This book presents a cross-disciplinary approach to smart grids, offering an invaluable basis for understanding their complexity and potential, and for discussing their technical, legal, economic, societal, psychological and security aspects. Smart grids are a complex phenomenon involving new, active roles for consumers and prosumers, novel social, political and cultural practices, advanced ICT, new markets, security of supply issues, the informational turn in energy, valuation of assets and investments, technological innovation and (de)regulation. Furthermore, smart grids offer new interfaces, in turn creating hybrid fields: with the increasing use of electric vehicles and electric transportation, smart grids represent the crossroads of energy and mobility. While the aim is to achieve more sustainable production, transportation and use of energy, the importance of smart grids actually has less to do with electricity, heat or gas, and far more with transforming the infrastructure needed to deliver energy, as well as the roles of its owners, operators and users. The immediate goal is to contribute positively to a sustainable world society. The chapters are revised and expanded texts based upon lectures delivered at the Groningen Energy Summer School 2014. Questions for further discussion at the end of each chapter highlight the key themes that emerge. The book offers an indispensable resource for researchers, professionals and companies in the power supply industry, and for students seeking to broaden and deepen their understanding of smart grids.

This volume collects the edited and reviewed contributions presented in the 6th iTi Conference in Bertinoro, covering fundamental and applied aspects in turbulence. In the spirit of the iTi conference, the volume has been produced after the conference so that the authors had the possibility to incorporate comments and discussions raised during the meeting. In the present book the contributions have been structured according to the topics : I Theory II Wall bounded flows III Particles in flows IV Free flows V Complex flows The volume is dedicated to the memory of Prof. Konrad Bajer who prematurely passed away in Warsaw on August 29, 2014.

This book focuses on the basic control and filtering synthesis problems for discrete-time switched linear systems under time-dependent switching signals. Chapter 1, as an introduction of the book, gives the backgrounds and motivations of switched systems, the definitions of the typical time-dependent switching signals, the differences and links to other types of systems with hybrid characteristics and a literature review mainly on the control and filtering for the underlying systems. By summarizing the multiple Lyapunov-like functions (MLFs) approach in which different requirements on comparisons of Lyapunov function values at switching instants, a series of methodologies are developed for the issues on stability and stabilization, and l2-gain performance or tube-based robustness for l disturbance, respectively, in Chapters 2 and 3. Chapters 4 and 5 are devoted to the control and filtering problems for the time-dependent switched linear systems with either polytopic uncertainties or measurable time-varying parameters in different sense of disturbances. The asynchronous switching problem, where there is time lag between the switching of the currently activated system mode and the controller/filter to be designed, is investigated in Chapter 6. The systems with various time delays under typical time-dependent switching signals are addressed in Chapter 7.

This new edition provides an up-to-date coverage of important theoretical models in the scheduling literature as well as significant scheduling problems that occur in the real world. It again includes supplementary material in the form of slide-shows from industry and movies that show implementations of scheduling systems. The main structure of the book as per previous edition consists of three parts. The first part focuses on deterministic scheduling and the related combinatorial problems. The second part covers probabilistic scheduling models; in this part it is assumed that processing times and other problem data are random and not known in advance. The third part deals with scheduling in practice; it covers heuristics that are popular with practitioners and discusses system design and implementation issues. All three parts of this new edition have been revamped and streamlined. The references have been made completely up-to-date. Theoreticians and practitioners alike will find this book of interest. Graduate students in operations management, operations research, industrial engineering, and computer science will find the book an accessible and invaluable resource. Scheduling - Theory, Algorithms, and Systems will serve as an essential reference for professionals working on scheduling problems in manufacturing, services, and other environments.

The book investigates stability theory in terms of two different measure, exhibiting the advantage of employing families of Lyapunov functions and treats the theory of a variety of inequalities, clearly bringing out the underlying theme. It also demonstrates manifestations of the general Lyapunov method, showing how this technique can be adapted to various apparently diverse nonlinear problems. Furthermore it discusses the application of theoretical results to several different models chosen from real world phenomena, furnishing data that is particularly relevant for practitioners. Stability Analysis of Nonlinear Systems is an invaluable single-sourse reference for industrial and applied mathematicians, statisticians, engineers, researchers in the applied sciences, and graduate students studying differential equations.

