Mathematical and Computational Modeling

With Applications in Natural and Social Sciences, Engineering, and the Arts

Roderick Melnik

• Publisher: Wiley
• Year: 2015
• ISBN:
  9781118853856
• Edition: 1
• Topic: Mathematics
• Subtopic: Mathematics General
• Index: Mathematics

Mathematical and Computational Modeling

Illustrates the application of mathematical and computational modeling in a variety of disciplines

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-theart 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. The book 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, and 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.

SECTION 1
INTRODUCTION

1
UNIVERSALITY OF MATHEMATICAL MODELS IN UNDERSTANDING NATURE, SOCIETY, AND MAN-MADE WORLD

RODERICK MELNIK

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” [11]. 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 the same time to philosophy and to physics, and it is for these two neighbors that we work” [11]. Today, the quantitative analysis as an essential tool in the mathematics arsenal, along with associated mathematical, statistical, and computational models, advances knowledge in pretty much every domain of human endeavor. The quantitative-analysis-based models are now rooted firmly in the application areas that were only recently (by historical account) considered as non-traditional for conventional mathematics. This includes, but not limited to, life sciences and medicine, user-centered design and soft engineering, new branches of arts, business and economics, social, behavioral, and political sciences.

Recognition of universality of mathematical models in understanding nature, society, and man-made world is of ancient origin too. Already Pythagoras taught that in its deepest sense the reality is mathematical in nature. The origin of quantification of science goes back at least to the time of Pythagoras’ teaching that numbers provide a key to the ultimate reality. The Pythagorean tradition is well reflected in the Galileo statement that “the Book of Nature is written in the language of mathematics.” Today, we are witnessing the areas of mathematics applications not only growing rapidly in more traditional natural and engineering sciences but also in social and behavioral sciences as well. It should be noted that the term “universality” is also used in the literature in different, more specific and narrow contexts. For example, in statistical mechanics, universality is the observation that there are properties for a large class of systems that are independent of the dynamical details of the system. A pure mathematical definition of a universal property is usually given based on representations of category theory. Another example is provided by computer science and computability theory where the word “universal” is usually applied to a system which is Turing complete. There is also a universality principle, a system property often modeled by random matrices. These concepts are useful for corresponding mathematical or statistical models and are subject of many articles (see, e.g., [2–7,14,16] and references therein). For example, the authors of Ref. [2] discuss universality classes for complex networks with possible applications in social and biological dynamic systems. A universal scaling limit for a class of Ising-type mathematical models is discussed in Ref. [6]. The concept of universality of predictions is discussed in Ref. [14] within the Bayesian framework. Computing universality is a subject of discussions in Ref. [3], while universality in physical and life sciences are discussed in Refs. [7] and [5], respectively. Given a brief historical account demonstrating the intrinsic presence of models in human knowledge from the dawn of civilizations, “universality” here is understood in a more general, Aristotle’s sense: “To say of what is, that it is not, or of what is not, that it is, is false; while to say of what is, that it is, and of what is not, that it is not, is true.” The underlying reason for this universality lies with the fact that models are inherently linked to algorithms. From the ancient times till now, human activities and practical applications have stimulated the development of model-based algorithms. If we note that abstract areas of mathematics are also based on models, it can be concluded that mathematical algorithms have been at the heart of the development of mathematics itself. The word “algorithm” was derived from Al-Khwarizmi (c. 780 –c. 850), a mathematician, astronomer and geographer, whose name was given to him by the place of his birth (Khwarezm or Chorasmia). The word indicated a technique with numerals. Such techniques were present in human activities well before the ninth century, while specific algorithms, mainly stimulated by geometric considerations at that time, were also known. Examples include algorithms for approximating the area of a given circle (known to Babylonians and Indians), an algorithm for calculating π by inscribing and then circumscribing a polygon around a circle (known to Antiphon and Bryson already in the fifth century B.C.), Euclid’s algorithm to determine the greatest common divisor of two integers, and many others. Further development of the subject was closely interwoven with applications and other disciplines. It led to what in the second part of the twentieth century was called by E. Wigner as “the unreasonable effectiveness of mathematics in the natural sciences.” In addition to traditional areas of natural sciences and engineering, the twentieth century saw an ever increasing role of mathematical models in the life and environmental sciences too. This development was based on earlier achievements. Indeed, already during the 300 B.C., Aristotle studied the manner in which species evolve to fit their environment. His works served as an important stepping stone in the development of modern evolutionary theories, and his holistic views and teaching that “the whole is more than the sum of its parts” helped the progress of systems science in general and systems biology in particular. A strong growth of genetics and population biology in the twentieth century effectively started from the rediscovery of G. Mendel’s laws in 1900 (originally published in 1865–1866), and a paramount impetus for this growth to be linked with mathematical models was given by R. A. Fisher’s Fundamental Theorem of Natural Selection in 1930. This result was based on a partial differential equation (PDE), expressing the rate of fitness increase for any living organism. Mathematical models in other areas of life sciences were also developing and included A. J. Lotka and V. Volterra’s predator–prey systems (1925–1931), A. A. Malinovsky’s models for evolutionary genetics and systems analysis (1935), R. Fisher and A. Kolmogorov equation for gene propagation (1937), A. L. Hodgkin and A. F Huxley’s equations for neural axon membrane potential (1952), to name just a few. New theories, such as self-organization and biological pattern formation, have appeared, demonstrating the powerful cross-fertilization between mathematics and the life sciences (see additional details in Ref. [1]). More recently, the ready availability of detailed molecular, functional, and genomic data has led to the unprecedented development of new data-driven mathematical models. As a result, the tools of mathematical modeling and computational experiment are becoming largely important in today’s life sciences. The same conclusion applies to environmental, earth, and climate sciences as well. Based on the data since 1880, by now we know that global warming has been mostly caused by the man-made world with its emission from the burning of fossil fuels, environmental pollution, and other factors. In moving forward, we will need to solve many challenging environmental problems, and the role of mathematical and computational modeling in environmental, earth, and climate sciences will continue to increase [13].

