1) for some reason, I forgot to put divisor symbols on page 3. the very last line of page 3 should have theta values pi/2 and 3pi/2, as opposed to pi2 and 3pi2. (obv.). Also on page 3 in section 2, the radius of the aster is abs(sec) DIVIDED BY (1+yaddayadda).
2) I did a horrible job of defining the heat kernel. The implicit definition I gave (via heat kernel of As) was (laplacian)times(function squared). Aside from correcting lack of an explicit definition itself, I have to also amend the implicit definition I gave. Via "Heat Kernels on Manifolds, Graphs, and Fractals" by Alexander Grigor'yan, the heat kernel (on a manifold) should be (laplacian wrt to some borel measure)times(function). Since As exists in Real plane (& Real 3-space as the hyperbolic octahedron) , the borel measure is ("well" or "naturally"?) defined (measure is simply defined, geometrically/differentially, by unit norms as measure) , and so I amend my definition to comply with the definition given by Grigor'yan.
I'd also like to make a note of the fact that I have had no help, guidance, or counsel with any professor or student in writing this or any other paper. This is all purely original research in that sense (whether or not someone has already done this exact same research is not known to me). There are mistakes, I'm sure. And I should be held accountable for those mistakes, but only as accountable as any other undergraduate doing research in an autodidactic manner. I don't mean that to sounds condescending or conceited, for what it's worth.