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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How to determine the fractal dimension of a random walk in 2D?

I have to determine the fractal dimension of a random walk in 2D. How can I do that? I am first supposed to picture the random walk for 2D as in the following site. However, I am also asked to ...
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Inequality on the uniform norm of a conformal differentiable transformation

I'm currently working on a question on chaotic dynamics on fractals. Particularly if we have an iterated function system of conformal differentiable contraction mappings {$f_1$,...,$f_n$} we denote ...
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Induction over Rham piecewise linear function

The following exercise is inspired in Daubechies - Lagarias: Two-Scale Difference Equations, Existence and Global Regularity of Solutions. Define the following functions: $$f_0(x)= \left\{ \begin{...
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Newton fractals with nonzero measure?

The german wikipedia article about Newton fractals currently contains the following claim: Überraschenderweise kann die Julia-Menge (das Newtonfraktal) auch positives Maß in der Ebene haben: das ...
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Calculating the fractal dimension of a Romanesco Broccoli

It has been quite a while since I've had some introductions to fractal geometry, but only on special cases, e.g., the Sierpinski triangle and other self similar fractals. Now I have challenged myself ...
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What is the dimension of the polar fractal $\mathrm{r=\sum_{n=1}^\infty \frac{sgn(sec(n\theta))}{n}}$?

Here is what the fractal looks like on desmos; you can change the width of the function for more detail desmos fractal This is the inspiration for the question. I will use this inverse tangent ...
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Determining the fractal dimension of the series $\sum_{n=1}^{\infty} \sin{(n^2x)^2/n^3}$

I've been trying to determine the fractal dimension of the infinite sine series $f\sum_{n=1}^{\infty} \frac{\sin{(n^2x)^2}}{n^3} $. OWN ATTEMPTS: My first attempt was to attempt to show that it is NOT ...
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Is there an operator valued mandelbrot set?

Here's a thing I literally dreamed of last night :D Let $B=\mathcal{B}(\ell^2(\mathbb{N}))$, or for that matter some other unital C*-algebra. Let $M_B\subset B$ be the set of operators $c\in B$ for ...
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Is it possible to solve differential equations on a fractal?

If we let $M$ be the Mandelbrot set on $\mathbb{R}^2$ (specifically the set of points $(x,y)$ such that $x+yi\in M$). I was wondering what happens if we have a ideal drum whose shape is that of the ...
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Usage of Egoroff's Theorem in Fractal Geometry

While reading Fractal Geometry by K.Falconer, I have came across a few applications of the Egoroffs Theorem in some proofs. But in none of them, the theorem is used in any easy format. Below I mention ...
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Koch snowflake is nowhere differentiable

In stein's real analysis, Prove that the von Koch curve $t \mapsto \mathcal{K}^\ell(t), 1/4 < \ell \le 1/2$ is continuous but nowhere differentiable. [Hint: If $\mathcal{K}'(t)$ exists for some $t$...
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Are there any optimized root to tip space-filling fractal trees?

I've been looking for fractals optimized for specific purposes, and one of the simplest is a space-filling tree that minimizes the distance needed to travel from the root to any one of the tips/ends. ...
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Why is it enough to consider limits as $𝛿\to 0$ through $\{𝛿_k\}$ such that $𝛿_{k+1} \ge c𝛿_k$ for some $0 < c < 1$ to find $\dim_B F$?

To set the stage, let me recall the definition of the box-counting dimension of a set $F \subset \mathbb R^n$. The lower and upper box-counting dimensions of a subset $F$ of $ℝ^n$ are given by $$\...
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Does every point in the Koch curve belong to one of its approximating polygons?

So, pretty self-explanatory title: I was wondering wether if every point that the Koch curve has does necesarilly belong to one of its approximating polygons. Just in case, on the picture below, the ...
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Box-counting dimension: $\lim_{\delta \to 0} N_{\delta}(F)\delta^{s} = \infty$ when $s < \dim_{M}(F)$ and another similar limit.

I want to understand these two limits: $\lim_{\delta \to 0} N_{\delta}(F)\delta^{s} = \infty$ when $s < \dim_{M}(F)$ $\lim_{\delta \to 0} N_{\delta}(F)\delta^{s} = 0$ when $s > \dim_{M}(F)$ ...
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Do fractals necessarily have to be self-similar? What is the definition of a fractal? Is this figure a fractal? [duplicate]

I am doing a dissertation in Materials Engineering. I obtained the following images regarding crystal growth: In the literature they usually call these structures "fractals" and calculate ...
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Integration of function using Hausdorff measure of Cantor Set

I am learning geometry on fractal shapes, along with how fractional calculus can relate to said geometry. At the moment I am trying to understand integration over a Hausdorff measure. According to ...
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How to calculate Assouad Dimension of sets

I am trying to work with sets $F$ as subsets of $\mathbb{R}^n$ and their Assouad dimension. I am a bit stuck on how to apply the definition as it asks that We find the infimum of possible $\alpha$ We ...
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Can the fractal dimension of a coastline be less than 1?

