Complex Nonlinearity: Chaos, Phase Transitions, Topology Change and Path Integrals is a book about prediction & control of general nonlinear and chaotic dynamics of high-dimensional complex systems of various physical and non-physical nature and their underpinning geometro-topological change. The book starts with a textbook-like expose on nonlinear dynamics, attractors and chaos, both temporal and spatio-temporal, including modern techniques of chaos-control. Chapter 2 turns to the edge of chaos, in the form of phase transitions (equilibrium and non-equilibrium, oscillatory, fractal and noise-induced), as well as the related field of synergetics. While the natural stage for linear dynamics comprises of flat, Euclidean geometry (with the corresponding calculation tools from linear algebra and analysis), the natural stage for nonlinear dynamics is curved, Riemannian geometry (with the corresponding tools from nonlinear, tensor algebra and analysis). The extreme nonlinearity - chaos - corresponds to the topology change of this curved geometrical stage, usually called configuration manifold. Chapter 3 elaborates on geometry and topology change in relation with complex nonlinearity and chaos. Chapter 4 develops general nonlinear dynamics, continuous and discrete, deterministic and stochastic, in the unique form of path integrals and their action-amplitude formalism. This most natural framework for representing both phase transitions and topology change starts with Feynman's sum over histories, to be quickly generalized into the sum over geometries and topologies. The last Chapter puts all the previously developed techniques together and presents the unified form of complex nonlinearity. Here we have chaos, phase transitions, geometrical dynamics and topology change, all working together in the form of path integrals. The objective of this book is to provide a serious reader with a serious scientific tool that will enable them to actually perform a competitive research in modern complex nonlinearity. It includes a comprehensive bibliography on the subject and a detailed index. Target readership includes all researchers and students of complex nonlinear systems (in physics, mathematics, engineering, chemistry, biology, psychology, sociology, economics, medicine, etc.), working both in industry/clinics and academia.

Quantum Neural Computation is a graduate-level monographic textbook. It presents a comprehensive introduction, both non-technical and technical, into modern quantum neural computation, the science behind the fiction movie Stealth. Classical computing systems perform classical computations (i.e., Boolean operations, such as AND, OR, NOT gates) using devices that can be described classically (e.g., MOSFETs). On the other hand, quantum computing systems perform classical computations using quantum devices (quantum dots), that is devices that can be described only using quantum mechanics. Any information transfer between such computing systems involves a state measurement. This book describes this information transfer at the edge of classical and quantum chaos and turbulence, where mysterious quantum-mechanical linearity meets even more mysterious brain's nonlinear complexity, in order to perform a super-high-speed and error-free computations. This monograph describes a crossroad between quantum field theory, brain science and computational intelligence.

