Questions tagged [dimension-theory-analysis]
For questions about topological or metric notions of dimension, including the Lebesgue, small and large inductive, Hausdorff, and packing dimensions, etc. and for questions regarding fractal geometry and analysis. Use [linear-algebra] instead for questions about the dimension of vector spaces. Also note [dimension-theory-algebra].
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Embedded subshift in $[0,1]$ has dimension $0$?
I was wondering whether most embeddings of one-dimensional subshifts have zero Hausdorff dimension? Given a finite alphabet $\mathcal{A}= \{ 0,...,d-1 \}$ and $\Omega\subseteq \mathcal{A}^\mathbb{N}$, ...
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Is there a nice class of topological spaces satisfying the "finite sum theorem" for the small inductive dimension?
The small inductive dimension $\operatorname{ind}X\in\{-1,0,\ldots\,\infty\}$ of a topological space $X$ is defined inductively, as follows.
If $X=\varnothing$, then $\operatorname{ind}X=-1$.
If for ...
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Reverse Holder functions
For a function defined on $[0,1]\subset \mathbb{R}$, we call it reverse-Holder with exponent $\alpha >0$ if there exists some $c > 0$ such that
$$|f(x) - f(y)| \geq c|x-y|^\alpha. $$
The ...
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A 4-dimensional orange peeling problem: Can you disassemble a 4D hypersphere into a 3D object?
Where I am coming from: Similar to if you peel an orange, you cannot project the 2D surface of the Earth onto a 2D plane without curves. However, I’m pretty sure it’s possible to take the ...
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How to prove $V$ is a linear space and find $\dim V$?
Let $f(x)$ be a cubic real polynomial with a leading coefficient of $1$.
Let $\alpha$ be all the complex roots of $f(x)$.
$$V=\{y\,|\,y=g(\alpha),g(x) \text{ is a real polynomial}\}$$
Prove that $V$ ...
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Local dimension of stationary measures for iterated function systems with an expanding map
This question was previously posted on MathOverflow.
Consider the iterated function system (IFS) $X_n$ on $I = [0,1] $generated by the functions $\Phi = \{f_1,f_2,f_3\}$ and the probability vector $P =...
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Can we extend the box counting dimension to unbounded sets?
The box counting dimension of a set, $X$ in a metric space looks at boxes of dimension $\varepsilon$ and how many, $N(\varepsilon)$ are needed to covert $X$ and then says
$$dim_{box}(X) = \lim_{\...
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Proof of the convergence part of Jarník's Theorem
Let $\psi: \mathbb{N} \to \mathbb{R}^+$ be some function and denote by $W(\psi)$ the set of numbers $x \in [0,1]$ for which $$\left\lvert x - \frac{p}{q} \right\rvert < \frac{\psi(q)}{q} \quad \...
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Converting the volumetric flow rate, into 2D
Suppose there is a fluid flowing in a tube with some flow rate. Let's say: $10cm^3s^{-1}$.
Now suppose I'm going to model this problem in 2D, by assuming that the tube is a rectangle. I was wondering ...
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Question regarding the cohomological dimension of an open subset in a compact , connected, homogenous , metric ANR space .
$\mathbf {The \ Problem \ is}:$ Let, $X$ be a compact , connected homogeneous metric ANR (absolute neighbourhood retract) has cohomological dimension $\operatorname{dim}_G =n$ with respect to an ...
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Does there exist a function $f:\mathbb{R}\to\mathbb{R}$ where all points in the graph of f cannot be approximated by other points in $f$? [duplicate]
Does there exist an explicit function $f:\mathbb{R}\to\mathbb{R}$, such if $X$ is the graph of $f$ (i.e., $X=\left\{(x,f(x)):x\in\mathbb{R}\right\}$) then for each point $p$ on the graph of $f$, I ...
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Conditional Expectation of a mathematical function
Suppose $(X,d)$ is a metric space. If set $A\subseteq X$, let $H^{\alpha}$ be the $\alpha$-dimensional Hausdorff measure on $A$, where $\alpha\in[0,+\infty)$ and $\text{dim}_{\text{H}}(A)$ is the ...
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Question about Minkowski dimension
I'm learning Minkowski and Hausdorff dimensions to study Brownian motion right now, and I'm trying to understand the reasoning behind the Minkowski dimension of the set $(0,1,1/2, 1/3,\ldots)$ being $...
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What is the "content" of a box?
Let $(X,d)$ be a metric space. If set $A\subseteq X$, let $H^{\alpha}$ be the $\alpha$-dimensional Hausdorff measure on $A$, where $\alpha\in[0,2]$ and $\text{dim}_{\text{H}}(A)$ is the Hausdorff ...
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Intuition behind the Sum Theorem in dimension theory
In the Menger-Urysohn dimension, for any two subspaces $A, B$ of $X$, there is
$$
\operatorname{dim}(A\cup B)\leqslant \operatorname{dim}(A)+\operatorname{dim}(B)+1
$$
However, if $A, B$ are closed ...
