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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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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43 views

Trying to prove $\text{Hdim}(\bigcup X_i)=\sup_i \text{Hdim}(X_i)$

Suppose $X=\bigcup_i X_i$ is a countable union. I'm trying to prove a statement which wikipedia says follows directly from the definition of Hausdorff Dimension: ...
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275 views

Looking for Cover's hubris-busting ${\mathbb R}^{N\gg3}$ counterexamples

In lecture 4 of his Introduction to Dynamical Linear Systems course, right after interpreting the inner product in ${\mathbb R}^N$ in terms of the cosine of the subtended angle, Stanford's Stephen ...
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90 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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155 views

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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103 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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138 views

Transcendence degree of $K[X_1,X_2,\ldots,X_n]$

Let $K$ be field. How do I proof that transcendence degree of $K[X_1,X_2,\ldots,X_n]$ is $n$? The set $\{X_1,X_2,\ldots,X_n\}$ is algebraically independent over $K$. So, I have to show that every ...
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228 views

Hausdorff dimension of a smooth manifold

I read a book about fractal stating that without proof: every $m$-dimension $(m<n)$ smooth manifold $M$ in $\mathbb{R}^n$ has Hausdorff dimension $m$. How can we prove it?
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2answers
179 views

Krull dimension of the injective hull of residue field

Let $(R,\mathfrak{m})$ be a noetherian local ring, and $E=E_R(R/\mathfrak{m})$ the injective hull of $R/\mathfrak{m}$. What do we know about the Krull dimension of $E$? Thank you.
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Why should Gaussian noise have fractal dimension of 1.5?

In a paper I'm trying to understand, the following time series is generated as "simulated data": $$Y(i)=\sum_{j=1}^{1000+i}Z(j) \:\:\: ; \:\:\: (i=1,2,...,N)$$ where $Z(j)$ is a Gaussian noise with ...
3
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n points and (n-1) dimensions

2 points make a line -1D 3 points at most make a plane -2D similary can we find $n$ points making up a $n-1$-dimensional object in $\mathbb{R}^n$?
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Proof that the Hausdorff Dimension of Liouville Numbers is zero.

I am studying a proof that the set $L$ of Liouville numbers in $[0,1]$ has Hausdorff dimension zero. $L=\lbrace x\in [0,1]: \forall n\in \mathbb{N}, \exists p,q\in \mathbb{Z}, q>1, \text{and such ...
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1answer
109 views

Planar cross section of leech lattice?

I was wondering what any planar cross sections of the leech lattice would look like. I don't know much about this topic at all, I'm just quite curious. Is there any way to find out?
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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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1answer
202 views

Linear Algebra Question ( rank of matrix )

Let $\bf A$ be an $m \times n$ matrix. If $\bf P$ and $\bf Q$ are invertible $m \times m$ and $n \times n$ matrices, respectively prove $\operatorname{rank}(\mathbf{PA}) = ...
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Prime ideals of height less than the dimension

Let $A$ be a noetherian local ring with maximal ideal $\mathfrak{m}$ of height $d$, and suppose $\mathfrak{p}_1, \ldots, \mathfrak{p}_s$ are prime ideals of height $i - 1 < d$. It's quite clear ...
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Related questions on the Hausdorff dimension and local dimension of a Cantor set

Suppose $I$ is an interval on a Cantor tree with $m$ children $I'$, each of length $\varepsilon$. I have that $\sum_{\hat I\text{ is a child of }I}|\hat I |^s=|I|^s \Rightarrow ...
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776 views

Why is it that I cannot imagine a tesseract?

I try hard to "visualise" (say "imagine") a tesseract but I can't. Why is it that I can't? This may be a question for a scholar of some other discipline and not for a mathematician, e.g. ...
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1answer
97 views

Cutting a $n$-dimensional cubic cake

Given a cubic cake, defined as $\{(x,y,z)|0\leq x,y,z\leq 1\}$. We cut it by the planes $p_1\leftrightarrow x=y$ $p_2\leftrightarrow y=z$ $p_3\leftrightarrow x=z$. How many pieces will we have ...
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Which definition of dimension came first?

In my algebraic geometry class, the dimension of an affine variety $X=V(I)$ was defined as the supremum of the length of chains of prime ideals in the coordinate ring $R=k[x_1,\ldots,x_n]/\sqrt{I}$, ...
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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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143 views

If the special fiber of a flat morphism is reduced, then any other fiber is reduced?