Modern Geometrical Machinery; 1 .1 Introduction; 1 .2 Smooth Manifolds; 1.2.1 Intuition Behind a Smooth Manifold; 1.2.2 Definition of a Smooth Manifold; 1.2.3 Smooth Maps Between Manifolds; 1.2.4 (Co)Tangent Bundles of a Smooth Manifold; 1.2.5 Tensor Fields and Bundles of a Smooth Manifold; 1.2.6 Lie Derivative on a Smooth Manifold; 1.2.7 Lie Groups and Associated Lie Algebras; 1.2.8 Lie Symmetries and Prolongations on Manifolds;1.2.9 Riemannian Manifolds; 1.2.10 Finsler Manifolds; 1.2.11 Symplectic Manifolds; 1.2.12 Complex and Kahler Manifolds; 1.2.13 Conformal Killing-Riemannian Geometry; 1.3 Fibre Bundles; 1.3.1 Intuition Behind a Fibre Bundle; 1.3.2 Definition of a Fibre Bundle;1.3.3 Vector and Affine Bundles; 1.3.4 Principal Bundles; 1.3.5 Multivector-Fields and Tangent-Valued Forms; 1.4 Jet Spaces; 1.4.1 Intuition Behind a Jet Space; 1.4.2 Definition of a 1-Jet Space; 1.4.3 Connections as Jet Fields; 1.4.4 Definition of a 2-Jet Space; 1.4.5 Higher-Order Jet Spaces; 1.4.6 Jets in Mechanics;1.4.7 Jets and Action Principle; 1.5 Path Integrals: Extending Smooth Geometrical Machinery; 1.5.1 Intuition Behind a Path Integral; 1.5.2 Path Integral History; 1.5.3 Standard Path-Integral Quantization; 1.5.4 Sum over Geometries/Topologies; 1.5.5 TQFT and Stringy Path Integrals; 2 Dynamics of High-Dimensional Nonlinear Systems; 2.1 Mechanical Systems; 2.1.1 Autonomous Lagrangian/Hamiltonian Mechanics; 2.1.2 Non-Autonomous Lagrangian/Hamiltonian Mechanics; 2.1.3 Semi-Riemannian Geometrical Dynamics; 2.1.4 Relativistic and Multi-Time Rheonomic Dynamics; 2.1.5 Geometrical Quantization; 2.2 Physical Field Systems; 2.2.1 n-Categorical Framework; 2.2.2 Lagrangian Field Theory on Fibre Bundles; 2.2.3 Finsler-Lagrangian Field Theory; 2.2.4 Hamiltonian Field Systems: Path-Integral Quantization; 2.2.5 Gauge Fields on Principal Connections; 2.2.6 Modern Geometrodynamics; 2.2.7 Topological Phase Transitions and Hamiltonian Chaos; 2.2.8 Topological Superstring Theory; 2.2.9 Turbulence and Chaos Field Theory; 2.3 Nonlinear Control Systems; 2.3.1 The Basis of Modern Geometrical Control;2.3.2 Geometrical Control of Mechanical Systems;2.3.3 Hamiltonian Optimal Control and Maximum Principle; 2.3.4 Path-Integral Optimal Control of Stochastic Systems; 2.4 Human-Like Biomechanics; 2.4.1 Lie Groups and Symmetries in Biomechanics; 2.4.2 Muscle-Driven Hamiltonian Biomechanics; 2.4.3 Biomechanical Functors; 2.4.4 Biomechanical Topology; 2.5 Neurodynamics; 2.5.1 Microscopic Neurodynamics and Quantum Brain; 2.5.2 Macroscopic Neurodynamics; 2.5.3 Oscillatory Phase Neurodynamics;2.5.4 Neural Path-Integral Model for the Cerebellum; 2.5.5 Intelligent Robot Control; 2.5.6 Brain-Like Control Functor in Biomechanics; 2.5.7 Concurrent and Weak Functorial Machines; 2.5.8 Brain-Mind Functorial Machines; 26 Psycho-Socio-Economic Dynamics; 2.6.1 Force-Field Psychodynamics; 2.6.2 Geometrical Dynamics of Human Crowd; 2.6.3 Dynamical Games on Lie Groups; 2.6.4 Nonlinear Dynamics of Option Pricing; 2.6.5 Command/Control in Human-Robot Interactions; 2.6.6 Nonlinear Dynamics of Complex Nets; 2.6.7 Complex Adaptive Systems: Common Characteristics; 2.6.8 FAM Functors and Real-Life Games; 2.6.9 Riemann-Finsler Approach to Information Geometry; 3 Appendix: Tensors and Functors; 3.1 Elements of Classical Tensor Analysis; 3.1.1 Transformation of Coordinates and Elementary Tensors; 3.1.2 Euclidean Tensors; 3. 1 .3 Tensor Derivatives on Riemannian Manifolds; 3.1.4 Tensor Mechanics in Brief; 3.1.5 The Covariant Force Law in Robotics and Biomechanics; 3.2 Categories and Functors; 3.2.1 Maps; 3.2.2 Categories; 3.2.3 Functors; 3.2.4 Natural Transformations; 3.2.5 Limits and Colimits; 3.2.6 The Adjunction; 3.2.7 ri-Categories; 3.2.8 Abelian Functorial Algebra; References; Index.

Human-Like Biomechanics is a comprehensive introduction into modern geometrical methods to be used as a unified research approach in two apparently separate and rapidly growing fields: mathematical biomechanics and humanoid robotics. The term human-like biomechanics is used to denote this unified modelling and control approach to humanoid robotics and mathematical biomechanics, based on theoretical mechanics, differential geometry and topology, nonlinear dynamics and control, and path-integral methods. From this geometry-mechanics-control modelling perspective, "human" and "humanoid" means the same. This unified approach enables both design of humanoid systems of immense complexity and prediction/prevention of subtle neuro-musculo-skeletal injuries. This approach has been realized in the form of the world-leading human-motion simulator with 264 powered degrees of freedom, called Human Biodynamics Engine (developed in Defence Science & Technology Organisation, Australia).

