Fractals exhibit similar patterns at increasingly small scales called self similarity, also known as expanding symmetry or unfolding symmetry. Each piece is split into 4 pieces each piece is smaller by factor of 3. We also prove that the similarity dimension is the same as the hausdor dimension of a fractal if the iterated function system associated to the fractal satis es the open set condition. Fractal dimension fd is the most important parameter of fractal geometry for it reveals the hierarchical relationship of fractal objects at different scales. This paper investigates the weierstrass type function in the padic local field k p whose graph is a repelling set of a discrete dynamical system, and proves that there exists a linear connection between the orders of the padic calculus and the dimensions of the. However, before this relationship is stated and proved, the necessary background needs to be provided. It is not hard to show that this series converges for all x. In mathematics, the weierstrass function is an example of a realvalued function that is continuous everywhere but differentiable nowhere.
Though there is a simple formula for the hausdorff dimension of the graph which is widely accepted, it has not been rigorously proved to hold. These novel ideas might be useful to establish the box. Pdf on the fractional integrals of the weierstrass functions. In mathematics, the weierstrass function is an example of a pathological realvalued function on the real line. Fractals and the weierstrassmandelbrot function rosehulman.
Novel approach to calculation of box dimension of fractal. A method to calculate fractal dimensions of the weierstrass. Parametric studies have been performed to analyze the dependencies between the contact properties and the scaleindependent fractal parameters. Massopust, in fractal functions, fractal surfaces, and wavelets second edition, 2016. First, the cuckoo search cs algorithm is adopted as an optimization algorithm to extract the. Indeed, by the mass distribution principle, it implies that the hausdorff dimension is at least d. The regularity of the test results of the weierstrass mandelbrot function isconsistent with this observation andthesystematic errorobserved infig. This paper first introduce the fractional integrals and the fractional derivatives of a fractal function.
Fractals and the weierstrassmandelbrot function pdf. Generalized weierstrass mandelbrot function model for actual stocks markets indexes with nonlinear characteristics article pdf available in fractals 232. The function has the property of being continuous everywhere but differentiable nowhere. On the hausdorff dimension of the graph of the weierstrass function.
On the box dimension of graph of harmonic functions on the sierpi. Moreover, since the partial sums are continuous as nite sums of continuous functions, their uniform limit fis also continuous. This type of fractal, which was described by an iteration. The graph of rew or im w has hausdorffbesicovitch fractal dimension d. The present paper investigates the fractal structure of fractional integrals of weierstrass functions. In this section we introduce the reader to commonly used fractal dimensions. Note that the weierstrass functions have the form where is. Estimation of the fractal dimension of weierstrass. A version of weierstrass function presented here is based on an infinite sum of cosine curves, and a general case is graph of c x against x, where in this case a 8 and b 0. Fractal dimensions and singularities of the weierstrass type functions tianyou hu and kasing lau abstract.
More generally, we consider the following function. A fractal is a shape that is selfsimilar small parts of a fractal look like the. Dynamical variation of weierstrassmandelbrot function in. Dimension of the graphs of the weierstrass type functions. The connections of our results with related objects known in the literature, including girgensohn functions, fractal interpolation functions and weierstrass functions, are established. Restrictions and extensions of continuous functions in regards to fractal dimensions are also investigated. If you are an experimentalist, you try to measure the fractal dimension of things in nature. Fractal dimension and singularities of weierstrass type. One of these is the fractal characterization of rough sur.
One of these is the fractal characterization of rough surfaces, which has gained ample acceptance in the area of contact mechanics. First, we discuss the similarity dimension, applicable to the class of. Fractals appear the same at different levels, as illustrated in successive magnifications of the mandelbrot set. We find that the fractal basin boundary is expressed by a superposition of the weierstrass function and the modified weierstrass function defined by eq. Fractal dimensions and singularities of the weierstrass. This function is weierstrassmadelbrot fractal function which is introduced as follow. In this paper, we investigate a generalized weierstrassmandelbrot function wmf model with two nonlinear characteristics. Like some fractals, the function exhibits selfsimilarity. Baranski, on the dimension of graphs of weierstrass type functions with rapidly growing frequencies, nonlinearity fractals and the w eierstrass mandelbrot function anthony zaleski 0 0 new jersey institute of technology, usa follow this and additional works at.
