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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Fractal dimension of rank-size distribution

I analysed the size of genres reported by the metadata of my music collection. There are a few very large ones (like "Electronic"), and very many tiny ones, with a power-law distribution. ...
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Proof that plane has small inductive dimension 2?

By definition, the small inductive dimension $\operatorname{ind}(\emptyset) = -1$ and, recursively, the small inductive dimension $\operatorname{ind}(X)$ of a nonempty topological space $X$ is the ...
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Can the correlation dimension (CD) say something about the required number of equations that yield a time-series?

My question falls into the field of topology, the estimation of the correlation dimension (CD) in phase space. I have a one-dimensional time-series (a list of numerical sampling/data points) that I ...
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If it exists, can cross sections of a real tesseract appear to us in 3D space completely different than Schlegel diagram?

We as human beings cannot comprehend how an object looks like in spatial dimensions higher than 3, it is in fact unimaginable. Yet, in mathematics we are able to project analogues of objects from ...
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Finding smallest possible dimension of a vector space

I am trying to find the smallest possible dimension of a vector space V, which has 3 subspaces $S_1, S_2, S_3$. These subspaces have the dimensions of: $dim(S_1) = 4dim(S_2) = 6dim(S_3)$. It is also ...
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$\lim_{R\to\infty} R^{-n} \mathcal L^n(\{x\in B(0,R): |\hat\mu(x)| > |x|^{-s/2}\}) = 0$ if $I_s(\mu) < \infty$

$\mathcal M(\Bbb R^n)$ denotes the set of all Borel measures $\mu$ on $\Bbb R^n$ with $0 < \mu(\Bbb R^n) < \infty$ and compact support $\operatorname{spt}\mu \subset \Bbb R^n$. Suppose $\mu\in \...
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Find a nowhere differentable functions with Hausdorff dimension one

I wonder that whether there exist a nowhere differentable continuous function with its graph in $\mathbb{R}^2$ has Hausdorff dimension $1$. A result about Weierstrass's function is that $\sum_{k=1}^{\...
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Existence of a holomorphic function when the plane cancels a Hausdorff dimension 1 set

(a) Show that there does not exist a holomorphic function $f$ on $\mathbb C \backslash \{1, -1\}$ so that $$f'(z) = \frac{1}{(z^2-1)^{2019}} \text{ for all } z \in \mathbb C \backslash \{1, -1\}.$$ (b)...
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Hausdorff dimension of sets with positive Lebesgue measure

I am reading Hausdorff Dimension, Its Properties, and Its Surprises by Dierk Schleicher. Among the elementary properties of the Hausdorff dimension, the last one is: If $X\subset \Bbb R^n$ has finite ...
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How to find the size of an ϵ-net of a vector space?

Taken from paper "A Universal Law of Robustness via isoperimetry" by Bubeck and Sellke. Theorem 3. Let $\mathcal{F}$ be a class of functions from $\mathbb{R}^{d} \rightarrow \mathbb{R}$ and ...
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Removing a set of zero $(d-2)-$Hausdorff measure from a $(d-1)-$dimensional connected surface does not destroy connectedness?

Suppose $A\subseteq \mathbb{R}^d$ is connected and open while its complement $A^c$ is connected too and has non-empty interior. The answers to this question imply that $\partial A$ is then connected ...
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On the countable sum theorem

I am trying to understand the above theorem. I get most of the proof but I have trouble understanding why the functions $g_i$ were introduced. I mean why can't we just define $g:X\rightarrow S^n$ by $...
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Hausdorff dimension $\dim_H (S) = \inf \{d \ge 0 \mid H^d (S) < +\infty\}$

I'm reading about Hausdorff dimesion. Could you confirm if my understanding (through the proof of the proposition) is correct? Let $(E, \rho)$ be a metric space. Let $d \in [0, +\infty)$ and $\delta&...
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Horiozontal throw dimensions analysis

I have a question about dimensions analysis of a horizontal throw: A ball is thrown from origin where the $x$-axis points in the horizontal direction and $y$-axis vertically upwards. The ball is ...
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Why does the volume of a unit hypersphere increase and then decrease with the dimension?