This volume provides an introduction to and overview of the emerging field of interconnected networks which include multilayer or multiplex networks, as well as networks of networks. Such networks present structural and dynamical features quite different from those observed in isolated networks. The presence of links between different networks or layers of a network typically alters the way such interconnected networks behave - understanding the role of interconnecting links is therefore a crucial step towards a more accurate description of real-world systems. While examples of such dissimilar properties are becoming more abundant - for example regarding diffusion, robustness and competition - the root of such differences remains to be elucidated. Each chapter in this topical collection is self-contained and can be read on its own, thus making it also suitable as reference for experienced researchers wishing to focus on a particular topic.

This book is devoted to Slime mould Physarum polycephalum, which is a large single cell capable for distributed sensing, concurrent information processing, parallel computation and decentralized actuation. The ease of culturing and experimenting with Physarum makes this slime mould an ideal substrate for real-world implementations of unconventional sensing and computing devices The book is a treatise of theoretical and experimental laboratory studies on sensing and computing properties of slime mould, and on the development of mathematical and logical theories of Physarum behavior. It is shown how to make logical gates and circuits, electronic devices (memristors, diodes, transistors, wires, chemical and tactile sensors) with the slime mould. The book demonstrates how to modify properties of Physarum computing circuits with functional nano-particles and polymers, to interface the slime mould with field-programmable arrays, and to use Physarum as a controller of microbial fuel cells. A unique multi-agent model of slime is shown to serve well as a software slime mould capable for solving problems of computational geometry and graph optimization. The multiagent model is complemented by cellular automata models with parallel accelerations. Presented mathematical models inspired by Physarum include non-quantum implementation of Shor's factorization, structural learning, computation of shortest path tree on dynamic graphs, supply chain network design, p-adic computing and syllogistic reasoning. The book is a unique composition of vibrant and lavishly illustrated essays which will inspire scientists, engineers and artists to exploit natural phenomena in designs of future and emergent computing and sensing devices. It is a 'bible' of experimental computing with spatially extended living substrates, it spanstopics from biology of slime mould, to bio-sensing, to unconventional computing devices and robotics, non-classical logics and music and arts.

This fascinating, colourful book offers in-depth insights and first-hand working experiences in the production of art works, using simple computational models with rich morphological behaviour, at the edge of mathematics, computer science, physics and biology. It organically combines ground breaking scientific discoveries in the theory of computation and complex systems with artistic representations of the research results. In this appealing book mathematicians, computer scientists, physicists, and engineers brought together marvelous and esoteric patterns generated by cellular automata, which are arrays of simple machines with complex behavior. Configurations produced by cellular automata uncover mechanics of dynamic patterns formation, their propagation and interaction in natural systems: heart pacemaker, bacterial membrane proteins, chemical rectors, water permeation in soil, compressed gas, cell division, population dynamics, reaction-diffusion media and self-organisation. The book inspires artists to take on cellular automata as a tool of creativity and it persuades scientists to convert their research results into the works of art. The book is lavishly illustrated with visually attractive examples, presented in a lively and easily accessible manner.

