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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59 views
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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77 views
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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19 views
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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30 views
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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52 views
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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25 views
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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38 views
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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123 views
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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11 views
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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28 views
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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43 views
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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21 views
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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32 views
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 ...
