Author: jamespatewilliamsjr

My whole legal name is James Pate Williams, Jr. I was born in LaGrange, Georgia approximately 70 years ago. I barely graduated from LaGrange High School with low marks in June 1971. Later in June 1979, I graduated from LaGrange College with a Bachelor of Arts in Chemistry with a little over a 3 out 4 Grade Point Average (GPA). In the Spring Quarter of 1978, I taught myself how to program a Texas Instruments desktop programmable calculator and in the Summer Quarter of 1978 I taught myself Dayton BASIC (Beginner's All-purpose Symbolic Instruction Code) on LaGrange College's Data General Eclipse minicomputer. I took courses in BASIC in the Fall Quarter of 1978 and FORTRAN IV (Formula Translator IV) in the Winter Quarter of 1979. Professor Kenneth Cooper, a genius poly-scientist taught me a course in the Intel 8085 microprocessor architecture and assembly and machine language. We would hand assemble our programs and insert the resulting machine code into our crude wooden box computer which was designed and built by Professor Cooper. From 1990 to 1994 I earned a Bachelor of Science in Computer Science from LaGrange College. I had a 4 out of 4 GPA in the period 1990 to 1994. I took courses in C, COBOL, and Pascal during my BS work. After graduating from LaGrange College a second time in May 1994, I taught myself C++. In December 1995, I started using the Internet and taught myself client-server programming. I created a website in 1997 which had C and C# implementations of algorithms from the "Handbook of Applied Cryptography" by Alfred J. Menezes, et. al., and some other cryptography and number theory textbooks and treatises.

Eigenfunctions of a Nonlinear Second Order Ordinary Differential Equation Initial Value Eigenvalue Problem by James Pate Williams, Jr.

Solutions of a nonlinear second order ordinary differential equation initial value eigenvalue problem:

y”(x) = x * x * y(x) + n * n * y(x) * y(x) for all x in [0, 1]
y(0) = 0
y'(0) = 1
n in [0, 1, 2, …]

I graphed the first five eigenfunctions.

Graphs for n = 0 to n = 4

Planetary Precession by James Pate Williams, Jr.

There are three classic theoretical tests of Albert Einstein’s  Theory of General Relativity: the perihelion precession of Mercury, the other Solar System planets, and the planetoid Pluto, the bending of light by massive bodies, and the gravitational red shift. I recently wrote a C# program for displaying the exaggerated Rosette motion of theoretical planets (Schwarzschild’s solution to Einstein’s general relativity field equation that admit the existence of black holes). I also wrote a C++ program to calculate planetary precession values that agree with experimental results.

Precession.cpp (c) James Pate Williams, Jr. August 2022

This program calculates the planetary precessions of the planets in our solar system. Some of the  equations and data are from “General Relativity” by Hans Stephani 1982 page 103 and the following websites. Also, two calculations of the mass of the Sun are exhibited, along with my weight on different planets:

https://nssdc.gsfc.nasa.gov/planetary/factsheet/

https://farside.ph.utexas.edu/teaching/celestial/Celestialhtml/node44.html

https://imagine.gsfc.nasa.gov/features/yba/CygX1_mass/gravity/sun_mass.html

https://socratic.org/questions/how-do-you-calculate-the-mass-of-the-sun-m-sun-using-kepler-s-third-law-t-2-4-pi

https://en.wikipedia.org/wiki/Surface_gravity

https://www.schoolsobservatory.org/discover/quick/weight/https://physicscalc.com/physics/escape-velocity-calculator/#:~:text=Steps%20to%20Find%20Escape%20Velocity%201%20Obtain%20the,the%20double%20the%20result%20is%20the%20escape%20velocity

Planetary Precession C++ Program Output

Rosette Motion 1

Rosette Motion 2
Rosette Motion 3
Rosette Motion 4
Rosette Motion 5

Rosette Motion 6

Rosette Motion 7

Blog Entry Monday April 4, 2022

This morning as I was leaving the gym at about 2:00 AM a fellow gym member made the following remark to me: “I hear you are a scientist” and I responded “yes, a computer scientist and computer or software engineer”. Later as I was headed towards my car, this same other gym member pointed towards a planet or star in the early morning sky and asked, “what is the object in the sky he was pointing towards was?”. I did note answer his question on the spot and now I offer the following theories. Well, if we had a high-resolution telescope and/or a spectroscope we could differentiate a planet from a star due to the chemical composition of the object. There is a theory posed by Alpher, Bethe, and Gamov which states the lightest elements in the universe, namely hydrogen and helium were formed during the Big Bang and the other 115 elements were either made by humans in nuclear physics laboratories (breeder nuclear reactors or other means) or the rest of the chemical elements were synthesized in the stars. Stars are classified by the Alpher, Bethe, and Gamov scheme:

Star Type Classification / NASA | Kaggle

https://blogs.scientificamerican.com/basic-space/httpblogsscientificamericancombasic-space20110802on-the-origin-of-chemical-elements/

Asymptotic giant branch – Wikipedia

Blog Entry Tuesday March 29, 2022

This application’s code was translated from vanilla C to C#. The C code is from the treatise “A Numerical Library for Scientists and Engineers” by H. T. Lau “Chapter 5: Analytic Problems – Runge-Kutta 5th order no derivatives in the right hand side”. The first test case is from the preceding tome.

RK2 Test Case
Micheal Penn’s Nonlinear ODE Data Table

I found an interesting differential second order nonlinear initial value equation on Michael Penn’s voluminous YouTube website: Michael Penn – YouTube. The equation is y”(x) = -exp(y). The solution is illustrated in the following table and graph:

Micheal Penn’s Nonlinear ODE Graph