Math Reading

Fractals and chaos are currently generating excitement across various scientific and medical disciplines. Biomedical investigators, graduate students, and undergraduates are becoming increasingly interested in applying fractals and chaos (nonlinear dynamics) to a variety of problems in biology and medicine. This accessible text lucidly explains these concepts and illustrates their uses with examples from biomedical research. The author presents the material in a very unique, straightforward manner which avoids technical jargon and does not assume a strong background in mathematics. The text uses a step-by-step approach by explaining one concept at a time in a set of facing pages, with text on the left page and graphics on the right page; the graphics pages can be copied directly onto transparencies for teaching. Ideal for courses in biostatistics, fractals, mathematical modeling of biological systems, and related courses in medicine, biology, and applied mathematics, Fractals and Chaos Simplified for the Life Sciences will also serve as a useful resource for scientists in biomedicine, physics, chemistry, and engineering.

Painter Perry Hall uses a set of experimental techniques that draw upon the organizing principles found in nature. In his Decalcomania paintings, he uses pressure to shape complex networks of interlocking lines and ridges of paint into organic compositions that evoke the notion of 'growing' a painting. In his Livepaintings (time-based paintings), he stimulates paint with temperature changes, vibration, turbulence, and various substances, transforming paint flows into compositions he captures onto video. In the Sound Drawings, sound waves are channeled into a vessel containing oil or acrylics: Hall paints by changing the qualities of the sound, in effect, 'playing' the paint like a musical instrument. His innovative paintings are a fascinating and beautiful collaboration with "material intelligence" and a meditation on the dynamics found in nature. Over 190 full-color reproductions. For more information and images of Hall's paintings: http://www.lovebrain.net/paintings

In recent years there has been an explosive growth in the study of physical, biological, and economic systems that can be profitably studies using densities. Because of the general inaccessibility of the mathematical literature to the nonspecialist, little diffusion of the applicable mathematics into the study of these "chaotic" systems has taken place. This book will help bridge that gap. To show how densities arise in simple deterministic systems, the authors give a unified treatment of a variety of mathematical system generating densities, ranging from one-dimensional discrete time transformations through continuous time systems described by integro-partial-differential equations. Examples have been drawn from many fields to illustrate the utility of the concepts and techniques presented, and the ideas in this book should thus prove useful in the study of a number of applied sciences. The authors assume that the reader has a knowledge of advanced calculus and differential equations. Basic concepts from measure theory, ergodic theory, the geometry of manifolds, partial differential equations, probability theory and Markov processes, and stochastic integrals and differential equations are introduced as needed. Physicists, chemists, and biomathematicians studying chaotic behavior will find this book of value. It will also be a useful reference or text for mathematicians and graduate students working in ergodic theory and dynamical systems.

This book develops deterministic chaos and fractals from the standpoint of iterated maps, but the method of analysis and choice of emphasis make it very different from all other books in the field. It is written to provide the reader with an introduction to more recent developments, such as weak universality, multifractals, and shadowing, as well as to older subjects such as universal critical exponents, devil's staircases, and the Farey tree. Throughout the book the author uses a fully discrete method, a "theoretical computer arithmetic," because finite (but not fixed) precision is a fact of life that cannot be avoided in computation or in experiment. This approach leads to a more general formulation in terms of symbolic dynamics and to the idea of weak universality. The author explains why continuum analysis, computer simulations, and experiments form three entirely distinct approaches to chaos theory. In the end, the connection is made with Turing's ideas of computable numbers. It is explained why the continuum approach leads to predictions that are not necessarily realized in computations or in nature, whereas the discrete approach yields all possible histograms that can be observed or computed.

During the last decade the well-established tools of statistical physics have been successfully applied to an increasing number of biological phenomena. It is a fruitful approach to systems characterized by fluctuations and/or a large number of very similar units, and such systems are common in biology, whether it be the individuals in the codons of a genetic code or the behavioral responses of macromolecules to thermal fluctuations. This book is thus able to cover a wide range of phenomena, including fractal pattern formation, group motion in organisms from bacteria to humans, or the mechanisms by which fluctuations are rectified in the cell's molecular machinery. This book provides a summary of the majority of recent approaches and concepts born in the study of biological phenomena involving collective behavior and random perturbation, as well as presenting some of the most important new results to specialist researchers.

An exciting new way of teaching chaos in dynamical systems to undergraduates

This book presents elements of the theory of chaos in dynamical systems in a framework of theoretical understanding coupled with numerical and graphical experimentation. The theory is developed using only elementary calculus and algebra, and includes dynamics of one- and two-dimensional maps, periodic orbits, stability and its quantification, chaotic behavior, and bifurcation theory of one-dimensional systems. There is an introduction to the theory of fractals, with an emphasis on the importance of scaling, and a concluding chapter on ordinary differential equations. The accompanying software, written in Java, enables students to carry out their own quantitative experiments on a variety of nonlinear systems, including the analysis of fixed points of compositions of maps, calculation of Fourier spectra and Lyapunov exponents, and box counting for two-dimensional maps. It also provides for visualizing orbits, final state and bifurcation diagrams, Fourier spectra and Lyapunov exponents, basins of attractions, three-dimensional orbits, Poincar sections, and return maps.

Fractals for the Classroom breaks new ground as it brings an exciting branch of mathematics into the classroom. The book is a collection of independent chapters on the major concepts related to the science and mathematics of fractals. Written at the mathematical level of an advanced secondary student, Fractals for the Classroom includes many fascinating insights for the classroom teacher and integrates illustrations from a wide variety of applications with an enjoyable text to help bring the concepts alive and make them understandable to the average reader. This book will have a tremendous impact upon teachers, students, and the mathematics education of the general public. With the forthcoming companion materials, including four books on strategic classroom activities and lessons with interactive computer software, this package will be unparalleled.

This textbook is intended to supplement the classical theory of uni- and multivariate splines and their approximation and interpolation properties with those of fractals, fractal functions, and fractal surfaces. This synthesis will complement currently required courses dealing with these topics and expose the prospective reader to some new and deep relationships. In addition to providing a classical introduction to the main issues involving approximation and interpolation with uni- and multivariate splines, cardinal and exponential splines, and their connection to wavelets and multiscale analysis, which comprises the first half of the book, the second half will describe fractals, fractal functions and fractal surfaces, and their properties. This also includes the new burgeoning theory of superfractals and superfractal functions. The theory of splines is well-established but the relationship to fractal functions is novel. Throughout the book, connections between these two apparently different areas will be exposed and presented. In this way, more options are given to the prospective reader who will encounter complex approximation and interpolation problems in real-world modeling. Numerous examples, figures, and exercises accompany the material.