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21 July 2014
Dear Bob,
For the past several years, I've been loading lots of information onto the Internet regarding our work on non-Newtonian calculus (NNC) and related matters. This effort has turned out to be worthwhile, because lately, there have been far more applications of NNC than ever before. Apparently some people have seen the stuff I've put on the Internet, and have found it useful.
One of the most exciting areas of NNC application is fractal geometry, a subject created in the 1970s by Benoit Mandelbrot, a brilliant maverick mathematician/scientist whose work was, like ours, at first dismissed as pointless. Now fractal geometry is being used practically everywhere, e.g., in biology, chemistry, physics, economics, engineering, psychology, physiology, and medicine. Fractal geometry, unlike Euclidean geometry, is concerned with roughness. In fact, fractal geometry is a new mathematical tool for investigating roughness.
Here's where NNC comes in. I've learned from various sources on the Internet that the bigeometric calculus is being used in fractal geometry. In fact, the bigeometric derivative is now used to analyze the rate of change of the roughness of a fractal over time.
"Random fractals, a quintessentially 20th century idea, arise as natural models of various physical, biological (think your mother's favorite cauliflower dish), and economic (think Wall Street, or the Horseshoe Casino) phenomena, and they can be characterized in terms of the mathematical concept of fractional dimension. Surprisingly, their time evolution can be analyzed by employing a non-Newtonian calculus [the bigeometric calculus] ..."
- Professor Wojbor Woyczynski, Case Western Reserve University, USA; from an abstract to his seminar at Case Western Reserve University on 03 April 2013.
"Many natural phenomena, from microscopic bacteria growth, through macroscopic turbulence, to the large scale structure of the Universe, display a fractal character. For studying the time evolution of such "rough" objects, the classical, "smooth" Newtonian calculus is not enough."
- Professor Wojbor Woyczynski, Case Western Reserve University, USA; from an abstract to his seminar "Non-Newtonian calculus for the dynamics of random fractal structures" at The Ohio State University on 22 April 2011, and at Cleveland State University on 02 May 2012.
"Together with a small, highly focused research team, Ostoja-Starzewski is working across disciplines to unite methods from solid mechanics, advanced continuum mechanics, statistical physics and mathematics. Some of the specific mathematical theories they use include probability theory and non- Newtonian calculus. These approaches allow them to focus on different fractal structures ... "
- Professor Martin Ostoja-Starzewski, University of Illinois at Urbana-Champaign, USA; from the 2013 media-upload "The inner workings of fractal materials", University of Illinois at Urbana-Champaign.
Another important area of NNC application is biomedical image analysis. Of course, biomedical imaging provides physicians and researchers with invaluable noninvasive-methods for examining living tissues. And biomedical imaging is used so extensively that some of its nomenclature has become commonplace: "X-ray", "ultrasound", "MRI", "CAT-scan", "PET-scan". Remarkably, it turns out that the geometric calculus is a useful tool for biomedical image analysis.
"We advocate the use of an alternative calculus in biomedical image analysis, known as multiplicative (a.k.a. non-Newtonian) calculus. ... The purpose of this article is to provide a condensed review of multiplicative calculus and to illustrate its potential use in biomedical image analysis." (The expression "multiplicative calculus" refers here to the geometric calculus.)
- Professors Luc Florack and Hans van Assen, both of Eindhoven University of Technology in The Netherlands; from their article "Multiplicative calculus in biomedical image analysis", Journal of Mathematical Imaging and Vision, Springer, 2012.
"Multiplicative calculus provides a natural framework in problems involving positive images and positivity preserving operators. In increasingly important, complex imaging frameworks, such as diffusion tensor imaging, it complements standard calculus in a nontrivial way. The purpose of this article is to illustrate the basics of multiplicative calculus and its application to the regularization of positive definite matrix fields." (The expression "multiplicative calculus" refers here to the geometric calculus.)
- Professors Luc Florack, Eindhoven University of Technology in The Netherlands; from his article "Regularization of positive definite matrix fields based on multiplicative calculus", Scale Space and Variational Methods in Computer Vision, Lecture Notes in Computer Science, Springer, 2012.
"This work presents a new operator of non-Newtonian type which [has] shown [to] be more efficient in contour detection [in images with multiplicative noise] than the traditional operators. ... In our view, the work proposed in (Grossman and Katz, 1972) stands as a foundation ... ."
- Professors Marco Mora, Fernando Córdova-Lepe, and Rodrigo Del-Valle, all of Universidad Católica del Maule in Chile; from their article "A non-Newtonian gradient for contour detection in images with multiplicative noise", Pattern Recognition Letters, International Association for Pattern Recognition, Elsevier, 2012.
"In our proseminar we'll learn some of the most exciting non-linear calculations and consider the applications for which they may be of particular interest. The applications range from rates of return and other growth processes to highly active areas of digital image processing."
- Professor Joachim Weickert (Gottfried Wilhelm Leibniz Prize winner), Saarland University in Germany; from his description of his non-Newtonian calculus course "Analysis beyond Newton and Leibniz", Saarland University, 2012.
In addition to fractal geometry and biomedical image analysis, I've come across many other areas of NNC application, some of which we anticipated years ago:
growth/decay analysis (e. g., in economics and biology), dynamical systems and chaos theory, finance (e.g., rates of return), the theory of elasticity in economics, marketing, wave theory in physics, the economics of climate change, signal processing, atmospheric temperature, information technology, pathogen counts in treated water, actuarial science, tumor therapy in medicine, materials science/engineering, demographics, differential equations (including multiplicative Lorenz systems and Runge-Kutta methods), calculus of variations, finite-difference methods, averages of functions, means of two positive numbers, weighted calculus, meta-calculus, approximation theory, least-squares methods, multivariable calculus, complex analysis, functional analysis, probability theory, utility theory, Bayesian analysis, stochastics, and decision making.
The non-Newtonian calculi most often used are the geometric and bigeometric calculi, which, of course, is not surprising.
As you well know, for many years lots of people, especially various pure mathematicians, claimed that our work was useless. But, despite their discouraging and sometimes arrogant comments, we always knew that NNC has considerable potential for application in science, engineering, and mathematics. -- And we were right!!
From:
Mike
.