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Weakly Nonlocal Solitary Waves and Beyond-All-Orders Asymptotics: Generalized Solitons and Hyperasymptotic Perturbation Theory represents the first thorough examination of weakly nonlocal solitary waves, which are just as important in applications as their classical counterparts. The book describes a class of waves which radiate away from the core of the disturbance but are nevertheless very long-lived nonlinear disturbances. Specific examples are provided in the areas of water waves, particle physics, meteorology, oceanography, fiber optics pulses and dynamical systems theory. For many species of nonlocal solitary waves the radiation is exponentially small in 1/epsilon where epsilon is a perturbation parameter, thus lying `beyond-all-orders'. A second theme is the description of hyperasymptotic perturbation theory and other extensions of standard perturbation methods. These methods have been developed for the computation of exponentially small corrections to asymptotic series. A t hird theme involves the use of Chebyshev and Fourier numerical methods to compute solitary waves. Special emphasis is given to steadily-translating coherent structures, a difficult numerical problem even today. A fourth theme is the description of a large number of non-soliton problems in quantum physics, hydrodynamics, instability theory and others where `beyond-all-order' corrections arise and where the perturbative and numerical methods described earlier are essential. Later chapters provide a thorough examination of matched asymptotic expansions in the complex plane, the small denominator problem in Poincaré-Linstead (`Stokes') expansions, multiple scale expansions in powers of the hyperbolic secant and tangent functions and hyperasymptotic perturbation theory. This book will be of special interest to applied mathematicians, fluid dynamicists in mechanical and aeronautical engineering, electrical engineers interested in fiber optics, quantum chemists and atomic and particle physi cists.

Mathematics. --- Global analysis (Mathematics) --- Mathematics. --- Geography. --- Analysis. --- Applications of Mathematics. --- Earth Sciences, general. --- Fluid- and Aerodynamics.

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(NOTES)This text focuses on the topics which are an essential part of the engineering mathematics course:ordinary differential equations, vector calculus, linear algebra and partial differential equations. Advantages over competing texts: 1. The text has a large number of examples and problems - a typical section having 25 quality problems directly related to the text. 2. The authors use a practical engineering approach based upon solving equations. All ideas and definitions are introduced from this basic viewpoint, which allows engineers in their second year to understand concepts that would otherwise be impossibly abstract. Partial differential equations are introduced in an engineering and science context based upon modelling of physical problems. A strength of the manuscript is the vast number of applications to real-world problems, each treated completely and in sufficient depth to be self-contained. 3. Numerical analysis is introduced in the manuscript at a completely elementary calculus level. In fact, numerics are advertised as just an extension of the calculus and used generally as enrichment, to help communicate the role of mathematics in engineering applications. 4.The authors have used and updated the book as a course text over a 10 year period. 5. Modern outline, as contrasted to the outdated outline by Kreysig and Wylie. 6. This is now a one year course. The text is shorter and more readable than the current reference type manuals published all at around 1300-1500 pages.

Engineering mathematics. --- Mathématiques de l'ingénieur --- Mathématiques de l'ingénieur --- Engineering mathematics --- Applied mathematics. --- Mathematical analysis. --- Analysis (Mathematics). --- Computer mathematics. --- Mathematical and Computational Engineering. --- Applications of Mathematics. --- Analysis. --- Computational Mathematics and Numerical Analysis. --- Computer mathematics --- Electronic data processing --- Mathematics --- 517.1 Mathematical analysis --- Mathematical analysis --- Engineering --- Engineering analysis

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The exponential increase in the use of MCMC methods and the corre­ sponding applications in domains of even higher complexity have caused a growing concern about the available convergence assessment methods and the realization that some of these methods were not reliable enough for all-purpose analyses. Some researchers have mainly focussed on the con­ vergence to stationarity and the estimation of rates of convergence, in rela­ tion with the eigenvalues of the transition kernel. This monograph adopts a different perspective by developing (supposedly) practical devices to assess the mixing behaviour of the chain under study and, more particularly, it proposes methods based on finite (state space) Markov chains which are obtained either through a discretization of the original Markov chain or through a duality principle relating a continuous state space Markov chain to another finite Markov chain, as in missing data or latent variable models. The motivation for the choice of finite state spaces is that, although the resulting control is cruder, in the sense that it can often monitor con­ vergence for the discretized version alone, it is also much stricter than alternative methods, since the tools available for finite Markov chains are universal and the resulting transition matrix can be estimated more accu­ rately. Moreover, while some setups impose a fixed finite state space, other allow for possible refinements in the discretization level and for consecutive improvements in the convergence monitoring.

Convergence --- Markov processes --- Monte Carlo method --- Convergence. --- Markov processes. --- Monte Carlo method. --- Discretization (Mathematics) --- Engineering & Applied Sciences --- Applied Mathematics --- Applied mathematics. --- Engineering mathematics. --- Applications of Mathematics. --- Engineering --- Engineering analysis --- Mathematical analysis --- Mathematics --- Mathematical models. --- Analysis, Markov --- Chains, Markov --- Markoff processes --- Markov analysis --- Markov chains --- Markov models --- Models, Markov --- Processes, Markov --- Stochastic processes --- Functions

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Quantum mechanics. Quantumfield theory --- Mathematical physics --- Quantum theory --- Mathematical physics. --- Quantum physics. --- Applied mathematics. --- Engineering mathematics. --- Functional analysis. --- Matrix theory. --- Algebra. --- Theoretical, Mathematical and Computational Physics. --- Quantum Physics. --- Applications of Mathematics. --- Functional Analysis. --- Linear and Multilinear Algebras, Matrix Theory. --- Mathematics --- Mathematical analysis --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Engineering --- Engineering analysis --- Quantum dynamics --- Quantum mechanics --- Quantum physics --- Physics --- Mechanics --- Thermodynamics --- Physical mathematics --- Quantum theory.

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Stochastic processes --- Nonparametric statistics --- Estimation theory --- Nonparametric statistics. --- Stochastic processes. --- Estimation theory. --- Statistique non-paramétrique --- Processus stochastiques --- Théorie de l'estimation --- Probabilities. --- Applied mathematics. --- Engineering mathematics. --- Probability Theory and Stochastic Processes. --- Applications of Mathematics. --- Engineering --- Engineering analysis --- Mathematical analysis --- Probability --- Statistical inference --- Combinations --- Mathematics --- Chance --- Least squares --- Mathematical statistics --- Risk --- Estimating techniques --- Distribution-free statistics --- Statistics, Distribution-free --- Statistics, Nonparametric --- Random processes --- Probabilities --- Statistique non paramétrique. --- Processus stochastiques. --- Statistique non paramétrique.