It is well known that the volume of an $n$-ball of radius $R$ is $$V_{n}(R)=\frac{\pi^{n/2}}{\Gamma(1+n/2)}R^n.$$ Graphically, the volume as a function of dimension looks like We see how the volume ...
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Prove that if $\mathcal H^1(F) = 0$ for $F\subset \Bbb R$, then $\Bbb R\setminus F$ is dense

Prove that if $\mathcal H^1(F) = 0$ for $F\subset \Bbb R$, then $\Bbb R\setminus F$ is dense. I need to prove the above result to understand Proposition $3.5.$ in Falconer's Fractal Geometry. $\...
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Is it possible to translate points on a $3D$ sphere to a $4D$ hypersphere

Would it be possible to translate coordinates on a $3D$ sphere, say a set of coordinates on earth, to a point on a hypersphere? I'm not sure on the possibilities of this but I cannot see why it wouldn'...
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Subsets of $\mathbb{R}^n$ with "mixed" dimension

In $\mathbb{R}^3$, consider the set $S$ which is the union of the $xy$-plane with the $z$-axis. What would be the dimension of $S$? More generally, in $\mathbb{R}^n$, assume that the set $X$ has ...
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About refinements of order at most $d+1$

I am a little confused about the definition of covering dimension of a topological space. In particular, when defining the dimension of a space $X$, we take the minimum over all natural numbers such ...
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Is a refinement of an open cover also a cover?

I am trying to understand the above proposition. So according to the definition of $\text{dim}\, X$, $\mathcal{W}$ is a refinement of the open cover. What I don't understand is why is it that any ...
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optical geometry- the apparent size formulization

In trying to find how large something appears in relationship to how tall it is and how far away it is, I was provided with 2inverse tangent(height/2*distance)=angle theta. I was also told separately ...
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Calculation of the efficacy rate of a drug

I am participating in a mathematical biology project. I would like to discuss the following problem: Let A be a drug such that $x_{o}$ chemical units of it kills $12\text{%}$ of $y$ cells per day, I ...
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Does UMAP preserve the dimensional units of differences?

I'm using Uniform Manifold Approximation and Projection (UMAP) to reduce the dimensionality of feature vectors that have $N\gg1$ components. Each component is valued as a positive, real measure of ...
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Dimension of Zariski closure of the image of an algebraic map

Let $\phi: \mathbb{R}^m \rightarrow \mathbb{R}^n$ be a polynomial map with rational coefficients. Let $M$ be the Zariski closure of the image of $\phi$ (in the complex space $\mathbb{C}^n$). Then the ...
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On topological definitions of zero-dimensionality

There are many different, and not necessarily equivalent, definitions of zero-dimensionality for a topological space. Here are two examples: Def. 1: A topological space is zero-dimensional if every ...
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Topological Dimension of Quotient Space

For a topological space $\mathcal X$ and a quotient space $\mathcal Y$ of $\mathcal X$, Wikipedia's page on quotient spaces says that the topological dimension of $\mathcal Y$ can be more than that of ...
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Does non-zero and finite Hausdorff dimension imply non-zero and finite Hausdorff measure?

Is it possible that a set $A$ has Hausdorff dimension $d=\mathrm{dim}_H (A)\in (0, \infty) $ but $ H^d (A)\notin (0, \infty) $? In other words, positive and finite Hausdorff dimension but its ...
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How to calculate Information Dimension

I am trying to implement this paper by Tian Bian and Yong Deng. In this paper, after applying probability they have gotten information entropy values $l_a(r)=(1.3741,0.6930,0.6385,0) , (r=1,2,3,4)$ ...
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Is there a topos-theoretic notion of "dimension"?

It seems like almost any "topological" phenomenon has a generalization to toposes. For instance, in Site Characterizations for Geometeric Properties of Toposes, Olivia Caramello shows how we ...
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Heuristic 'characterization' of Hausdorff dimension

I was wondering whether there is a heuristic simpler way to understand the saying that a compact $E\subseteq \mathbb{R}^d$ satisfies $\dim(E)=\alpha\in (0,d)$? Or maybe saying that $\mathcal{H}^{\...
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Union of one-dimensional family of straight lines has zero measure

Let $\gamma:[0,1]\to \mathbb S^2$ be a smooth curve. Consider the following set $$ \bigcup_{t\in [0,1]}\mathrm{span}(\gamma(t))\subset \mathbb R^3 $$ My question is: does this set have Hausdorff ...
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Motivation for the concept of dimension

I am currently learning about Hausdorff measure and dimension. In several of the resources I have encountered, the author provides several different definitions of dimension, and they are often ...
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Is there a property of compact metric spaces with intermediate dimension in the Gromov-Hausdorff space?

