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Questions tagged [fractal-analysis]

Fractal analysis involves the study measures on fractals and their applications. This includes the study of self-similar integration and differential equations on fractals.

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Python/R library for fractal analysis

There are several methods for fractal/multifractal analysis such as DFA, MFDFA, Wavelet-leaders, power spectrum exponent, etc. I am going to analysis signals using these methods (or at least most ...
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Hölder exponent and its relation to multifractality

Suppose that for a discrete set of x=0,1,2,...,N, I have f(x). They show multi-fractality properties: $P_q = \sum_{x=1}^N f(x)^{2q} \propto N^{-\tau}$ $\alpha = \frac{d\tau}{dq}$ $f(\alpha) = q \...
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“Funny Integral” over the Cantor Set

I was thinking about integrals and how one might generalize them to be able to integrate over fractals rather than just over intervals. For example, consider the cantor set $C$. Let us assume that $$\...
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Does uniform distribution over the Sierpinsky triangle exist?

I'm learning measure theory, I'll make some statements I'm not very sure about. Please correct me if I'm wrong. This problem from a job ad states: Let $(X_1,X_2)$ be uniform over the unit ...
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Derivative of recursively defined fractal function

I have recently defined a function on the unit interval $[0,1]$ recursively. The definition is based on repeated subdivision of the interval in two equally-sized parts. The definition is as follows: ...
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Covariance matrix of uniform distribution over the Sierpinski triangle

Let $(X_1, X_2)$ be uniform over the unit Sierpinski triangle (represented in Cartesian coordinates). What is its covariance matrix? This is a question I saw in a jobs ad. I would love some leads on ...
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Why does my results of multifractal analysis seem wrong? [closed]

I use method of moment to calculate multifractal spectrum of an 2D image or grid data. Figure 1 is the distribution of the data.Figure 2 is my result of multifractal spectrum from the following Matlab ...
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Show that this mass distribution measure on the middle-third Cantor set satisfies this identity

For a mass distribution $\mu$ defined on the middle-third Cantor set (with left intervals $E_1$ assigned mass $P_1$ and right intervals masses of $P_2$ with the ratios kept this way throughout the ...
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A question involving decimal expansions

Let $E_k$ denote those numbers in $[0, 1]$ whose decimal expansions do not contain the digit 5 in the first $k$ decimal places. Can someone explain me why $E_k$ may be regarded as a union of $9^k$ ...
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Is there a particularly good function form for this curve?

This curve shape seems to appear in various natural phenomena: Do you recognize it? Do you know a specific function form that could match it or approximate it closely?
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Please, explain me the ambiguity on this definition

Here is a definition on a paper named "Calculus on fractal subsets of real line" Page 4: Definition 2 A subdivision $P_{[a,b]}$ (or just P) of the interval $[a,b], a<b$ is a finite set of points $...
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Integral over Julia Set (Is my math correct?)

So I was answering this question about whether or not the Julia Set was self-similar in a known way. Of course it is, and that got me thinking. Even though the self similarity is nonlinear, what if ...
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Change of Variables for Integral over a Fractal

We define the integral of a function $f(x,y)$ over a fractal $F$ to be, $$(1) \quad \int_F f(x,y) \ d\mu(x,y)$$ Where $\mu$ is the normalized Hausdorff measure. Expressed another way, we have, $$\...
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Can the fractal dimension of a surface be less than 2?

I have two surfaces represented as raster images with heights as grayscale values. One is natural landscape elevations; the other is just distance from a line. I have computed Minkowsky D = 2 - H ...
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About moments of inertia, integrals and fractal dimensions.

As can be seen in the David Morin's Classical Mechanics, there are some scaling strategies in order to calculate the moments of inertia of certain fractals, for example, the Cantor's set has a moment ...
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What is asymptotics of this oscillatory double sum? (Fractal Dimension problem)

The term Gibbs Phenomenon refers to the peculiar way Fourier Series behave at sharp changes in a function's value. However, this problem becomes particularly annoying to deal with when trying to ...
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Fractal signal analysis

What kind of results can be proven about continuous functions $f: \mathbb{R} \rightarrow \mathbb{R}$ which present some sort of fractal behaviour / self-similarity? Do you have some textbook ...
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Calculate moment of inertia of Koch snowflake

That's just a fun question. Please, be creative. Suppose having a Koch snowflake. The area inside this curve is having the total mass $M$ and the length of the first iteration is $L$ (a simple ...
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Integral of a function over the Koch Curve. Is it rigourous enough?

(I want to investigate the validity of this approach, as I already know this is the correct result) I present a proof that $$\int_{K} (x+y) \ \mu(x,y)={{9+\sqrt 3} \over 18}$$ Where the region of ...
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What is the integral of $e^{a \cdot x+b \cdot y}$ evaluated over the Koch Curve

What is $$\int_{K} e^{a \cdot x+ b \cdot y} \mu(x,y)$$ where $K$ is the Koch curve and $\mu(x,y)$ is a uniform measure look here. Attempt: I can evaluate the integral numerically and I have derived ...
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This one weird trick integrates fractals. But does it deliver the correct results?

It occurs to me that people most likely already know how to explicitly integrate over fractals, but my method (edit: seems to have been highlighted out in a paper, see comments) seems to vastly ...
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Distance and Coordinates in fractional dimensions and the creation of functions with non-integral numbers of paramters.

Background: The Euclidean distance between two points in $n$ dimensions, where $n$ is a positive integer, and position can be described by a vector is given by... $$D_E=\left(\sum_{k=1}^n \left((x_k)^...
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Change of Variables for Hausdorff Measure

(Read bounty text for answering question) Let $H^{m}$ be the $m$-dimensional Hausdorff measure. Let $D$ be a linear transformation matrix. Consider the change of measure formula: $$ \int\limits_{A} ...