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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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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38 views

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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16 views

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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9 views

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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25 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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1answer
50 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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98 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
42 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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20 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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18 views

Proving that $\dim_M{K_{\lambda,\gamma}}=\dim_HK_{\lambda,\gamma}$

Define $K_{\lambda,\gamma}$ to be the attractor of the IFS $$\bigg\{\bigg(\matrix {\lambda&0\\0&\gamma}\bigg)\bigg(\array{x\\y}\bigg),\bigg(\matrix ...
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49 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
48 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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1answer
49 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
23 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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1answer
39 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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33 views

Proving that plane - cantor - set contains an interval

Let $C_\lambda$ the cantor set which is defined by the IFS $\{\lambda x,\lambda x+(1-\lambda)\}$ and also let $E=C_\lambda\times C_\lambda$. Denote the orthogonal projection of the set from the ...
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1answer
29 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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13 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 ...
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25 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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257 views

Is there a notion in mathematics saying that, in a sense, all finite dimensions are actually infinite dimensional?

So then every ordered pair or triplet and so on would be actually represented by an infinite sequence of numbers, and what we think of as 3 dimensions would mean that the point has an infinite number ...
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35 views

Hausdorff dimension of a ball

Let $\{f_1,\dots,f_m\}$ be an IFs and $E_n$ be the associated self similar set. It's known that $E_n$ is a union of disjoint balls $B(x_i,R\cdot r^n)$ (balls with same radius but not the same ...
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1answer
44 views

Restrictions of maps between projective varieties.

Let $f\colon X\to Y$ be a surjective algebraic map between two projective $k$-varieties, where $k$ is algebraically closed. Let $n=\dim(X),\,m=\dim(Y)$. Suppose furthermore that X,Y are irreducible. ...
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1answer
55 views

Hausdorff dimension of $\lim_{n\to\infty}\sin(2^nx)$

Calculate the Hausdorff dimension,$\dim_H$ of $$S=\{x\in(0,1):\lim_{n\to \infty}\sin2^nx=0\}$$ By definition We need to find the minimal $\alpha$ s.t $\sum_{i\in I}|U_i|^\alpha$ is minimal where ...
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202 views

The unit square has dimension two?

Show that the Lebesgue Covering Dimension of the unit square $I^2$ with $I=[0,1]$ is two. I know that a compact subset of Euclidean space $\Bbb R^n$ has Lebesgue dimension at most $n$. So it ...
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1answer
32 views

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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1answer
15 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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1answer
31 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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48 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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32 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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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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35 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
23 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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22 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
41 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
59 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
47 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
28 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
28 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
98 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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68 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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43 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
41 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
68 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
36 views

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
48 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
22 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
37 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
78 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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44 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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42 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 ...