Pseudocolor in Pure and Applied Mathematics, a Free on-Line e-Book with Source Code

Douglas Youvan

ISBN 978-0-615-43573-2

From the Youvan Foundation, a nonprofit 501(c)3 educational charity in formation.

Pseudocolor is very useful in the representation of computational output that involves complex data and phenomena - including numerical tensors and digital images.  Many of the examples in this e-book are pedagogical in nature and useful for introducing students to higher levels of Mathematics. Our full source code is published in Mathematica ".nb" and Adobe ".PDF" formats; the latter has fully executed graphics embedded.  We have used a very limited vocabulary of Mathematica functions so this code can be read easily and ported to other languages.  As such, the simple syntax of Mathematica can serve as a flow chart for other implementation.  Our 'pure' examples include work in matrix algebra,  fractals, tuples, permutations, transcendental numbers, statistics, computation speed, and P=NP.  Our applied examples include work in biology, molecular biology, biotechnology, chemical kinetics, energy conversion, thermodynamics, spectroscopy, signal processing, image processing, digital imaging spectroscopy, graphics.

As a way of introducing the idea of using pseudocolor to represent mathematical concepts and applications, three exemplary graphics are shown below.  These graphics come from the fields of aerodynamics (Savonius rotor), biology (genetic code), and pure math (Tuples).  Following these examples, you will find a Forward and Table of Contents with hyperlinks to explanations and source code.  Currently, 16 examples are completed, 4 more are in progress, and as many as 60 more examples are planned.  Please contact the author if you have any comments - as this is a work in progress.  Blue text indicates active hyperlinks.


       Savonius Waves.



      Pseudoinverse of the Genetic Code.



    Tuple Imagery.



With the advent of inexpensive computers running with gigahertz speed and  gigabyte RAM, it appears that issues of compiled code and concise memory management might become concepts of the past for most applications.  Such statements always look funny in hindsight, after computers become still faster and cheaper.  That makes a high-level language such as Mathematica even more attractive as a future platform.  A developer using a platform such as Mathematica can take advantage of the work of a larger group, such as Wolfram Research, to free themselves from monolithic operating systems.  The developer is then free to code and spend most of their time in logic rather than in ever-changing fundamentals underlying developer studios.  We also anticipate that Mathematica code is easy to read - even without flowcharts - and that it can be easily ported to another language.

It is important to visualize mathematics in order to learn.  For example, calculus is more easily learned when it is combined with analytical geometry.  One sees derivatives as the slope of functions, and one can picture the concept of an integral as simply the area under a functional curve.  Our sense of vision can not be neglected in the path to learning higher math.  That is why physicists are often the people that advance math - they actually see what they are doing.  A century ago, quantum mechanics began to show us the picture of atoms in the form of wave functions that graph as easily understood electron clouds.  Without this visualization, it would be very difficult to appreciate the new, underlying mathematics.  In electromagnetism and optics, theorems are best visualized as vectors pointing away from surfaces.  Many of us have learned math through physics.

Biotechnology is also increasing important in applied and pure mathematics.  One seeks to understand our 3 billion nucleotide genome and the interaction of drugs in terms of combinatorial possibilities.  This opens up the entire field of discrete mathematics and combinations to the applied mathematician for use in biotech. Rapid prototyping of new algorithms in these fields is very important.  That is aided by high level languages that can be used by an individual researcher or a sole practitioner.

Images and source code from this e-book are also made available to the public for any lawful use (worldwide) via our PD-Self deposition at Wikimedia.  As of 2008, the public domain SAGE platform will run Mathematica computer code and display, for example, using the Microsoft Internet Explorer.

Completed Examples:

  1. Temporal Color Changes in a Stack of Images Containing Randomly Generated Colors
  2. Pseudocoloring Ordinary Photographs to Highlight Grayscale Changes
  3. 300 Different Pseudocolor Schemes via One RGB Orthonormalization
  4. Counterfeit Detection Based on Ink Color
  5. Random Planes Passing Through a Schroedinger Cube
  6. Anti-Color, Real-Color, and Super-Color in Tuple-Images (See also, a new e-book, in progress, "Tuple Imagery", 2012.
  7. Morphological Stability in Anti-Color and Super-Color Diffusion
  8. Synergy and Antagonism for Colors in Diffusion
  9. Chambered Detonation of Super-Colors
  10. Pixel Accretion Yielding a Dissipative Structure
  11. Pseudocolored Stereo-Pair of a 3-D Lissajous Function
  12. Grayscale Rendering of Perpendicular Half-Planes
  13. Colorizing a Fractal with a One Dimensional Fractal
  14. Rendering 3-D Depth in an RGB Image Using Nested Subsets and Permutations 
  15. Similarity (SSD) Sorting of Signals Displayed in Contour Plots
  16. Selecting Pixels from an Image Stack that Match Temporal RGB Functions
  17. Singular Value Decomposition (SVD) of Amino Acid Hydropathy in the Genetic Code (PDF; nb)
  18. Rapid PseudoInverse for Scroll Matrices (PDF; nb)
  19. Savonius Rotor Traces (PDF; nb) uploaded 20120114

Future Work

About the author ... 

In hindsight, perhaps my most important publication is used in Example 17: "Genetic Coding Algorithms for Engineering Membrane Proteins"

(c) Youvan Foundation, 2006.  Last updated on 20140208


Pure and Applied Discrete Mathematics, a Free on-Line e-Book with Source Code

Douglas Youvan


In Progress

In this second mathematics e-book, I change emphasis from pseudocolor to discrete mathematics.  For now, I will be publishing both PDF and nb versions of the code; later, I will come back and write descriptions of each program.  From the title of each example and comment statements within the code, you should be able to deduce my use of the code and possible uses you might have.  We begin work in Version 9 of Mathematica, with an update to Version 10 expected soon.   Publishing began May 22, 2014 from code already completed.  I expect this book to take a few years to finish.

Einstein and Lorentz Play Marbles.  In this short story, I use relativity to reduce an N! problem to N^2 both in computation time and scaling.  Einstein's time dilation equation for the Twin Paradox is identified as a solution to P v NP.

20.  An Exocentric Derivative for Any Object (PDF; .nb; data)

21.  (PDF; .nb)

22.  (PDF; .nb)

23.  (PDF; .nb)

24.  (PDF; .nb)

25.  (PDF; .nb)

26.  (PDF; .nb)

27.  (PDF; .nb)