Im currently working on trajectory systems and forgot what tan() does and I kinda am embarrassed about that. Also I'm awful at trigonometry apparently. Haven't thought about it in a couple decades!
Hi all, I've started the arxiv submission process of my first author paper in the general math category, but it needs an endorser. Apparently the endorser must be someone who has published 2 papers earlier than 2 months ago and less than 5 years ago, in the general math category. Please let me know if you can. Thanks in advance! #math#arxiv#papers
I just realized that all perfect squares mod 9 can only be 0, 1, 4, 7, but I can't find an easier proof than by exhaustion (square all numbers 0 to 8, mod 9). Is there a more elegant proof of this?
mod 11 has a wider choice (0, 1, 3, 4, 5, 9), but I wonder how good of a “perfect square detector” they can be together. Of course if either proof (by 9s and by 11s) fails, it's not a perfect square, but how many “not perfect square” are perfect squares both mod 9 and mod 11?
I have a question about the aperiodic spectre tile (or the hat/turtle).
I know that the proof of aperiodicity works by showing that the tiles must fit together in a hierarchical structure that eventually repeats itself at a larger scale. But the larger units aren't literally scaled copies of the spectre. I also know that there is some freedom as to how you draw the edges of the spectre.
Is there a way you can draw the edges that allows you to literally use spectres to cover a larger copy of themselves? If so, is this way of doing it unique?
@OscarCunningham I'm pretty sure you can transform the hats' HTPF metatile system into a form where each higher-order metatile exactly covers a set of metatiles of the next order down. (Use the 'converged' metatile shapes; use a non-overlapping version of the expansion rules; do some horrible limiting thing that fractalises all the metatile edges.)
But then you still have four different fractally-shaped metatiles, and no way to decompose those into individual hats that are all congruent.
@OscarCunningham in fact, here's the paper I vaguely remembered seeing but couldn't put my hands on yesterday, which does pretty much what I said. https://arxiv.org/abs/2305.05639, diagrams on pages 7 and 8.
I said this to a room full of people years ago and it turned out to be controversial, so what the heck I'll post it here:
Science results and math theorems should not be named after people, and we should undertake to rename any that currently are. We should prioritize renaming results or theorems named after white men and other privileged categories of people, with special attention to cases where a privileged person accepted or was assigned credit for work a less-privileged person did.
"Mathematics must subdue the flights of our reason; they are the staff of the blind; no one can take a step without them; and to them and experience is due all that is certain in physics." – Voltaire (1694-1778) #quote#mathematics#math#maths
[ \sum_{n=0}^{\infty} {\frac{n^4}{n!}}=15e ]
This is strange enough to provoke wonder, but simple enough to serve as an entry-point to an interesting generalization.
@phonner@johncarlosbaez@paulmasson The “symbolic method” (as in the lovely "Analytic Combinatorics" book by Flajolet and Sedgewick) gives a slick (after you buy into it, i.e. the background) proof of Dobiński's formula. I wrote a quick post about it here a while ago; don't know how understandable it is: https://shreevatsa.net/post/permutations-dobinski/
#kdtree and ball trees seem cool, but require full knowledge of the thing I'm searching for. What if it's 7 dimensional and I only know 4 of the values?
I feel like a "parallel kd tree" with a separate binary index on each dimension would work better here.
Reduce depth. Allow unspecified values. It'd also be a snap to create and search each dim in parallel.