Mathematical models and algorithms have become essential for many professionals in other areas, including sociologists, financial analysts, political scientists, public administration workers, and the governments [12], with this list continuing to grow. Our discussion would be incomplete if we do not mention here a deep connection between mathematics and the arts. Ancient civilizations, including Egyptians, Mesopotamians, and Chinese, studied the mathematics of sound, and the Ancient Greeks investigated the expression of musical scales in terms of the ratios of small integers. They considered harmony as a branch of science, known now as musical acoustics. They worked to demonstrate that the mathematical laws of harmonics and rhythms have a fundamental character not only to the understanding of the world but also to human happiness and prosperity. While a myriad of examples of the intrinsic connection between mathematics and the arts are found in the Ancient World, undoubtedly the Renaissance brought an enriched rebirth of classical ancient world cultures and mathematical ideas not only for better understanding of nature but also for the arts. Indeed, painting three-dimensional scenes on a two-dimensional canvas presents just one example where such a connection was shown to be critical. Not only philosophers, but artists too, were convinced that the whole universe, including the arts, could be explained with geometric and numerical techniques. There are many examples from the Renaissance period where painters were also mathematicians, with Piero della Francesca (c.1415–1492) and Leonardo da Vinci (1452–1519) among them. Nowadays, the arts and mathematics are closely interconnected, continuing to enrich each other. There are many architectural masterpieces and paintings that have been preserved based on the implementation of sophisticated mathematical models. Efficient computer graphics algorithms have brought a new dimension to many branches of the modern arts, while a number of composers have incorporated mathematical ideas into their works (and the golden ratio and Fibonacci numbers are among them). Musical applications of number theory, algebra, and set theory, among other areas of mathematics, are well known.

