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.
The functions to be calculated are the exclusive or (XOR) function, the eight input and output identity function, and four non-linear three input and one output functions. I used the logistic function that represents 0 as 0.1 and 1 as 0.9. The three input functions use 8 hidden units, a tolerance of 1.0e-9, and maximum number of epochs as 1,000,000. The backpropagation feed-forward neural network algorithm is from Tom M. Mitchell’s classic textbook “Machine Learning” which has a copyright of 1997.
The n-queens problem is a constraint satisfaction problem. The object of the puzzle is to arrange n queens on a n x n chessboard, so that no two queens are attacking one another. Remember the chess queen can move any number of spaces diagonally, horizontally, or vertically. My brute force solver computes n factorial possible solutions and checks each configuration to see if it solves the n-queens puzzle. The n-queens problem is thought to be NP-Complete which means that solutions are found by a non-deterministic polynomial or exponential algorithm. My solutions seem to require 2^n seconds to find all solutions of the problem for n = 8 to 12. I generate n-factorial candidate solutions and test each candidate to see if it solves the n-queens problem.
I became interested in attempting to predict temperatures in the state of Georgia way back in the day, i.e., the date was November 1 – 4, 2009. I found a neat website for the temperatures from January 1895 to December 2001. Unfortunately, the website no longer exists online, however, I saved the data to an extinct PC of mine and a USB solid state drive. I used two methods to attempt predictions of the annual temperatures from January 2002 to December 2025. The first algorithm is polynomial least squares curve fitting [1]. The second method is a radial basis function neural network [2] [3] that is trained by an evolutionary hill-climber of my design and implementation [4] [5]. The applications were written in one of my favorite computer programming languages, namely, C# [6].
Methodologies
I created a polynomial least squares dynamic-link library (DLL) using Gaussian elimination with pivoting and an inverse matrix calculation utilizing an upper-triangular matrix [7]. At first the driver application was capable of fitting a 1-degree to 100-degrees polynomial. I found that my algorithm was only valid for a 1-degree to 76-degrees fitting polynomial. I used degrees: 1, 4, 8, 16, 32, 64, and 76.
Predicted Statewide Georgia Annual Average Temperatures in Degrees Fahrenheit
Poly Degree
1
4
8
16
32
64
76
Model
Year
T (F)
T (F)
T (F)
T (F)
T (F)
T (F)
T (F)
Average
2002
64.2
63.1
63.8
63.6
62.4
63.0
62.9
63.3
2003
64.2
63.2
63.6
63.4
64.1
63.7
63.7
63.7
2004
64.2
63.3
63.4
63.4
64.5
63.9
64.0
63.8
2005
64.2
63.3
63.4
63.5
64.2
63.9
64.0
63.8
2006
64.2
63.4
63.4
63.6
63.8
63.7
63.8
63.7
2007
64.2
63.5
63.4
63.7
63.3
63.6
63.6
63.6
2008
64.2
63.6
63.4
63.7
63.0
63.4
63.4
63.5
2009
64.2
63.6
63.4
63.7
62.9
63.3
63.3
63.5
2010
64.1
63.7
63.5
63.7
62.9
63.3
63.2
63.5
2011
64.1
63.8
63.6
63.6
63.1
63.3
63.2
63.5
2012
64.1
63.8
63.7
63.6
63.3
63.4
63.3
63.6
2013
64.1
63.9
63.7
63.6
63.6
63.5
63.5
63.7
2014
64.1
63.9
63.8
63.6
63.9
63.6
63.6
63.8
2015
64.1
64.0
63.9
63.6
64.1
63.8
63.8
63.9
2016
64.1
64.0
63.9
63.6
64.3
63.9
63.9
64.0
2017
64.1
64.1
64.0
63.7
64.4
64.0
64.1
64.1
2018
64.1
64.1
64.1
63.7
64.4
64.1
64.2
64.1
2019
64.1
64.2
64.1
63.9
64.4
64.2
64.2
64.2
2020
64.1
64.2
64.2
64.0
64.3
64.3
64.3
64.2
2021
64.1
64.3
64.2
64.1
64.2
64.3
64.3
64.2
2022
64.1
64.3
64.2
64.2
64.2
64.3
64.3
64.2
2023
64.1
64.3
64.3
64.3
64.1
64.3
64.3
64.2
2024
64.0
64.4
64.3
64.5
64.0
64.3
64.3
64.3
2025
64.0
64.4
64.3
64.6
64.0
64.3
64.3
64.3
The second method was a radial basis function neural network that was trained by an evolutionary hill-climber of my design and implementation. I used 8, 16, 32, 64 basis functions. The population of the hill-climber was 16 and generations 262,144.