The book contains six Chapters and an Appendix. The first Chapter is an Introduction, giving a brief review of mathematical techniques to be used in the text. The second Chapter develops geometrical basis of human-like biomechanics, while the third Chapter develops its mechanical basis, mainly from generalized Lagrangian and Hamiltonian perspective. The fourth Chapter develops topology of human-like biomechanics, while the fifth Chapter reviews related nonlinear control techniques. The sixth Chapter develops covariant biophysics of electro-muscular stimulation. The Appendix consists of two parts: classical muscular mechanics and modern path integral methods, which are both used frequently in the main text. The whole book is based on the authors own research papers in human-like biomechanics. "

This is a graduate level monographic textbook in the field of Computational Intelligence. It presents a modern dynamical theory of the computational mind, combining cognitive psychology, artificial and computational intelligence, and chaos theory with quantum consciousness and computation. The book introduces to human and computational mind, comparing and contrasting main themes of cognitive psychology, artificial and computational intelligence.

This book represents a comprehensive introduction into both conceptual and rigorous brain and cognition modelling. It is devoted to understanding, prediction and control of the fundamental mechanisms of brain functioning. The reader will be provided with a scientific tool enabling him or her to perform a competitive research in brain and cognition modelling. This is a graduate level monographic textbook.

This graduate-level textbook is devoted to understanding, prediction and control of high-dimensional chaotic and attractor systems of real life. The objective is to provide the serious reader with a serious scientific tool that will enable the actual performance of competitive research in high-dimensional chaotic and attractor dynamics. From introductory material on low-dimensional attractors and chaos, the text explores concepts including Poincare's 3-body problem, high-tech Josephson junctions, and more."

Quantum Neural Computation is a graduate level monographic textbook. It presents a comprehensive introduction, both non-technical and technical, into modern quantum neural computation, the science behind the fiction movie Stealth. Classical computing systems perform classical computations (i.e., Boolean operations, such as AND, OR, NOT gates) using devices that can be described classically (e.g., MOSFETs). On the other hand, quantum computing systems perform classical computations using quantum devices (quantum dots), that is devices that can be described only using quantum mechanics. Any information transfer between such computing systems involves a state measurement. This book describes this information transfer at the edge of classical and quantum chaos and turbulence, where mysterious quantum-mechanical linearity meets even more mysterious brain s nonlinear complexity, in order to perform a super high speed and error free computations. This monograph describes a crossroad between quantum field theory, brain science and computational intelligence."

This is a graduate level monographic textbook in the field of Computational Intelligence. It presents a modern dynamical theory of the computational mind, combining cognitive psychology, artificial and computational intelligence, and chaos theory with quantum consciousness and computation. The book introduces to human and computational mind, comparing and contrasting main themes of cognitive psychology, artificial and computational intelligence.

This book represents a comprehensive introduction into both conceptual and rigorous brain and cognition modelling. It is devoted to understanding, prediction and control of the fundamental mechanisms of brain functioning. The reader will be provided with a scientific tool enabling him or her to perform a competitive research in brain and cognition modelling. This is a graduate level monographic textbook.

This graduate level textbook is devoted to understanding, prediction and control of high dimensional chaotic and attractor systems of real life. The objective is to provide the serious reader with a serious scientific tool that will enable the actual performance of competitive research in high dimensional chaotic and attractor dynamics. From introductory material on low-dimensional attractors and chaos, the text explores concepts including Poincare s 3-body problem, high-tech Josephson junctions, and more.

Human-Like Biomechanics is a comprehensive introduction into modern geometrical methods to be used as a unified research approach in two apparently separate and rapidly growing fields: mathematical biomechanics and humanoid robotics. The term human-like biomechanics is used to denote this unified modelling and control approach to humanoid robotics and mathematical biomechanics, based on theoretical mechanics, differential geometry and topology, nonlinear dynamics and control, and path-integral methods. From this geometry-mechanics-control modelling perspective, "human" and "humanoid" means the same. This unified approach enables both design of humanoid systems of immense complexity and prediction/prevention of subtle neuro-musculo-skeletal injuries. This approach has been realized in the form of the world-leading human-motion simulator with 264 powered degrees of freedom, called Human Biodynamics Engine (developed in Defence Science & Technology Organisation, Australia).

