FRACTAL DIMENSION OF CERTAIN CONTINUOUS FUNCTIONS OF UNBOUNDED VARIATION

Fractals ◽  
2017 ◽  
Vol 25 (01) ◽  
pp. 1750009 ◽  
Author(s):  
Y. S. LIANG ◽  
W. Y. SU
Keyword(s):  
Fractal Dimension ◽  
Continuous Function ◽  
Bounded Variation ◽  
Closed Interval ◽  
Fractal Dimensions ◽  
Continuous Functions ◽  
One Dimensional ◽  
Closed Intervals ◽  
Bounded Variation Functions

Continuous functions on closed intervals are composed of bounded variation functions and unbounded variation functions. Fractal dimension of continuous functions with bounded variation must be one-dimensional (1D). While fractal dimension of continuous functions with unbounded variation may be 1 or not. Certain continuous functions of unbounded variation whose fractal dimensions are 1 have been mainly investigated in the paper. A continuous function on a closed interval with finite unbounded variation points has been proved to be 1D. Furthermore, we deal with continuous functions which have infinite unbounded variation points and part of them have been proved to be 1D. Certain examples of 1D continuous functions which have uncountable unbounded variation points have been given in the present paper.

DEFINITION AND CLASSIFICATION OF ONE-DIMENSIONAL CONTINUOUS FUNCTIONS WITH UNBOUNDED VARIATION

Fractals ◽  
2017 ◽  
Vol 25 (05) ◽  
pp. 1750048 ◽  
Author(s):  
Y. S. LIANG
Keyword(s):  
Fractal Dimension ◽  
Bounded Variation ◽  
Continuous Functions ◽  
Functions Of Bounded Variation ◽  
One Dimensional ◽  
Nondifferentiable Functions ◽  
Differentiable Functions ◽  
Closed Intervals ◽  
Bounded Variation Functions

The present paper mainly investigates the definition and classification of one-dimensional continuous functions on closed intervals. Continuous functions can be classified as differentiable functions and nondifferentiable functions. All differentiable functions are of bounded variation. Nondifferentiable functions are composed of bounded variation functions and unbounded variation functions. Fractal dimension of all bounded variation continuous functions is 1. One-dimensional unbounded variation continuous functions may have finite unbounded variation points or infinite unbounded variation points. Number of unbounded variation points of one-dimensional unbounded variation continuous functions maybe infinite and countable or uncountable. Certain examples of different one-dimensional continuous functions have been given in this paper. Thus, one-dimensional continuous functions are composed of differentiable functions, nondifferentiable continuous functions of bounded variation, continuous functions with finite unbounded variation points, continuous functions with infinite but countable unbounded variation points and continuous functions with uncountable unbounded variation points. In the end of the paper, we give an example of one-dimensional continuous function which is of unbounded variation everywhere.

THE CLASSIFICATION OF ONE-DIMENSIONAL CONTINUOUS FUNCTIONS AND THEIR FRACTIONAL INTEGRAL

Fractals ◽  
2018 ◽  
Vol 26 (05) ◽  
pp. 1850063 ◽  
Author(s):  
XING LIU ◽  
JUN WANG ◽  
HE LIN LI
Keyword(s):  
Fractal Dimension ◽  
Continuous Function ◽  
Bounded Variation ◽  
Fractional Integral ◽  
Continuous Functions ◽  
Functions Of Bounded Variation ◽  
One Dimensional

This paper mainly discusses the continuous functions whose fractal dimension is 1 on [Formula: see text]. First, we classify continuous functions into unbounded variation and bounded variation. Then we prove that the fractal dimension of both continuous functions of bounded variation and their fractional integral is 1. As for continuous functions of unbounded variation, we solve several special types. Finally, we give the example of one-dimensional continuous function of unbounded variation.

COMPOSITION OF ONE-DIMENSIONAL CONTINUOUS FUNCTIONS AND THEIR RIEMANN–LIOUVILLE FRACTIONAL INTEGRAL

Fractals ◽  
2019 ◽  
Vol 27 (04) ◽  
pp. 1950065
Author(s):  
BIN YU ◽  
TAO ZHANG ◽  
LEI YAO ◽  
WEI ZHAO
Keyword(s):  
Fractal Dimension ◽  
Fractional Derivative ◽  
Bounded Variation ◽  
Fractional Integral ◽  
Continuous Functions ◽  
Box Dimension ◽  
Important Conclusion ◽  
One Dimensional ◽  
Dimension One

In this paper, we make research on composition of continuous functions with Box dimension one of bounded variation or unbounded variation on [Formula: see text]. It has been proved that one-dimensional continuous functions must be one of functions with bounded variation, or functions with finite unbounded variation points, or functions with infinite unbounded variation points on [Formula: see text]. Based on discussion of one-dimensional continuous functions, fractal dimension, such as Box dimension, of Riemann–Liouville (R-L) fractional integral of those functions have been calculated. We get an important conclusion that Box dimension of R-L fractional integral of any one-dimensional continuous functions of any positive orders still is one. R-L fractional derivative of certain one-dimensional continuous functions has been explored elementary.

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

Fractals ◽  
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.