This book contains the proceedings of the Sixth National Conference of the Italian Systems Society. The title, Towards a post-Bertalanffy Systemics, aims to underline the need for Systemics and Systems Science to generalize theoretically concepts related to complexity (the great enemy of Bertalanffy Systemics). Hopefully this goal should be achieved by working in an inter-disciplinary and trans-disciplinary fashion, using systemic concepts arising from various disciplines and from the original, or Bertalanffy Systemics, as well. The interdisciplinary nature of the original Systemics and its power of generalization were given, overall, by the fact that the problems and solutions of one discipline become problems and solutions for another. Today, the modeling and interpretation of multidisciplinary approaches and representations makes easier to recognize these interconnections. The context, however, has changed dramatically. Of course, the challenge is still to find theoretical generalizations and applications, even where we have a lot of specificities, but we know very little on how to combine them. We cannot, however, simply replace the old with the new, but we must introduce strategies to recognize, represent, model and act on new levels, combining multiple representations, functions and emergence. In many disciplines this has been already done, and inevitably well, since targets and projects are well specified and oriented. The challenge is to do it for Systemics, with the vocations of cultural and theoretical generalization. Examples of new issues introduced by such theoretical disciplinary improvements, dealt with by many disciplines, include the study of mesoscopic or middle-way level, of multiple and dynamic coherence, of equivalence/non-equivalence, of fractality, of networks, of non-causality, of non-invasiveness, of non-prescribability, of non-separability, of quasi properties, of symmetry properties, of topological dynamics, as well as of quantum theories and concepts. The conference was devoted to identifying, discussing and understanding possible interrelationships of theoretical disciplinary improvements, recognized as having prospective fundamental roles for a new post-Bertalanffy Systemics. The latter should be able to deal with problems related to complexity in a generalized way. In this context the inter-disciplinarity should consists, for instance, in a disciplinary reformulation of problems, as from algebraic to geometrical, from military to political, from biological to chemical, while the trans-disciplinarity should be related to the study of such reformulations and their properties. The Italian Systems Society (AIRS) was founded in the 1996. The AIRS is a network of academicians, scientists, researchers and professionals involved in Systemics. A partial list of disciplines represented is: Architecture Biology Economics Education Engineering Mathematics Neurosciences Medicine Music Philosophy Psychology Physics. Previous conferences had as open lecturers professors Arecchi, Haken, Klir, and Kauffman. The proceedings have been published as: 1. Minati, G., (ed.), (1998), Proceedings of the first Italian Conference on Systemics, Apogeo Scientifica, Milan, Italy. 2. Minati, G., and Pessa, E., (eds.) (2002), Emergence in Complex Cognitive, Social and Biological Systems. Kluwer, New York. 3. Minati, G., Pessa, E., and Abram, M., (eds.), (2006), Systemics of Emergence: Research and Applications. Springer, New York. 4. Minati, G., Abram, M. and Pessa, E., (eds.), (2009), Processes of emergence of systems and systemic properties. Towards a general theory of emergence. World Scientific, Singapore. 5. Minati, G., Abram, M. and Pessa, E., (eds.), (2012), Methods, Models, simulations and approaches - towards a general theory of change. World Scientific, Singapore.

This self-contained text develops a Markov chain approach that makes the rigorous analysis of a class of microscopic models that specify the dynamics of complex systems at the individual level possible. It presents a general framework of aggregation in agent-based and related computational models, one which makes use of lumpability and information theory in order to link the micro and macro levels of observation. The starting point is a microscopic Markov chain description of the dynamical process in complete correspondence with the dynamical behavior of the agent-based model (ABM), which is obtained by considering the set of all possible agent configurations as the state space of a huge Markov chain. An explicit formal representation of a resulting "micro-chain" including microscopic transition rates is derived for a class of models by using the random mapping representation of a Markov process. The type of probability distribution used to implement the stochastic part of the model, which defines the updating rule and governs the dynamics at a Markovian level, plays a crucial part in the analysis of "voter-like" models used in population genetics, evolutionary game theory and social dynamics. The book demonstrates that the problem of aggregation in ABMs - and the lumpability conditions in particular - can be embedded into a more general framework that employs information theory in order to identify different levels and relevant scales in complex dynamical systems

The book addresses the control issues such as stability analysis, control synthesis and filter design of Markov jump systems with the above three types of TPs, and thus is mainly divided into three parts. Part I studies the Markov jump systems with partially unknown TPs. Different methodologies with different conservatism for the basic stability and stabilization problems are developed and compared. Then the problems of state estimation, the control of systems with time-varying delays, the case involved with both partially unknown TPs and uncertain TPs in a composite way are also tackled. Part II deals with the Markov jump systems with piecewise homogeneous TPs. Methodologies that can effectively handle control problems in the scenario are developed, including the one coping with the asynchronous switching phenomenon between the currently activated system mode and the controller/filter to be designed. Part III focuses on the Markov jump systems with memory TPs. The concept of -mean square stability is proposed such that the stability problem can be solved via a finite number of conditions. The systems involved with nonlinear dynamics (described via the Takagi-Sugeno fuzzy model) are also investigated. Numerical and practical examples are given to verify the effectiveness of the obtained theoretical results. Finally, some perspectives and future works are presented to conclude the book.

This book addresses the processes of stochastic structure formation in two-dimensional geophysical fluid dynamics based on statistical analysis of Gaussian random fields, as well as stochastic structure formation in dynamic systems with parametric excitation of positive random fields f(r,t) described by partial differential equations. Further, the book considers two examples of stochastic structure formation in dynamic systems with parametric excitation in the presence of Gaussian pumping. In dynamic systems with parametric excitation in space and time, this type of structure formation either happens - or doesn't! However, if it occurs in space, then this almost always happens (exponentially quickly) in individual realizations with a unit probability. In the case considered, clustering of the field f(r,t) of any nature is a general feature of dynamic fields, and one may claim that structure formation is the Law of Nature for arbitrary random fields of such type. The study clarifies the conditions under which such structure formation takes place. To make the content more accessible, these conditions are described at a comparatively elementary mathematical level by employing ideas from statistical topography.

