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Randomness is ubiquitous in nature. Random drivers are generally considered a source of disorder in environmental systems. However, the interaction between noise and nonlinear dynamics may lead to the emergence of a number of ordered behaviors (in time and space) that would not exist in the absence of noise. This counterintuitive effect of randomness may play a crucial role in environmental processes. For example, seemingly 'random' background events in the atmosphere can grow into larger instabilities that have great effects on weather patterns. This book presents the basics of the theory of stochastic calculus and its application to the study of noise-induced phenomena in environmental systems. It will be an invaluable reference text for ecologists, geoscientists and environmental engineers interested in the study of stochastic environmental dynamics.

Geophysical prediction --- Random noise theory. --- Environmental sciences --- Environmental science --- Science --- Gaussian noise --- Noise, Random --- Statistical communication theory --- Stochastic processes --- Uncertainty (Information theory) --- Geophysics --- Prediction, Geophysical --- Forecasting --- Mathematics.

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This book investigates the impact of noise upon the emergence and sustenance of patterns. "Patterns" loosely refers to coherent spatial structures, including fronts, as well as temporal patterns. The crucial role of nonlinearities is highlighted and expanded upon in the context of dynamical system frameworks. The author's familiarity with chaos theory, statistical physics and nonlinear science is reflected in the highly interdisciplinary character of the text. Model equations and experiments taken from fluid dynamics, semiconductor devices, biophysics and statistical mechanics complement theor

Stochastic processes. --- Random noise theory. --- Stability. --- Dynamics --- Mechanics --- Motion --- Vibration --- Benjamin-Feir instability --- Equilibrium --- Gaussian noise --- Noise, Random --- Statistical communication theory --- Stochastic processes --- Uncertainty (Information theory) --- Random processes --- Probabilities

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Random noise theory --- Nanostructured materials --- Nonlinear systems --- Nonlinear Dynamics --- Nanostructured materials. --- Nonlinear systems. --- Random noise theory. --- Nonlinear Dynamics. --- Gaussian noise --- Noise, Random --- Systems, Nonlinear --- Nanomaterials --- Nanometer materials --- Nanophase materials --- Nanostructure controlled materials --- Nanostructure materials --- Ultra-fine microstructure materials --- Non-linear Dynamics --- Non-linear Models --- Chaos Theory --- Models, Nonlinear --- Chaos Theories --- Dynamics, Non-linear --- Dynamics, Nonlinear --- Model, Non-linear --- Model, Nonlinear --- Models, Non-linear --- Non linear Dynamics --- Non linear Models --- Non-linear Dynamic --- Non-linear Model --- Nonlinear Dynamic --- Nonlinear Model --- Nonlinear Models --- Theories, Chaos --- Theory, Chaos --- Statistical communication theory --- Stochastic processes --- Uncertainty (Information theory) --- System theory --- Microstructure --- Nanotechnology --- Fractals --- Engineering --- Life Sciences --- Material Science and Metallurgy --- Telecommunications Technology --- Biomedical Engineering --- Electronics --- Biology --- Wireless Communications

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The book presents the necessary mathematical basis to obtain and rigorously use likelihoods for detection problems with Gaussian noise. To facilitate comprehension the text is divided into three broad areas – reproducing kernel Hilbert spaces, Cramér-Hida representations and stochastic calculus – for which a somewhat different approach was used than in their usual stand-alone context. One main applicable result of the book involves arriving at a general solution to the canonical detection problem for active sonar in a reverberation-limited environment. Nonetheless, the general problems dealt with in the text also provide a useful framework for discussing other current research areas, such as wavelet decompositions, neural networks, and higher order spectral analysis. The structure of the book, with the exposition presenting as many details as necessary, was chosen to serve both those readers who are chiefly interested in the results and those who want to learn the material from scratch. Hence, the text will be useful for graduate students and researchers alike in the fields of engineering, mathematics and statistics.

Mathematical Statistics --- Mathematics --- Physical Sciences & Mathematics --- Random noise theory. --- Gaussian noise --- Noise, Random --- Mathematics. --- Functional analysis. --- Information theory. --- Probabilities. --- Probability Theory and Stochastic Processes. --- Functional Analysis. --- Information and Communication, Circuits. --- Statistical communication theory --- Stochastic processes --- Uncertainty (Information theory) --- Distribution (Probability theory. --- Math --- Science --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities --- Communication theory --- Communication --- Cybernetics --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk