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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Proof of an inequality about set in four-dimensional space

It is needed to prove that for set $A \subset N^4$ $3\log|A|\le \log|\pi_{1,2,3}(A)|+\log|\pi_{1,2,4}(A)|+\log|\pi_{1,3,4}(A)|+\log|\pi_{2,3,4}(A)|$ where $\pi_{i,j,k}(A)$ is a projection of $A$ on ...
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1answer
23 views

an example of when Hausdorff and box-counting dimensions are equal?

I am new to fractals and dimension theory, so please excuse any errors in my understanding. For a set $F$, let $dim_b (F)$ be the box counting dimension of $F$, and $dim_H (F)$ be the Hausdorff ...
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Distance and Coordinates in fractional dimensions and the creation of functions with non-integral numbers of paramters.

Background: The Euclidean distance between two points in $n$ dimensions, where $n$ is a positive integer, and position can be described by a vector is given by... $$D_E=\left(\sum_{k=1}^n ...
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41 views

Intuitive meaning of fractal dimension.

I'm studying M. Barnsley's book 'Fractals Everywhere', but I'm stuck in the chapter 'Fractal Dimension'. Suppose $(X, d)$ is a complete metric space and let $A \in \mathcal{H}(X)$ be a nonempty ...
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316 views

Dimension theory “based on $\mathbb R^n$”

This question is somewhat vague, so please be gentle with me. I want to know if there is some definition of topological dimension that has $\mathbb R^n$ as a "paradigm", something like 'A nice ...
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32 views

Must a subset of $\mathbb{R}^n$ of dimension $n$ contain a neighborhood?

Let $X\subset\mathbb{R}^n$ be a connected subspace with Lebesgue covering dimension equal to $n$. Must there be a point $x\in X$ and a neighborhood $N$ of $x$ such that $x\in N\subseteq X$?
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36 views

Proof of fractal dimension of Thomae's function

Thomae's function is defined to be $0$ if x is irrational. Its defined to be $1 \over q$ where $x={p \over q}$ in lowest terms and $q \gt 0$. Its measure is $0$ since the set of rational numbers is ...
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61 views

The existence of a measure of finite energy implies a lower bound on Hausdorff dimension

What is the significance of $\mu(x)=0$ and the use of continuity this proof? I am not quite sure about the general direction in the second paragraph.
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122 views

How does one estimate the Hausdorff measure for arbitrary fractals, and does the constant c in $N=c\epsilon^d$ provide a good estimate?

Background: When one finds the fractal dimension of a fractal in real life, they will generally use the relation $N=c\epsilon^d$ to do so. However, the constant c is almost always neglected in ...
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What would be an example such that cardinality of linearly independent set is greater than the rank?

Let $R$ be a commutative ring and $M$ be a free $R$-module. Since $R$ is commutative, $R$ has IBN property, hence the rank of $M$ is uniquely well-defined. So set $n:=rnk(M)$. Let $A$ be an ...
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32 views

Dimension Theorem Corollary

Let $V$ and $W$ be vector spaces with $\dim V = \dim W$. If $T : V → W$ is linear then $T$ is one-to-one if and only if $T$ is onto. But this is true only when the dimensions of $V$ and $W$ are ...
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90 views

Dimension of local rings on scheme of finite type over a field.

In chapter III Hartshorne seems to be using without proof or mention a theorem on the dimensions of local rings of schemes of finite type over a field. I know that for an integral scheme of finite ...
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23 views

Finding dimension of vector spaces

Let V be a vector space and W a subspace of V . Let q : V → V/W be defined by q(v) = v + W for v ∈ V. a) Prove that q : V → V/W is a linear transformation which is onto and show that N(q) = W. b) ...
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18 views

Higher dimension leads to different optimization?

It appears z= f(x,y) has a global max/min at another particular (x,y). Using only one independent variable $x$ at fixed $y$ i.e., for z = f(x) I get another max/min point for $x$ optimum point ...
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1answer
39 views

Isn't three-dimensional really three-directional?

