Maybe I'll just show my algebraic ineptitude, but how do you know that g' restores "everything else"? I get that gg' = I, of course, but gT*g'? Rubik cube algebra isn't commutative, is it?
The part that is important is T has a very specific effect, such as rotating three corner pieces, that doesn’t change most of the cube. Then you pick a g that doesn’t affect those pieces either (except in the way that you want) so that gTg’ does what you want.
It doesn’t work in general for any arbitrary T of course since it’s not commutative as you noted. This is more of a cubing thing than an algebra thing.
You are right that it is not really an algebra thing. It's a bijection between finite sets thing. If you apply a bijective mapping from set A onto set B, then apply some permutation to B, and then apply the inverse mapping, you get a permutation of A that has the same structure as the permutation you applied to B.
By "same structure" I mean that if written as a product of disjoint cycles has the same number and sizes of cycles.
Abstract algebra is probably the first place most people would encounter it though, in the context of conjugates in groups.
Let's step away from the cube and consider permutations on a set of N items, labeled 1, 2, 3, ..., N. In the example below N is at least 9. We start out with item 1 at position 1, item 2 at position 2, and so on.
Suppose we want to move what is at 4 => 8, 8 => 9, and 9 => 4, leaving everything else unchanged but the only permutation we know that moves 3 items in a cycle leaving everything else unchanged moves 1 => 2, 2 => 3, and 3 => 1. This latter permutation we'll call T.
Consider any permutation g that does 4 => 1, 8 => 2, and 9 => 3. It may or may not move other things. The inverse of g, g', does 1 => 4, 2 => 8, and 3 => 9.
Let's work out what happens if we do gTg'.
First let is just look at items 4, 8, and 9, which are at positions 4, 8, and 9, respectively. g takes those to positions 1, 2, and 3. Then T takes positions 1, 2, 3 to 2, 3, 1, so what we have in positions 1, 2, 3 is items 9, 4, and 8 in that order. Finally g' takes what is in positions 1, 2, and 3 to 4, 8, and 9, so we end up with item 9 at 4, item 4 at 8, and item 8 at 9.
So for positions 4, 8, and 9, gTg' does 4 => 8, 8 => 9, and 9 => 4.
g necessarily had to move whatever was in 1, 2, and 3 out of the way to make room for 4, 8, and 9, so we have to consider what happens to what was at 1, 2, 3. Let's just look at 1. It has to go somewhere. Call that position p. So g does 1 => p, where p is not 1, 2, or 3. g' does p => 1.
T only moves things in 1, 2, 3, so after g moves whatever was originally at 1 to p, T leaves it along. g' then does p => 1, putting it back where it came from. So we see that gTg' does not move 1. Same reasoning applies for 2 and 3.
If g moves any x else other than 1, 2, 3, 4, 8, 9 we can use the same argument. x => y for some y that is not 1, 2, or 3. T does not move y. g' does y => x, putting x back where it came from.
Thus we can conclude that gTg' only moves 4, 8, and 9.
Another way you can visualize why it works when applied to the cube is by cheating a bit. First do g legitimately by actually doing the moves on your cube. Then instead of actually doing the moves for T just repaint the faces on the cubies that T permutes so it looks like you did T. Then legitimately do g'.
Since you haven't actually done T the only actual moves you have done are gg' which as you've noted is of course I, so everything is back where it started but with some of the faces on some of the cubies repainted. The repainted faces are exactly those that would have ended up moved if you had actually done T. The faces not repainted are those that would be undisturbed by gTg'.
The key here is that all these mappings are finite, one to one, and onto (bijective if we want to get fancy). If you apply such a mapping from A to B, then permute k elements in B, and then apply the inverse mapping, you end up with k elements permuted in A.
It's dead simple and obvious in retrospect, but I don't think I would have ever thought of that! I've always wondered if there's a lineage to this set of definitions. Does anyone know?
> In 1973, professor Sir James Lighthill was asked by the UK Parliament to evaluate the state of AI research in the United Kingdom. His report, now called the Lighthill report, criticized the utter failure of AI to achieve its "grandiose objectives." He concluded that nothing being done in AI couldn't be done in other sciences. He specifically mentioned the problem of "combinatorial explosion" or "intractability", which implied that many of AI's most successful algorithms would grind to a halt on real world problems and were only suitable for solving "toy" versions.[15]
> The report led to the complete dismantling of AI research in England.[15] AI research continued in only a few universities (Edinburgh, Essex and Sussex).
In retrospect it was at least arguably the right call, no? Suspend most research and resume when available processing power was orders of magnitude greater, a process that was independently driven by demands other than (and much greater than) AI research.
Playing devil's advocate only slightly, maybe particle physics should similarly pare down to a bare maintenance level of research (or even mostly teaching) for a few centuries until we can harness much higher energies.
They were right; modern ML doesn’t use any of the ideas the previous generation of AI people were pursuing. It turns out no expert system is a match for just doing a bunch of matrix multiplications.
No bike paths, just rural roads. Generally without any center line, even. We gave our kids mobile phones early on so they could stay in touch while being independent. It is common for kids in town to go fishing or sailing small boats on their own on a large pond in town. Lots of autonomy.
Yes, this was, and still is, normal. Some people do drive their kids to school. But kids are still very independent. It's a small town with spread out houses. The general store, library, after school sports, and the town beach, in warm weather, are the choices of things to do. Things have changed a bit. It's more upscale. Farm dogs don't roam about. Hidden fences are installed most places. That makes life easier for cyclists.
"Karl Marx was right, socialism works, it is just that he had the wrong species"
It seems that what Marx called socialism is basically communism.
By 1888, Marxists employed socialism in place of communism as the latter had come to be considered an old-fashioned synonym for socialism. It was not until after the Bolshevik Revolution that socialism was appropriated by Vladimir Lenin to mean a stage between capitalism and communism.
> It seems that what Marx called socialism is basically communism.
Many people (including me) would say that there isn't any meaningful difference between the two, whether we are talking about Marx's time or any time since. I don't know whether Wilson thought there was any meaningful difference between the two; as far as I know nobody asked him.
Is there any forecast of where the equilibrium happen? Say, if you're in a 1000-people closed society where cryptocurrency is the only thing used to buy stuff, what's the economic calculation for the long-term energy consumption? Seems like there should be papers about that...