The book contains six Chapters and an Appendix. The first Chapter is an Introduction, giving a brief review of mathematical techniques to be used in the text. The second Chapter develops geometrical basis of human-like biomechanics, while the third Chapter develops its mechanical basis, mainly from generalized Lagrangian and Hamiltonian perspective. The fourth Chapter develops topology of human-like biomechanics, while the fifth Chapter reviews related nonlinear control techniques. The sixth Chapter develops covariant biophysics of electro-muscular stimulation. The Appendix consists of two parts: classical muscular mechanics and modern path integral methods, which are both used frequently in the main text. Thewhole book is based on the authorsa (TM) own research papers in human-like biomechanics.

Geometrical Dynamics of Complex Systems is a graduate-level monographic textbook. Itrepresentsacomprehensiveintroductionintorigorousgeometrical dynamicsofcomplexsystemsofvariousnatures. By'complexsystems', inthis book are meant high-dimensional nonlinear systems, which can be (but not necessarily are) adaptive. This monograph proposes a uni?ed geometrical - proachtodynamicsofcomplexsystemsofvariouskinds: engineering, physical, biophysical, psychophysical, sociophysical, econophysical, etc. As their names suggest, all these multi-input multi-output (MIMO) systems have something in common: the underlying physics. However, instead of dealing with the pop- 1 ular 'soft complexity philosophy', we rather propose a rigorous geometrical and topological approach. We believe that our rigorous approach has much greater predictive power than the soft one. We argue that science and te- nology is all about prediction and control. Observation, understanding and explanation are important in education at undergraduate level, but after that it should be all prediction and control. The main objective of this book is to show that high-dimensional nonlinear systems and processes of 'real life' can be modelled and analyzed using rigorous mathematics, which enables their complete predictability and controllability, as if they were linear systems. It is well-known that linear systems, which are completely predictable and controllable by de?nition - live only in Euclidean spaces (of various - mensions). They are as simple as possible, mathematically elegant and fully elaborated from either scienti?c or engineering side. However, in nature, no- ing is linear. In reality, everything has a certain degree of nonlinearity, which means: unpredictability, with subsequent uncontrollability.

Complex Nonlinearity: Chaos, Phase Transitions, Topology Change and Path Integrals is a book about prediction & control of general nonlinear and chaotic dynamics of high-dimensional complex systems of various physical and non-physical nature and their underpinning geometro-topological change.

The book starts with a textbook-like expose on nonlinear dynamics, attractors and chaos, both temporal and spatio-temporal, including modern techniques of chaos-control. Chapter 2 turns to the edge of chaos, in the form of phase transitions (equilibrium and non-equilibrium, oscillatory, fractal and noise-induced), as well as the related field of synergetics. While the natural stage for linear dynamics comprises of flat, Euclidean geometry (with the corresponding calculation tools from linear algebra and analysis), the natural stage for nonlinear dynamics is curved, Riemannian geometry (with the corresponding tools from nonlinear, tensor algebra and analysis). The extreme nonlinearity - chaos - corresponds to the topology change of this curved geometrical stage, usually called configuration manifold. Chapter 3 elaborates on geometry and topology change in relation with complex nonlinearity and chaos. Chapter 4 develops general nonlinear dynamics, continuous and discrete, deterministic and stochastic, in the unique form of path integrals and their action-amplitude formalism. This most natural framework for representing both phase transitions and topology change starts with Feynman's sum over histories, to be quickly generalized into the sum over geometries and topologies. The last Chapter puts all the previously developed techniques together and presents the unified form of complex nonlinearity. Here we have chaos, phase transitions, geometrical dynamics and topology change, all working together in the form of path integrals.

The objective of this book is to provide a serious reader with a serious scientific tool that will enable them to actually perform a competitive research in modern complex nonlinearity. It includes a comprehensive bibliography on the subject and a detailed index. Target readership includes all researchers and students of complex nonlinear systems (in physics, mathematics, engineering, chemistry, biology, psychology, sociology, economics, medicine, etc.), working both in industry/clinics and academia.

