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Archive for the ‘Differential Equations’ Category

On Shifted Legendre Polynomial Coefficients

March 27th, 2017 No comments

So here it is: the shifted Legendre polynomial coefficients are:

 a(k-1,n) = a_{k-1}^n = (-1)^k k \left( \prod_{m \neq 1} \frac{m-1}{m-n-1} \prod_{m \neq k} \frac{m-n-1}{m-k} \prod_{m \neq n} \frac{m-n-k}{m-n-1} \right)

with k \leq n and m = 1 \ldots n, so that

\tilde{P}_n(x) = \sum_{i = 0}^n a_i \cdot x^i

I'd love to hear your ideas of a proof.  Mine is a very particular thorny meticulous one. I may be off on the sign, but the absolute value is right. Let me know!

On Naturally Arising Differential Equations

October 18th, 2016 No comments

So if you have been following the argument a bit, it turns out that

 p(x,y,t) = \alpha^{t-1} \mathbf{P}_x(x) \cdot \mathbf{P}_y(y) + f_n^*(x)

is the starting at time t = 1 transition probability propagator of a probability distribution, say c_0(x) at t=0, in the interval x = 0 to 1.  A question that I tried to answer was how zeros are propagated via the propagator or at the probability distribution, which lead to theorems that I dubbed "Shadow Casting" because, under that context, it turns out that a zero, if found on the propagator, remained in place until infinity, and via the propagator it appears on the probability distribution we want to propagate as well (therefore casting a "shadow").  I hadn't thought of the following approach until recently, and I haven't worked it out completely, but it connects to the theory of Ordinary Differential Equations which seems interesting to me. Here's the argument:

Suppose we focus on p(x,y,1) for the time being, and wish to find the zeros on the transition probability surface.  Thus we seek p(x,y,1) = 0 and suppose y(x) is an implicit function of x. We have

 p(x,y,1) = 0 = \mathbf{P}_x(x) \cdot \mathbf{P}_y(y(x)) + f_n^*(x)

 Now let \mathbf{P}_y(y) is a collection derived from y(x), so that, for example,

 \mathbf{P}_y(y(x)) = \left[ \begin{array}{c} y(x) \\ y^{\prime}(x) \\ \vdots \\ y^{n-1}(x) \end{array} \right]

and I think we have successfully created a link to ODEs.  To find the zeros on the surface (and other time surfaces of the propagator) we stick in the correct \alpha and solve, using the familiar methods (solve the homogeneous equation and the particular solution via sin-cos-exponential solutions, variation of parameters, power series, etc.).

I'm working out the specificities, for example including the constraints we know on f_n^*(x) or \mathbf{P}_x(x).  Perhaps this approach will help us broaden the spectrum of differential equations we can solve, by linking via Shadow Casting.

It may seem basic, but I think there is some untapped power here.

Additionally, I have been working on clarifying some thoughts on polynomials that converge in area in the interval from 0 to 1, but all those details tend to be a bit drab and I keep having trouble focusing.  Nevertheless, there is a lot of clarity that I have been able to include, and it is now in the newest draft of "Compendium".  By the way, I renamed it. It is now "Compendium of Claims and Proofs on How Probability Distributions Transform".  There's still soooo much more to do.

Here it is! part-i-v28

On Shadow Casting

February 26th, 2016 2 comments

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]

October 9th, 2014 No comments

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

Latest version of Compendium

February 9th, 2014 No comments

I am including here the latest version of the Compendium (v17).

Part I v17