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Link between Hausdorff measure of a set and Hausdorff measure of a sphere
I'm reading a proof given by Edward Szpilrajn found here and it says, among other things, that if $A$ is a non-empty set, and if $m\in\mathbb{N}$ are such that
$$(*)\quad \mathcal{H}^{m+1}(A) = 0 ,$$ ...
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Hausdorff Dimension of an Intersection
Let $E$ and $F$ be two non-empty subsets of $\mathbb{R}^n$ for some $n \in \mathbb{N}$.
I'm wondering if there's a simple formula or bound for the Hausdorff dimension of $E \cap F$ in terms of the ...
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Prove Ker$T=($Im$T^*)^\bot$ and (Ker$T^*$)$^\bot$ = Im$T$. Deduce $\dim$Im$T=\dim$Im$T^*$.
Let $V$ be a finite dimensional inner product space and $T:V\to V.$
Prove Ker$T=($Im($T^*))^\bot$ and $($Ker$T^*)^\bot$ = Im$T$.
Deduce dimIm$T$ = dimIm$T^*$.
I have written a proof but I'm very ...
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Will the upper bound $b=2$ work in a simplicial proof of Brouwer's Fixed Point Theorem, Hurewicz' and Wallman's "Dimension Theory"?
Background:
I am self studying Hurewicz' and Wallman's 'Dimension Theory'. My question concerns pp. 37-39 there within. The theorem A) below is equivalent to Brouwer's FPT, hence the title. I will ...
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Does the Box-Counting Dimension always exist after applying a Lipschitz mapping?
When you have a bi-Lipschitz mapping and the Box-Counting Dimension of $A$ is defined, then the Box-Counting Dimension of $f(A)$ ist also defined (and the same as the B.-Dim. of $A$).
We can also ...
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Marstrand's slicing theorem - lower bound and equality
Consider a fractal $\mathcal{S}\subset\mathbb{R}^2$ (e.g., a Koch snowflake) and a 1-dim line $\ell$. Marstrand's slicing theorem states that
$$dim_H(\mathcal{S}\cap\ell)\leqslant dim_H(\mathcal{S})-1$...
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Naïve definition of a measure on a fractal
Let $K\subset \mathbb R^2$ be a compact fractal of Hausdorff dimension $1<d<2$. I want to define a natural measure on $K$.
One option would be to use the so-called Hausdorff measure $\mathcal H^...
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hypersphere packing and Shannon capacity
Shannon claims that "We can also send at the
rate $C$ with arbitrarily small $\epsilon$" [1], where
$C=Wlog_2(\frac{P+N}{N})$
$C$ is the channel capacity in bits/second
W is bandwidth
P is ...
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Products of spaces of covering dimension zero
Recall that a space has covering dimension zero if every open cover of it can be refined to a disjoint open cover.
Question 1. Does the product of two spaces of covering dimension zero have covering ...
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Hausdorff dimension of the Koch curve
It is well known that the Hausdorff dimension of the Koch curve is $\log_34\approx1.26>1$. Thus by the property of Hausdorff measure, $\forall1<p<1.2$, the $p$-Hausdorff measure of the Koch ...
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How to calculation the Hausdorff dimension of this set?
It's about Exercise 1.19, Fractals in Probability and Analysis?
Suppose $S\subset\mathbb{N}$, and we are given $E,F\subset\{0,1,2\}$. Define $B_S=\{x=\sum_{k=1}^\infty x_k2^{-k}\}$ where $x_k\in E$ ...
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Box dimension of a perturbed set in the interval $[0,1]$
I've asked a couple questions about box dimensions (also called the Minkowski-Bouligand dimensions) recently. My main goal is to better understand the box dimension of discrete sets of points within ...
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How can I find the box dimension of the infinite union $\cup_{m=1}^\infty \{ \frac{1}{n^m}\mid n=1,2,3,\ldots \}$?
I recently learned of the concept of box dimension (also called the Minkowski-Bouligand dimension) as a way of measuring the dimension of a set (with use in describing the non-integer dimensions of ...
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Reference request: dimension of convex hull of n random points
Let $X$ be an i.i.d. random sample of $n$ points in $\mathbb{R}^d$ from a probability distribution that is absolutely continuous with respect to the $d$-dimensional Lebesgue measure. Assume $n>d$. ...
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How many 2D objects fit into a 3D object?
Hoe many times can you stack 2D objects before it becomes 3D?
I assume stacking 2-dimensional planes alone the 3rd dimension would never actually stack, as along the 3rd dimension, the 2-dimensional ...
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How can I find the box dimension of $\{ \frac{1}{n^m}\mid n=1,2,3,\ldots \}$ for a fixed $m$?
I recently learned of the concept of box dimension (also called the Minkowski-Bouligand dimension) as a way of measuring the dimension of a set (with use in describing the non-integer dimensions of ...
- 1,544
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Linear application on finite dimensional product space
Consider the set of probability measure $M(X\times Y)$ on $X\times Y$ with $X$ and $Y$ finite dimensional spaces (for example euclidean or discrete space).
Consider the application $f:M(X\times Y)\to \...
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Why does different nondimensionalizations give different results? Although the results should be the same.
I have some problems with the non-dimensionalization of the Hamiltonian of motion in a Coulomb field.
