oly   193
Learn economics and see that investment and consumption levels (as percentages) depend only marginally on age and existing net worth and mostly on your risk preferences and utility function.
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11 weeks ago by nhaliday
gn.general topology - Pair of curves joining opposite corners of a square must intersect---proof? - MathOverflow
In his 'Ordinary Differential Equations' (sec. 1.2) V.I. Arnold says "... every pair of curves in the square joining different pairs of opposite corners must intersect".

This is obvious geometrically but I was wondering how one could go about proving this rigorously. I have thought of a proof using Brouwer's Fixed Point Theorem which I describe below. I would greatly appreciate the group's comments on whether this proof is right and if a simpler proof is possible.

...

Since the full Jordan curve theorem is quite subtle, it might be worth pointing out that theorem in question reduces to the Jordan curve theorem for polygons, which is easier.

Suppose on the contrary that the curves A,BA,B joining opposite corners do not meet. Since A,BA,B are closed sets, their minimum distance apart is some ε>0ε>0. By compactness, each of A,BA,B can be partitioned into finitely many arcs, each of which lies in a disk of diameter <ε/3<ε/3. Then, by a homotopy inside each disk we can replace A,BA,B by polygonal paths A′,B′A′,B′ that join the opposite corners of the square and are still disjoint.

Also, we can replace A′,B′A′,B′ by simple polygonal paths A″,B″A″,B″ by omitting loops. Now we can close A″A″ to a polygon, and B″B″ goes from its "inside" to "outside" without meeting it, contrary to the Jordan curve theorem for polygons.

- John Stillwell
nibble  q-n-a  overflow  math  geometry  topology  tidbits  intricacy  intersection  proofs  gotchas  oly  mathtariat  fixed-point  math.AT  manifolds  intersection-connectedness
october 2017 by nhaliday
Best Topology Olympiad ***EVER*** - Affine Mess - Quora
Most people take courses in topology, algebraic topology, knot theory, differential topology and what have you without once doing anything with a finite topological space. There may have been some quirky questions about such spaces early on in a point-set topology course, but most of us come out of these courses thinking that finite topological spaces are either discrete or only useful as an exotic counterexample to some standard separation property. The mere idea of calculating the fundamental group for a 4-point space seems ludicrous.

Only it’s not. This is a genuine question, not a joke, and I find it both hilarious and super educational. DO IT!!
nibble  qra  announcement  math  geometry  topology  puzzles  rec-math  oly  links  math.AT  ground-up  finiteness  math.GN
october 2017 by nhaliday
What is the best way to parse command-line arguments with Python? - Quora
- Anders Kaseorg

Use the standard optparse library.

It’s important to uphold your users’ expectation that your utility will parse arguments in the same way as every other UNIX utility. If you roll your own parsing code, you’ll almost certainly break that expectation in obvious or subtle ways.

Although the documentation claims that optparse has been deprecated in favor of argparse, which supports more features like optional option arguments and configurable prefix characters, I can’t recommend argparse until it’s been fixed to parse required option arguments in the standard UNIX way. Currently, argparse uses an unexpected heuristic which may lead to subtle bugs in other scripts that call your program.

consider also click (which uses the optparse behavior)
q-n-a  qra  oly  best-practices  programming  terminal  unix  python  libraries  protocol  gotchas  howto  pls  yak-shaving  integration-extension
august 2017 by nhaliday
Links 6/15: URLing Toward Freedom | Slate Star Codex
Why do some schools produce a disproportionate share of math competition winners? May not just be student characteristics.

My post The Control Group Is Out Of Control, as well as some of the Less Wrong posts that inspired it, has gotten cited in a recent preprint article, A Skeptical Eye On Psi, on what psi can teach us about the replication crisis. One of the authors is someone I previously yelled at, so I like to think all of that yelling is having a positive effect.

A study from Sweden (it’s always Sweden) does really good work examining the effect of education on IQ. It takes an increase in mandatory Swedish schooling length which was rolled out randomly at different times in different districts, and finds that the districts where people got more schooling have higher IQ; in particular, an extra year of education increases permanent IQ by 0.75 points. I was previously ambivalent about this, but this is a really strong study and I guess I have to endorse it now (though it’s hard to say how g-loaded it is or how linear it is). Also of note; the extra schooling permanently harmed emotional control ability by 0.5 points on a scale identical to IQ (mean 100, SD 15). This is of course the opposite of past studies suggest that education does not improve IQ but does help non-cognitive factors. But this study was an extra year tacked on to the end of education, whereas earlier ones have been measuring extra education tacked on to the beginning, or just making the whole educational process more efficient. Still weird, but again, this is a good experiment (EDIT: This might not be on g)
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march 2017 by nhaliday

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