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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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Relation between local homeomorphism and homological dimension

For a given topological space $X$ (one can assume a simplicial complex if required), define it's homological dimension $\operatorname{hdim}(X)$ as the largest integer $n$ such that $H_n(X,A)\ne 0$ for ...
Rainy's user avatar
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41 views

Regarding the Boundary of Bounded Non-empty Open Sets of $\mathbb{R}^n$

I tend to imagine bounded open sets of $\mathbb{R}^n$ as collections of nice slightly punctured blobs so I came up with these questions to check whether my biases are actually correct. If $\emptyset\...
wakewi's user avatar
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-1 votes
0 answers
57 views

Can we find numerosities of fractals?

Numerosity is a measure of a set that conforms to the Euclid's principle (the whole is begger than a part). It also can be seen as a generalization of Lebesgue and Housdorff measures, where not only ...
Anixx's user avatar
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1 vote
1 answer
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Are there compact metric spaces with Hausdorff measure equal to 0 or infinty

I am looking for examples of non-empty metric spaces which are compact with Hausdorff dimension $\alpha$, and have either its $\alpha$-dimensional Hausdorff measure equal to $0$ or its $\alpha$-...
Cosine's user avatar
  • 412
2 votes
1 answer
34 views

What does it mean to be $(M, s)$-homogeneous?

I'm trying to get a handle on the Assouad dimension and Wikipedia says that for a metric space $(X, \zeta)$, The Assouad dimension of $X$, $d_A(X)$, is the infimum of all $s$ such that $(X, \zeta)$ ...
roundsquare's user avatar
  • 1,575
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0 answers
28 views

Hausdorff dimension if Hausdorff measure is infinity

I am reading the article "Cyclic Functions in Lp(R), 1≤p<∞" from Joseph M. Rosenblatt and Karen L. Shuman. In Theorem 5 they calculate the Hausdorff dimension of the set E, which is a ...
tf101097's user avatar
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41 views

What is the dimension of this set of points?

The following set of points in $\mathbb{R}^n$ is full-dimensional ($n$-dimensional): $$\{(x_1,\ldots,x_n)| 0\leq x_i \leq 1 \text{ for all }i\in[n] \}$$ What is the dimension of the following set - is ...
Erel Segal-Halevi's user avatar
1 vote
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15 views

Hausdorff dimension of neighbourhood of Jordan domain

Let $D \subseteq \mathbb{R}^2$ be a bounded domain such that $\partial D$ is a Jordan curve, and for $\varepsilon > 0$ let $D^\varepsilon = \{ x \in \mathbb{R}^2 : \text{dist}(x,D) < \varepsilon ...
Julius's user avatar
  • 1,633
1 vote
1 answer
59 views

Iterated function system with a fractional number of contractions

Iterated function systems can be used to generate fractals. One starts off with a simple geometric figure and applies the IFS infinitely many times to obtain a fractal. For example, in the case of the ...
Artur Wiadrowski's user avatar
5 votes
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159 views

Does $X \times \mathbb{R} \simeq X$ hold for infinite dimension inner product space $X$?

The $X$ is infinite dimension real inner product space, not restricting it as a Hilbert space. This question has troubled my friend and me a lot of days. It is obviously true when $X$ is infinite ...
CTuser_103's user avatar
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16 views

Properties of disjoint n-intervals whose finite union is n-interval

First let's define what an n-interval is: $$I = \left[a_1, ..., b_1\right] \times ... \times \left[a_n, ..., b_n\right] \subseteq \mathbb{R}^n$$ Now suppose we have $I_1, \dots, I_m$ n-intervals with ...
A.Lugini's user avatar
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0 answers
52 views

Can a finite dimensional topological space have no subspace with finite Lebesgue covering dimension

I wonder if it's possible for a topological space $(M, \tau)$ where $\tau$ is the collection of all open subsets of $M$, to have a finite Lebesgue covering dimension, while having no proper subspace ...
Bastam Tajik's user avatar
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41 views

Calculating the Hausdorff Dimension of specific profinite groups

I'm trying to understand how to calculate the Hausdorff dimension of some profinite groups and came across the following paper (here) which defines the Hausdorff dimension for a closed subgroup $H \...
Flo's user avatar
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1 vote
1 answer
51 views

Zero-dimensional space with multiple objects

I am unsure if this belongs to math or philosophy. Let's say there's 0-dimensional space, however multiple objects exist within in, occupying the same "spot". If multiple objects exist, is ...
SigTerm's user avatar
  • 115
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58 views

Locally bi-Lipschitz map preserves Hausdorff dimension

I was able to prove that bi-Lipschitz function $f: \mathbb R^n \rightarrow \mathbb R^m$ preserves Hausdorff dimension. In particular, the following lemma was useful: Let $E \subseteq \mathbb R^k$ be ...
Luke's user avatar
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1 vote
1 answer
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Example of covering dimension of a finite space

Consider $X=\lbrace 1,2,3\rbrace\newcommand{\dim}{\operatorname{dim}}$ endowed with the topology $T$ generated by $\lbrace \lbrace 1,2\rbrace,\lbrace 2,3\rbrace\rbrace$. It is claimed (in Exercise 2....
user21251713's user avatar
1 vote
0 answers
64 views

On the equivalence of two definitions of cohomological dimension for locally compact topological spaces.