The Hamiltonian has a following form:
$$H=-\frac{\hbar^2}{2\mu^*} \Delta_r-\frac{e^2}{\epsilon_0 r}...
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How do I find the dimension of a space with a closed partition with empty interior?
I am not a mathematician (so please be kind :) but I need to understand this problem in relation to my work in a different field.
Let $\mathcal X,\mathcal Y,\mathcal Z$ be topological spaces with sets ...
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Definition of the Fourier dimension of a set and sets with zero Fourier dimension
Mattila defines in Fourier Analysis and Hausdorff Dimension the Fourier dimension of a set $A\subset\mathbb{R}^n$ to be
$\mathrm{dim_F}A = \sup\{s\leq n: \exists \mu \in \mathcal{M}(A):|\hat{\mu}(x)| ...
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What's the intuitive meaning of this relation between volumes of $n$-balls and umbral calculus?
The volume of an $n$-ball of radius $1$ is $$V_{n}={\frac {\pi ^{n/2}}{\Gamma {\bigl (}{\tfrac {n}{2}}+1{\bigr )}}}.$$
The functional equation of Riemann zeta function is $${\displaystyle \pi ^{-{s \...
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Inequality involving minimal cardinality of open subcovers
I am working on the following exercise, where I am not sure if the claim actually holds since I may have found a (simple) counter example. Maybe I am missing something?
Let $T \colon X \to X$ be a ...
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Krull dimension greater than or equal to small inductive dimension for Noetherian topological spaces
I am trying to prove Krull dimension and the small inductive dimension coincide for any Noetherian topological space $X$. The inequality Krull$(X) \le$ ind$(X)$ holds for all topological spaces. It is ...
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Does this space exist?
Does the space $[0,1]^r$ exist with $r\in\mathbb R$?
And the Hausdorff dimension is $r$
Thank you very much
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Why do almost all points in the unit interval have Kolmogorov complexity 1?
I am reading
Jin-yi Cai, Juris Hartmanis, On Hausdorff and topological dimensions of the Kolmogorov complexity of the real line, Journal of Computer and System Sciences, Volume 49, Issue 3, December ...
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Is the filled Sierpinski Curve (Gasket, Square...) a fractal? [closed]
This is regarding the area enclosed by the Sierspinski curve as shown white(!) in the added image. I get a Hausdorff dimension of this shape equal to 2, which is the same as the topological dimension. ...
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Question about Topological Manifolds
Suppose we have a topological manifold $M$ so by definition we have that,
(1) $M$ is Hausdorff
(2) $M$ is second countable
(3) $M$ is locally euclidean
I have a question but I don't know if it is the ...
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How do we know the hyperbolic plane is 2 dimensional?
Consider a disk in the hyperbolic plane with radius $R$, the area of this circle is given by $2 \pi \sinh(R) = 2\pi \frac{e^{R} - e^{-R}}{2}$
Usually an argument that the disk is 2-dimensional goes by ...
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Line element, metric tensor, integral and the sphere in fractional dimensions
I have a question regarding fractional calculus, namely, what is the line element "$$ds^2 = g^{\mu\nu} dx_\mu dx_\nu$$ in fractional "n" dimensions?
I am aware of some formulas that ...
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Help me make sense of t-Hausdorff measures
I am so confused about the definition of Hausdorff measures. In the book I'm following, it is defined as such:
Given our set F, we define the quantity $$H_{\delta}^t=\inf\{\sum_{i=1}^{\infty}|U_i|^t:\...
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Are there two computable numbers in the unit interval that map to the same point under a space-filling curve?
A space-filling plane curve is a continuous surjective function from the unit interval $[0,1]$ to the unit square. Netto's theorem gives that continuous bijections preserve dimension, so a space-...
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Counterexample to sum theorem for small inductive dimension ind?
Whereas the small inductive dimension $\operatorname{ind}(X)$ agrees with the covering dimension $\operatorname{cov}(X)$ when $X$ is a separable metrizable space, that is no longer the case for more ...
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Idea behind "distance measure"
From Mattila's Fourier Analysis and Hausdorff Dimension (page 59):
Theorem 4.6(a): If $A \subset \mathbb{R}^n$ is Borel and its dimension is greater than $(n+1)/2$, then the interior of $D(A):=\{\|x-...
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Calculating the Hausdorff dimension of a self-affine set in $\mathbb{R}^2$.
Consider the iterated function system defined by these two affine contractions:
$f_1:\mathbb{R}^2 \to \mathbb{R}^2$,
$f_2:\mathbb{R}^2 \to \mathbb{R}^2$,
$f_1 \left( x,y\right) = \left( \dfrac{x}{2},\...
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If $S$ is a convex subset of $\mathbb R^d$, then the Hausdorff dimension of the boundary of $S$ is at most $d-1$
Let $S$ be a subset of $\mathbb R^d$. Let $H^k$ be the $k$-dimensional Hausdorff measure and $\dim_H (S)$ the Hausdorff dimension of $S$, i.e.,
$$
\dim_H (S) := \inf \{k \in [0, +\infty) \mid H^k (S) =...
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