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Results tagged with gm.general-mathematics
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user 2926
Questions about mathematics which don't fall into the other arXiv categories. If you have a general question about mathematics but it is not research level, it's off-topic but it might be welcomed on Mathematics Stack Exchange.
12
votes
Accepted
Is there any monoid in which the product of two non-invertible elements could be invertible?
Well, why not?
Let $\oplus_{n \in \mathbb{N}} k$ be a direct sum of countably many copies of a 1-dimensional space over a field $k$; the direct sum affords a standard basis $e_i = (0, \ldots, 0, 1, …
- 53.3k
34
votes
Is amateur research in mathematics viable?
As someone who has gone through a in-some-ways similar experience, I don't think you would be ostracized, and yes, you absolutely could still publish work. You might need people to endorse your work, …
18
votes
17 camels trick
Evaluation of definite integrals via the method of differentiation under the integral sign has this character. This probably needs no introduction, but the idea is to view the integrand $f(x)$ of $\in …
21
votes
A search for theorems which appear to have very few, if any hypotheses
The theorem of Nash and Tognoli says that any compact smooth manifold is diffeomorphic to a nonsingular real algebraic set.
10
votes
Basic results with three or more hypotheses
A compact convex subset of $\mathbb{R}^n$ with nonempty interior is homeomorphic to the $n$-dimensional ball.
17
votes
Where do surreal numbers come from and what do they mean?
Not numbers exactly, but certainly Conway's games (and related game-like structures) have aroused plenty of interest in certain category theorists and logicians, as they can be used to give categorica …
- 53.3k
17
votes
Accepted
What is the term for combining functions $f_1,f_2,\dots,f_n$ into a tuple $(f_1,\dots,f_n)$?
I was encouraged to make my comment an answer:
In the case $n = 2$, I would call it the pairing. Similarly, one has "tripling", "quadrupling", and so in general one might call it the ($n$-)tupling o …
- 53.3k
9
votes
Where can square roots come from when they are not distances?
Fixed points of fractional linear transformations $x \mapsto \frac{ax + b}{cx + d}$, where $a, b, c, d$ are integers, are usually quadratic surds. Equivalently, values of periodic regular continued fr …
6
votes
Are there proofs that you feel you did not "understand" for a long time?
I didn't feel I had really grokked Euler's magnificent identity
$$e^{ix} = \cos(x) + i \sin(x)$$
for a very long time. The first time I saw this formula, I think I was fourteen, and it was one of …
8
votes
Individual mathematical objects whose study amounts to a (sub)discipline?
The braid group.
The Monster group.
The Steenrod algebra.
The representation ring of the symmetric group.
28
votes
Dimensional Analysis in Mathematics
This may be somewhat obliquely along the lines you are asking about, but I think it's interesting enough that it deserves to be made public.
My friend James Dolan has been developing with a number o …
- 53.3k
4
votes
Are there proofs that you feel you did not "understand" for a long time?
The recent answer by coudy reminds me that for a very long time I didn't really grok the proof of Minkowski's inequality from Hölder's inequality, even though it's fairly simple. It just looks like a …
19
votes
How to refer to a “theorem” that you have shown to be wrong
In my opinion, it would be a bad idea to label statements known to be false as theorems. If you really want to do this, maybe you could put inverted commas around the word "theorem", to indicate you e …
12
votes
Proof synopsis collection
I don't know that I have a favorite proof-synopsis, but here's one I like which is a little different from the way most people prove it.
Proposition. Let $A$ be a real symmetric $n \times n$ matrix. …
14
votes
Sophisticated treatments of topics in school mathematics
Another categorical example: the laws of arithmetic, as applied to the arithmetic of finite integers, are ultimately explicable by the fact that the category of finite sets $\mathbf{Fin}$ is a cartesi …