This occured to me while writing a mathematics/physics library. Given a 3d shape such as a cube, since it provides a simple demonstration, you have three commonly used measures: length, width, and ...
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26 views

Dimension of a LOTS in $\mathbb{R}^2$

Equip $\mathbb{R}^2$ with the lexicographic order $(x_1,y_1) < (x_2,y_2)$ iff $x_1 < x_2$ or $x_1 = x_2 \Rightarrow y_1 < y_2$. Consider the induced order topology generated by the subbase ...
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74 views

Prove that this condition is true on an Zariski open set

Why is there an Zariski open set of $P\in GL(n,\mathbb{R})$ such that $P\,\text{diag}(1,\dots,1,-1)P^{-1}$ can be conjugated by a diagonal matrix $D$ to get an orthogonal matrix? Note that ...
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48 views

Example of affine variety with infinite dimension

It is a typical question in Algebraic Geometry to find a Noetherian topological space with infinite dimension: we can take $X=[0,1]$ and check that. However, it is not so obvious for me to: Find ...
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38 views

Length of a vector in 3 dimension spaces

while I'm studying principal-component analysis, I have misunderstanding about principal component, I was think that it's orthogonal and has a single point,which I mean by single points is if we have ...
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43 views

Explicit formula for IFS fractal dimesnion

Is there an explicit formula for finding the box counting dimension of an arbitrary IFS fractal, such as the IFS fern or any other random IFS fractal? If not, is there at least a summation, or ...
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37 views

Integral over Fractals with respect to fractal dimension

I understand that there is type of integral with respect to measures that can return values when evaluated over an integral. But is there an Integral that returns d dimensional volume where d is the ...
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714 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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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 non-trivial ...
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49 views

Invariant dimension property and a ring epimorphism

In Hungerford's Algebra, p. 186, the Proposition 2.11 says Let $f:R\to S$ be a nonzero epimorphism of rings with identity. If $S$ has the invariant dimension property (IDP), then so does $R$. It is ...
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1answer
34 views

A question on Lebesgue's covering dimension

Roughly, a compact, Hausdorff space $X$ has covering dimension $\leqslant n$ if each finite cover $\mathcal{U}$ of $X$ can be refined by a cover $\mathcal{V}$ such that each point $x\in K$ belongs to ...
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78 views

Wheel Graphs and Dimension of Embeddings

I'd like to preface this by saying this is the tip of the iceberg for an optional question for a summer REU program application, so if you think asking this question is in bad taste, let me know and I ...
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67 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 $\mathbb{R}^n$?
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111 views

Evaluating the limit for the Minkowski content of $F_{\alpha}=\{0,1,\frac{1}{2^{\alpha}},\frac{1}{3^{\alpha}},\frac{1}{4^{\alpha}},\dots\}$.

The Minkowski content is defined as $\displaystyle M_{\beta}(A)=\lim_{\delta \rightarrow 0} \frac{\mu(A_{\delta})}{(2\delta)^{1-\beta}}$ where $0 < \beta < 1$, $A \subset \mathbb{R}$, and ...
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43 views

prove the subspace is equal to L1∩L2

Let $L1,L2$ be subspaces of $V$ and $dim(L1+L2)=1+L1 \cap L2$ Show that $L1 \subseteq L2$ or $L2 \subseteq L1$ and $L1+L2= L1$ or $L2$ it's has something to do with if $L1,L2 \ne L1 \cap L2$ ...
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460 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 ...
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Can these characterisations of finite dimensionality be proven equivalent without using a basis?

I was wondering about how to define "finite dimensional" without talking about bases. Two possibilities occurred to me: Say $V$ is finite dimensional if the canonical inclusion $V\hookrightarrow ...
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Compute the dimesion of these subsets of $\mathbb{R}^4$

I need to compute the codimension of the subsets $S_1,S_2,S_3\subseteq\mathbb{R}^4$ given by \begin{align} S_1&=\{(a,b,c,d)\in\mathbb{R}^4:ad-bc\neq0,\},\\ ...
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Infimum of fractal dimensions left by almost-disjoint-disk covering of plane

Answers to this question show that it is not possible to cover the plane by almost-disjoint closed disks. As note in the comments, the Appollonian Gasket leaves a set of fractal dimension ...
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640 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 $\dim$ ...
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27 views

To prove existence of an open set of functions

I am trying to prove the following: In $C(X,Y)$ with $X=[0,1]$ and $Y$ of finite dimension $K$, $C(X,Y)$ having the topology of uniform convergence, for any $K$ finite there exists an open set of ...
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47 views

