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Noncommutative mathematics is a significant new trend of mathematics. Initially motivated by the development of quantum physics, the idea of 'making theory noncommutative' has been extended to many areas of pure and applied mathematics. This book is divided into two parts. The first part provides an introduction to quantum probability, focusing on the notion of independence in quantum probability and on the theory of quantum stochastic processes with independent and stationary increments. The second part provides an introduction to quantum dynamical systems, discussing analogies with fundamental problems studied in classical dynamics. The desire to build an extension of the classical theory provides new, original ways to understand well-known 'commutative' results. On the other hand the richness of the quantum mathematical world presents completely novel phenomena, never encountered in the classical setting. This book will be useful to students and researchers in noncommutative probability, mathematical physics and operator algebras.

Probabilities. --- Quantum theory. --- Potential theory (Mathematics) --- Green's operators --- Green's theorem --- Potential functions (Mathematics) --- Potential, Theory of --- Mathematical analysis --- Mechanics --- Quantum dynamics --- Quantum mechanics --- Quantum physics --- Physics --- Thermodynamics --- Probability --- Statistical inference --- Combinations --- Mathematics --- Chance --- Least squares --- Mathematical statistics --- Risk

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This book elaborates on the asymptotic behaviour, when N is large, of certain N-dimensional integrals which typically occur in random matrices, or in 1+1 dimensional quantum integrable models solvable by the quantum separation of variables. The introduction presents the underpinning motivations for this problem, a historical overview, and a summary of the strategy, which is applicable in greater generality. The core  aims at proving an expansion up to o(1) for the logarithm of the partition function of the sinh-model. This is achieved by a combination of potential theory and large deviation theory so as to grasp the leading asymptotics described by an equilibrium measure, the Riemann-Hilbert approach to truncated Wiener-Hopf in order to analyse the equilibrium measure, the Schwinger-Dyson equations and the boostrap method to finally obtain an expansion of correlation functions and the one of the partition function. This book is addressed to researchers working in random matrices, statistical physics or integrable systems, or interested in recent developments of asymptotic analysis in those fields.

Mathematics. --- Potential theory (Mathematics). --- System theory. --- Probabilities. --- Mathematical physics. --- Physics. --- Mathematical Physics. --- Probability Theory and Stochastic Processes. --- Potential Theory. --- Complex Systems. --- Mathematical Methods in Physics. --- Statistical Physics and Dynamical Systems. --- Math --- Science --- Distribution (Probability theory. --- Statistical physics. --- Physics --- Mathematical statistics --- Physical mathematics --- Green's operators --- Green's theorem --- Potential functions (Mathematics) --- Potential, Theory of --- Mathematical analysis --- Mechanics --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities --- Statistical methods --- Mathematics --- Dynamical systems. --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics --- Dynamical systems --- Kinetics --- Mechanics, Analytic --- Force and energy --- Statics --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Risk --- Potential theory (Mathematics) --- Dynamics.

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A unified methodology for categorizing various complex objects is presented in this book. Through probability theory, novel asymptotically minimax criteria suitable for practical applications in imaging and data analysis are examined including the special cases such as the Jensen-Shannon divergence and the probabilistic neural network. An optimal approximate nearest neighbor search algorithm, which allows faster classification of databases is featured. Rough set theory, sequential analysis and granular computing are used to improve performance of the hierarchical classifiers. Practical examples in face identification (including deep neural networks), isolated commands recognition in voice control system and classification of visemes captured by the Kinect depth camera are included. This approach creates fast and accurate search procedures by using exact probability densities of applied dissimilarity measures. This book can be used as a guide for independent study and as supplementary material for a technically oriented graduate course in intelligent systems and data mining. Students and researchers interested in the theoretical and practical aspects of intelligent classification systems will find answers to: - Why conventional implementation of the naive Bayesian approach does not work well in image classification? - How to deal with insufficient performance of hierarchical classification systems? - Is it possible to prevent an exhaustive search of the nearest neighbor in a database?

Mathematics. --- Pattern recognition. --- Potential theory (Mathematics). --- System theory. --- Mathematical optimization. --- Machinery. --- Optimization. --- Pattern Recognition. --- Machinery and Machine Elements. --- Systems Theory, Control. --- Complex Systems. --- Potential Theory. --- Databases. --- Database searching. --- Data mining. --- Automatic classification. --- Algorithmic knowledge discovery --- Factual data analysis --- KDD (Information retrieval) --- Knowledge discovery in data --- Knowledge discovery in databases --- Mining, Data --- Data base searching --- Database search strategies --- Search strategies in databases --- Searching databases --- Data banks --- Data bases --- Databanks --- Database systems --- Automatic indexing --- Classification --- Information storage and retrieval systems --- Database searching --- Electronic information resource searching --- Computer files --- Electronic information resources --- Optical pattern recognition. --- Engineering. --- Systems theory. --- Green's operators --- Green's theorem --- Potential functions (Mathematics) --- Potential, Theory of --- Mathematical analysis --- Mechanics --- Construction --- Industrial arts --- Technology --- Optical data processing --- Pattern perception --- Perceptrons --- Visual discrimination --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- Maxima and minima --- Operations research --- Simulation methods --- System analysis --- Systems, Theory of --- Systems science --- Science --- Machinery --- Machines --- Manufactures --- Power (Mechanics) --- Mechanical engineering --- Motors --- Power transmission --- Design perception --- Pattern recognition --- Form perception --- Perception --- Figure-ground perception --- Philosophy --- Curious devices