Symmetry And Complexity

Cosmic evolution leads from symmetry to complexity by symmetry breaking and phase transitions. The emergence of new order and structure in nature and society is explained by physical, chemical, biological, social and economic self-organization, according to the laws of nonlinear dynamics. All these dynamical systems are considered computational systems processing information and entropy. Are symmetry and complexity only useful models of science or are they universals of reality? Symmetry and Complexity discusses the fascinating insights gained from natural, social and computer sciences, philosophy and the arts. With many diagrams and pictures, this book illustrates the spirit and beauty of nonlinear science. In the complex world of globalization, it strongly argues for unity in diversity.

Symmetry and complexity are the focus of a selection of outstanding papers, ranging from pure Mathematics and Physics to Computer Science and Engineering applications. This collection is based around fundamental problems arising from different fields, but all of them have the same task, i.e. breaking the complexity by the symmetry. In particular, in this Issue, there is an interesting paper dealing with circular multilevel systems in the frequency domain, where the analysis in the frequency domain gives a simple view of the system. Searching for symmetry in fractional oscillators or the analysis of symmetrical nanotubes are also some important contributions to this Special Issue. More papers, dealing with intelligent prognostics of degradation trajectories for rotating machinery in engineering applications or the analysis of Laplacian spectra for categorical product networks, show how this subject is interdisciplinary, i.e. ranging from theory to applications. In particular, the papers by Lee, based on the dynamics of trapped solitary waves for special differential equations, demonstrate how theory can help us to handle a practical problem. In this collection of papers, although encompassing various different fields, particular attention has been paid to the common task wherein the complexity is being broken by the search for symmetry.

This Brief is an essay at the interface of philosophy and complexity research, trying to inspire the reader with new ideas and new conceptual developments of cellular automata. Going beyond the numerical experiments of Steven Wolfram, it is argued that cellular automata must be considered complex dynamical systems in their own right, requiring appropriate analytical models in order to find precise answers and predictions in the universe of cellular automata. Indeed, eventually we have to ask whether cellular automata can be considered models of the real world and, conversely, whether there are limits to our modern approach of attributing the world, and the universe for that matter, essentially a digital reality.

The principle of local activity explains the emergence of complex patterns in a homogeneous medium. At first defined in the theory of nonlinear electronic circuits in a mathematically rigorous way, it can be generalized and proven at least for the class of nonlinear reaction-diffusion systems in physics, chemistry, biology, and brain research. Recently, it was realized by memristors for nanoelectronic device applications. In general, the emergence of complex patterns and structures is explained by symmetry breaking in homogeneous media, which is caused by local activity. This book argues that the principle of local activity is really fundamental in science, and can even be identified in quantum cosmology as symmetry breaking of local gauge symmetries generating the complexity of matter and forces in our universe. Applications are considered in economic, financial, and social systems with the emergence of equilibrium states, symmetry breaking at critical points of phase transitions and risky acting at the edge of chaos./a

MATHEMATICAL MACROEVOLUTION IN DIATOM RESEARCH Buy this book to learn how to use mathematics in macroevolution research and apply mathematics to study complex biological problems. This book contains recent research in mathematical and analytical studies on diatoms. These studies reflect the complex and intricate nature of the problems being analyzed and the need to use mathematics as an aid in finding solutions. Diatoms are important components of marine food webs, the silica and carbon cycles, primary productivity, and carbon sequestration. Their uniqueness as glass-encased unicells and their presence throughout geologic history exemplifies the need to better understand such organisms. Explicating the role of diatoms in the biological world is no more urgent than their role as environmental and climate indicators, and as such, is aided by the mathematical studies in this book. The volume contains twelve original research papers as chapters. Macroevolutionary science topics covered are morphological analysis, morphospace analysis, adaptation, food web dynamics, origination-extinction and diversity, biogeography, life cycle dynamics, complexity, symmetry, and evolvability. Mathematics used in the chapters include stochastic and delay differential and partial differential equations, differential geometry, probability theory, ergodic theory, group theory, knot theory, statistical distributions, chaos theory, and combinatorics. Applied sciences used in the chapters include networks, machine learning, robotics, computer vision, image processing, pattern recognition, and dynamical systems. The volume covers a diverse range of mathematical treatments of topics in diatom research. Audience Diatom researchers, mathematical biologists, evolutionary and macroevolutionary biologists, paleontologists, paleobiologists, theoretical biologists, as well as researchers in applied mathematics, algorithm sciences, complex systems science, computational sciences, informatics, computer vision and image processing sciences, nanoscience, the biofuels industry, and applied engineering.

This book is a printed edition of the Special Issue "Symmetry in Vision" that was published in Symmetry