Introduction in real analysis, the classical weierstrass function is with. The topography of surface leads to imperfect contact which making the real contact area only a fraction of the nominal area , 14. In this chapter, the extendedboundarycondition method ebcm is employed with the weierstrass mandelbrot wm fractal function to model the surface and solve the electromagneticscattering problem. Fractal dimensions of graph of weierstrasstype function and local holder exponent spectra. Most researches focus on the systems as parameters fixed, such as calculations of its different fractal dimensions or the statistical characteristics of its generalized form and so on. Quantifying fractals stanley10 has outlined the program of the practicing scientist who wants to study fractals. It is also an example of a fourier series, a very important and fun type of series. Fractal dimensions and singularities of the weierstrass type. Citeseerx the hausdorff dimension of graphs of weierstrass. The exact box dimension for such functions many important cases is established. On the weierstrassmandelbrot fractal function proceedings. Limitations of fractal dimension estimation algorithms. Mandelbrot function wmf model with two nonlinear characteristics. Weierstrass function wikimili, the best wikipedia reader.
Fractal mathematics using the weierstrass mandelbrot wm function has spread to many fields of science and engineering. Many real complex phenomena are related with weierstrass mandelbrot function wmf. Mar 12, 2018 this paper aims to propose a semianalytical model to investigate the elasticplastic contact between fractal rough surfaces. Fractal modeling of elasticplastic contact between three. Calculation of fractal dimension chaos and timeseries analysis.
This is enough to conclude that the hausdorff dimension of the graph of f is d. On the other hand, it is easy to check that f is a \c2d\ function which implies that the hausdorff dimension is at most d see for example theorem 8. Karl weierstrass 1872 published a function equation 1 which graph was a fractal gure 1. We prove that there exists some linear connection between the order of the fractional calculus and the dimensions of the graphs of the weierstrass function, and we show this linear relation in figs. Mcdonald, di raction of fractal optical elds by simple apertures, in proceedings of the european optical society annual meeting, berlin, germany, 1519 september 2014 online. Basic theory the mathematical history of fractals began with mathematician karl weierstrass in 1872 who introduced a weierstrass function which is continuous everywhere but differentiable nowhere 4. A new type of fractal measures xs 1 function is introduced.
The weierstrass nowhere differentiable function, and functions constructed from similar infinite series, have been studied often as examples of functions whose. The weierstrass mandelbrot randomized function wmrf has been widely applied to characterize rough surface profiles with a specified fractal dimension d. The weierstrass nowhere differentiable function, and functions constructed from similar infinite series, have been studied often as examples of functions whose graph is a fractal. Request pdf estimation of the fractal dimension of weierstrass mandelbrot function based on cuckoo search methods the weierstrass mandelbrot randomized function wmrf has been widely applied. Fractals and the weierstrassmandelbrot function pdf paperity. So it is significant to propose a mathematical model with these characteristics. Fractals, hausdorff dimension, weierstrass function.
Dimension of the graphs of the weierstrass type functions mimuw. We need to point out that, although the result itself achieved in the present paper is interesting, the new technique and method should be emphasized. In this study, we proposed a novel approach to determine the d. Considerable effort has been put to cal culate the hausdorffbesicovitch dimension of. The dimension defined by this measure is closer to the hausdorff dimension than the other. He was preceded by bolzano in 1830, cellerier also 1830, and riemann 1862. Fractal dimensions of graph of weierstrasstype function and. Fractal dimensions of graph of weierstrasstype function. Built by replacing each end segment with a cross segment scaled by a factor of 5 12, consisting of 3 new segments, as illustrated in the inset images generated with fractal generator for imagej.
Oct 10, 2016 in this paper, we obtain the explicit value of the hausdorff dimension of the graphs of the classical weierstrass functions, by proving absolute continuity of the srb measures of the associated solenoidal attractors. Spectral analysis of the weierstrassmandelbrot function. On the connection between the order of fractional calculus. The hausdorff dimension of the weierstrass function. The weierstrass nowhere differentiable function has been studied often as example of functions whose graphs are fractals in. The following sequence shows the curve as an increasing number of cosine terms are added together. Citeseerx document details isaac councill, lee giles, pradeep teregowda. The weierstrass function math 104 proof of theorem. Novel approach to calculation of box dimension of fractal functions 7179 construct a graph of the weierstrass function, taking the parameters 0. Using the formalism of iterated function systems, we obtain a sharp result on the hausdorff dimension in terms of the corresponding fractal structures. Moreover, real systems always change according to different environments, so that to study the dynamical behavior of these. However, the question of determining the hausdorff dimension of the classical weierstrass function w.
