# Programming in Delphi

Old (Fortran/Basic/Turbo Pascal) programs have been ported to Borland Delphi 3/5/6:

The Trilogy, reported in this section, consists (indeed) of three mutually related Projects:

1. Contour Thinning, without theory but with executable
This replaces the deprecated pixel thinning
2. Fuzzy Frenet Analysis, with theory and executable
3. Fuzzy Lissajous Analysis upgraded, with theory and executable
4. Last but not least: accompanying B&W bitmap files
Usage: Make a directory. Unzip in that directory and Run the 'exe' files.
All Source code has been made available too, for the serious researcher.

Is it possible to entirely automate the art of Balancing a Chemical Equation ? The answer has found to be affirmative.
And somewhat suprising too ! Because the method used in my program is an integer version of the Least Squares Method. (The latter has also been employed for solving the problem of Ideal Flow around a circular cylinder, as has been formulated in 1976 by Th.E. Labrujère)

The gist of the method is as follows. In the chemical equation, let M be the number of Elements and N be the number of unknown coefficients. Then, mathematically speaking, we have to solve a system of M linear equations with N unknowns. Applying the Least Sqares Method results in a more convenient system: of N equations with N unknowns. The load vector of this system is zero. We know, in addition, that the outcome is an array of integers. And any multiple of the outcome must be considered as another mathematical solution. Because the equations are mutually dependent, the last equation can be cancelled. Furthermore, the last coefficient is set to unity, for the moment being, and the accompanying matrix coefficients are moved to the right hand side. Then the equations are solved by a direct method. The integer nature of the problem must preferrably be left intact. Therefore multiply everything with the determinant, instead of performing a division with it. At last, the coefficients are to be simplified, by dividing all of them with their GCD (Greatest Common Divisor).

Anyway, here comes:

• The Balancing of Chemical Equations (console version),
with executable, source code & data all zipped together
• Here is a sample input and accompanying output
Usage: Make a directory. Unzip in that directory and Run the 'exe' file.
Known Limitations:
• No fancy Windows
• No syntax checking on input
• Chemical formulas with parentheses, like Ca(PO3)2 , cannot be processed
• No equations with ions in it, like H+ , (OH)- and the like. Balancing of charges is not done.
Almost inevitably, an automated solution to the Balancing problem in Chemistry has also been found by others:
 P. Ramasami, A Concise Description of an Old Problem: Application of Matrices
to Obtain the Balancing Coefficients of Chemical Equations, Journal
of Mathematical Chemistry, Volume 34, Issue 1-2, July  2003, Pages 123 - 129


Kind of an update of "Sensible Densities", is (no longer) under development. It is placed here for a reference:

Sensible Densities have been found useful for carrying out Experiments with Prime Number Theory.

## The Art of Contouring

Kind of an update of my contouring techniques.
Contouring routines for functions defined at a triangular mesh are found in my source code archives for numerical work.
You must have the '
GRAFISCH.PAS' module in: For the (still far-fetched) purpose of character recognition and the like: