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1500 questions
51
votes
2 answers
A strengthening of the Cauchy-Schwarz inequality
Suppose $\mathbf{v},\mathbf{w} \in \mathbb{R}^n$ (and if it helps, you can assume they each have non-negative entries), and let $\mathbf{v}^2,\mathbf{w}^2$ denote the vectors whose entries are the squares of the entries of $\mathbf{v}$ and…
Nathaniel Johnston
- 5,755
51
votes
11 answers
Intuition and/or visualisation of Itô integral/Itô's lemma
Riemann-sums can e.g. be very intuitively visualized by rectangles that approximate the area under the curve.
See e.g. Wikipedia:Riemann sum.
The Itô integral has due to the unbounded total variation but bounded quadratic variation an extra term…
vonjd
- 5,875
51
votes
4 answers
Does anyone know an intuitive proof of the Birkhoff ergodic theorem?
For many standard, well-understood theorems the proofs have been streamlined to the point where you just need to understand the proof once and you remember the general idea forever. At this point I have learned three different proofs of the…
Paul Siegel
- 28,772
51
votes
8 answers
The "derived drift" is pretty unsatisfying and dangerous to category theory (or at least, to me)
I'm currently a young, not-so-young mathematician, finishing its second postdoc. I developed an interest for rather different topics in the last few years but constantly, slowly converged towards something that has to do with (but at this point I'm…
51
votes
2 answers
Is there winning strategy in Tetris ? What if Young diagrams are falling?
Question 1
Is there a winning strategy (algorithm to play infinitely) in Tetris,
or is there a sequence of bricks which is impossible to pack without holes?
Consider generalized Tetris with Young diagrams (for some $n$) are falling down.
Question 2…
Alexander Chervov
- 23,944
51
votes
10 answers
Mathematically interesting screensavers
A screensaver is a computer program that fills a computer screen with a moving pattern that eluminates each pixel for approximately the same proportion of time. Originally designed to prevent burn-in of computer screens based on cathode-ray tubes,…
Daniel Moskovich
- 21,887
51
votes
1 answer
What is Atiyah's topological formulation of the odd order theorem?
Here is a quote from an article by Daniel Gorenstein on the history of the classification of finite simple groups (available here).
During that year in Harvard, Thompson began his monumental classification of the minimal simple groups. He soon…
spin
- 2,781
51
votes
5 answers
Can $N^2$ have only digits 0 and 1, other than $N=10^k$?
Pablo Solis asked this at a recent 20 questions seminar at Berkeley. Is there a positive integer $N$, not of the form $10^k$, such that the digits of $N^2$ are all 0's and 1's?
It seems very unlikely, but I don't have a proof. It's easy to see that…
Scott Morrison
- 7,540
51
votes
7 answers
What does a projective resolution mean geometrically?
For R a commutative ring and M an R-module, we can always find a projective resolution of M which replaces M by a sequence of projective R-modules. But as R is commutative, we can consider the affine variety X=Spec R and the sheaf of modules…
Justin DeVries
- 1,046
51
votes
2 answers
Ring-theoretic characterization of open affines?
Background
Recall that, given two commutative rings $A$ and $B$, the set of morphisms of rings $A\to B$ is in bijection with the set of morphisms of schemes $\mathrm{Spec}(B)\to\mathrm{Spec}(A)$. Furthermore, we know that Spec$(A)$ has a base of…
Manny Reyes
- 5,132
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51
votes
3 answers
What is the purpose of the flat/fppf/fpqc topologies?
There have been other similar questions before (e.g. What is your picture of the flat topology?), but none of them seem to have been answered fully.
As someone who originally started in topology/complex geometry, the étale topology makes some sense…
Simon Rose
- 6,250
51
votes
4 answers
Why do Pell equations appear in Ramanujan's pi formulas?
While answering this MSE question about the Pell equation $x^2-29y^2=1$, I noticed that certain fundamental solutions appeared in Ramanujan's famous pi formula.
I. Given the fundamental unit $\displaystyle U_{29} =\tfrac{5+\sqrt{29}}{2}$…
Tito Piezas III
- 12,044
51
votes
4 answers
what-if.xkcd.com: stabbing (simply connected) regions on the 2-sphere with few geodesics
In the latest what-if Randall Munroe ask for the smallest number of geodesics that intersect all regions of a map. The following shows that five paths of satellites suffice to cover the 50 states of the USA:
A similar configuration where the lines…
Moritz Firsching
- 10,663
51
votes
2 answers
Polynomial with the primes as coefficients irreducible?
If $p_n$ is the $n$'th prime, let $A_n(x) = x^n + p_1x^{n-1}+\cdots + p_{n-1}x+p_n$. Is $A_n$ then irreducible in $\mathbb{Z}[x]$ for any natural number $n$?
I checked the first couple of hundred cases using Maple, and unless I made an error in the…
Tobias Kildetoft
- 2,488
51
votes
2 answers
$H^4(BG,\mathbb Z)$ torsion free for $G$ a connected Lie group
Recently, prompted by considerations in conformal field theory, I was lead to guess that for every compact connected Lie group $G$, the fourth cohomology group of it classifying space is torsion free.
By using the structure theory of connected Lie…
André Henriques
- 42,480