P.S. I should've mentioned that this result is best possible: The real projective spaces P^n for n > 1 cannot be immersed in R^k for k < 2n - g(n). —Dan
On Jun 1, 2016, at 7:48 PM, Dan Asimov <asimov@msri.org> wrote:
An interesting result in topology is that for n > 1 every compact smooth n-manifold can be immersed* in the Euclidean space of dimension 2n - g(n).
—Dan ________________________________________________ * An immersion is a smooth map between manifolds
f: M —> W
such that the derivative matrix Df(x) has rank = dim(M) at every x in M.
This was a conjecture since before 1960 and was finally proved in 1985 by Ralph Cohen.
On Jun 1, 2016, at 5:10 PM, Erich Friedman <erichfriedman68@gmail.com> wrote:
Let n be a positive integer. Let g(n) be the sum of the binary digits of n. I am looking for interesting facts involving g(n). Here are 3 that i have so far, in increasing difficulty to prove:
Theorem 1: The minimum number of integral powers of 2 to sum to n is g(n).
Theorem 2: The highest power of 2 to divide n! is 2^[ n - g(n) ].
Theorem 3: The number of odd entries in the nth row of Pascal's Triangle is 2^[ g(n) ].
Anyone know any others?