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Nov
17
revised Are there functions that satisfy $f(km)\bmod m=f(m)$ that are not of the form $m\mapsto n\bmod m$?
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Nov
17
revised A Characterization of the Tangent Function?
added 53 characters in body
Nov
17
asked Are there functions that satisfy $f(km)\bmod m=f(m)$ that are not of the form $m\mapsto n\bmod m$?
Nov
15
revised A Characterization of the Tangent Function?
added 251 characters in body
Nov
15
answered A Characterization of the Tangent Function?
Nov
14
comment Random Sampling of vectors on the Complex Unit Sphere
Ah, in that case, it's geometrically equivalent to the unit sphere in $\mathbb R^{2n}$ because $\mathbb C\cong\mathbb R^2$. So you just have to sample from a real unit sphere and interpret the result as a vector in $\mathbb C^n$. In fact, since it's a one-liner in Mathematica, here you go: ({1, I}.#) & /@ Partition[Normalize@RandomVariate[NormalDistribution[], 2 n], 2]
Nov
14
comment Random Sampling of vectors on the Complex Unit Sphere
Can you provide a definition of the complex unit sphere? I found a reference which said it is the subset of $\mathbb C^n$ such that $z_1^2+\cdots+z_n^2=1$, but such a set is unbounded and so I'm not sure it even makes sense to sample from it uniformly.
Nov
13
answered A Characterization of the Tangent Function?
Nov
13
comment Constraining mathematics to a subset of $\mathbb{R}$
See Why do we need the real numbers? Part IV discusses definable real numbers, which is what you are asking about.
Nov
13
comment A Characterization of the Tangent Function?
There are also two more trivial solutions $f=\pm\sqrt3$.
Nov
12
comment Matrix Multiplication Size Requirements
Yes, the problem is incorrect unless $m=n$.
Nov
12
comment How to visualize $ \mathcal P \ ( \ \mathcal P \ ( \ \mathbb R ^2 \ ) ) $?
How exactly does $\mathcal P(\mathbb R^2)$ contain text?
Nov
12
revised Sorting a list of points in 2-D clockwise
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Nov
12
comment Smallest circle enclosing three disjoint circles
I'll also leave this here for an arbitrary number of circles: Smallest-circle problem. Wikipedia only lists the problem of enclosing points, but the algorithms can typically be extended to enclose circles easily (I know Welzl's can, at least).
Nov
12
comment Smallest circle enclosing three disjoint circles
@Alex: Oh, right, my bad. The only interesting case is known as Apollonius's problem. :)
Nov
12
comment Sorting a list of points in 2-D clockwise
Nice overview of the possible ideas. Regarding point (2): The solution of a travelling salesman problem cannot intersect itself, because if you had an intersection that looked like X you could replace it by either || or = and reduce the total length. One of the two options will disconnect the tour into two disjoint cycles, but the other won't. Re (3): You don't need the full Delaunay triangulation. Its boundary is the same as the boundary of the convex hull of the points.
Nov
12
comment Smallest circle enclosing three disjoint circles
This is known as Apollonius's problem. The Wikipedia article lists a lot of different solution methods.
Nov
12
comment Can I assign a gravity field to an infinite grid of point masses?
+1: Computing the potential first is a good idea. If you add a constant to each potential contribution so it assigns say $(\frac12,\frac12)$ a potential of zero, it looks like the sum doesn't diverge: i.stack.imgur.com/m7Vu4.png (this is with point masses at $\{-100,99,\ldots,100,101\}^2$ and a $-1/r$ potential). I haven't proved absolute convergence though.
Nov
12
comment What is fleventy five?
Fleventy-five will be standard usage in the year 207̃012
Nov
11
comment Why using Gradient Descent?
If you want to minimize $f(x)=x^6/6-x^2/2+x$ by setting the derivative to zero, you have to solve $x^5-x+1=0$, which has no solution in terms of elementary functions.