Prove carpet has positive Hausdorff measure in its dimension

Given $D\subset\{0,1,2,\dots n-1\}\times\{0,\dots,m-1\}$, let $$K(D)=\{\sum_{k=1}^\infty(a_kn^{-k},b_km^{-k}):(a_k,b_k)\in D\forall k\}.$$ Show that if $D$ has uniform horizontal fibers (i.e. the ...
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1answer
109 views

Proving ineqalities for the similarity dimension

a. Let $K$ be the attractor of the IFS $\{f_1,\dots f_n\}$ which satisfies SSC (i.e $f_i(K)\cap f_j(K)=\emptyset\forall i\neq j$) where for all $i, c_i$ such that $ 1\le i\le n, \space ...
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1answer
58 views

Bounding dimension of IFS

Given the IFS $\{\frac x {2+x},\frac 2 {2+x}\}$ ($0\le x \le 1$) with attractor K prove that $0.53<\dim_HK<0.8$ I thought using the results from my last question by saying ...
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27 views

Vertex Figure of Leech Lattice

I'm doing some research on the newton number (kissing #) problem and it would be extremely useful to know the type and number of elements, particularly the facets, of the vertex figure formed from the ...
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639 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 ...
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1answer
334 views

If $(V,k)$ is a finite-dimensional vector space, then the space of all linear transformations on $V$ is finite dim and find its dim?

My issue with this is the only way I know how to prove it is to set $\dim V=n$, but then that wouldn't make sense because the second part is find the $\dim$. What I was thinking is using the ...
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73 views

Self-similar fractal dimension of unsymmetrial fractal

As far as I know, the following fractal has a self-similar fractal dimension of $D = -\log(3) / \log(1/2) = 1.5850$ But what is the fractal dimension of the following fractal (4 times the fractal ...
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50 views

Identity of dimensions related to cohomology group of projective space

Let $k$ be a field, $S=k[T_0,\ldots T_r]$, $\mathbb P=\mathbb P_k^r=\operatorname{Proj}S$ and $O$ a structure sheaf of $\mathbb P$. How can I show the identity $$\operatorname{dim}_kH^0(\mathbb ...
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1answer
60 views

Calculating a spread of $m$ vectors in an $n$-dimensional space

My question is regarding spreading $m$ vectors in an $n$ dimensional space such that the vectors are maximally distant from each other. For example, let us say I have a 2-D space, and 3 vectors, the ...
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1answer
47 views

Conditions so that Lebesgue Covering Dimension and “Usual” Dimension are Equal

The definition of covering dimension is as follows: The ply of a cover is the smallest number $n$ (if it exists) such that each point of the space belongs to at most n sets in the cover. A refinement ...
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449 views

Can a 2D person walking on a Möbius strip prove that it's on a Möbius strip?

Or other non-orientable surface, can a 2D walker on a non-orientable surface prove that the surface is non-orientable or does it always take an observer from a next dimension to prove that an entity ...
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782 views

Which sets are removable for holomorphic functions?

Let $\Omega$ be a domain in $\mathbb C$, and let $\mathscr X$ be some class of functions from $\Omega$ to $\mathbb C$. A set $E\subset \Omega$ is called removable for holomorphic functions of class ...
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54 views

Dimension of rectifiable curve

Suppose $\Gamma$ is a rectifiable curve (means a curve with finite length), I want to prove that the Hausdorff measure of the intersection of it with closed subset $A\subset \mathbb{R}$ is 0, i.e ...
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1answer
36 views

Fractal Dimension of $C_{\frac{1}{3}}\times[0,1]$

I wonder what is the dimension of the fractal set given by the product of the unit interval $[0,1]$ by the thirds-cantor-set ($C_\frac{1}{3}=\bigcap_n C_n$ where $C_0=[0,1],C_1=[0,\frac 1 3]\cup[\frac ...
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19 views

Extensions to higher dimensions by tensorization. Unitary DFT in 2D?

I have problem understanding the underlying concept of tensoration (if there is such term). Fist of all the unitary DFT is NxN. Is it 1D ? How does it look when we increase the dimension let say to ...