I am currently investigating the fractal dimension of the coastline of the island the Palm Jumeirah, including the crescent. Using the Hausdorff method I have reached an answer of 0.879. This is less ...
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Box-counting dimension (Minkowski Dimension) of {1, 1/4, 1/9, 1/16,...}

I want to determine the box-counting dimension (Minkowski Dimension) of the set $S=\{\frac{1}{n^2} \ | \ n \in \mathbb{Z}^+\}$. My first attempt was to first define my $\delta_n$. What I did was ...
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What fraction of a Sierpiński triangle is to the left of a given vertical line?

Let's say that we have a Sierpiński triangle of width 1 and height 1 I want to find the function $y(x)$ for $0<x<0.5$ that gives the fraction of the Sierpiński triangle contained in $[0,x]$ If ...
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Does the Mandelbrot set get smoother because of a rounding error

On many fractal explorer programs, after a certain level of zoom, the shapes begin to become smoother, as is shown. Is this a rounding error or an actual representation of the computation?
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what shape does this function $\delta_{x}(x)$ look like if it's a fractal what is it's fractal dimension?

what shape does this function $\delta_{x}(x)$ look like if it's a fractal what is its fractal dimension? so $\alpha$ is a real number and $\delta_{\alpha}(x)$ is a function based of $\alpha$ written ...
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Properties of the Cantor set constructed by removing just the middle point

Context: I am currently doing some exercises on the middle $\lambda$ Cantor set $C_\lambda$ (construction is similar to the usual Cantor set, but we remove the middle $\lambda$ proportion of the ...
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A set with infinity Hausdorff measure, but Hausdorff dimension $\frac{\log2}{\log3}$

I am going through the text 'Fractal Geometry: Mathematical Foundations and Applications' and came the following exercise: Give a set $B\subset\mathbb{R}$ that has Hausdorff dimension $s=\frac{\log2}{...
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Hausdorff measure of Cantor set

I am trying to find a set $A\subset\mathbb{R}$ with Hausdorff dimension $\log2/\log3=:s$ but has $H^s(A)=\infty$. I suspect this is the Cantor set, but im struggling to show that it has Hausdorff ...
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The Hausdorff measure of the unit interval

I am trying to calculate the Hausdorff measure of the unit interval. Here's my attempt: Fix $\epsilon>0$. consider the open balls $B(x,\epsilon)$ with $x\in[0,1]$. How many can cover the unit ...
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Can we cover a self similar set using another self similar set with smaller dimension?

Suppose we have two self similar sets $A$ and $B$ constructed as follows: Let $\varphi_i(x):=\frac{x+i}{5},i=0,1,2,3,4$ be similarities, then $A$ and $B$ are self similar sets with: $$A=\varphi_0(A)\...
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Showing that a Hölder map is also Hölder with a smaller Hölder parameter

Let $X\subset{\mathbb{R}^d}$ be bounded and let $f:X\rightarrow{}\mathbb{R}^d$ be an $\alpha$-Hölder map, that is to say: $$\|f(x)-f(y)\|_2\le{}C\|x-y\|_2^{\alpha},\text{ for some }\alpha>0\text{ ...
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Fractal construction and IFS (iterated function system): defining a specific construction with IFS

The setting: We consider the plane $\mathbb R^2$ with its canonical euclidean structure. The canonic base is written $(e_1,e_2) = ((1,0)^\top,(0,1)^\top)$. Let's consider $I = [a,b]e_1$ (with $a<b$...
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Are there two sets $X$ and $Y$ such that the following inequality for box dimension holds

I am trying to find sets $X$ and $Y$ s.t. $\dim_B(X\bigcup{}Y)>\max\{\dim_B(X),\dim_B(Y)\}$. At first I thought taking $X=[0,1]$ and $Y=\{10+1/n^2:n\ge{}1\}$ but I don't think that works. Is this ...
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Showing the map from the middle $\lambda$ cantor set to the $\nu$ cantor set is $\gamma$-hölder continuous

Let $C_\lambda$ and $C_\nu$ be the middle $\lambda$ and $\nu$ cantor sets, respectively. I want to show the map $\Pi_{\lambda,\nu}:C_\lambda\rightarrow{C_\nu}$ is $\gamma$-hölder continuous, with $\...
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Why the use of the term "snowflaking"?

I've seen a few places in the literature (in particular in fractal geometry) where we consider a metric space $(X,d)$, and then for $0 < \epsilon < 1$ define a new metric space $(X,d^{\epsilon})$...
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Hausdorff dimension for bounded set

Let $F$ be the set of of numbers $x\in [0,1]$ with base 3 expansions $0.a_1a_2...$ for which there exists an integer k such that $a_i\neq 1$ for all $i\geq k$. Find the Hausdorff dimension of $F$. ...
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How to find the box counting dimension of line segment [0,1]?

I uploaded this question and no one has answered because there were some flaws in the question but I have the text now. Please can some one explain how we arrive on example of line segment to have a ...
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Box counting dimension of line segment [0,1]?

I am not understanding this example from Kenneth Falconers book on Fractal geometry. can someone please explain how the inequalities follows from the definitions in the example given?
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Hausdorff dimension of the relative complement of a set contained in a $G_\delta$

We have the following result: Every set is contained in a $G_\delta$ set of the same Hausdorff dimension I was wondering how tight can this inclusion be made, complement-wise. Is true that: Let ...
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10 votes
1 answer
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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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2 votes
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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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4 votes
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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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1 vote
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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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2 votes
1 answer
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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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4 votes
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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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8 votes
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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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