Mathematical and Computational Modeling
With an emphasis on the interdisciplinary nature of mathematical and computational modeling, Mathematical and Computational Modeling: With Applications in the Natural and Social Sciences, Engineering, and the Arts features chapters written by well-known, international experts in these fields and presents readers with a host of state-of-the-art achievements in the development of mathematical modeling and computational experiment methodology. The book is a valuable guide to the methods, ideas, and tools of applied and computational mathematics as they apply to other disciplines such as the natural and social sciences, engineering, and technology. Mathematical and Computational Modeling: With Applications in the Natural and Social Sciences, Engineering, and the Arts also features: Rigorous mathematical procedures and applications as the driving force behind mathematical innovation and discovery Numerous examples from a wide range of disciplines to emphasize the multidisciplinary application and universality of applied mathematics and mathematical modeling Original results on both fundamental theoretical and applied developments in diverse areas of human knowledge Discussions that promote interdisciplinary interactions between mathematicians, scientists, and engineers Mathematical and Computational Modeling: With Applications in the Natural and Social Sciences, Engineering, and the Arts is an ideal resource for professionals in various areas of mathematical and statistical sciences, modeling and simulation, physics, computer science, engineering, biology and chemistry, industrial, and computational engineering. The book also serves as an excellent textbook for graduate courses in mathematical modeling, applied mathematics, numerical methods, operations research, and optimization.
Mathematical and Computational Modeling
UNIVERSALITY OF MATHEMATICAL MODELS IN UNDERSTANDING NATURE, SOCIETY, AND MAN-MADE WORLD
The MS2Discovery Interdisciplinary Research Institute, M2NeT Laboratory and Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario, Canada
1.1 HUMAN KNOWLEDGE, MODELS, AND ALGORITHMS
There are various statistical and mathematical models of the accumulation of human knowledge. Taking one of them as a starting point, the Anderla model, we would learn that the amount of human knowledge about 40 years ago was 128 times greater than in the year A.D. 1. We also know that this has increased drastically over the last four decades. However, most such models are economics-based and account for technological developments only, while there is much more in human knowledge to account for. Human knowledge has always been linked to models. Such models cover a variety of fields of human endeavor, from the arts to agriculture, from the description of natural phenomena to the development of new technologies and to the attempts of better understanding societal issues. From the dawn of human civilization, the development of these models, in one way or another, has always been connected with the development of mathematics. These two processes, the development of models representing the core of human knowledge and the development of mathematics, have always gone hand in hand with each other. From our knowledge in particle physics and spin glasses [4,6] to life sciences and neuron stars [1,5,16], universality of mathematical models has to be seen from this perspective.
Of course, the history of mathematics goes back much deeper in the dawn of civilizations than A.D . 1 as mentioned earlier. We know, for example, that as early as in the 6th-5th millennium B.C., people of the Ancient World, including predynastic Sumerians and Egyptians, reflected their geometric-design-based models on their artifacts. People at that time started obtaining insights into the phenomena observed in nature by using quantitative representations, schemes, and figures. Geometry played a fundamental role in the Ancient World. With civilization settlements and the development of agriculture, the role of mathematics in general, and quantitative approaches in particular, has substantially increased. From the early times of measurements of plots of lands and of the creation of the lunar calendar, the Sumerians and Babylonians, among others, were greatly contributing to the development of mathematics. We know that from those times onward, mathematics has never been developed in isolation from other disciplines. The cross-fertilization between mathematical sciences and other disciplines is what produces one of the most valuable parts of human knowledge. Indeed, mathematics has a universal language that allows other disciplines to significantly advance their own fields of knowledge, hence contributing to human knowledge as a whole. Among other disciplines, the architecture and the arts have been playing an important role in this process from as far in our history as we can see. Recall that the summation series was the origin of harmonic design. This technique was known in the Ancient Egypt at least since the construction of the Chephren Pyramid of Giza in 2500 BCE (the earliest known is the Pyramid of Djoser, likely constructed between 2630 BCE and 2611 BCE). The golden ratio and Fibonacci sequence have deep roots in the arts, including music, as well as in the natural sciences. Speaking of mathematics, H. Poincare once mentioned that "it is the unexpected bringing together of diverse parts of our science which brings progress" . However, this is largely true with respect to other sciences as well and, more generally, to all branches of human endeavor. Back to Poincare's time, it was believed that mathematics "confines itself at