Curvature-dependent energies: a geometric and analytical approach

2017 ◽  
Vol 147 (3) ◽  
pp. 449-503 ◽  
Author(s):  
Emilio Acerbi ◽  
Domenico Mucci
Keyword(s):  
Bounded Variation ◽  
Analytical Approach ◽  
Total Curvature ◽  
Representation Formula ◽  
Energy Functional ◽  
Continuous Functions ◽  
Explicit Representation ◽  
Functions Of Bounded Variation ◽  
One Dimensional ◽  
Relaxed Energy

We consider the total curvature of graphs of curves in high-codimension Euclidean space. We introduce the corresponding relaxed energy functional and prove an explicit representation formula. In the case of continuous Cartesian curves, i.e. of graphs cu of continuous functions u on an interval, we show that the relaxed energy is finite if and only if the curve cu has bounded variation and finite total curvature. In this case, moreover, the total curvature does not depend on the Cantor part of the derivative of u. We treat the wider class of graphs of one-dimensional functions of bounded variation, and we prove that the relaxed energy is given by the sum of the length and total curvature of the new curve obtained by closing the holes in cu generated by jumps of u with vertical segments.

The Gaussian distribution revisited

1996 ◽  
Vol 28 (2) ◽  
pp. 500-524 ◽  
Author(s):  
Carlos E. Puente ◽  
Miguel M. López ◽  
Jorge E. Pinzón ◽  
José M. Angulo
Keyword(s):  
Fractal Dimension ◽  
Probability Measure ◽  
Gaussian Distribution ◽  
Closed Interval ◽  
Fractal Dimensions ◽  
Cumulative Distribution ◽  
Probability Measures ◽  
New Construction

A new construction of the Gaussian distribution is introduced and proven. The procedure consists of using fractal interpolating functions, with graphs having increasing fractal dimensions, to transform arbitrary continuous probability measures defined over a closed interval. Specifically, let X be any probability measure on the closed interval I with a continuous cumulative distribution. And let fΘ,D:I → R be a deterministic continuous fractal interpolating function, as introduced by Barnsley (1986), with parameters Θ and fractal dimension for its graph D. Then, the derived measure Y = fΘ,D(X) tends to a Gaussian for all parameters Θ such that D → 2, for all X. This result illustrates that plane-filling fractal interpolating functions are ‘intrinsically Gaussian'. It explains that close approximations to the Gaussian may be obtained transforming any continuous probability measure via a single nearly-plane filling fractal interpolator.

The Gaussian distribution revisited

1996 ◽  
Vol 28 (02) ◽  
pp. 500-524 ◽  
Author(s):  
Carlos E. Puente ◽  
Miguel M. López ◽  
Jorge E. Pinzón ◽  
José M. Angulo
Keyword(s):  
Fractal Dimension ◽  
Probability Measure ◽  
Gaussian Distribution ◽  
Closed Interval ◽  
Fractal Dimensions ◽  
Cumulative Distribution ◽  
Probability Measures ◽  
New Construction

A new construction of the Gaussian distribution is introduced and proven. The procedure consists of using fractal interpolating functions, with graphs having increasing fractal dimensions, to transform arbitrary continuous probability measures defined over a closed interval. Specifically, let X be any probability measure on the closed interval I with a continuous cumulative distribution. And let f Θ,D :I → R be a deterministic continuous fractal interpolating function, as introduced by Barnsley (1986), with parameters Θ and fractal dimension for its graph D. Then, the derived measure Y = f Θ,D (X) tends to a Gaussian for all parameters Θ such that D → 2, for all X. This result illustrates that plane-filling fractal interpolating functions are ‘intrinsically Gaussian'. It explains that close approximations to the Gaussian may be obtained transforming any continuous probability measure via a single nearly-plane filling fractal interpolator.

SOME REMARKS ON FRACTIONAL INTEGRAL OF ONE-DIMENSIONAL CONTINUOUS FUNCTIONS

Fractals ◽  
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.

FRACTAL DIMENSIONS OF WEYL–MARCHAUD FRACTIONAL DERIVATIVE OF CERTAIN ONE-DIMENSIONAL FUNCTIONS

Fractals ◽  
2019 ◽  
Vol 27 (07) ◽  
pp. 1950114
Author(s):  
Y. S. LIANG ◽  
N. LIU
Keyword(s):  
Fractional Derivative ◽  
Fractal Dimensions ◽  
Continuous Functions ◽  
Box Dimension ◽  
One Dimensional ◽  
Dimension One

Fractal dimensions of Weyl–Marchaud fractional derivative of certain continuous functions are investigated in this paper. Upper Box dimension of Weyl–Marchaud fractional derivative of certain continuous functions with Box dimension one has been proved to be no more than the sum of one and its order.

Classical Analysis

2019 ◽  
pp. 51-69
Author(s):  
John Stillwell
Keyword(s):  
Continuous Function ◽  
Continuous Functions ◽  
Uniform Continuity ◽  
Cantor Set ◽  
Binary Trees ◽  
Real Numbers ◽  
Limit Concept ◽  
Closed Intervals ◽  
Basic Concepts

This chapter explores the basic concepts that arise when real numbers and continuous functions are studied, particularly the limit concept and its use in proving properties of continuous functions. It gives proofs of the Bolzano–Weierstrass and Heine–Borel theorems, and the intermediate and extreme value theorems for continuous functions. Also, the chapter uses the Heine–Borel theorem to prove uniform continuity of continuous functions on closed intervals, and its consequence that any continuous function is Riemann integrable on closed intervals. In several of these proofs there is a construction by “infinite bisection,” which can be recast as an argument about binary trees. Here, the chapter uses the role of trees to construct an object—the so-called Cantor set.

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