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Archive for March, 2016

On Proving Eigenvalue = 1 for Particular Surfaces

March 21st, 2016 No comments

So, as you have read here, I've been saying for quite a few years now that (bounded) smooth surfaces on \left[ 0, 1 \right]^2 \to \mathbb{R} have certain invariants when viewed from a particular perspective (eigenvalues, eigenfunctions).  There are particular surfaces which I dubbed Pasquali patches (for lack of a better word) and a particular construction, namely

 p(x,y) = f_1(x) g_1(y) + f_2(x) \frac{1-g_1(y) F_1}{F_2}

with F_2 = \int_0^1 f_2(x) \, dx \neq 0 which are very special in that they have lots of properties which are interesting and closely tied to probability theory.  I have now proven for this particular construction that it possesses two very specific eigenvalues given a particular operator "star" \star... one of which is \lambda_1 = 1, regardless of function choices for f_1(x), g_1(y) and almost arbitrary choice of f_2(x), which, by requirement needs F_2 \neq 0.  This mimics well known probability mathematics (except in the surface realm) and operator theory/linear algebra. I think of this as a very proud accomplishment.

Thusly, I have revamped the relevant sections in Compendium full of new and juicy recharacterizations in order to be able to do just this... particularly Section 2, the definition of Pasquali patches and Section 11.6, Relevant Generalizations as Applied to Pasquali Patches (where I have included such a proof).  Section 2 is now a lot tighter than it used to be, and I'm trying really hard to go over everything to close all the loopholes I've left so that Compendium isn't just notes but an actual... Volume of Mathematics or Book or something.

I am very close to a full eigenvalue theory which I will apply to a generalization of the Pasquali patch formula above, giving a multiplicity of calculable eigenvalues for, not just Pasquali patches, but any sufficiently well behaved surface on \left[ 0, 1 \right]^2 \to \mathbb{R} .

Part I v26