Recall the definition of the Hausdorff dimension of a subset $A$ of a metric space $\mathcal{M}$. (I'm not especially wedded to the Hausdorff dimension per se and I'd be interested in answers for any ...
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Lyapunov dimension

I have a nonlinear differential equation system composed of 4 equations. I calculated Lyapunov's dimension of each of the states to be a little bit over 3 (say 3.11, 3.1, 3.13, 3.14). How can I ...
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Fractal Dimension of Lag Plots

I have created a number of lag plots from a single time series, namely the EUR/USD exchange rate, with lag = 1,2,...,20. E.g. Lag Plot I want to get the Fractal Dimension between the original time ...
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Hausdorff dimension of a connected set in $\mathbb{R}^d$

I recently asked in this question, whether a set $A$ not being totally disconnected in $\mathbb{R}$ implies $\dim(A)=1$? The answer I was given, showed that the answer is yes by relatively simple ...
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Can a manifold have infinite Minkowski dimension?

If $M$ is a topological $n$-manifold, then I know that there is a metric $\rho$ (compatible with the topology) such that the Hausdorff dimension of $(M,\rho)$ is $n$ (see this question). Is this also ...
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What proportion of the volume of Gabriel's horn could be filled by its surface area?

Space filling curves imply that lower dimensional objects can fill a proportion of higher dimensional objects. Eg. in the case of the Hilbert curve and the 2D space it fills, this proportion is 1. ...
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Number of regular $n$-topes in $\mathbb{R}^n$ for $n\in\mathbb{R}$?

Ever since I've come across it, I have been puzzled by the sequence $(a_n)_{n\in\mathbb{N}_0}=(1,1,\infty,5,6,3,3,3,\cdots)$, describing the number of regular $n$-topes in $n$-dimensions, where $a(k)=...
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Is a random walk in 1.585-dimensional space transient or recurrent? [closed]

A drunk man will eventually find his way home, but a drunk bird may get lost forever. Will a drunk squirrel eventually climb down the 3-branch tree?
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Hausdorff dimension of a connected set in $\mathbb{R}$

I have been looking for certain properties of the Hausdorff dimension lately, and noticed that all the examples I know of in $\mathbb{R}$ are only totally disconnected spaces. So I was wondering ...
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Show that $\Bbb Q$ is zero-dimensional.

Show that $\Bbb Q$ is zero-dimensional. I’m not sure if I’m confusing something, but since $\Bbb Q$ is countable the set $\mathcal{B}=\{\{q\} : q \in \Bbb Q\}$ forms a basis for it? And by definiton ...
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Countable stability of Modified Box-Counting Dimension

The upper modified box-counting dimension of $F \subseteq{\mathbb{R}^n}$ is defined as $\overline{\dim}_{MB}F = \inf \left\{ \sup_{i} \overline{\dim}_{B}F_i; F \subseteq \bigcup_{i=1}^{\infty} F_i\...
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The Dimension of the Logistic Map (Fractal & Topological)

I'm a high school student doing my math IA on fractal dimensions. According to this Wikepedia page, the logistic map's correlation dimension, Hausdorff dimension, and information dimension are all ...
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8 votes
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Does a compact space contain a compact fractal of every dimension?

Let $X$ be a compact metric space with Hausdorff dimension $x \in \mathbb{R}$. Let $0 \leq w < x$. Does there exist a compact subspace $W \subset X$ with Hausdorff dimension $w$? If not, what are ...
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Why is the Hausdorff measure defined the way it is? And how can it be computed?

The Hausdorff measure of non-negative dimension $d$ of some set $S$ is: $$\mathscr{H}^d(S)=\lim_{\delta\to0^+}\inf\left\{\sum_{i=1}^\infty\left(\operatorname{diam}U_i\right)^d:0\le\operatorname{diam}...
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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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'Simple' reference for dynamically defined cantor sets

I have recently been made to believe, that a better understanding of dynamically defined cantor sets will be useful to compute Hausdorff dimension of a specific class of sets I'm interested in. ...
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Approximating an integral over Boundary of the Mandelbrot Set

It is a famous discovery by M Shishikura that the Hausdorff dimension of the Mandelbrot Set's boundary is 2. I would like to computationally approximate the following integral: $$I=\int_{\partial M} \...
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Is there a standard definition of fractal dimension?

There are many definitions/methods of calculating the fractal dimension (Hausdorff, Box-counting, Higuchi, Katz, Correlation, etc. dimensions) and I think the main one is the Hausdorff dimension $D$ ...
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