Fractals and the weierstrassmandelbrot function pdf. The exact box dimension for such functions many important cases is established. Fractal mathematics using the weierstrass mandelbrot wm function has spread to many fields of science and engineering. A comparative study of fractal dimension calculation methods. Quantifying fractals stanley10 has outlined the program of the practicing scientist who wants to study fractals. Like some fractals, the function exhibits selfsimilarity. Pdf on the fractional integrals of the weierstrass functions. He was preceded by bolzano in 1830, cellerier also 1830, and riemann 1862. Weierstrass functions are famous for being continuous everywhere, but differentiable nowhere. A new type of fractal measures xs, 1 function is introduced. The regularity of the test results of the weierstrass mandelbrot function isconsistent with this observation andthesystematic errorobserved infig. These novel ideas might be useful to establish the box. It is named after its discoverer karl weierstrass the weierstrass function has historically served the role of a pathological function, being the first published example 1872 specifically concocted to challenge the. Many real complex phenomena are related with weierstrass mandelbrot function wmf.
The topography of surface leads to imperfect contact which making the real contact area only a fraction of the nominal area , 14. Fractal basin boundary constructed by superposition of the. 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. Karl weierstrass 1872 published a function equation 1 which graph was a fractal gure 1. Nonanaliticity of w x is linked to the powerlaw behavior of its. It is also an example of a fourier series, a very important and fun type of series. The function has the property of being continuous everywhere but differentiable nowhere. On the hausdorff dimension of the graph of the weierstrass function. 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.
One of these is the fractal characterization of rough surfaces, which has gained ample acceptance in the area of contact mechanics. In the case of hiddenvariable fractal functions, there is an interesting relation between the box dimension of graph f k and the ambient space. The fractal dimension of the weierstrass type functions. The weierstrass nowhere differentiable function, and functions constructed from similar infinite series, have been studied often as examples. The following sequence shows the curve as an increasing number of cosine terms are added together. Fractal dimensions and singularities of the weierstrass type. We begin with the celebrated weierstrass function 10,11. Moreover, real systems always change according to different environments, so that to study the dynamical behavior of these. Diffraction of weierstrass scalar fractal waves by circular. Fractals exhibit similar patterns at increasingly small scales called self similarity, also known as expanding symmetry or unfolding symmetry. It is not hard to show that this series converges for all x.
Fractal dimensions of graph of weierstrasstype function and local h. However, the question of determining the hausdorff dimension of the classical weierstrass function w. In this section we introduce the reader to commonly used fractal dimensions. Dimension of the graphs of the weierstrass type functions mimuw. Di raction of weierstrass scalar fractal waves by circular. A new type of fractal measures xs 1 function is introduced. Fractals appear the same at different levels, as illustrated in successive magnifications of the mandelbrot set. The weierstrass nowhere differentiable function has been studied often as example of functions whose graphs are fractals in. Parametric studies have been performed to analyze the dependencies between the contact properties and the scaleindependent fractal parameters. 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. In mathematics, the weierstrass function is an example of a realvalued function that is continuous everywhere but differentiable nowhere. First, we discuss the similarity dimension, applicable to the class of. On the hausdorff dimension of the graph of a weierstrass type function. Dimension of the graphs of the weierstrasstype functions 5 let log logb 2 d.
It exhibits fractal properties and the box dimension of its graph is d w 2 2 j lnb lna. This function is weierstrassmadelbrot fractal function which is introduced as follow. The weierstrass function math 104 proof of theorem. He presented a function, to the prussian academy of sciences in 1872, which is continuous everywhere but di erentiable nowhere. Dimension of the graphs of the weierstrass type functions. Fractals, hausdorff dimension, weierstrass function. Mar 12, 2018 this paper aims to propose a semianalytical model to investigate the elasticplastic contact between fractal rough surfaces. The weierstrass graph graph of the weierstrass function. This type of fractal, which was described by an iteration.
Indeed, by the mass distribution principle, it implies that the hausdorff dimension is at least d. 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. So it is significant to propose a mathematical model with these characteristics. Though there is a simple formula for the hausdorff dimension of the graph which is widely accepted, it.
However, before this relationship is stated and proved, the necessary background needs to be provided. Last, we apply the theory we have developed to four examples. Fresnel di raction, fractal waves, weierstrass function. On the dimension of the graph of the classical weierstrass function.
Pdf the present paper investigates the fractal structure of fractional integrals of weierstrass functions. Fractal dimensions of graph of weierstrasstype function. On the connection between the order of fractional calculus. 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. In this paper we consider continuous real functions of the form f. In this paper, we investigate a generalized weierstrassmandelbrot function wmf model with two nonlinear characteristics. 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. Fractal dimensions of graph of weierstrasstype function and. Dynamical variation of weierstrassmandelbrot function in. Fractal dimensions and singularities of the weierstrass type functions tianyou hu and kasing lau abstract. Note that the weierstrass functions have the form where is. Introduction in real analysis, the classical weierstrass function is with. A method to calculate fractal dimensions of the weierstrass.
Like many other fractals, the function exhibits selfsimilarity. In this study, we proposed a novel approach to determine the d. On the weierstrassmandelbrot fractal function proceedings. Weierstrass function wikimili, the best wikipedia reader. 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. 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.