This book addresses fault detection and isolation topics from a computational perspective. Unlike most existing literature, it bridges the gap between the existing well-developed theoretical results and the realm of reliable computational synthesis procedures. The model-based approach to fault detection and diagnosis has been the subject of ongoing research for the past few decades. While the theoretical aspects of fault diagnosis on the basis of linear models are well understood, most of the computational methods proposed for the synthesis of fault detection and isolation filters are not satisfactory from a numerical standpoint. Several features make this book unique in the fault detection literature: Solution of standard synthesis problems in the most general setting, for both continuous- and discrete-time systems, regardless of whether they are proper or not; consequently, the proposed synthesis procedures can solve a specific problem whenever a solution exists Emphasis on the best numerical algorithms to solve the synthesis problems for linear systems in generalized state-space form (also known as descriptor systems) Development of general synthesis procedures relying on new computational paradigms, such as factorization-based design based on filter updating techniques and nullspace-based synthesis Availability of a comprehensive set of free accompanying software tools for descriptor systems, which allows readers to easily implement all synthesis procedures presented in the book and ensures that all results are reproducible This book is primarily intended for researchers and advanced graduate students in the areas of fault diagnosis and fault-tolerant control. It will also appeal to mathematicians with an interest in control-oriented numerics.

This book presents a systematic overview of cutting-edge research in the field of parametric modeling of personal income and wealth distribution, which allows one to represent how income/wealth is distributed within a given population. The estimated parameters may be used to gain insights into the causes of the evolution of income/wealth distribution over time, or to interpret the differences between distributions across countries. Moreover, once a given parametric model has been fitted to a data set, one can straightforwardly compute inequality and poverty measures. Finally, estimated parameters may be used in empirical modeling of the impact of macroeconomic conditions on the evolution of personal income/wealth distribution. In reviewing the state of the art in the field, the authors provide a thorough discussion of parametric models belonging to the " -generalized" family, a new and fruitful set of statistical models for the size distribution of income and wealth that they have developed over several years of collaborative and multidisciplinary research. This book will be of interest to all who share the belief that problems of income and wealth distribution merit detailed conceptual and methodological attention.

This volume presents recent advances in the field of matrix analysis based on contributions at the MAT-TRIAD 2015 conference. Topics covered include interval linear algebra and computational complexity, Birkhoff polynomial basis, tensors, graphs, linear pencils, K-theory and statistic inference, showing the ubiquity of matrices in different mathematical areas. With a particular focus on matrix and operator theory, statistical models and computation, the International Conference on Matrix Analysis and its Applications 2015, held in Coimbra, Portugal, was the sixth in a series of conferences. Applied and Computational Matrix Analysis will appeal to graduate students and researchers in theoretical and applied mathematics, physics and engineering who are seeking an overview of recent problems and methods in matrix analysis.

This book is primarily concerned with the computational aspects of predictability of dynamical systems - in particular those where observation, modeling and computation are strongly interdependent. Unlike with physical systems under control in laboratories, for instance in celestial mechanics, one is confronted with the observation and modeling of systems without the possibility of altering the key parameters of the objects studied. Therefore, the numerical simulations offer an essential tool for analyzing these systems. With the widespread use of computer simulations to solve complex dynamical systems, the reliability of the numerical calculations is of ever-increasing interest and importance. This reliability is directly related to the regularity and instability properties of the modeled flow. In this interdisciplinary scenario, the underlying physics provide the simulated models, nonlinear dynamics provides their chaoticity and instability properties, and the computer sciences provide the actual numerical implementation. This book introduces and explores precisely this link between the models and their predictability characterization based on concepts derived from the field of nonlinear dynamics, with a focus on the finite-time Lyapunov exponents approach. The method is illustrated using a number of well-known continuous dynamical systems, including the Contopoulos, Henon-Heiles and Roessler systems. To help students and newcomers quickly learn to apply these techniques, the appendix provides descriptions of the algorithms used throughout the text and details how to implement them in order to solve a given continuous dynamical system.

