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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329 views
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 ...
14
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5answers
204 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 ...
8
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4answers
231 views
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 ...
7
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1answer
139 views
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 $R=k[x_1,\ldots,x_n]/\sqrt{I}$, ...
7
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1answer
357 views
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 ...
7
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1answer
178 views
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 ...
6
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2answers
117 views
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 ...
4
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2answers
335 views
$\operatorname{height} \mathfrak{p} + \dim A / \mathfrak{p} = \dim A$
Let $A$ be a 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 $\dim$ refers ...
4
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3answers
400 views
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 ...
4
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4answers
92 views
$T:V\rightarrow W$ such that $N(T)=V'\subset V$ and $R(T)=W'\subset W$
How should one prove that there exists a linear map $T:V\rightarrow W$ such that $N(T)=V'\subset V$ and $R(T)=W'\subset W$ if $\dim(V')+\dim(W')=\dim(V)$, where $V$ and $W$ are finite-dimensional ...
4
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1answer
50 views
Dimension of subspace is greater then dimension of space
If $X$ is a topological space then it's (covering) dimension is defined as a minimal number $n$ such that for every finite open cover $\{U\}$ of $X$ there is a finite open cover $\{V\}$ of $X$ that ...
4
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1answer
65 views
algebraic geometry exercise: infinite subset is dense
A hypersurface $C \subset \mathbb{A}^2$ we call a plane curve.
Show that any infinite subset of an irreducible plane curve $C \subset \mathbb{A}^2$ is dense in $C.$
Note. We call hypersurface the ...
4
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1answer
115 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 $0$, ...
3
votes
2answers
380 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 ...
3
votes
1answer
140 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 ...
3
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1answer
74 views
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 ...
3
votes
1answer
116 views
Matrices to model 3D object
I'm toying around with an algorithm to determine placement of 3D objects into a larger 3D space. I immediately thought of using matrices.
It's been some years since my Linear Algebra courses. I was ...
3
votes
1answer
138 views
A detail in the proof of Auslander-Buchsbaum Theorem
I'm trying to understand the proof of a theorem (Auslander-Buchsbaum) which says that given a local ring $(R,m,K)$, where $m$ is the maximal ideal and $K$ the residue field, and a finitely generated ...
3
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2answers
113 views
Measuring Vitali sets
If $V$ is a vector space and $m$ is the counting measure, then $$\dim(V) = \inf \{m(U) : U \subset V, \text{ span}(U) = V\}.$$
Given a measure space $(V, \mathcal M, \mu)$ such that $V$ is a vector ...
3
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1answer
42 views
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 $\mathbb{R}^n$?
3
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1answer
72 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 ...
3
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1answer
62 views
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 ...
3
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0answers
71 views
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 ...
2
votes
2answers
228 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. ...
2
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1answer
62 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 ...
2
votes
2answers
54 views
Dimension of the corresponding eigenspace?
I'm studying for my linear exam and would appreciate any help for this practise question:
You are given that λ = 1 is an eigenvalue of A. What is the dimension of the corresponding eignspace?
A = ...
2
votes
1answer
73 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?
2
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1answer
37 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: ...
2
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1answer
94 views
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 $n$ ...
2
votes
1answer
141 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 ...
2
votes
1answer
48 views
Poles of formal power series (Hilbert-Poincaré series)
How are poles and orders of poles of formal power series defined?
The particular case, I am interested in, is the following definition from [Atiyah-Macdonald, Introduction to commutative algebra, ...
2
votes
1answer
127 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.
2
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1answer
138 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}) = ...
2
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0answers
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 ...
2
votes
1answer
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 ...
2
votes
1answer
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 ...
2
votes
0answers
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 $\pi$?
1
vote
2answers
126 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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vote
5answers
1k 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 ...
1
vote
1answer
120 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
$$\dim ...
1
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2answers
252 views
In what sense is a tesseract (shown) 4-dimensional?
This video and this image show a tesseract, which is a 4d cube:
In what sense is this cube 4 dimensional? Where is time? (commonly called the 4th dimension, although I realize here its probably ...
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1answer
52 views
+50
Reverse Hölder Continuity and Hausdorff dimension
Let $f$ be a function on $[0,1]$. Say that $f$ is reverse Hölder continuous of exponent $\beta > 0$ if there is a $C >0$ such that for any $s<t\in [0,1]$, there exists $s',t'\in [s,t]$ such ...
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vote
2answers
43 views
Odd valued dimensional number impossible to build?
Using numbers of the form $$\alpha_1+\alpha_2e_1+\alpha_3e_2+...+\alpha_ne_{n-1}$$
where $\alpha_n\in\Bbb R$ and for all $a≠b, \alpha_a≠\alpha_b$ with $e_n^2=-1$, can these numbers exist for an odd n? ...
1
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2answers
37 views
Dimension Recovery of $S \subset P_n(F)$
How is the subset of $P_n(F)$ consisting of all polynomials $f$ such that
$f(1) = 0$ a subspace of $P_n(F)$? What is the dimension of this subset?
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vote
1answer
48 views
Diagrammatic Representations: $\dim(Skew_{n\times n}(\mathbb{R}))+\dim(Sym_{n\times n}(\mathbb{R})) = \dim(M_{n\times n}(\mathbb{R}))$
SEE AUTHOR'S ANSWER BELOW
So I'm trying to derive the dimensions of both $Skew_{n\times n}(\mathbb{R})$ and $Sym_{n\times n}(\mathbb{R})$. I know that $\dim(M_{n\times n}(\mathbb{R}))=n^2$, but I ...
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1answer
104 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?
1
vote
1answer
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 ...
1
vote
0answers
51 views
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 ...
1
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0answers
97 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 ...
1
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0answers
48 views
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 ...