Our aim in the thesis is to estimate the fractal dimension of certain graphs related to the weierstrass functions. The graph of rew or im w has hausdorffbesicovitch fractal dimension d. Fractal dimensions of graph of weierstrasstype function and local holder exponent spectra. In real analysis, the classical weierstrass function is. Citeseerx document details isaac councill, lee giles, pradeep teregowda.
Restrictions and extensions of continuous functions in regards to fractal dimensions are also investigated. We find that the fractal basin boundary is expressed by a superposition of the weierstrass function and the modified weierstrass function defined by eq. 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. Each piece is split into 4 pieces each piece is smaller by factor of 3. However, it is difficult to accurately determine the specified d for the wmrf. Hausdor dimension of weierstrass graphs tobias j ager 4th bremen winter school dynamics, chaos and applications 1418 march 2016. 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. Citeseerx the hausdorff dimension of graphs of weierstrass. The hausdorff dimension of graphs of weierstrass functions. The weierstrassmandelbrot randomized function wmrf has been widely applied to characterize rough surface profiles with a specified fractal dimension d. This paper first introduce the fractional integrals and the fractional derivatives of a fractal function. The main aim of the present paper is to investigate the fractional derivatives of weierstrass function. Novel approach to calculation of box dimension of fractal.
Fractal modeling of elasticplastic contact between three. 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. A key point of the procedure is the property of the wm to be an almostperiodic function. Generalized weierstrass mandelbrot function model for actual stocks markets indexes with nonlinear characteristics article pdf available in fractals 232. 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. Fractal dimensions of graph of weierstrass type function and local holder exponent spectra.
Massopust, in fractal functions, fractal surfaces, and wavelets second edition, 2016. Spectral analysis of the weierstrassmandelbrot function. Limitations of fractal dimension estimation algorithms. A fractal is a shape that is selfsimilar small parts of a fractal look like the.
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. Fractal mathematics using the weierstrassmandelbrot wm function has spread to many. Fractal and hausdorff dimensions for systems of iterative. 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. Considerable effort has been put to cal culate the hausdorffbesicovitch dimension of. Fractal dimension and singularities of weierstrass type. Hausdorff dimension of the graphs of the classical.
Novel approach to calculation of box dimension of fractal functions 7179 construct a graph of the weierstrass function, taking the parameters 0. In mathematics, a fractal is a selfsimilar subset of euclidean space whose fractal dimension strictly exceeds its topological dimension. 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. On the box dimension of graph of harmonic functions on the sierpi. 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. 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. Moreover, since the partial sums are continuous as nite sums of continuous functions, their uniform limit fis also continuous. The connections of our results with related objects known in the literature, including girgensohn functions, fractal interpolation functions and weierstrass functions, are established. Pdf generalized weierstrassmandelbrot function model. Fractal dimension fd is the most important parameter of fractal geometry for it reveals the hierarchical relationship of fractal objects at different scales. Estimation of the fractal dimension of weierstrass. The dimension defined by this measure is closer to the hausdorff dimension than the other. Pdf generalized weierstrassmandelbrot function model for.
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. The hausdorff dimension of the weierstrass function. First, the cuckoo search cs algorithm is adopted as an optimization algorithm to extract the. This is enough to conclude that the hausdorff dimension of the graph of f is d. Pdf dimension preserving approximation semantic scholar. The graph of functions and its box and hausdorff dimension. The weierstrass mandelbrot randomized function wmrf has been widely applied to characterize rough surface profiles with a specified fractal dimension d. 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. More generally, we consider the following function. One of these is the fractal characterization of rough sur. If you are an experimentalist, you try to measure the fractal dimension of things in nature. Guidelines of the ebcm have been given in chapter 4. Apply the fractal approximation of weierstrass function. Fractal dimensions and singularities of the weierstrass.
The present paper investigates the fractal structure of fractional integrals of weierstrass functions. Calculation of fractal dimension chaos and timeseries analysis. Fractals and the weierstrassmandelbrot function pdf paperity. In mathematics, the weierstrass function is an example of a pathological realvalued function on the real line. Exact spectral moments and differentiability of the. Dimension of the graphs of the weierstrass type functions 5 let log logb 2 d. The following upper estimate of the box dimension of the graphs of the considered functions is a consequence of proposition 2. 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. Mandelbrot function wmf model with two nonlinear characteristics. 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.
Using the formalism of iterated function systems, we obtain a sharp result on the hausdorff dimension in terms of the corresponding fractal structures. None of the others published their work indeed, their examples were not discovered in their notes until after their deaths. The hausdorff dimension of graphs of weierstrass functions jstor. Fractals and the weierstrassmandelbrot function rosehulman. Oct 11, 2017 this is enough to conclude that the hausdorff dimension of the graph of f is d. The weierstrass nowhere differentiable function, and functions constructed from similar infinite series, have been studied often as examples of functions whose. Pdf we compute the hausdorff dimension of a twodimensional weierstrass function, related to lacunary hadamard gap power series, that has no l\evy. For the weierstrass type functions defined by wx aaigaix, where a 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 oscillation of the function. In 252627 28, the fractal fractional modeling and the weierstrass mandelbrot function have also proposed and can be used to improve the radar performance. 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.
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