The papers collected in this volume focus on new perspectives on individuals, society, and science, specifically in the field of socio-economic systems. The book is the result of a scientific collaboration among experts from "Alexandru Ioan Cuza" University of Iasi (Romania), "G. d'Annunzio" University of Chieti-Pescara (Italy), "University of Defence" of Brno (Czech Republic), and "Pablo de Olavide" University of Sevilla (Spain). The heterogeneity of the contributions presented in this volume reflects the variety and complexity of social phenomena. The book is divided in four Sections as follows. The first Section deals with recent trends in social decisions. Specifically, it aims to understand which are the driving forces of social decisions. The second Section focuses on the social and public sphere. Indeed, it is oriented on recent developments in social systems and control. Trends in quantitative theories and models are described in Section 3, where many new formal, mathematical-statistical tools for modelling complex social phenomena are presented. Finally, Section 4 shows integrative theories and models; particularly, it deals with the ethical, cultural and political approaches to social science, the pedagogical methods, and the relationship between literature, politics, religion and society. The book is addressed to sociologists, philosophers, mathematicians, statisticians, people interested in ethics, and specialists in the fields of communication, social, and political sciences.

This book presents recent research work on stochastic jump hybrid systems. Specifically, the considered stochastic jump hybrid systems include Markovian jump Ito stochastic systems, Markovian jump linear-parameter-varying (LPV) systems, Markovian jump singular systems, Markovian jump two-dimensional (2-D) systems, and Markovian jump repeated scalar nonlinear systems. Some sufficient conditions are first established respectively for the stability and performances of those kinds of stochastic jump hybrid systems in terms of solution of linear matrix inequalities (LMIs). Based on the derived analysis conditions, the filtering and control problems are addressed. The book presents up-to-date research developments and novel methodologies on stochastic jump hybrid systems. The contents can be divided into two parts: the first part is focused on robust filter design problem, while the second part is put the emphasis on robust control problem. These methodologies provide a framework for stability and performance analysis, robust controller design, and robust filter design for the considered systems. Solutions to the design problems are presented in terms of LMIs. The book is a timely reflection of the developing area of filtering and control theories for Markovian jump hybrid systems with various kinds of imperfect information. It is a collection of a series of latest research results and therefore serves as a useful textbook for senior and/or graduate students who are interested in knowing 1) the state-of-the-art of linear filtering and control areas, and 2) recent advances in stochastic jump hybrid systems. The readers will also benefit from some new concepts, new models and new methodologies with practical significance in control engineering and signal processing.

This book introduces and reviews recent advances in the field in a comprehensive and non-technical way by focusing on the potential of emerging citizen-science and social-computation frameworks, coupled with the latest theoretical and modeling tools developed by physicists, mathematicians, computer and social scientists to analyse, interpret and visualize complex data sets. There is overwhelming evidence that the current organisation of our economies and societies is seriously damaging biological ecosystems and human living conditions in the short term, with potentially catastrophic effects in the long term. The need to re-organise the daily activities with the greatest impact - energy consumption, transport, housing - towards a more efficient and sustainable development model has recently been raised in the public debate on several global, environmental issues. Above all, this requires the mismatch between global, societal and individual needs to be addressed. Recent advances in Information and Communication Technologies (ICT) can trigger important transitions at the individual and collective level to achieve this aim. Based on the findings of the collaborative research network EveryAware the following developments among the emerging ICT technologies are discussed in depth in this volume: * Participatory sensing - where ICT development is pushed to the level where it can support informed action at the hyperlocal scale, providing capabilities for environmental monitoring, data aggregation and mining, as well as information presentation and sharing. * Web gaming, social computing and internet-mediated collaboration - where the Web will continue to acquire the status of an infrastructure for social computing, allowing users' cognitive abilities to be coordinated in online communities, and steering the collective action towards predefined goals. * Collective awareness and decision-making - where the access to both personal and community data, collected by users, processed with suitable analysis tools, and re-presented in an appropriate format by usable communication interfaces leads to a bottom-up development of collective social strategies.

