Articles of dimension theory

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 maximally distant spread would look as follows: If I had 4 vectors in a 3D space it would look […]

Transcendence degree of $K$

Let $K$ be field. How do I proof that transcendence degree of $K[X_1,X_2,\ldots,X_n]$ is $n$? The set $\{X_1,X_2,\ldots,X_n\}$ is algebraically independent over $K$. So, I have to show that every subset of size greater than $n$ is algebraically dependent.

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 $M=P\,\text{diag}(1,\dots,1,-1)P^{-1}$ is involutory ($MM=I$) and $\det(M)=-1$. Possible starting point: Consider $$\{P\in GL_n(\mathbb R),\ D \text{ invertible and diagonal s.t. } DP\text{diag}(1,\dots,1,-1)P^{-1}D^{-1} \text{ is orthogonal}\}$$ In general, this […]

Hausdorff Dimension Calculation

This is a homework question that I don’t know where to start with it. Can somebody help me please? I am trying to work out the Hausdorff dimension of the set $\{0,1,\frac{1}{4},\frac{1}{9},…\}$. I have worked this sort of thing out for a couple of fractal examples, but never for a set of numbers and I […]

$T:V\rightarrow W$ such that $N(T)=V'\subset V$ and $R(T)=W'\subset W$

How should one prove that there exists a linear map $T:V\rightarrow W$ such that $N(T)=V’\subset V$ and $R(T)=W’\subset W$ if $\dim(V’)+\dim(W’)=\dim(V)$, where $V$ and $W$ are finite-dimensional vector spaces? My “answer” is just a guess really… It seems pocketed with holes. What do you think?

Canonical $\pi$ dimensional space?

Can we talk about a canonical space of dimension $\pi$? Is there anything like $\mathbb R^\pi$? Have anyone met any fractal of dimension $\pi$?

Box Dimension Example

I am trying to find the lower- and upper-box dimensions (and show that they are the same) of the set $A=\{0,1,\frac{1}{4},\frac{1}{9},\ldots\}=\{\frac{1}{n^{2}}:n\in\mathbb{Z}_{\geqslant0}\}\cup\{0\}$. My thinking: There are $k$ intervals of length $\frac{1}{k^{2}}$ at stage $k$ of the construction. So $$\dim_{B}(A)=\lim_{\varepsilon\to0}\frac{\log{N_{\varepsilon}(A)}}{-\log{\delta}}=\lim_{k\to\infty}\frac{\log{k}}{-\log{k^{2}}}=\lim_{k\to\infty}\frac{\log{k}}{2\log{k}}=0.5.$$ But this doesn’t feel right. I haven’t found the upper- and lower- limits, I have just kind […]

Topological manifolds (dimension)

I am taking an introductory course to topology and the professor defined a topological manifold of dimension $n$ if it is hausdorff and if for every point $x$ there exists an open set $U$ around $x$ such that $U$ is homeomorphic to $\mathbb{R}^n$. My question is: By the way she defined it, one could have […]

Calculation of the $s$-energy of the Middle Third Cantor Set

As the title suggests, I am trying to calculate the $s$-energy of the middle third Cantor set. I am reading Falconer’s Fractal Geometry book, available here: http://www.dm.uba.ar/materias/optativas/geometria_fractal/2006/1/Fractales/1.pdf and this is an exercise (exercise 4.9 on page 45 of the pdf – 68 of the book). First, we define the $s$-potential at a point $x\in\mathbb{R}^{n}$ as […]

Algorithm to generate an uniform distribution of points in the volume of an hypersphere/on the surface of an hypersphere.

I am searching two simple/efficient/generic algorithms to generate a uniform distribution of random points: in the volume of a n-dimensional hypersphere on the surface of a n-dimensional hypersphere knowing the dimension $n$, the center of the hypersphere $\vec{x}$ and its radius $r$. How to do that ?