The Polarization identity for quadratic forms like inner products is also known as the parallelogram law. It is very important and always a good teaching moment in a multi-variable calculus course. It is more than just an identity. It tells that if know “lengths”, then we can recover “angles”. All …
The number theory of quaternion integers has connections to many beautiful parts of mathematics. The quaternions themselves are selected out from all the structures in mathematics as the only associative real non-commutative division algebra and so play a rather unique role in the entire landscape of mathematics. Its unit sphere …
We have seen last time that describing many particles moving on a manifold is best done by keeping on each cell a tag telling how many particles there are there. In other words, we do not evolve a sequence of maps but a divisor . This is more effective if …
There are exactly 3 associative real normed division algebras as the Frobenius theorem from 1879 tells. Each of them produce natural Lie groups . Each of them produces natural dense sphere packings . Each of them produces natural rings , the ring of integers, the Eisenstein ring and the ring …
A nice thing about mathematics is that it has no dogmas, statements which have to be taken on good faith. Axioms come closest, but by nature also, they come with an honest warning that one can either accept them or not. Already Euclid fought with the parallel axiom. Today it …
Eugene Wigner in 1939 associated elementary particles with irreducible representations of groups, especially the Poincare group. In a first year algebra course, we learn about representations of finite groups and especially the symmetric group , where there are p(n) irreducible representations, where p(n) is the number of integer partitions of …
Happy new year 2024. Here is the code displayed on the right upper corner of the board written this morning when wondering how frequent the situation is that the year is divisible by the year modulo 1000 minus 1. This happens for 2024 as it is divisible by 23. The …
Is the dyadic group of integers natural in the sense that there is a metric space which forces the group structure on it?
A natural metric space defines a unique group structure on the same set such that all group operations are isometries. The integers are not natural but the dihedral numbers are. The half integers are in this sense more natural than the integers.
