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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Wormhole - How to model it?

I am trying to model wormhole between two points in 3D space, but do not know/understand how to do so. A concrete example: Think of something like a game where we have a 3D world with a size of ...
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1answer
226 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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Dimension of a topological space [duplicate]

I want to prove this fact that if X is a topological space which is covered by a family of open subsets {U_i} than dimX=supdimU_i One direction I can see that the RHS is less than or equal to the ...
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Why $\dim(K\cap N(I)) +1 \geq \dim(K)$ if the index of the index form equals 1?

Let $c:[0,1]\rightarrow O$ a geodesic, $O$ Riemann manifold and let $\mathcal{W}$ the space of piecewise smoothl normal vector fields $W(t)$ along $c$ mit $W(0)=W(1)=0$. $N(I)$ is the nullspace of ...
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423 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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130 views

Equivalence of definitions of Krull dimension of a module

I've seen two definitions of Krull dimension of a module $M$ over a (commutative) ring $R$, and their equivalance does not seem obvious: Matsumura on page 31 of his book Commutative Ring Theory ...
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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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62 views

Is the space of probability distributions an infinite dimensional space?

Is the space of probability distributions an infinite dimensional space? If so, would you explain how? This is a follow up question to an answer to this question.
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52 views

Krull dimension of localization

If $R$ is a commutative ring and $m$ a maximal ideal therein, then what are the conditions for the Krull dimension of $R$ equaling to the Krull dimension of $R_m$?
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38 views

Depth and kernel of multiplication map

If $A$ is a commutative $k$-algebra, $\mu: A\otimes_k A\rightarrow A$ is its multplication map and $I$ is the kernel of that map (viewed as an $A\otimes_k A$-module map) then what is the relationship ...
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56 views

What is the point of view when we say $\mathbb{R}^n$ has dimension $n$ in mathematical analysis?

I know there are several different ways to define the dimension. What is the correct point of view to say that $\mathbb{R}^n$ has dimension $n$ in mathematical analysis?
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77 views

Is a Möbius Strip in > 4 dimensions impossible?

I seem to remember reading, on a plaque in the math building at Penn State, that Möbius Strips are only possible in 3 and 4 dimensions. In higher dimensional spaces, a Möbius strip will use the extra ...
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1answer
64 views

Lebesgue covering dimension of $[0,1]$

Say, we define the Lebesgue covering dimension (LCD) like this: A set $S\in \mathbb R^n$ has LCD $d\in \mathbb N$ if and only if $d$ is the smallest natural number such that for any open cover ...
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1answer
81 views

Can an n dimensional object cover an n+1 dimensional object?

Is it possible for an n dimensional object to ever cover an n+1 dimensional object? For example, could a square ever cover a cube? Note: Definition of "cover" here means to completely cover the ...
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1answer
21 views

Showing that the upper packing dimension is the packing dimension

I cannot see how the first inclusion in this proof works. $P$ is the maximum number of disjoint $B(\epsilon/2)$ with centres in $A$ and the following will help. Moreover I cannot see how it ...
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1answer
15 views

Equalities for the Upper and Lower Minkowski dimension definition

In a Geometric Measure Theory textbook the following was written: I cannot see how any of these equalities hold and dont believe they are obvious. If they are relatively obvious could someone ...
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24 views

Find a basis and state its dimension of a $C$-vector space polynomial.

The $C$ vector space $V$ of polynomials $P(t) \in C[t]$ of degree at most $n$ and such that $P(a) = P'(a) = 0$ for $a \in C$ fixed. Indication : prove that $P(t) \in V \Leftrightarrow (t − a)^2$ ...
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68 views

Find a basis of $A = (\{1, \sin(x), (\cos)^2(x), (\sin)^2(x)1\})$ and determining its dimension.

We consider a space F(R,R) of functions of R in R. Let A = ({1, \sin(x), $\cos^2(x)$, $\sin^2(x)$}) Find a basis of the vector subspace of F(R,R) and determine its dimension. So I used the identity ...
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3answers
47 views

Why should anyone care about computing the Hausdorff Dimension?

I am trying to justify an thesis/investigation that will (hopefully) lead to a formula for computing a particular class of Cantor-like sets. The question that I have not satisfactorily answered, ...
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516 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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44 views

Hochschild dimension

I'm curious; if $A$ ia a commutative $k$-algebra over a field $k$ of global dimension $n$, then is its $A^e$-projective dimension $2n$ (this is also sometimes called the Hochschild cohomological ...
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46 views

Finding ker, im, dim of a linear transformation

1Ok, I am a student trying to wrap my head around some of these concepts and need help understanding how to approach some problems. Question: Let $\alpha:\mathbb{R}^3 \rightarrow \mathbb{R}^3$ be the ...
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2answers
26 views

Questions about the Nature of Chirality (with some focus on dimensionality)

Are all chiralities the same? (Not in the sense of "is the right hand the same as the left hand?" but in the sense of "is the way in which X is chiral the same (or negative of the same) as the way in ...
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31 views

Hausdorff dimension of computable real numbers

This might be a trivial question but do the computable numbers have a positive Hausdorff dimension?
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235 views

Best way to plot a 4 dimensional meshgrid

I have $4$ variables $X$, $Y$, $Z$ and $C$, and I want to plot these on a graph. Usually I would just plot the surface $X$, $Y$, $Z$ and then use color to represent the $4$th dimension, as shown ...
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Dimensionality and Subspace Existence: A Potential Outlet for Disquisition

The subset of $F^n$ consisting of all vectors $(a_1,a_2,\dots,a_n)$ such that $a_1+a_2+\cdots+a_n=0$ is a subspace of $F^n$ and its dimension is ...(?).... Initially, my intuition said the ...
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49 views

prove $\dim\mathbb{Z}[X_1,X_2]=3$ from first principles

Since $\dim R[X] =\dim R+1$ for any Noetherian ring $R$, the ring $\mathbb{Z}[X_1,X_2]$ must have dimension 3. But how can this be proved 'from first principles', i.e. without using any big theorems ...
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55 views

$\dim V + \dim (V^{\perp} + W) = \dim W + \dim (V + W^{\perp}).$

Show that if $V$ and $W$ are two subspaces of $\Bbb{R}^n$ it is true that $\dim V + \dim (V^{\perp} + W) = \dim W + \dim (V + W^{\perp}).$ I used some properties like $\dim (V^{\perp} + W) = \dim ...
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Fractals - when the number of seed shapes that can fit into the scaled-up copy is non-integer.

