The Theory of Complex System

This is a collection of exercises in the theory of analytic functions, with completed and detailed solutions. We wish to introduce the student to applications and aspects of the theory of analytic functions not always touched upon in a first course. Using appropriate exercises show the students some aspects of what lies beyond a first course in complex variables. We also discuss topics of interest for electrical engineering students (for instance, the realization of rational functions and its connections to the theory of linear systems and state space representations of such systems). Examples of important Hilbert spaces of analytic functions (in particular the Hardy space and the Fock space) are given. The book also includes a part where relevant facts from topology, functional analysis and Lebesgue integration are reviewed.

The 2nd edition of this book provides novel topics and studyies in boundaries of networks and Big Data Systems.The central theme of this book is the extent to which the structure of the free dynamical boundaries of a system controls the evolution of the system as a whole. Applying three orthogonal types of thinking - mathematical, constructivist and morphological, it illustrates these concepts using applications to selected problems from the social and life sciences, as well as economics.
In a broader context, it introduces and reviews some modern mathematical approaches to the science of complex systems.

Artificial neural networks possess several properties that make them particularly attractive for applications to modelling and control of complex non-linear systems. Among these properties are their universal approximation ability, their parallel network structure and the availability of on- and off-line learning methods for the interconnection weights.

This book applies system theory to analyze the operation and structure of the complex earth surface system, including the interactions between society and nature that cause environmental degradation and threats to human populations. The possible ways to harmonize the operation of a global society as a complex system using the United Nation sustainable development goals are investigated, as well as the major efforts currently implemented to achieve this objective and why many are unsuccessful.

Research into social systems is challenging due to their complex nature. Traditional methods of analysis are often difficult to apply effectively as theories evolve over time. This can be due to a lack of appropriate data, or too much uncertainty. It can also be the result of problems which are not yet understood well enough in the general sense so that they can be classified, and an appropriate solution quickly identified. Simulation is one tool that deals well with these challenges, fits in well with the deductive process, and is useful for testing theory.