# 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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### Doubts about Hausdorff measure.

I am studying real analysis and I have some problems in understanding properties of Hausdorff measure. Let $\mathcal{E}_\delta$ be collection of subsets of $\mathbb{R}^N$ whose diameter is less then ...
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### Dimension of graphs (Differential Geometry)

I have a rather basic question about the dimension of a graph mapped by the exponential map. The problem is I can't really visualize it. It starts like that: Let $M$ be a smooth manifold of ...
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### Injective endomorphism and Hilbert dimension

Let $\mathcal{O}$ be a finitely-generated $K$-algebra where $K$ is a field and let $M$ be a finitely-generated $\mathcal{O}$-module. For every good filtration \$0 = M_0 \subset M_1 \subset M_2 \subset ...
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### Confusion related to curse of dimensionality in k nearest neighbor

I have this confusion related to curse of dimensionality in k nearest neighbor search. It says that as the number of dimensions are higher I need to cover more space to get the same number of ...
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### Codimensionality: On Cardinality of Linear Equations

How does the codimension of a subspace give the number of linear equations needed to define the subspace?
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### Dimensions of Matrices Range (equalities).

I’d like to find range equalities. Considering the following: $$A=B+C \\ A=B.C^T \\ A=[ B^T C^T ]^T \\$$ I would like to find the function $f$ for each equality above.  dim( R(A) ) = f( R(B) , ...
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### Dimension of coefficents in a density equation

The density throughout a composite material is given by $T(x, y, z) = Axy^2 + Bxz^3 + Cy^2z^3,$ where $x$, $y$ and $z$ are the cartesian coordinates of the position inside the material. (a) Find the ...
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### A short proof for $\dim(R[T])=\dim(R)+1$?

If $R$ is a commutative ring, it is easy to prove $\dim(R[T]) \geq \dim(R)+1$. For noetherian $R$, we have equality. Every proof I'm aware of uses quite a bit of commutative algebra and nontrivial ...
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### Topological dimension of a countable dense set

I'm reading a (dynamical systems) paper in which topological dimension figures. In my situation, I'm trying to compute the topological dimension of the subset of the $d$-dimensional torus consisting ...
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### Defining the Rank of a Projective Module

I am trying to understand the definition of rank for a projective module over a noncommutative ring. The definition I am using is: A sufficient condition for the rank of a free module over a ring ...
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### Canonical $\pi$ dimensional space?

Can we talk about a canonical space of dimension $\pi$? Is there anything like $\mathbb R^\pi$? Have anyone met any fractal of dimension $\pi$?
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### Buckingham Π-Theorem

I'm about to conduct some experiments concerning a welding process. To prepare for this I wanted to do a dimensional analysis of the process. So I read a lot about the Π-Theorem and I was able to ...
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### Hilbert (polynomial) dimension and dimension of a support of a module

$\newcommand{\Supp}{\mathrm{Supp}}$ $\newcommand{\Ann}{\mathrm{Ann}}$ Let $X$ be an affine algebraic variety (over a field $K$, can assume it is algebraicaly closed), $M$ a finitely generated ...
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### Disjointness of stars in a simplicial complex in $\ell_2$

Definitions Let's consider a full $n$-dimensional simplicial complex $C$ in $\ell_2(X)$. By that I mean a set of all functions $f:X \to[0,1]$ such that $\sum_{x\in X}f(x)=1$ and there are at most ...
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### Cubic interpolation in arbitrary dimension?

Consider a $N$-dimensional space discretized with a regular cubic grid of $n^N$ cubes, each cube containing the value of a function $f$ in its center. How to correctly interpolate $f(x, y, z)$ using ...
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### How to apply other than trivial dimensions?

I only used natural dimensions so far but I understand there could be Negative dimension with application e.g. dimension -2 definition and usage Non-integer dimension with application e.g. dimension ...
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### Separation of Euclidean Space

Consider a finite collection $\mathcal{H}$ of hyperplanes of $\mathbb{R}^n$ that have a common line. Given some $A \subseteq \mathbb{R}^n$ that is homeomorphic to a subset of $\bigcup\mathcal{H}$, ...
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### Density of a multifractal distribution

I am trying to grasp the concept of density of a multifractal. So I start from the simple case of a line. Let's assume I have a uniform distribution of points on a line and I center a cubic box in the ...
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### Covering dimension of the union

Let $\{{A_n}\}$ be the closed subsets of $X$, such that ${A_n} \subset \operatorname{Int}{A_{n + 1}}$ and $\cup {A_n} = X$, if $A_1$ and all $\operatorname{cl}(A_n-A_{n-1})$ have the covering ...
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