I've heard people say (for eg. here) that the dimension of fractal patterns (particularly, in this question, Lindenmayer fractals) can be formulated as follows: $$D=\frac{\ln N}{\ln S}$$ Where $N$ ...
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35 views

Covering dimension of a compact metric space

I would like to see the proof of the following fact (references appreciated). A compact metric space $X$ has covering dimension $\leqslant n$ if and only if there is a continuous surjection $\pi ...
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1answer
61 views

Hausdorff dimension mathces Box-counting dimension

I need to compute the Hausdorff dimension of certain sets using a computer and, to date, my approach has been to use a Box-counting algorithm, for I once read that the Hausdorff dimension of an ...
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1answer
26 views

relationship between number of polynomials and dimension of the space.

If p1,p2,...,pk are linearly independent polynomials in Pn, a mathematical relationship between k and n is: k<=n. If the k will be more than n, the set of polynomials can not be linearly ...
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dimension of the sum of two planes in $\mathbb{R}^3$

$\dim( U + V)=\dim( U) + \dim(V) - \dim( U \cap V)$ where U and V are linear subspaces and $U+V$ is algebraic sum (the least linear space that contains both $U$ and $V$). In case of two planes in ...
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1answer
204 views

Cantor set exercise

This is an exercise from Abbott's real analysis book. It's exercise 3.4.4.(b) on page 93. I couldn't find a definition of ''dimension'' in the book. The only thing I could find is something on page ...
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44 views

Examples and counterexamples in dimension theory

I am looking for examples of topological spaces that are interesting from a dimension theoretical point of view - in particular I am looking for these three notions of dimension: Little inductive ...
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1answer
36 views

Hausdorff dimension of $\mathrm{R}^d$

I assume the Hausdorff Dimension of $\mathrm{R}^d$ is $d$.. To prove this I guess one hast to prove these two statements: the $\alpha$-dimensional Hausdorff measure of $\mathrm{R}^d$ is $0$ for $d ...
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26 views

Equal fields and dimension

Let $K \subseteq L$ be fields. Show that K=L if and only if $dim_k L=1$ I know that I will have to show both directions of the implication. I'm very new to the topic of fields so I'm still ...
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1answer
30 views

Solid Ball Bearing 4 Dimensional?

Is a solid ball bearing a 4 dimensional object, but all we see is one particular level surface? When we look at a solid ball bearing, we can only see the outside of it, we cannot see inside even ...
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2answers
136 views

Can four lines be perpendicular

my question is a little weird, but it bothers me to not find an answer. Since in 2D two lines can be perpendicular and in 3D three lines can be perpendicular, how about 4D? Is it possible to have ...
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2answers
47 views

Topological dimension and derham cohomological dimension

If G is a compact complex manifold then does the topological dimension bound the deRham cohomological dimension below? By derham cohomological dimension, I mean the largest extended natrual number ...
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1answer
43 views

Question on dimensions

It is well known that a line has 2 points. A square has 4 lines. A cube has 6 squares. A tesseract has 8 cubes. And continuing there will be 10, 12, 14, etc. Why does the sequence increase by 2? ...
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73 views

Hausdorff dimension is less than box counting dimension?

I have been asked to prove that for a bounded set $F\subset\mathbb{R}^n$, $dim_H F\le \underline{dim}_B F \le \overline{dim}_B F$ where $dim_H F$ is the Hausdorff dimension, $\underline{dim}_B ...
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531 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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The 1-dimensional Hausdorff measure of a curve in the plane

For a set $X\subseteq\mathbb{R}^2$, let $H^1(X)$ be its 1-dimensional Hausdorff measure. Suppose $X$ is a regular curve (say, a graph of a continuous function $f:\mathbb{R}\to\mathbb{R}$). In that ...
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49 views

laplace-beltrami and angular momentum operator

Im trying to understand this formula: In $\mathbb{R}^n$ we say that the laplacian can be seen as (radial part and angular part). : $$\triangle=(\frac{\partial^2}{\partial ...
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1answer
45 views

How to visualize four dimensional tic-tac-toe?

I have played three dimensional tic-tac-toe with three players before, and we had no problem visualizing it. We drew three layers on a sheet of paper and just remembered all the different ways you ...
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On the definition of (small) inductive dimension

A regular topological space $X$ has inductive dimension smaller or equal to n if and only if: ($n=-1$) $X=\emptyset$; ($n>-1$) The space $X$ has a base of opens $\mathscr{B}$ such that, for all ...
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3k 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 non-trivial ...
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75 views

Hausdorff content and Hausdorff measure

I am dealing with the Hausdorff dimension and I came across two different ways of defining this dimension. This question is possibly related to Hausdorff Measure and Hausdorff Dimension but the ...
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34 views

Question on density of a set

I am reading a paper on complex dynamics and Hausdorff dimension, and there is a result that I can't prove. I have the following situation. For each $k=1,2,...$, we denote $E_k$ a finite collection ...