# 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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### 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 ... 69 views

### 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 ...
1 vote
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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 ...
1 vote
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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 ...
1 vote
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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 ...
1 vote
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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 ...
1 vote
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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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### 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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### 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 ...
The Hausdorff measure of non-negative dimension $d$ of some set $S$ is: $$\mathscr{H}^d(S)=\lim_{\delta\to0^+}\inf\left\{\sum_{i=1}^\infty\left(\operatorname{diam}U_i\right)^d:0\le\operatorname{diam}... 0 votes 0 answers 75 views ### What is the dimension of the polar fractal \mathrm{r=\sum_{n=1}^\infty \frac{sgn(sec(n\theta))}{n}}? Here is what the fractal looks like on desmos; you can change the width of the function for more detail desmos fractal This is the inspiration for the question. I will use this inverse tangent ... 1 vote 0 answers 58 views ### 'Simple' reference for dynamically defined cantor sets I have recently been made to believe, that a better understanding of dynamically defined cantor sets will be useful to compute Hausdorff dimension of a specific class of sets I'm interested in. ... 3 votes 0 answers 97 views ### Approximating an integral over Boundary of the Mandelbrot Set It is a famous discovery by M Shishikura that the Hausdorff dimension of the Mandelbrot Set's boundary is 2. I would like to computationally approximate the following integral:$$I=\int_{\partial M} \...
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 $D$ ...