While algorithms and models have always been central in the development of mathematical sciences, providing an essential links to the applications, their importance has been drastically amplified in the computer age, where the role of mathematical modeling and computational experiment in understanding nature and our world becomes paramount.

1.2 LOOKING INTO THE FUTURE FROM A MODELING PERSPECTIVE

Although on a historical scale electronic computers belong to a very recent invention of humans, the first computing operations were performed from ancient times by people themselves. From abacus to Napier’s Bones, from the Pascaline to the Leibnitz’s Stepped Reckoner, from the Babbage’s Difference (and then Analytic) Engine to the Hollerith’s Desk invention as a precursor to IBM, step by step, we have drastically improved our ability to compute. Today, modern computers allow us to increase productivity in intellectual performance and information processing to a level not seen in the human history before. In its turn, this process leads to a rapid development of new mathematics-based algorithms that are changing the entire landscape of human activities, penetrating to new and unexpected areas. As a result, mathematical modeling expands its interdisciplinary horizons, providing links between different disciplines and human activities. It becomes pervasive across more and more disciplines, while practical needs of human activities and applications, as well as the interface between these disciplines, human activities, mathematics and its applications, stimulate the development of state-of-the-art new methods, approaches, and tools. A special mention in this context deserves such areas as social, behavioral, and life sciences. The ever expanding range of the two-way interaction between mathematical modeling and these disciplines indicates that this interaction is virtually unlimited. Indeed, taking life sciences as an example, the applications of mathematical algorithms, methods, and tools in drug design and delivery, genetic mapping and cell dynamics, neuroscience, and bionanotechnology have become ubiquitous. In the meantime, new challenges in these disciplines, such as sequencing macromolecules (including those already present in biological databases), provide an important catalyst for the development of new mathematics, new efficient algorithms, and methods [1]. Euclidian, non-Euclidian, and fractal geometries, as well as an intrinsic link between geometry and algebra highlighted by R. Descartes through his coordinate system, have all proved to be very important in these disciplines, while the discovery of what is now known as the Brownian motion by Scottish botanist R. Brown has revolutionized many branches of mathematics. Game theory and the developments in control and cybernetics were influenced by the developments in social, behavioral, and life sciences, while the growth of systems science has provided one of the fundamentals for the development of systems biology where biological systems are considered in a holistic way [1]. There is a growing understanding that the interactions between different components of a biological system at different scales (e.g., from the molecular to the systemic level) are critical. Biological systems provide an excellent example of coupled systems and multiscale dynamics. A multiscale spatiotemporal character of most systems in nature, science, and engineering is intrinsic, demonstrating complex interplay of its components, well elucidated in the literature (e.g., [8,9,13] and references therein). In life sciences, the number of such examples of multiscale coupled systems and associated problems is growing rapidly in many different, albeit often interconnected, areas. Some examples are as follows:

• Complex biological networks, genomics, cellular systems biology, and systems biological approaches in other areas, studies of various organs, their systems, and functions;
• Brain dynamics, neuroscience and physiology, developmental biology, evolution and evolutionary dynamics of biological games;
• Immunology problems, epi...

Table of contents

• COVER
• TITLE PAGE
• TABLE OF CONTENTS
• LIST OF CONTRIBUTORS
• PREFACE
• SECTION 1: INTRODUCTION
• SECTION 2: ADVANCED MATHEMATICAL AND COMPUTATIONAL MODELS IN PHYSICS AND CHEMISTRY
• SECTION 3: MATHEMATICAL AND STATISTICAL MODELS IN LIFE AND CLIMATE SCIENCE APPLICATIONS
• SECTION 4: MATHEMATICAL MODELS AND ANALYSIS FOR SCIENCE AND ENGINEERING
• SECTION 5: MATHEMATICAL METHODS IN SOCIAL SCIENCES AND ARTS
• INDEX
• PURE AND APPLIED MATHEMATICS
• END USER LICENSE AGREEMENT