Mathematical / Computational / Theoretical physics Books

Book SynopsisOur book introduces a method to evaluate the accuracy of trend estimation algorithms under conditions similar to those encountered in real time series processing. This method is based on Monte Carlo experiments with artificial time series numerically generated by an original algorithm. The second part of the book contains several automatic algorithms for trend estimation and time series partitioning. The source codes of the computer programs implementing these original automatic algorithms are given in the appendix and will be freely available on the web. The book contains clear statement of the conditions and the approximations under which the algorithms work, as well as the proper interpretation of their results. We illustrate the functioning of the analyzed algorithms by processing time series from astrophysics, finance, biophysics, and paleoclimatology. The numerical experiment method extensively used in our book is already in common use in computational and statistical physics.Table of ContentsDiscrete stochastic processes and time series.- Trend definition.- Finite AR(1) stochastic process.- Monte Carlo experiments. - Monte Carlo statistical ensembles.- Numerical generation of trends.- Numerical generation of noisy time series.- Statistical hypothesis testing.- Testing the i.i.d. property.- Polynomial fitting.- Linear regression.- Polynomial fitting.- Polynomial fitting of artificial time series.- An astrophysical example.- Noise smoothing.- Moving average.- Repeated moving average (RMA).- Smoothing of artificial time series.- A financial example.- Automatic estimation of monotonic trends.- Average conditional displacement (ACD) algorithm.- Artificial time series with monotonic trends.- Automatic ACD algorithm.- Evaluation of the ACD algorithm.- A paleoclimatological example.- Statistical significance of the ACD trend.- Time series partitioning.- Partitioning of trends into monotonic segments.- Partitioning of noisy signals into monotonic segments.- Partitioning of a real time series.- Estimation of the ratio between the trend and noise.- Automatic estimation of arbitrary trends.- Automatic RMA (AutRMA).- Monotonic segments of the AutRMA trend.- Partitioning of a financial time series.

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Book SynopsisMean field approximation has been adopted to describe macroscopic phenomena from microscopic overviews. It is still in progress; fluid mechanics, gauge theory, plasma physics, quantum chemistry, mathematical oncology, non-equilibirum thermodynamics. spite of such a wide range of scientific areas that are concerned with the mean field theory, a unified study of its mathematical structure has not been discussed explicitly in the open literature. The benefit of this point of view on nonlinear problems should have significant impact on future research, as will be seen from the underlying features of self-assembly or bottom-up self-organization which is to be illustrated in a unified way. The aim of this book is to formulate the variational and hierarchical aspects of the equations that arise in the mean field theory from macroscopic profiles to microscopic principles, from dynamics to equilibrium, and from biological models to models that arise from chemistry and physics.Table of Contents

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Book SynopsisODE/IM Correspondence.- Exact WKB Analysis and TBA Equations.- Massive ODE/IM Correspondence.- Integrable Models and Functional Relations.

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Book Synopsis.- Foundations of Computational Thermokinetics in RPUF..- Modeling Techniques..- Key Factors in Computational Modeling of RPUF..- Implications and Future Outcomes.

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Book SynopsisChapter 1 Introduction.- Chapter 2 Density Functional Theory.- Chapter 3  Anharmonic Phonon Theory.- Chapter 4 Modern Theory and Machine Learning of Polarization.- Chapter 5 Dielectric Properties of Strongly Anharmonic TiO2.- Chapter 6 Dielectric Properties of Liquid Alcohols and Its Polymers.- Chapter 7 Conclusion.

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Book SynopsisMathematics and Statistics.- Physics.- Chemistry.- Biology.- Engineering Sciences.- Artificial Intelligence.

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Book SynopsisPreface.- Chapter 1. Survey on the Prandtls Equations and Related Boundary Layer Equations.- Chapter 2. Global Well-posedness of Solutions to the 2D Prandtl-Hartmann Equations in Analytic Framework.- Chapter 3. Local Existence of Solutions to the 2D Prandtl Equations in A Weighted Sobolev Space.- Chapter 4. Local Well-posedness of Solutions to the 2D Mixed Prandtl Equations in A Sobolev Space Without Monotonicity or Lower Bound.- Chapter 5. Local Well-posedness of Solutions to 2D Magnetic Prandtl Model in the Prandtl-Hartmann regime.- Chapter 6. Local Existence of Solutions to 3D Prandtl Equations with a Special Structure.- Bibliography.

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Book SynopsisChapter 1. Introduction to Chemical transformations in far from equilibrium systems.- Chapter 2. A brief introduction to vectors spaces: succinct but pertinent summary for scientists.- Chapter 3. White noise probability spaces (Hermite polynomials and functions and their use in defining Weiner Chaos expansion).- Chapter 4. Introduction to Skorohod integration and Malliavian derivativespractical interpretations.- Chapter 5. Introduction to Wick Product and its algebra (analytical solutions to Wick product driven stochastic differential equations; Hermite transformations).- Chapter 6. Numerical solutions to stochastic chemical reactions.- Chapter 7. Stochastic coupled reactions systems: Numerical solutions.- Chapter 8. Modelling chiral symmetry breaking and stability in a noisy environment using Wick productsA case study.

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Book SynopsisTable of Contents1. Introduction 2. The Basis of Monte Carlo 3. Pseudorandom Number Generators 4. Sampling, Scoring, and Precision 5. Variance Reduction Techniques 6. Markov Chain Monte Carlo 7. Inverse Monte Carlo 8. Linear Operator Equations 9. The Fundamentals of Neutral Particle Transport 10. Monte Carlo Simulation of Neutral Particle Transport 11. Monte Carlo Applications

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Book SynopsisEquations of Mathematical Diffraction Theory focuses on the comparative analysis and development of efficient analytical methods for solving equations of mathematical diffraction theory. Following an overview of some general properties of integral and differential operators in the context of the linear theory of diffraction processes, the authors provide estimates of the operator norms for various ranges of the wave number variation, and then examine the spectral properties of these operators. They also present a new analytical method for constructing asymptotic solutions of boundary integral equations in mathematical diffraction theory for the high-frequency case.Clearly demonstrating the close connection between heuristic and rigorous methods in mathematical diffraction theory, this valuable book provides you with the differential and integral equations that can easily be used in practical applications.Table of ContentsSome Preliminaries from Analysis and the Theory of Wave Processes. Integral Equations of Diffraction Theory for Obstacles in Unbounded Medium. Wave Fields in a Layer of Constant Thickness. Analytical Methods for Simply Connected Bounded Domains. Integral Equations in Diffraction by Linear Obstacles. Short-Wave Asymptotic Methods on the Basis of Multiple Integrals. Inverse Problems of the Short-Wave Diffraction. Ill-Posed Equations of Inverse Diffraction Problems for Arbitrary Boundary. Numerical Methods for Irregular Operator Equations.

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