See: Math Notebook
I'm writing up my solutions to math exercises that I have done to increase my understanding.
To Do
- Show a bijection between involutions and standard tableau.
- I misplaced my bijective proof of standard tableau and paths on Pascal's triangle, find it or redo it.
- Show that {$SU(2)$} is closed (and thus compact, because it is bounded).
- Show that {$SL(2,\mathbb{C})$} is an analytic manifold.
- Show that {$SU(2)$} is a differentiable manifold but not analytic.
- Show that {$SL(2,\mathbb{R})$} is a differentiable manifold but not analytic.
- Show that the complex solutions {$x$} and {$y$} to the equation {$x^2+y^2=1$} form a sphere with two holes.
- Learn how to calculate the hyperarea of an n-sphere and the hypervolume of the n-ball.
Need to Write Up
- {$SU(N)$} is bounded. Multiplying by the conjugate transpose shows that, for each row, and each column, the sum of squares equal 1, thus for each entry the magnitude is at most 1.
- {$SL(N)$} is not bounded. The determinant is 1, thus the volume is fixed, thus the vectors can be arbitrary large in any dimension.
Solved