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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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Global dimension of the center

Let $R$ be a ring. Must the global dimension of the centre $Z(R)$ of the ring $R$ always be atmost that of $R$ itself? I mean is it generally true that: $D(Z(R)) \leq D(R)$ (where D is the global ...
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14 views

Drawing (multifractals) of any specified dimension

How could anyone draw or plot fractals with a given Haussdorf non-intenger dimension? Examples: D=0.5 D=1/3
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23 views

Find $\dim_H \operatorname{proj}_\theta E$ where is the circular Cantor set in the complex plane

$\newcommand{\proj}{\operatorname{proj}}$ $\dim_H(\cdot)$ is the Hausdorff dimension of a set. Let us denote the line through the origin in $\mathbb{R}^2$ which makes an angle of $\theta$ with the ...
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1answer
32 views

Hausdorff Dimension for Brownian motion over [0,1]

I am trying to calculate Hausdorff dimension for the trajectory of Brownian motion over $[0,1]$. I read the book of Morters and Peres and know that the dimension will be $\frac{3}{2}$. I tried to use ...
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27 views

Hausdorff dimension of closed intervals is not changed under $f(x)$

Let $f(x)=x^2$. Prove that for any $E\subseteq\mathbb{R}$ the dimension of image is not changed i.e $$\dim_HE=\dim_H(f(E))$$ Any set in $\mathbb{R}$ can be represnted as a countable union of ...
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70 views

Von Neumann and Hausdorff continuous dimensions are related?

Von Neumann in his book Continuous Geometry introduced (in a suitable lattice) a dimension function that has a continuous range. The definition of a dimension function is axiomatic: see Continuous ...
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26 views

Showing that $\mathcal{H}^s$ is Borel regular (assuming we know already know that $\mathcal{H}^s$ is measure)

I am trying to show that $\mathcal{H}^s$ (s-dimensional Hausdorff measure) is Borel regular. I am using the defintion $\mathcal{H}^s_{\delta}(F)=inf \Bigg\{ \sum_{i=1}^{\infty}|V_i|^s : \{V_i\} \text{ ...
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1answer
20 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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21 views

$\ell_2^\mathbb{Q}$ as countable cartesian product

Since countable cartesian product of $0$-dimensional spaces (metrizable, separable) is $0$-dimensional, why $\ell_2^\mathbb{Q}$ being $1$-dimensional doesn't contradict that statement? ...
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1answer
32 views

Are the following quotient spaces finite dimensional?

If we take $\mathbb{F}[x]$ to be the set of all polynomials over the field $\mathbb{F}$, $E$ to be the subset of all such even polynomials, $N$ to be the set of these polynomials that have degree less ...
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1answer
44 views

Subspaces and annihilators

I am trying to show this question. My understanding of annihilators is that for a vector space $V$ over $K$, with $S$ being a subset, the annihilator of $S$ is the subspace $S^0$ of linear functions ...
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1answer
44 views

Showing that a precursor to the packing measure (on $\mathbb{R}^n$) is not a measure

I am trying to prove the highlighted sentence. What countable dense sets should I consider? and how am I trying to prove this is not a measure? I am using the usual definition of a measure (and do ...
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1answer
18 views

Showing the equivalence of different definitions of the box-counting dimension

I am trying to prove the statment "by taking logarithms...". $\lim\limits_{\delta \rightarrow 0} \frac{log(N_{4\delta}(F))}{log(\frac{1}{4 \delta})} \leq \lim\limits_{\delta \rightarrow 0} ...
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1answer
26 views

alternative solution for $\dim R[T]\geq\dim R+1$.

please give some comments regarding my proof to the $$\dim R[T]\geq\dim R+1$$ Let $R$ be a noetherian ring, and let $n\geq 0$ by induction we only have to show that $\dim R[T]=\dim R+1$. It is easy ...
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2answers
81 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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2answers
63 views

Proof that $[v, Tv, T²v, … , T^n v]$ is a basis for $V$ ($dim(V)=n$)?

Let $T:V\to V$ be a linear map from a finite dimensional vector space over a field $F$ to itself. Assume $[v,Tv,T²v,...]$ spans $V$ for some $v \in V$. Don't know at all how to prove that $[v, Tv, ...
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1answer
32 views

Hausdorff dimension of a Modified Cantor like set

Suppose you have the unit interval $[0,1]$. For the first iteration you remove the segment $(1/5,3/5)$. So you are left with two intervals of lengths $1/5$ and $2/5$. You now repeat the process on the ...
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1answer
30 views

Relationship between the Hausdorff dimension and the Box-counting dimension

In Fractal Geometry by Falconer the author writes: If $1<\mathcal H^s(F)=\lim_{\delta\to0}\mathcal H_\delta^s(F)$ then $\log N_\delta(F)+s\log\delta>0$ if $\delta$ is sufficiently small. ...
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1answer
56 views

Compute the transcendence degree (transcendence degree and tensor products)

$\DeclareMathOperator{\quot}{Quot}\DeclareMathOperator{\tr}{tr}$ Let $I_1$ and $I_2$ be nontrivial ideals in $\mathbb C[x_1,\ldots,x_k]$ and $\mathbb C[y_1,\ldots,y_m]$, respectively. Define $$ R_1 ...
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1answer
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Problem with Hausdorff dimension

Let $\varepsilon>0,s\geq0$ and $C\subseteq \mathbb{R}^d$ be randomly given. Now define: $$ \mathcal{H}^s_\varepsilon(C)= \inf\biggl\{\,\sum^\infty_{n=1}(\rho (A_n))^s\biggm| C\subseteq ...
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1answer
39 views

Does interval spacing effect Hausdorff dimension of Cantor set?

Let $C=\bigcap_{j=0}^{2^n}C_j$, $C_0=[0,1]$, and the intervals in the construction of each stage of $C_j$ consists of removing the center 1/3 from the $j-1$ stage intervals. In other words, the ...
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1answer
20 views

Questions about open balls and convergence in Hilbert space

So, I've started reading about dimension theory and currently am dealing with a lemma which is used in a proof of $\dim(\mathcal{\ell}^{2}_{\mathbb{Q}})=1$. This lemma says that convergence of a ...
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1answer
28 views

What is the (algebraic) dimension of the dual of a vector space?

Let $V$ be a vector space over the field $\mathbb F (=\mathbb R$ or$ \mathbb C$). If $\alpha$ is the dimension of $V$, for some cardinal number $\alpha$, I want to know, what is the dimension of ...
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1answer
64 views

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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43 views

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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39 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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58 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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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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67 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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1answer
86 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
75 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
26 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
22 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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38 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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3answers
53 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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0answers
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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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47 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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3answers
59 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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31 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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1answer
34 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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60 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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56 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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0answers
40 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
168 views

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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1answer
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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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1answer
29 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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2answers
52 views

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 ...