## On Riding the Wave

So here it is, the pinnacle of my research effort thus far. I'll start by the definitions:

Definition 1. Let $f(x,y)$ and $g(x,y)$ be surfaces so that $f,g \colon [0,1] \times [0,1] \to \mathbb{R}$. The star operator $\star \colon [0,1]^2 \times [0,1]^2 \to [0,1]^2$ takes two surfaces and creates another in the following way:

with the central transformation being defined by $\diamond \colon [0,1]^2 \times [0,1]^2 \to [0,1]^2$

and the last transformation that takes $h(x,z) \rightsquigarrow h(x,y)$ we will call $j \colon [0,1]^2 \to [0,1]^2$. Thus

Definition 2. Define a continuous, bounded surface $p(x,y)$, with $p \colon [ 0, 1 ] \times [ 0, 1 ] \to \mathbb{R}^+ \cup \{0\}$,  and let $\int_0^1 p(x,y) \, dx = 1$ be true regardless of the value of $y$.    In other words, integrating such surface with respect to $x$ yields the uniform  probability distribution $u(y)$, $u \colon [0,1] \to \{1\}$. We will call this a strict Pasquali patch, and such is intimately related to probability notions.  With $p \colon [ 0, 1 ] \times [ 0, 1 ] \to \mathbb{R}$, we have a more general definition for a Pasquali patch.

Construction 1. Let $p(x,y) = \sum_{i = 1}^n f_i(x) \cdot g_i(y) = \mathbf{f}(x) \cdot \mathbf{g}(y)$, a function which consists of a finite sum of pairs of functions of $x$ and $y$. In the spirit of conciseness, we omit the transpose symbology, thusly understanding the first vector listed in the dot product as a row vector and the second vector as a column vector.  Then $p(x,y)$ is a Pasquali patch provided

and $F_n \neq 0$. Thus, we may choose $n-1$ arbitrary functions of $x$,  $n-1$ arbitrary functions of $y$, an $n$th function of $x$ so that $F_n \neq 0$, and

We may write the normalized version as:

and again observe that the unit contribution to the integral of the Pasquali patch is provided by $f_n^*(x)$, so that $F_n^* = 1$.

Claim 1. Pasquali patches constructed as by Construction 1 are closed.

Proof. To make the proof clear, let us relabel the normalized version of Construction 1 as

with $\int_0^1 \mathbf{P}_x(x) \, dx = 0$ so as to manipulate the equation more simply, and

with $F_n^* = 1$ and $H_n^* = 1$. Then

with $\alpha = \int_0^1 \mathbf{P}_x(1-y) \cdot \mathbf{Q}_y(y) \, dy$, $\beta = \int_0^1 \mathbf{P}_x(1-y) \cdot \mathbf{1} \, dy = 0$, and $\gamma = \int_0^1 f_n^*(1-y) \cdot \mathbf{Q}_y(y) \, dy$. We're not too concerned of the form of the resultant star product as much as its structure. Observe

can be folded back into function vectors $\mathbf{R}_x(x)$ and $\mathbf{R}_y(y)$. Thus the structure of Construction 1 functions is preserved when we multiply one by another, showing closure. Of course the property of Construction 1 being Pasquali patches means $r(x,y)$ is closed under that property, and so is a Pasquali patch also, as can be seen when we integrate across $x$:

and the unit contribution is given by $h_n^*(x)$. \qed

Claim 2. Pasquali patches constructed as by Construction 1 have powers:

with

and

Proof by Induction. First, using the formula observe

and the second power

which is exactly what we expect from the definition of Construction 1 and Claim 1. Next, let us assume that the formula works and

Let us examine $p_{k+1}(x,y) = p_k(x,y) \star p(x,y)$

term by term upon dotting. The first dot the first term is:

The first dot the last term is:

The last dot the first term is:

The last dot the last term is:

The middle dot the first term is:

Finally, the middle dot the last term vanishes:

Putting all this information together we get:

\qed

Claim 3. It follows that

and $\lvert \alpha \rvert < 1$, $\gamma$ bounded, both conditions necessary and sufficient to establish that such a limiting surface indeed exists (convergence criterion). Furthermore, we check that this is indeed a Pasquali patch.

Proof. To reach a steady state limit,

Next, the steady state limit must be solely functions of $x$, so the functions of $y$ must vanish at the limit. Thus, it follows that $\lvert \alpha \rvert < 1$. We have now established bounds on $\alpha$, which happen to be exactly the radius of convergence of the geometric series:

and

gives the desired result:

As a check, we integrate across $x$ to corroborate the definition of Pasquali patch:

Claim 4.

is the eigenfunction corresponding to eigenvalue $\lambda = 1$ of all Construction 1 functions, through each power independently.

Proof. An eigenfunction $e(x)$ has the property

where the eigenfunction's   corresponding eigenvalue is $\lambda$.  The claim is more ambitious, and we will show that $p_\infty(x) \star p_n(x,y) = 1 \cdot p_\infty(x)$ for any $n \in \mathbb{Z}^+$.  The left-hand side is

Observe the first term dotted with the middle and last term produce $\beta$ which annihilates the results, so that the only relevant term is the first dot the first:

The second term dot the first produces:

The second term and the second:

and the second by the last term gives

Factoring gives

The parenthetical part of this last formulation is equivalent to

within the bounds already established for $\alpha$, and the result of the star product is

as we wanted to show. \qed

Claim 5.

$A$ is a constant, is the eigenfunction corresponding to eigenvalue $\lambda = \alpha$ of

Proof. The eigenfunction equation is suggestive of what we must do to prove the claim: $e(x) \star p(x,y) = \lambda e(x)$. We must show that, starring the eigenfunction with $p(x,y)$, we obtain $\alpha$ times the eigenfunction.

Thus:

\qed

Part I v27

## On Proving Eigenvalue = 1 for Particular Surfaces

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

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

Well, time to shake things up a bit! I have decided to change a few things on my blog, including the theme.

The latex plugin broke with the new WP auto-installation features and I haven't been writing on here much, but I've kept going on new Theorems whenever I can.  One of my latest successes, of which I am very proud of, is Shadow Casting.  It is in this new version of my Compendium of Claims and Proofs.  I will be trying to fix how the equations display in the meantime...

Part I v25

## More On Convergent-in-Area Polynomials in the Interval [0,1]

So remember last time I wrote about the rich algebraic structure of polynomials that have convergent area in the interval [0,1]? Turns out there is a lot that can be done, even more than I imagined.  I have already begun to sketch the properties out and have a long ways to go still, but at least I have defined operations on these polynomials that I think can be very useful.

Here's my newest version of Compendium where I have added these things.

Part I v23