0
votes
0answers
41 views

Why does distance lose meaning in high-dimensional space?

I'm working on an algorithm that clusters points in extremely high-dimensional space (thousands, if not more). However, I came across this wikipedia page: ...
1
vote
0answers
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 ...
6
votes
1answer
141 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 $N$ ...
0
votes
2answers
116 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 ...
0
votes
1answer
23 views

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 ...
7
votes
2answers
227 views

What does it mean to be $0.9-$Dimension?

We can visualize $1\mathrm D$, $2\mathrm D$, $3\mathrm D$ and we can think of a higher $M-\mathrm {Dimension}$ where $M\in \mathbb N^+$as a vector. But I recently learned that there are non integer ...
6
votes
2answers
195 views

Can a 2D person walking on a Möbius strip prove that it's on a Möbius strip?

Or other non-orientable surface, can a 2D walker on a non-orientable surface prove that the surface is non-orientable or does it always take an observer from a next dimension to prove that an entity ...
0
votes
2answers
173 views

Proof of the following: How many $(n-2)$ dimensional faces from a corner of a hypercube

I asked a question earlier regarding the number of $(n-2)$ dimensional faces exiting a corner of an $n$ dimensional hypercube. (For example the number of points in a corner of a square, or the number ...
4
votes
2answers
199 views

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 ...
2
votes
5answers
3k views

Orthogonal planes in n-dimensions

In three dimensions two planes are orthogonal when their normal vectors are orthogonal (their inner product is zero). For example, planes $xy$ and $xz$ are orthogonal because the vectors $\hat{z}$ and ...