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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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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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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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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61 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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70 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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63 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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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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19 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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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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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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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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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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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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44 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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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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48 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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$\dim V + \dim (V^{\perp} + W) = \dim W + \dim (V + W^{\perp}).$

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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33 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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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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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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20 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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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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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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29 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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129 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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41 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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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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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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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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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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44 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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159 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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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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67 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 ...
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33 views

Number of touching triplets in hypersphere packing contact graph

I would like to know what is curently known about the number $N$ of touching-triplets involving a given vertex $V$ in the contact-graph of $d$-hypersphere packings. For $d=2$ and $d=3$, one has $N = ...
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How can I understand is the picture $2D$ or $3D$

I can not understand is this picture 2D or 3D .what is the rule or condition to be a 2D or 3D picture.How can I understand that?please help me?
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How to imagine a fractal dimension?

I am interested in fractal dimensions and more or less familiar with the self similarity, boxcounting and Hausdorff dimension. If someone asked me “What is a fractal dimension?”, my answer would be ...
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225 views

Easy visualizations of small countable ordinals

The ordinal number $\omega^2$ can be visualized as $\omega$-many copies of $\omega$. Likewise, the ordinal number $\omega^3$ can be visualized as $\omega^2$-many copies of $\omega$, arranged as ...
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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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Can correlation dimension of an attractor exceed the dimension of the space?

Here is the definition of the correlation dimension: http://en.wikipedia.org/wiki/Correlation_dimension Is there a proof that the correlation dimension cannon exceed the dimension of the space?
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Principal ideal domain is universally catenary …

... actually, even more general statement is true: Theorem. Every regular ring is universally catenary. (see for example Algebraic Geometry by Qing Liu, Corollary 2.16, Chapter 8) Though, the ...
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91 views

Hausdorff dimensions of smooth but non-rectifiable curves

Smooth curves of finite length have a Hausdorff dimension of 1. How about smooth but non-rectifiable (i.e. infinite-length) curves? Are they also of Hausdorff dimension 1, or does it depend on the ...
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Behavior of Hausdorff dimension under homeomorphisms

Let $X$ and $Y$ be metric spaces, $f : X\rightarrow Y$ a homeomorphism. Denote by $\dim_{\mathcal H}$ the Hausdorff dimension. I know that it is possible that $\dim_{\mathcal H} Y < \dim_{\mathcal ...
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199 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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the dimension problem of complex projection

Is it true that $\operatorname{dim }H^{0}(P^{n},P(T^{*}P^{n}))>0$? That is, is there a global holomorphic section? Here $P^{n}$ is $n$-dimensional complex projection space and ...