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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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 ...
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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 ...