This book is a printed edition of the Special Issue "Evaluation of Systems’ Irregularity and Complexity: Sample Entropy, Its Derivatives, and Their Applications across Scales and Disciplines" that was published in Entropy

Multiscale entropy (MSE) measures to evaluate the complexity of time series by taking into account the multiple time scales in physical systems were proposed in the early 2000s. Since then, these approaches have received a great deal of attention and have been used in a wide range of applications. Multivariate approaches have also been developed. The algorithms for an MSE approach are composed of two main steps: (i) a coarse-graining procedure to represent the system’s dynamics on different scales and (ii) the entropy computation for the original signal and for the coarse-grained time series to evaluate the irregularity for each scale. Moreover, different entropy measures have been associated with the coarse-graining approach, each one having its advantages and drawbacks. In this Special Issue, we gathered 24 papers focusing on either the theory or applications of MSE approaches. These papers can be divided into two groups: papers that propose new developments in entropy-based measures or improve the understanding of existing ones (9 papers) and papers that propose new applications of existing entropy-based measures (14 papers). Moreover, one paper presents a review of cross-entropy methods and their multiscale approaches.

Multiscale entropy (MSE) measures to evaluate the complexity of time series by taking into account the multiple time scales in physical systems were proposed in the early 2000s. Since then, these approaches have received a great deal of attention and have been used in a wide range of applications. Multivariate approaches have also been developed. The algorithms for an MSE approach are composed of two main steps: (i) a coarse-graining procedure to represent the system's dynamics on different scales and (ii) the entropy computation for the original signal and for the coarse-grained time series to evaluate the irregularity for each scale. Moreover, different entropy measures have been associated with the coarse-graining approach, each one having its advantages and drawbacks. In this Special Issue, we gathered 24 papers focusing on either the theory or applications of MSE approaches. These papers can be divided into two groups: papers that propose new developments in entropy-based measures or improve the understanding of existing ones (9 papers) and papers that propose new applications of existing entropy-based measures (14 papers). Moreover, one paper presents a review of cross-entropy methods and their multiscale approaches.

In each generation, scientists must redefine their fields: abstracting, simplifying and distilling the previous standard topics to make room for new advances and methods. Sethna's book takes this step for statistical mechanics - a field rooted in physics and chemistry whose ideas and methods are now central to information theory, complexity, and modern biology. Aimed at advanced undergraduates and early graduate students in all of these fields, Sethna limits his main presentation to the topics that future mathematicians and biologists, as well as physicists and chemists, will find fascinating and central to their work. The amazing breadth of the field is reflected in the author's large supply of carefully crafted exercises, each an introduction to a whole field of study: everything from chaos through information theory to life at the end of the universe.

One of the basic tenets of science is that deterministic systems are completely predictable-given the initial condition and the equations describing a system, the behavior of the system can be predicted 1 for all time. The discovery of chaotic systems has eliminated this viewpoint. Simply put, a chaotic system is a deterministic system that exhibits random behavior. Though identified as a robust phenomenon only twenty years ago, chaos has almost certainly been encountered by scientists and engi neers many times during the last century only to be dismissed as physical noise. Chaos is such a wide-spread phenomenon that it has now been reported in virtually every scientific discipline: astronomy, biology, biophysics, chemistry, engineering, geology, mathematics, medicine, meteorology, plasmas, physics, and even the social sci ences. It is no coincidence that during the same two decades in which chaos has grown into an independent field of research, computers have permeated society. It is, in fact, the wide availability of inex pensive computing power that has spurred much of the research in chaotic dynamics. The reason is simple: the computer can calculate a solution of a nonlinear system. This is no small feat. Unlike lin ear systems, where closed-form solutions can be written in terms of the system's eigenvalues and eigenvectors, few nonlinear systems and virtually no chaotic systems possess closed-form solutions.

In Complexity and Postmodernism, Paul Cilliers explores the idea of complexity in the light of contemporary perspectives from philosophy and science. Cilliers offers us a unique approach to understanding complexity and computational theory by integrating postmodern theory (like that of Derrida and Lyotard) into his discussion. Complexity and Postmodernism is an exciting and an original book that should be read by anyone interested in gaining a fresh understanding of complexity, postmodernism and connectionism.

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