When taking a penalty kick in soccer, the kicker must chose left or right.
And before he kicks, the goalkeeper must also decide left or right, because there is no time to see where the ball is going.
Because the kicker is right footed however, he kicker kicks better to one side than the other. So we have four probabilities:
- goal kick left keeper jumps left
- goal kick right keeper jumps right
- goal kick left keeper jumps right. Note that it is possible that this won't be a goal, even though the keeper is nowhere near the ball, as the ball might just miss the goal by a bit.
- kick right and keeper jumps left. Analogous to above
Not to be confused with algebra over a field, which is a particular algebraic structure studied within algebra.
As mentioned at Human Compatible by Stuart J. Russell (2019), game theory can be seen as the part of artificial intelligence that deas with scenarios where multiple intelligent agents are involved.
Applications of power, we have to remember it is there to notice how awesome it is!
- lightning
- motors
- sending nad receiving communication signals
- computers, which in turn can do computations and improved communication
These are basically technically minded people that Ciro Santilli feels have similar interests/psychology to him, and who write too much for their own good:
- cat-v.org
- gwern.net. Dude's a bit overly obsessed with the popup preview though! "new Wikipedia popups (this 7th implementation enables recursive WP popups)" XD
- settheory.net by Sylvain Poirier
- HyperPhysics
- Orange Papers
Maybe one day these will also be legendary, who knows:
Another category Ciro admires are the "computational physics visualization" people, these people will go to Heaven:
Related:
Institution led:
- www.biology.arizona.edu/ The Biology Project
Other mentions:
- arngren.net/ lots of images of toys and gear with descriptions in Norwegian
Ciro Santilli intends to move his beauty list here little by little: github.com/cirosantilli/mathematics/blob/master/beauty.md
The most beautiful things in mathematics are results that are:
- simple to state but hard to prove:
- Fermat's Last Theorem
- number of unknown rationality, e.g. is rational?
- transcendental number conjectures, e.g. is transcendental?
- basically any conjecture involving prime numbers:
- many combinatorial game questions, e.g.:
- surprising results: we had intuitive reasons to believe something as possible or not, but a theorem shatters that conviction and brings us on our knees, sometimes via pathological counter-examples. General surprise themes include:Lists:
- classification of potentially infinite sets like: compact manifolds, etc.
- problems that are more complicated in low dimensions than high like:
- generalized Poincaré conjectures. It is also fun to see how in many cases complexity peaks out at 4 dimensions.
- classification of regular polytopes
- unpredictable magic constants:
- why is the lowest dimension for an exotic sphere 7?
- why is 4 the largest degree of an equation with explicit solution? Abel-Ruffini theorem
- undecidable problems, especially simple to state ones:
- mortal matrix problem
- sharp frontiers between solvable and unsolvable are also cool:
- attempts at determining specific values of the Busy beaver function for Turing machines with a given number of states and symbols
- related to Diophantine equations:
- applications: make life easier and/or modeling some phenomena well, e.g. in physics. See also: explain how to make money with the lesson
Good lists of such problems Lists of mathematical problems.
Whenever Ciro Santilli learns a bit of mathematics, he always wonders to himself:Unfortunately, due to how man books are written, it is not really possible to reach insight without first doing a bit of memorization. The better the book, the more insight is spread out, and less you have to learn before reaching each insight.
Am I achieving insight, or am I just memorizing definitions?
Because the people who are crazy enough to think they can change the world are the ones who do.
Created by Dr. Rod Nave from Georgia State University, where he worked from 1968 after his post-doc in North Wales on molecular spectroscopy.
While there is value to that website, it always feels like it falls a bit too short as too "encyclopedic" and too little "tutorial-like". Most notably, it has very little on the history of physics/experiments.
Ciro Santilli likes this Rod, he really practices some good braindumping, just look at how he documented his life in the pre-social media Internet dark ages: hyperphysics.phy-astr.gsu.edu/Nave-html/nave.html
The website evolved from a HyperCard stack, as suggested by the website name, mentioned at: hyperphysics.phy-astr.gsu.edu/hbase/index.html.
Shame he was too old for CC BY-SA, see "Please respect the Copyright" at hyperphysics.phy-astr.gsu.edu/hbase/index.html.
exhibits.library.gsu.edu/kell/exhibits/show/nave-kell-hall/capturing-a-career has some good photo selection focused on showing the department, and has an interview.
Kell hall is a building of GSU that was demolished in 2019: atlanta.curbed.com/2020/1/31/21115980/gsu-georgia-state-atlanta-kell-hall-demolition-park-library-north
This is a good book. It is rather short, very direct, which is a good thing. At some points it is slightly too direct, but to a large extent it gets it right.
The main goal of the book is to basically to build the Standard Model Lagrangian from only initial symmetry considerations, notably the Poincaré group + internal symmetries.
The book doesn't really show how to extract numbers from that Lagrangian, but perhaps that can be pardoned, do one thing and do it well.
50,000,000x Magnification by AlphaPhoenix (2022)
Source. There are unlisted articles, also show them or only show them.