functions ◽

Box Dimension ◽

Hölder Condition ◽

Hölder Continuous ◽

Lipschitz Continuous ◽

Liouville Fractional Derivative

In the present paper, fractal dimension and properties of fractional calculus of certain continuous functions have been investigated. Upper Box dimension of the Riemann–Liouville fractional integral of continuous functions satisfying the Hölder condition of certain positive orders has been proved to be decreasing linearly. If sum of order of the Riemann–Liouville fractional integral and the Hölder condition equals to one, the Riemann–Liouville fractional integral of the function will be Lipschitz continuous. If the corresponding sum is strictly larger than one, the Riemann–Liouville fractional integral of the function is differentiable. Estimation of fractal dimension of the derivative function has also been discussed. Finally, the Riemann–Liouville fractional derivative of continuous functions satisfying the Hölder condition exists when order of the Riemann–Liouville fractional derivative is smaller than order of the Hölder condition. Upper Box dimension of the function has been proved to be increasing at most linearly.

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RELATIONSHIP BETWEEN UPPER BOX DIMENSION OF CONTINUOUS FUNCTIONS AND ORDERS OF WEYL FRACTIONAL INTEGRAL

Fractals ◽

10.1142/s0218348x20502230 ◽

2021 ◽

Author(s):

H. B. GAO ◽

Y. S. LIANG ◽

W. XIAO

Keyword(s):

Fractal Dimension ◽

Continuous Function ◽

Bounded Variation ◽

Fractional Integral ◽

Continuous Functions ◽

Fractional Integrals ◽

Box Dimension ◽

Hölder Continuous ◽

Dimension One ◽

Weyl Fractional Integral

In this paper, we mainly investigate relationship between fractal dimension of continuous functions and orders of Weyl fractional integrals. If a continuous function defined on a closed interval is of bounded variation, its Weyl fractional integral must still be a continuous function with bounded variation. Thus, both its Weyl fractional integral and itself have Box dimension one. If a continuous function satisfies Hölder condition, we give estimation of fractal dimension of its Weyl fractional integral. If a Hölder continuous function is equal to 0 on [Formula: see text], a better estimation of fractal dimension can be obtained. When a function is continuous on [Formula: see text] and its Weyl fractional integral is well defined, a general estimation of upper Box dimension of Weyl fractional integral of the function has been given which is strictly less than two. In the end, it has been proved that upper Box dimension of Weyl fractional integrals of continuous functions is no more than upper Box dimension of original functions.

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Dimensions of Fractional Brownian Images

Journal of Theoretical Probability ◽

10.1007/s10959-021-01131-3 ◽

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.

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SOME REMARKS ON FRACTIONAL INTEGRAL OF ONE-DIMENSIONAL CONTINUOUS FUNCTIONS

Fractals ◽

10.1142/s0218348x2050005x ◽

2020 ◽

Vol 28 (01) ◽

pp. 2050005

Author(s):

JIA YAO ◽

YING CHEN ◽

JUNQIAO LI ◽

BIN WANG

Keyword(s):

Continuous Function ◽

Fractional Integral ◽

Continuous Functions ◽

Box Dimension ◽

Positive Order ◽

One Dimensional ◽

Katugampola Fractional Integral ◽

Hadamard Fractional Integral

In this paper, we make research on Katugampola and Hadamard fractional integral of one-dimensional continuous functions on [Formula: see text]. We proved that Katugampola fractional integral of bounded and continuous function still is bounded and continuous. Box dimension of any positive order Hadamard fractional integral of one-dimensional continuous functions is one.

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The horizon problem for prevalent surfaces

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s030500411100048x ◽

2011 ◽

Vol 151 (2) ◽

pp. 355-372 ◽

Cited By ~ 9

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.

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THE RELATIONSHIP BETWEEN FRACTAL DIMENSIONS OF BESICOVITCH FUNCTION AND THE ORDER OF HADAMARD FRACTIONAL INTEGRAL

Fractals ◽

10.1142/s0218348x20501285 ◽

2020 ◽

Vol 28 (07) ◽

pp. 2050128

Author(s):

BIN WANG ◽

WENLONG JI ◽

LEGUI ZHANG ◽

XUAN LI

Keyword(s):

Fractional Integral ◽

Fractal Dimensions ◽

Packing Dimension ◽

Box Dimension ◽

Dimension Formula ◽

The Relationship ◽

Hadamard Fractional Integral

In this paper, we mainly research on Hadamard fractional integral of Besicovitch function. A series of propositions of Hadamard fractional integral of [Formula: see text] have been proved first. Then, we give some fractal dimensions of Hadamard fractional integral of Besicovitch function including Box dimension, [Formula: see text]-dimension and Packing dimension. Finally, relationship between the order of Hadamard fractional integral and fractal dimensions of Besicovitch function has also been given.

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

Mathematical Problems in Engineering ◽

10.1155/2017/2095896 ◽

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.

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