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  • Cover image of <span class="hlFld-Title">Calculus for the Natural Sciences</span>

    Calculus for the Natural Sciences

    Published: 2023

    This textbook is intended for a two-semester course on calculus of one variable. The target audience is comprised of students in biology, chemistry, mathematics, physics, and related disciplines, as well as professionals in these areas. It grew out of the Symbiosis Project at East Tennessee State University.

  • Cover image of <span class="hlFld-Title">Nonlinear Time Scale Systems in Standard and Nonstandard Forms</span>

    Nonlinear Time Scale Systems in Standard and Nonstandard Forms

    Published: 2014

    Singular perturbation theory plays a significant role in analysis and design of control laws. It presents control engineers with the advantage of model reduction that simplifies both the software and hardware implementation of control algorithms. Through this book, our goal is to present the reader with control design techniques that extend these benefits to a larger class of systems, specifically to those that are not traditionally controlled through this methodology. Our aim in doing so is to show the reader that some classes of nonminimum phase control problems can be actively controlled in real-time through use of singular perturbation methods.

  • Cover image of <span class="hlFld-Title">Solving Polynomial Systems Using Continuation for Engineering and Scientific Problems</span>

    Solving Polynomial Systems Using Continuation for Engineering and Scientific Problems

    Published: 2009

    This is an introduction to “polynomial continuation,” which is used to compute the solutions to systems of polynomial equations. The book shows how to solve practical problems but maintains an elementary mathematical perspective. The first two chapters illustrate most of the important concepts and numerical processes, using only high-school mathematics and some simple computer programs. Since 1987, when the book was first published, the field has advanced through many developments, noted below. Still, I have been gratified that students continue to take the trouble to let me know that they have found this book a useful starting point.

    The concrete, empirical, and conversational style of this book arose from my experiences as a mathematician at General Motors working with engineers. They sometimes were not convinced by proofs, but their mechanical intuition responded well to numerical experiments that demonstrated the “feel” of the concepts. I learned at this time the paradox that a verbal explanation can defeat its purpose, if made to a tactile person, no matter how correct the explanation, no matter how skilled the person. Consequently, the language in this book stays as basic as possible for as long as possible: subscripts are not used until the end of Chapter 2, the only “spaces” referenced are Euclidean, and no concept or computer code is more general than needed. Yet, the mathematical facts are complete and proven, not necessarily when they are introduced, but eventually. For this, I have relied on a few results from differential topology, avoiding more abstract mathematics.

    Carrying out the numerical exercises using the Fortran code provided is important to the learning experience of the book.