In a unified form, this monograph presents fundamental results on the approximation of centralized and decentralized stochastic control problems, with uncountable state, measurement, and action spaces. It demonstrates how quantization provides a system-independent and constructive method for the reduction of a system with Borel spaces to one with finite state, measurement, and action spaces. In addition to this constructive view, the book considers both the information transmission approach for discretization of actions, and the computational approach for discretization of states and actions. Part I of the text discusses Markov decision processes and their finite-state or finite-action approximations, while Part II builds from there to finite approximations in decentralized stochastic control problems. This volume is perfect for researchers and graduate students interested in stochastic controls. With the tools presented, readers will be able to establish the convergence of approximation models to original models and the methods are general enough that researchers can build corresponding approximation results, typically with no additional assumptions.

This monograph is an exposition of a novel method for solving inverse problems, a method of parameter estimation for time series data collected from simulations of real experiments. These time series might be generated by measuring the dynamics of aircraft in flight, by the function of a hidden Markov model used in bioinformatics or speech recognition or when analyzing the dynamics of asset pricing provided by the nonlinear models of financial mathematics. Dynamic Systems Models demonstrates the use of algorithms based on polynomial approximation which have weaker requirements than already-popular iterative methods. Specifically, they do not require a first approximation of a root vector and they allow non-differentiable elements in the vector functions being approximated. The text covers all the points necessary for the understanding and use of polynomial approximation from the mathematical fundamentals, through algorithm development to the application of the method in, for instance, aeroplane flight dynamics or biological sequence analysis. The technical material is illustrated by the use of worked examples and methods for training the algorithms are included. Dynamic Systems Models provides researchers in aerospatial engineering, bioinformatics and financial mathematics (as well as computer scientists interested in any of these fields) with a reliable and effective numerical method for nonlinear estimation and solving boundary problems when carrying out control design. It will also be of interest to academic researchers studying inverse problems and their solution.

This book provides international perspective for those studying or working in the security domain, from enforcement to policy. It focuses on non-traditional threats in a landscape that has been described as transnational in nature and incorporates natural disasters, gang violence, extremism and terrorism, amongst other issues. Chapters provide innovative thinking on themes including cyber security, maritime security, transnational crime, human security, globalization and economic security. Relevant theoretical frameworks are presented and readers are expertly guided through complex threats, from matters pertaining to health security which pose threats not only to humans but also have significant national security implications, to issues regarding critical infrastructure vulnerability and the complexity of understanding terrorist operations. Authors reveal how emerging uncertainties regarding global critical infrastructure and supply chain security, food security, and health security are linked to the notion of human security. Security professionals, policy makers and academics will all gain from the insights, strategies and perspectives in this book. It builds understanding of the deepening and broadening domain of security studies and provides a valuable reference text for courses on security studies and international relations.

The book covers nonlinear physical problems and mathematical modeling, including molecular biology, genetics, neurosciences, artificial intelligence with classical problems in mechanics and astronomy and physics. The chapters present nonlinear mathematical modeling in life science and physics through nonlinear differential equations, nonlinear discrete equations and hybrid equations. Such modeling can be effectively applied to the wide spectrum of nonlinear physical problems, including the KAM (Kolmogorov-Arnold-Moser (KAM)) theory, singular differential equations, impulsive dichotomous linear systems, analytical bifurcation trees of periodic motions, and almost or pseudo- almost periodic solutions in nonlinear dynamical systems.

This brief examines mathematical models in nonsmooth mechanics and nonregular electrical circuits, including evolution variational inequalities, complementarity systems, differential inclusions, second-order dynamics, Lur'e systems and Moreau's sweeping process. The field of nonsmooth dynamics is of great interest to mathematicians, mechanicians, automatic controllers and engineers. The present volume acknowledges this transversality and provides a multidisciplinary view as it outlines fundamental results in nonsmooth dynamics and explains how to use them to study various problems in engineering. In particular, the author explores the question of how to redefine the notion of dynamical systems in light of modern variational and nonsmooth analysis. With the aim of bridging between the communities of applied mathematicians, engineers and researchers in control theory and nonlinear systems, this brief outlines both relevant mathematical proofs and models in unilateral mechanics and electronics.

This book focuses on approximations under the presence of ordinary and fractional smoothness, presenting both the univariate and multivariate cases. It also explores approximations under convexity and a new trend in approximation theory -approximation by sublinear operators with applications to max-product operators, which are nonlinear and rational providing very fast and flexible approximations. The results presented have applications in numerous areas of pure and applied mathematics, especially in approximation theory and numerical analysis in both ordinary and fractional senses. As such this book is suitable for researchers, graduate students, and seminars of the above disciplines, and is a must for all science and engineering libraries.