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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Multifractal scalling exponent function \tau(q) for a Cauchy r.v.

The following function is called Multifractal scalling exponent function $\tau(q)$ $$ \tau(q)=\lim_{n\rightarrow\infty}\log_{n}\sum_{k=1}^{n}\left|\mathscr{H}\left(\left[\tfrac{k-1}{n},\tfrac{k}{n}\...
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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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Construct 0 Hausdorff dimensional sets s.t. there product has Hausdorff dimension 1

I'm trying to find sets $E,F\subset\mathbb{R}$ s.t. $\dim_H(E)=\dim_H(F)=0$ but $\dim_H(E\times{}F)=1$. This seems impossible. The only examples of sets with dimension $0$ I can think of are ...
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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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55 views

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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Optimization over the Koch snowflake

Let $C$ be the Koch snowflake and let $f:\mathbb{R}^2\to\mathbb{R}$ be a nice function, say analytic. Is there any hope of finding the extrema of $f$ along $C$? $C$ is nowhere differentiable and I ...
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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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Are there interesting general properties held by sets with small Hausdorff dimension?

Let us focus on the plane. I am wondering whether the sets with small (but non zero) Hausdorff dimension share some interesting properties. Let $\epsilon\in(0,1)$ and assume that a set $F\subset \...
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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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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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67 views

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} ...