$\mathbf{The \ Problem \ is}:$ Let $X$ is a locally compact, separable metric space. Let $G$ be an abelian group. Now I came across two definitions of cohomological dimension of $X.$ One is the usual ...
Rabi Kumar Chakraborty's user avatar
0 votes
0 answers
45 views

How do I prove that a connected Manifold of dimension m can't have another dimension? [duplicate]

So I've started studying differentiable manifolds, and I came across the following problem: Problem: Assume that $\mathbb{R}^n$ and $\mathbb{R}^m$ are homeomorphic iff $m=n$. Let $M$ be a connected ...
Bruno Dias's user avatar
2 votes
0 answers
42 views

Is this the small (or large) inductive dimension?

I recently learned about the inductive dimension, but the formal definition is sometimes still a bit inaccessible. I have through experimentation however come upon my own definition for dimension ...
Carlyle's user avatar
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6 votes
1 answer
131 views

Is the Hausdorff measure of this disk equal to 0?

Consider a topological space $D^n$ homeomorphic to the closed unit ball of $\mathbb{R}^n$. Suppose that $D^n$ is equipped with a topogically compatible metric that is intrinsic. Is the $(n+1)$-...
mathplayer's user avatar
0 votes
0 answers
24 views

Can sets of points have dimensions that differ from point to point?

I have thought about something interesting, in regards to dimension and Euclidean space $\mathbb{R}^n$. To take a simple case, consider $\mathbb{R}^3$, and take the set which is the union of the $xy$-...
user107952's user avatar
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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 $\Omega\subseteq \mathcal{A}^\mathbb{N}$, ...
Keen-ameteur's user avatar
  • 7,805
1 vote
0 answers
17 views

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 ...
Noiril's user avatar
  • 590
1 vote
0 answers
71 views

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 ...
Bihu Duo's user avatar
  • 568
2 votes
1 answer
95 views

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 ...
sreysus's user avatar
  • 751
0 votes
1 answer
75 views

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 $V$ ...
Peter Pan's user avatar
1 vote
0 answers
46 views

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 =...
Matheus Manzatto's user avatar
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0 answers
48 views

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_{\...
roundsquare's user avatar
  • 1,575
1 vote
0 answers
38 views

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 \...
EndothermicIntegral's user avatar
0 votes
0 answers
36 views

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 ...
Charith's user avatar
  • 1,626
1 vote
0 answers
65 views

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 ...
Rabi Kumar Chakraborty's user avatar
2 votes
0 answers
100 views

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 ...
Arbuja's user avatar
  • 1
0 votes
0 answers
25 views

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 ...
Arbuja's user avatar
  • 1
0 votes
1 answer
71 views

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 $...
EzBots's user avatar
  • 303
0 votes
0 answers
95 views

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 ...
Arbuja's user avatar
  • 1
0 votes
0 answers
61 views

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 ...
Eugene Zhang's user avatar
  • 16.9k
1 vote
1 answer
161 views

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 ...
EndothermicIntegral's user avatar
1 vote
0 answers
66 views

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 ...
MathStudent101's user avatar
1 vote
1 answer
72 views

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 ...
Derek H.'s user avatar
  • 353
1 vote
0 answers
36 views

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 ...
herrjo's user avatar
  • 11
0 votes
1 answer
270 views

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$...
corey979's user avatar
  • 393
3 votes
1 answer
262 views

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^...
Matheus Manzatto's user avatar
1 vote
2 answers
133 views

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 ...
Zhen Lin's user avatar
  • 90.9k
0 votes
0 answers
141 views

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 ...
Liyang Shao's user avatar
0 votes
0 answers
76 views

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 $x_k\in E$ ...
Nekomiya Kasane's user avatar
1 vote
0 answers
33 views

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 ...
user196574's user avatar
  • 1,846
1 vote
0 answers
111 views

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 ...
user196574's user avatar
  • 1,846
1 vote
0 answers
19 views

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 $n>d$. ...
12345's user avatar
  • 187
0 votes
1 answer
419 views

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 ...
Adithya's user avatar
  • 11
3 votes
1 answer
349 views

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 ...
user196574's user avatar
  • 1,846

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