# Tagged Questions

In general topology, dimension theory studies various notions of dimension defined for topological spaces, for example Lebesgue covering dimension, small and large inductive dimension or Hausdorff dimension. In commutative algebra, dimension can be defined for commutative rings.

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### Intersection/conjunction matrix and transforms?

I'm looking for a method to understand particular types of inclusion/intersection between objects (sets). Say I have a collection of sets, and between every pair of sets [A,B] I can derive a number ...
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### Making equations dimensionless

I have a equation of motion for a forced pendulum show below $${d^2\theta\over dt^2} = -{g\over L}\sin\theta + C\cos\theta\sin(Dt)$$ $L=10$ cm, $C=2\ \hbox{s}^{-2}$ and $D=5\ \hbox{s}^{-1}$. I am ...
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### Finding the similarity dimension of a variation of the Cantor Set.

If we take the Cantor set and instead of removing the interval $[1/3, 2/3]$, we remove the open interval $[x,1-x]$, with $0<x<1/2$, will the similarity dimension change? What I think is that we ...
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### Is there a simple proof that the covering dimension of the complement of $Q^2$ is one?

It can be shown that dim $R^2\setminus Q^2$ is 1 since this set has no interior. Is there a simple direct proof that does not use the equivalence of the small inductive dimension with the covering ...
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### Is a 2D object on a plane that gets curved still considered 2D or is it now 3D?

Imagine a 2D square drawn on a flat membrane. The square has length and width, but no depth (obviously; it's a square). Now take said membrane and place it with the middle of the square centered atop ...
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### (quasi)coherent rings for which $\dim R[T]\neq \dim R+1$

What are some some examples of (quasi)coherent rings for which $\dim R[T]\neq \dim R+1$? Why (hopefully geometrically) should we not always have equality? Notation. Let $I,J$ be two ideals of ...
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### Showing an equation satisfies laplace equation

I'm completely lost with the laplace equation I've searched different explanations of it on google and on this website and nothing is helping explain it. The question I was given is: Show that the ...
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### Lipschitz continuous one-to-one mapping from subset $K\subset\mathbb{R}^n$ of positive measure to $\mathbb{R}^{n-1}$

Let $f:\mathbb{R}^n\to \mathbb{R}^{n-1}$ and $K\subseteq \mathbb{R}^n$ be a set of positive Lebesgue measure. What kind of regularity do we have to impose on $f$ (e.g., $C^1$, Lipschitz) to conclude ...
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### Continuous one-to-one mapping from a subset $K \subset \mathbb{R}^n$ of positive measure to $\mathbb{R}^{n-1}$

Let $f\colon \mathbb{R}^n \to \mathbb{R}^{n-1}$ be a continuous function and $K\subset \mathbb{R}^n$ a subset of positive Lebesgue measure. Is it possible that $f$ is one-to-one on $K$? If $K$ ...
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### How to characterize the dimension of a manifold using homology?

This might be a trivial question but I'm a physicist, not a mathematician. For me, the n dimensional euclidean space is n dimensional as a vector space. I have heard however that there more intrinsic ...
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### Singular points of one-to-one mapping on rectifiable set

Let $E$ be an $s$-rectifiable set in $\mathbb{R}^n$ of positive $s$-dimensional Hausdorff measure $H^s(E)>0$. The original question I have is: Can there exist a one-to-one Lipschitz function ...
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### Hausdorff dimension of homeomorphic compact metric spaces

Are there examples of homeomorphic compact metric spaces of different Hausdorff dimension? If yes, are there sufficient conditions on the spaces which would imply the equality of Hausdorff ...