Fractal analysis involves the study measures on fractals and their applications. This includes the study of self-similar integration and differential equations on fractals.
Questions tagged [fractal-analysis]
79 questions
24
votes
3 answers
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 equilateral triangle of side $L$). Assume the density…
Blex
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20
votes
2 answers
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 a method to integrate $e^x$ over some cantor sets,…
Zach466920
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10
votes
1 answer
"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
$$\int_C dx=1$$
for our funny, not-yet-well-defined…
Franklin Pezzuti Dyer
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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 integration is the Koch Curve, and $\mu(x,y)$ is…
Zach466920
- 8,419
9
votes
2 answers
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 simplify the process (So even a comparative layman like…
Zach466920
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8
votes
1 answer
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 where, for surface area A and cell size s at…
J Kelly
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7
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1 answer
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 Mandelbrot set, and more generally if we have means…
tox123
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6
votes
2 answers
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 solving it.
Carl Patenaude Poulin
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6
votes
1 answer
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:
$$
\int\limits_{A} f(Dx) \; dH^{m}(x) = \int\limits_{ D A} f(y) \;…
Appliqué
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6
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0 answers
Distance and Coordinates in fractional dimensions and the creation of functions with non-integral numbers of parameters.
Background: The Euclidean distance between two points in $n$ dimensions, where $n$ is a positive integer, and position can be described by a vector is given by...
$$D_E=\left(\sum_{k=1}^n \left((x_k)^2 \right) \right)^{1/2}$$
where $x_k$ is an entry…
Zach466920
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5
votes
0 answers
Are there fractal "manifolds"?
Differentiable manifolds are a generalization of the local geometry of Euclidean space. In fact, every differentiable manifold of dimension $m$ is locally diffeomorphic to the Euclidean space of the same dimension.
On the other hand, in general,…
Davius
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5
votes
1 answer
About moments of inertia, integrals and fractal dimensions.
As can be seen in the David Morin's Classical Mechanics, there are some scaling strategies in order to calculate the moments of inertia of certain fractals, for example, the Cantor's set has a moment of inertia of $I_C = \displaystyle…
Rafa
- 576
4
votes
1 answer
Moran's theorem Open set condition Koch Curve
I am currently studying some measure theoretic fractal geometry, and I am trying to learn how to use Moran theorem. The statement I currently have is:
Moran's theorem:
if $F_1,...,F_N : \mathbb{R}^d \to\mathbb{R}^d $ such that $|F_i(x) - F_i(y)| =…
Samael Manasseh
- 778
4
votes
1 answer
Why is it enough to consider limits as $\to 0$ through $\{_k\}$ such that $_{k+1} \ge c_k$ for some $0 < c < 1$ to find $\dim_B F$?
To set the stage, let me recall the definition of the box-counting dimension of a set $F \subset \mathbb R^n$.
The lower and upper box-counting dimensions of a subset $F$ of $ℝ^n$ are given by
$$\underline{\dim}_B F = \underline{\lim}_{\delta\to 0}…
stoic-santiago
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4
votes
0 answers
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 you could exploit it to integrate over a normalized…
Zach466920
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