This book gives a comprehensive introduction into geometrical methods of modern biomechanics. This book is designed for a rigorous, one--semester course at the graduate level. The intended audience includes mechanical, control and biomedical engineers (with stronger mathematical background), mathematicians, physicists, computer scientists and mathematical biologists, as well as all researchers and technical professionals interested in modelling and control of biomechanical systems and humanoid robots.

The 'Handbook of Geometrical Methods for Scientists and Engineers' is an undergraduate applied mathematics text, compiled as a collection of concepts and formulas of modern geometrical and topological methods designed for use in science and engineering. These geometrical methods are currently being used for modelling complex systems in theoretical physics, chemistry and biology, non-linear dynamics and non-linear control, as well as mathematically -- enriched human sciences (medicine, psychology, sociology and economics). This book contains an easy-to-follow essence of geometrical and topological methods for modelling complex dynamical systems, extracted from our five graduate -- level monographs (including over 2000 cited references in total): 1. Geometrical Dynamics of Complex Systems: A Unified Modelling Approach to Physics, Control, Biomechanics, Neurodynamics and Psycho- Socio-Economical Dynamics. Springer, 2006; 2. Complex Dynamics: Advanced System Dynamics in Complex Variables, Springer, 2007; 3. Applied Differential Geometry: A Modern Introduction. World Scientific,2007; 4. Complex Nonlinearity: Chaos, Phase Transitions, Topology Change and Path Integrals, Springer, 2008; and 5. Quantum Leap: From Dirac and Feynman, Across the Universe, to Human Body and Mind. World Scientific, Singapore, 2008. The only necessary background for efficient understanding and using of the Handbook is standard Engineering Mathematics IB (namely, Calculus and Linear Algebra).

Mathematics of Autonomy provides solid mathematical foundations for building useful Autonomous Systems. It clarifies what makes a system autonomous rather than simply automated, and reveals the inherent limitations of systems currently incorrectly labeled as autonomous in reference to the specific and strong uncertainty that characterizes the environments they operate in. Such complex real-world environments demand truly autonomous solutions to provide the flexibility and robustness needed to operate well within them.This volume embraces hybrid solutions to demonstrate extending the classes of uncertainty autonomous systems can handle. In particular, it combines physical-autonomy (robots), cyber-autonomy (agents) and cognitive-autonomy (cyber and embodied cognition) to produce a rigorous subset of trusted autonomy: Cyber-Physical-Cognitive autonomy (CPC-autonomy).The body of the book alternates between underlying theory and applications of CPC-autonomy including 'Autonomous Supervision of a Swarm of Robots' , 'Using Wind Turbulence against a Swarm of UAVs' and 'Unique Super-Dynamics for All Kinds of Robots (UAVs, UGVs, UUVs and USVs)' to illustrate how to effectively construct Autonomous Systems using this model. It avoids the wishful thinking that characterizes much discussion related to autonomy, discussing the hard limits and challenges of real autonomous systems. In so doing, it clarifies where more work is needed, and also provides a rigorous set of tools to tackle some of the problem space.

The book Complexity and Control: Towards a Rigorous Behavioral Theory of Complex Dynamical Systems is a graduate-level monographic textbook, intended to be a novel and rigorous contribution to modern Complexity Theory.This book contains 11 chapters and is designed as a one-semester course for engineers, applied and pure mathematicians, theoretical and experimental physicists, computer and economic scientists, theoretical chemists and biologists, as well as all mathematically educated scientists and students, both in industry and academia, interested in predicting and controlling complex dynamical systems of arbitrary nature.

New Trends in Control Theory is a graduate-level monographic textbook. It is a contemporary overview of modern trends in control theory. The introductory chapter gives the geometrical and quantum background, which is a necessary minimum for comprehensive reading of the book. The second chapter gives the basics of classical control theory, both linear and nonlinear. The third chapter shows the key role that Euclidean group of rigid motions plays in modern robotics and biomechanics. The fourth chapter gives an overview of modern quantum control, from both theoretical and measurement perspectives. The fifth chapter presents modern control and synchronization methods in complex systems and human crowds. The appendix provides the rest of the background material complementary to the introductory chapter. The book is designed as a one-semester course for engineers, applied mathematicians, computer scientists and physicists, both in industry and academia. It includes a most relevant bibliography on the subject and detailed index.