Predicted Statewide Georgia Annual Average Temperatures in Degrees Fahrenheit
Basis
8
16
32
64
Model
Year
T (F)
T (F)
T (F)
T (F)
Average
2002
65.4
64.2
65.4
65.2
65.1
2003
65.5
64.2
65.4
65.2
65.1
2004
65.5
64.2
65.5
65.3
65.1
2005
65.5
64.2
65.5
65.3
65.1
2006
65.6
64.2
65.5
65.3
65.2
2007
65.6
64.2
65.6
65.3
65.2
2008
65.6
64.2
65.6
65.4
65.2
2009
65.7
64.2
65.6
65.4
65.2
2010
65.7
64.2
65.6
65.4
65.2
2011
65.7
64.2
65.7
65.4
65.3
2012
65.7
64.2
65.7
65.5
65.3
2013
65.8
64.2
65.7
65.5
65.3
2014
65.8
64.2
65.8
65.5
65.3
2015
65.8
64.3
65.8
65.5
65.4
2016
65.9
64.3
65.8
65.6
65.4
2017
65.9
64.3
65.8
65.6
65.4
2018
65.9
64.3
65.9
65.6
65.4
2019
66.0
64.3
65.9
65.6
65.5
2020
66.0
64.3
65.9
65.7
65.5
2021
66.0
64.3
66.0
65.7
65.5
2022
66.0
64.3
66.0
65.7
65.5
2023
66.1
64.3
66.0
65.7
65.5
2024
66.1
64.3
66.1
65.8
65.6
2025
66.1
64.3
66.1
65.8
65.6
I trust the polynomial fitting results more than the radial basis function neural network values. The polynomial fitting had mean square errors < 1 whereas the radial basis function neural network had mean square errors between 1 and 9. The general trends were increasing temperatures which are in line with the theorized global warming.
References
[1]
H. T. Lau, A Numerical Library in C for Scientists and Engineers, Boca Raton: CRC Press, 1995.
[2]
T. M. Mitchell, Machine Learning, Boston: WCB McGraw-Hill, 1997.
[3]
A. P. Engelbrecht, Computational Intelligence An Introduction, Hoboken: John Wiley and Sons, 2002.
[4]
Z. Michalewicz, Genetic Algorithms + Data Structures = Evolutionary Programs 3rd Edition, Berlin: Springer, 1999.
[5]
D. B. Fogel, Evolutionary Computation Toward a New Philosophy of Machine Intelligence, Piscataway: IEEE Press, 2000.
[6]
C. Petzold, Programming Windows with C#, Redmond: Microsoft Press, 2002.
[7]
S. D. Conte and C. d. Boor, Elementary Numerical An Algorithmic Approach, New York: McGraw-Hill Book Company, 1980.
This is another attempt to reproduce the United States Navy’s Ordnance Pamphlet 770: https://eugeneleeslover.com/USN-GUNS-AND-RANGE-TABLES/OP-770-1.html which contains ballistic tables for the battleship USS Iowa (BB-61) artillery (16-inch/50 caliber) and is dated October 1941. My C# Windows desktop application is capable of calculating the elevation from range table which has the columns range in yards, angle of elevation in degrees and minutes, positive angle of fall in degrees and minutes, time of flight in seconds, apogee also called summit in feet, striking velocity in feet per second, and energy in foot pound force. Three corrections can be applied to the trajectory: trunnion height in feet, acceleration of gravity correction, and the curvature of the Earth correction (Vincenty calculation). The first image below is the ballistic settings interface. The second image is the uncorrected table. The third image is the application of a trunnion height of 32 feet. The fourth image is the curvature of the Earth correction. The fifth image is the trunnion height of 32 feet and Vincenty corrections. It is to be noticed that the striking velocity and kinetic energy are the only non-monotonically increasing or decreasing data fields.
A few days ago, I finished my Revell 1:144 scale Apollo 11 and Saturn V plastic model. The kit had 82 parts and I used every one of the pieces. I deviated from the original paint scheme since the original black and white pattern is too reminiscent of Wehner von Braun’s World War II terror weapon, the V-2 also known as Retaliation Weapon 2. The V-2 was the first liquid propelled ballistic missile. It used ethanol as the fuel and liquid oxygen as the oxidizing agent.
This is my project for Professor Kai Chang’s Comp 640 Advanced Computer Graphics at Auburn University in 1999. We used the Open Graphics Library also known as OpenGL. It is a hybrid C/C++ computer application.
I became interested in attempting to predict average annual temperatures in the state of Georgia way back in the day. I found a neat website for temperatures for January 1895 to December 2001. Unfortunately, the website no longer exists online, however, I saved the data to an extinct PC of mine and a USB solid state drive. I used several methods to attempt predictions of the annual temperatures from 2002 to 2025. The first algorithm is polynomial least squares utilizing a 32-degree polynomial and a 75-degree polynomial. The application was written in one of my favorite programming languages, namely, C#.
Solution to the differential equation: d2y/dx2 = -0.25 * x * x * y – a * y which is valid for all real and complex numbers. We examine the real solutions to the second order ordinary differential equations. See “Handbook of Mathematical Functions” by Milton Abramowitz and Irene A. Stegun, Chapter 19. Parabolic Cylinder functions. Page 686 Equation 19.2.7 Recurrence formula and 19.2.5 Series solutions.
Numerical Explorations 