Progress on a #polynomial signals representation project:

This is a new/old idea that makes specific promises, has some research, but doesn't have a lot of practical radio development or useful applications (yet).

Well, that's what we're here for!

There's repository started over at @OpenResearchIns ns@micro.blog

at https://github.com/OpenResearchInstitute/Polynomial_Signals_Representation

Trying to put together a coherent tutorial so that ordinary radio enthusiasts can get some traction.

#opensource #digital #radio #hamradio

I learned from Jon Brett about physicist David #Bohm 's colleague Basil Hiley https://en.wikipedia.org/wiki/Basil_Hiley Their distinction between implicate and explicate order seems to appear in my study of #orthogonal #Sheffer #polynomial s. The Sheffer constraint of exponentiality yields an implicate order of #partition of a set https://www.math4wisdom.com/wiki/Exposition/20221122SpaceBuilders whereas orthogonality yields 5 possible explicate orders upon #measurement. Also curious how they use #CliffordAlgebra ,real and #symplectic. #BottPeriodicity ?

Another #complex #polynomial video

#complex #polynomial plot video

Another #complex #plot animation of a #polynomial (degree 7)

Roots of a #complex #polynomial

Roots of a #complex #polynomial

Faster #polynomial operations in newly release version of #mathematica

https://writings.stephenwolfram.com/2022/12/the-latest-from-our-rd-pipeline-version-13-2-of-wolfram-language-mathematica/#dramatically-faster-polynomial-operations

Seen on TBBBS:

1. Consider a random #polynomial with degree "n"

\[
P(z) = a_n z^n + ... + a_1 z + a_0
\]

where the coefficients are random complex Gaussian numbers.

Choose the degree "n" large.

2. Compute the #complex roots and plot them.

3. Wonder why they concentrate on the #UnitCircle

From here:

https://twitter.com/i/status/1600132699800289280

@andresvillaveces The statement: "there are infinitely many 𝑛∈ω s.t. 𝑛²+𝑛+41 is a prime" is called Bunyakovsky conjecture. It seems that this conjecture is still open (𝑛²+𝑛+41 is the polynomial studied independently by Euler and Legendre with the remarkable property that its values for all 𝑛=0,1,...,39 are prime).
#polynomial #primenumber

#Mathologer : « #Lill's method, an unexpectedly simple and highly visual way of finding solutions of #polynomial #equations (using #turtles and #lasers). After introducing the method I focus on a couple of stunning applications: pretty ways to solve quadratic equations with ruler and compass and cubic equations with #origami, #Horner's form, #SyntheticDivision and a newly discovered incarnation of #Pascal's famous triangle. »
https://www.youtube.com/watch?v=IUC-8P0zXe8
http://www.qedcat.com/misc/lill_method/
#Math