Suppose $R=\mathbb{C}[x_1,\ldots, x_n]$ is a polynomial ring with $I$ being an ideal of $R$. Let $I'$ be an ideal of $R[t]$. If $R[t]/I'$ is flat as a $\mathbb{C}[t]$-module and over $0$, ...
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Krull dimension of $\mathbb{C}[x_1, x_2, x_3, x_4]/\left< x_1x_3-x_2^2,x_2 x_4-x_3^2,x_1x_4-x_2 x_3\right>$

Krull dimension of a ring $R$ is the supremum of the number of strict inclusions in a chain of prime ideals. Question 1. Considering $R = \mathbb{C}[x_1, x_2, x_3, x_4]/\left< x_1x_3-x_2^2,x_2 ...
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For an $n$-dimensional object, how many types of holes are possible?

Update 2012-06-06: At some point I'll attempt to answer my own question by using a dual-fluid model that places the dimensionality and connectivity of "solids" and "holes" on an equal footing. With ...
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240 views

Does a basis span even linear systems?

I understood basis as a set of vector $v_{1},v_{2},...,v_{n}$ as the set whose linear combination will span the entire vector space say $ \mathbb R^{n}$ which makes perfect sense in intuitive terms. ...
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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

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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841 views

Hausdorff measure for Lebesgue measurable sets?

As I never had a course which dealt with Hausdorff measures and every time I heard about Hausdorff measure I was only thinking using my intuition what that should be. So I decided to take a look at ...
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171 views

On Krull dimension of $M/(0 :_{M} \mathfrak{m}^t)$ module

Let $(R,\mathfrak{m})$ be a commutative Noetherian local ring and $M$ is an $R$-module. There is an non-negative integer $t$ such that $M/(0 :_{M} \mathfrak{m}^t)$ is finitely generated. Then $$\dim ...
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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 ...
3
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1answer
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Hilbert function on ideal generated by linear forms.

This is a slight extension of a remark a read a few days ago. Let $K$ be a field, and let $A=K[X_0,\dots,X_N]$ be a polynomial ring, which is graded in the standard way (the elements of degree $n$ ...
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Has the notion of having a complex amount of dimensions ever been described? And what about negative dimensionality?

The notion of having a number $a \in \mathbb{R}_{\geq 0} $ associated to any metric space is described by the definition of a "Hausdorff Dimension". I was wondering if work has been done on spaces ...
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2answers
164 views

Dimension of a subset

For a closed subset $Y$ of a space $X$ we have the following inequlity of topological (covering) dimensions: $$\dim{Y} \leq \dim{X}$$ (assuming at least one of those is finite). I have two questions ...
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Fractional versions of euclidean space?

This is going to be a somewhat vague question, but I'll be happy if you indulge me. Euclidean space $\mathbb{R}^n$ is equipped with a lot of nice (algebraic, metric, topological,...) structure and ...
2
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231 views

Degrees of freedom

Grateful if someone could tell me whether my rationale is correct: If I impose $m$ constraints on $\mathbb R^n$ (where $n<\infty$), then the set has $n-m$ degrees of freedom. Hence this subspace ...
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In what sense is a tesseract (shown) 4-dimensional?

This video and this image show a tesseract, which is a 4d cube: In what sense is this cube 4 dimensional? Where is time? (commonly called the 4th dimension, although I realize here its probably ...
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5answers
3k views

Orthogonal planes in n-dimensions

In three dimensions two planes are orthogonal when their normal vectors are orthogonal (their inner product is zero). For example, planes $xy$ and $xz$ are orthogonal because the vectors $\hat{z}$ and ...
3
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1answer
161 views

3D - 1D = 2D? Doing arithmetic with dimension

If you consider $\mathbb{R}^3$ and a one-dimensional space curve, by "removing" the curve from $\mathbb{R}^3$ you are left with a space that is still three dimensional, for an appropriate definition ...
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501 views

$\operatorname{height} \mathfrak{p} + \dim A / \mathfrak{p} = \dim A$

Let $A$ be an integral domain of finite Krull dimension. Let $\mathfrak{p}$ be a prime ideal. Is it true that $$\operatorname{height} \mathfrak{p} + \dim A / \mathfrak{p} = \dim A$$ where $\dim$ ...