Algebra 1 and pre-algebra worksheets are something that all would love to have been able to dominate in high school. Sometimes they do not dominate it there and that's when it could be a bit difficult.
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Loops and their multiplication groups
A thread in 15 parts
(0/15)
Recall that a quasigroup (Q,*) is a set Q with a binary operation * such that for each a,b in Q, the equations a*x=b and y*a=b have unique solutions x,y. Groups are quasigroups and this property is usually one of the first things proved in elementary group theory.
(1/15)
Note that we don't assume associativity of *!
A loop is a quasigroup with an identity element. The story of why they are called loops is an interesting one and may even be true, but I will save it for another day. I am going to focus on loops in this thread.
(2/15)
Natural examples of nonassociative loops:
- The nonzero octonions under multiplication
- The sphere S^7 under octonion multiplication
- I have discussed other examples
For each x in a loop Q, define the left & right translations L_x, R_x : Q->Q by L_x(y)=xy and R_x(y)=yx. These mappings are permutations of Q. The composition L_x L_y of two left translations is not necessarily a left translation because Q is not necessarily associative.
(4/15)
A thread in 15 parts
(0/15)
Recall that a quasigroup (Q,*) is a set Q with a binary operation * such that for each a,b in Q, the equations a*x=b and y*a=b have unique solutions x,y. Groups are quasigroups and this property is usually one of the first things proved in elementary group theory.
(1/15)
Note that we don't assume associativity of *!
A loop is a quasigroup with an identity element. The story of why they are called loops is an interesting one and may even be true, but I will save it for another day. I am going to focus on loops in this thread.
(2/15)
Natural examples of nonassociative loops:
- The nonzero octonions under multiplication
- The sphere S^7 under octonion multiplication
- I have discussed other examples
Rethinking Vector Addition
— Michael Kinyon (@ProfKinyon) December 1, 2020
or
How I Learned to Stop Worrying and Love Nonassociativity
A thread in 29 tweets
(0/28)
For each x in a loop Q, define the left & right translations L_x, R_x : Q->Q by L_x(y)=xy and R_x(y)=yx. These mappings are permutations of Q. The composition L_x L_y of two left translations is not necessarily a left translation because Q is not necessarily associative.
(4/15)
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1/Politics thread time.
To me, the most important aspect of the 2018 midterms wasn't even about partisan control, but about democracy and voting rights. That's the real battle.
2/The good news: It's now an issue that everyone's talking about, and that everyone cares about.
3/More good news: Florida's proposition to give felons voting rights won. But it didn't just win - it won with substantial support from Republican voters.
That suggests there is still SOME grassroots support for democracy that transcends
4/Yet more good news: Michigan made it easier to vote. Again, by plebiscite, showing broad support for voting rights as an
5/OK, now the bad news.
We seem to have accepted electoral dysfunction in Florida as a permanent thing. The 2000 election has never really
To me, the most important aspect of the 2018 midterms wasn't even about partisan control, but about democracy and voting rights. That's the real battle.
2/The good news: It's now an issue that everyone's talking about, and that everyone cares about.
3/More good news: Florida's proposition to give felons voting rights won. But it didn't just win - it won with substantial support from Republican voters.
That suggests there is still SOME grassroots support for democracy that transcends
4/Yet more good news: Michigan made it easier to vote. Again, by plebiscite, showing broad support for voting rights as an
5/OK, now the bad news.
We seem to have accepted electoral dysfunction in Florida as a permanent thing. The 2000 election has never really
Bad ballot design led to a lot of undervotes for Bill Nelson in Broward Co., possibly even enough to cost him his Senate seat. They do appear to be real undervotes, though, instead of tabulation errors. He doesn't really seem to have a path to victory. https://t.co/utUhY2KTaR
— Nate Silver (@NateSilver538) November 16, 2018