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27
votes
Accepted
Does $A-A=\mathbb Q$ hold for $A=\{x^4+y^4:\ x,y\in\mathbb Q\}$?
According to Tito Piezas's website $x^4+y^4-(z^4+t^4) = N$,
There is an identity
$((2a+b)c^3d)^4 + (2ac^4-bd^4)^4 - (2ac^4+bd^4)^4 - ((2a-b)c^3d)^4 = a(2bcd)^4$
where $b = c^8-d^8$, for arbitrary {$a, …
- 1,567
13
votes
Why might André Weil have named Carl Ludwig Siegel the greatest mathematician of the 20th ce...
Recently I was digging through the meeting minutes for IAS and chanced upon a report prepared for Siegel to justify hiring him. The relevant pages are 13-18. In this report, it is stated that "we deci …
- 107
7
votes
1
answer
465
views
Realizability for constructive Zermelo-Fraenkel set theory
$ \def \CZF {\mathbf {CZF}}
\def \IZF {\mathbf {IZF}}
\def \A {\mathcal A}
\def \then {\mathrel \rightarrow}
\def \r {\mathrel \Vdash}
\DeclareMathOperator \V V $
In "Realizability for Constructive Ze …
- 21.3k
6
votes
Accepted
Connected graphs that are not 2 connected
See
Kemkes, Graeme; Sato, Cristiane M.; Wormald, Nicholas, Asymptotic enumeration of sparse $2$-connected graphs, Random Struct. Algorithms 43, No. 3, 354–376 (2013). Zbl 1273.05127. PDF
and referen …
- 1,446
39
votes
5
answers
38k
views
The letter $\wp$; Name & origin?
Do you think the letter $\wp$ has a name? It may depend on community - the language, region, speciality, etc, so if you don't mind, please be specific about yours. (Mainly I'd like to know the Englis …
- 103
9
votes
Recent breakthroughs with applied origins
If you include mathematical physics (string theory, QFT) to "applied mathematics", the breakthroughs are too numerous to list them here. Statistical mechanics is especially productive in its influence …
- 91.8k
15
votes
Examples of mathematical theories that are naturally written in exotic logics
It is up for debate how "natural" their approach is, but Smullyan and Fitting use modal logic to develop forcing, in their book, Set Theory and the Continuum Problem. See Michael Weiss's notes for a d …
- 82.6k
9
votes
Accepted
Beyond Presburger Arithmetic
From a paper by Françoise Point, "On the expansion $(N, +; 2^x)$ of Presburger arithmetic," I learned of a much more general result of Semenov, “Logical theories of one-place functions on the set of n …
- 4,713
8
votes
When is 2 qualitatively different from 3?
The Ising model is NP-complete for $d \geq 3$ but not $d \leq 2$.
The Ising Model Is NP-Complete. SIAM News, Volume 33, Number 6:
The three-dimensional spin glass model for the standard square lattic …
- 2,203
9
votes
Different ways of thinking about the derivative
Marsden & Weinstein use the "method of exhaustion" to define the derivative without limits.
- 2,203
5
votes
The word problem in braid groups
For a very nice (if slightly typesetting-challenged) survey of this subject, see Patrick Dehornoy's survey.
- 33.8k
97
votes
6
answers
18k
views
Why didn't Vladimir Arnold get the Fields Medal in 1974?
As you all probably know, Vladimir I. Arnold passed away yesterday. In the obituaries, I found the following statement (AFP)
In 1974 the Soviet Union opposed Arnold's award of the Fields Medal, the m …
- 13
11
votes
Accepted
Double category of algebras, lax and colax morphisms of algebras
The double category of pseudoalgebras, lax and colax morphisms is defined in §5.4 of Grandis and Paré's Multiple categories of generalised quintets. As far as I'm aware, it is the only reference for t …
- 10.5k
16
votes
The unproved formulas of Ramanujan
Bruce Berndt has claimed that all the claims in Ramanujan's "Lost Notebook" have been proved, with a solution to the the final problem being published by Berndt, Li, and Zaharescu in J. London Math. S …
- 82.6k
47
votes
Most interesting mathematics mistake?
Not just a great mistake, but also a great documentation of a mistake: Stallings's How not to prove the Poincaré Conjecture.1 (I think this paper is my answer to every community-wiki question.)
1Here …
- 12.9k