The graph of functions and its box and hausdorff dimension. Apply the fractal approximation of weierstrass function. Nonanaliticity of w x is linked to the powerlaw behavior of its. Pdf we compute the hausdorff dimension of a twodimensional weierstrass function, related to lacunary hadamard gap power series, that has no l\evy. Di raction of weierstrass scalar fractal waves by circular. Our aim in the thesis is to estimate the fractal dimension of certain graphs related to the weierstrass functions. Fractal mathematics using the weierstrassmandelbrot wm function has spread to many. Pdf the present paper investigates the fractal structure of fractional integrals of weierstrass functions.
The weierstrass nowhere differentiable function, and functions constructed from similar infinite series, have been studied often as examples. A key point of the procedure is the property of the wm to be an almostperiodic function. Like many other fractals, the function exhibits selfsimilarity. The hausdorff dimension of graphs of weierstrass functions jstor.
A comparative study of fractal dimension calculation methods. The weierstrass graph graph of the weierstrass function. Pdf generalized weierstrassmandelbrot function model for. Exact spectral moments and differentiability of the. Fresnel di raction, fractal waves, weierstrass function. The fractal dimension of the weierstrass type functions. Fractal and hausdorff dimensions for systems of iterative. It exhibits fractal properties and the box dimension of its graph is d w 2 2 j lnb lna. In this paper we consider continuous real functions of the form f. None of the others published their work indeed, their examples were not discovered in their notes until after their deaths. The main aim of the present paper is to investigate the fractional derivatives of weierstrass function.
We begin with the celebrated weierstrass function 10,11. However, it is difficult to accurately determine the specified d for the wmrf. The weierstrass nowhere di erentiable function, and functions constructed from similar in nite series, have been studied often as examples of functions whose graph is a fractal. Pdf dimension preserving approximation semantic scholar. Dimension of the graphs of the weierstrasstype functions 5 let log logb 2 d. The weierstrassmandelbrot randomized function wmrf has been widely applied to characterize rough surface profiles with a specified fractal dimension d. In mathematics, a fractal is a selfsimilar subset of euclidean space whose fractal dimension strictly exceeds its topological dimension. A new type of fractal measures xs, 1 function is introduced. Last, we apply the theory we have developed to four examples. In 252627 28, the fractal fractional modeling and the weierstrass mandelbrot function have also proposed and can be used to improve the radar performance.
Fractal dimensions of graph of weierstrass type function and local holder exponent spectra. The following upper estimate of the box dimension of the graphs of the considered functions is a consequence of proposition 2. In real analysis, the classical weierstrass function is. This demonstration helps the user to visualize the jaggedness of an irregular line, time series, or signature by allowing its generation at different apparent fractal dimensions in the range 1 to 2 using the mandelbrot weierstrass function as the model. Fractal dimensions of graph of weierstrasstype function and local h. In this paper, we obtain the explicit value of the hausdorff dimension of the graphs of the classical weierstrass functions, by proving absolute continuity of the srb measures of the associated solenoidal attractors. Baranski, on the dimension of graphs of weierstrasstype functions with rapidly growing frequencies, nonlinearity fractals and the w eierstrass mandelbrot function anthony zaleski 0 0 new jersey institute of technology, usa follow this and additional works at. Though there is a simple formula for the hausdorff dimension of the graph which is widely accepted, it. Fractal basin boundary constructed by superposition of the. We propose a new method for calculating fractal dimension df of a signal yt, based on coefficients, mean absolute values of its nth order derivatives consecutive finite differences for sampled signals.
Skaigjx, where x 1, 0 function of order greater than a, it is shown that the dimension of the graph of w equals to 2 a, this conclusion is also equivalent to certain rate of the local. In the case of hiddenvariable fractal functions, there is an interesting relation between the box dimension of graph f k and the ambient space. On the dimension of the graph of the classical weierstrass function. Oct 11, 2017 this is enough to conclude that the hausdorff dimension of the graph of f is d. On the hausdorff dimension of the graph of a weierstrass type function. Hausdorff dimension of the graphs of the classical. Dimension of the graphs of the weierstrass type functions 5 let log logb 2 d.
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