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Ergodicity of Markov Processes via Nonstandard Analysis
The Markov chain ergodic theorem is well-understood if either the time-line or the state space is discrete. However, there does not
exist a very clear result for general state space continuous-time Markov processes. Using methods from mathematical logic and
nonstandard analysis, we introduce a class of hyperfinite Markov processes-namely, general Markov processes which behave like finite
state space discrete-time Markov processes. We show that, under moderate conditions, the transition probability of hyperfinite Markov
processes align with the transition probability of standard Markov processes. The Markov chain ergodic theorem for hyperfinite Markov
processes will then imply the Markov chain ergodic theorem for general state space continuous-time Markov processes.
eBook
The regularity of general parabolic systems with degenerate diffusion
The aim of the paper is twofold. On one hand the authors want to present a new technique called $p$-caloric approximation, which is a proper generalization of the classical compactness methods first developed by DeGiorgi with his Harmonic Approximation Lemma. This last result, initially introduced in the setting of Geometric Measure Theory to prove the regularity of minimal surfaces, is nowadays a classical tool to prove linearization and regularity results for vectorial problems. Here the authors develop a very far reaching version of this general principle devised to linearize general degenerate parabolic systems. The use of this result in turn allows the authors to achieve the subsequent and main aim of the paper, that is, the implementation of a partial regularity theory for parabolic systems with degenerate diffusion of the type $\\partial_t u - \\mathrm{div} a(Du)=0$, without necessarily assuming a quasi-diagonal structure, i.e. a structure prescribing that the gradient non-linearities depend only on the the explicit scalar quantity.
eBook
Extremes and recurrence in dynamical systems
Written by a team of international experts, Extremes and Recurrence in Dynamical Systems presents a unique point of view on the mathematical theory of extremes and on its applications in the natural and social sciences. Featuring an interdisciplinary approach to new concepts in pure and applied mathematical research, the book skillfully combines the areas of statistical mechanics, probability theory, measure theory, dynamical systems, statistical inference, geophysics, and software application. Emphasizing the statistical mechanical point of view, the book introduces robust theoretical embedding for the application of extreme value theory in dynamical systems. Extremes and Recurrence in Dynamical Systems also features:
• A careful examination of how a dynamical system can serve as a generator of stochastic processes
• Discussions on the applications of statistical inference in the theoretical and heuristic use of extremes
• Several examples of analysis of extremes in a physical and geophysical context
• A final summary of the main results presented along with a guide to future research projects
• An appendix with software in Matlab ® programming language to help readers to develop further understanding of the presented concepts
Extremes and Recurrence in Dynamical Systems is ideal for academics and practitioners in pure and applied mathematics, probability theory, statistics, chaos, theoretical and applied dynamical systems, statistical mechanics, geophysical fluid dynamics, geosciences and complexity science. VALERIO LUCARINI, PhD, is Professor of Theoretical Meteorology at the University of Hamburg, Germany and Professor of Statistical Mechanics at the University of Reading, UK. DAVIDE FARANDA, PhD, is Researcher at the Laboratoire des science du climat et de l'environnement, IPSL, CEA Saclay, Université Paris-Saclay, Gif-sur-Yvette, France. ANA CRISTINA GOMES MONTEIRO MOREIRA DE FREITAS, PhD, is Assistant Professor in the Faculty of Economics at the University of Porto, Portugal. JORGE MIGUEL MILHAZES DE FREITAS, PhD, is Assistant Professor in the Department of Mathematics of the Faculty of Sciences at the University of Porto, Portugal.
MARK HOLLAND, PhD, is Senior Lecturer in Applied Mathematics in the College of Engineering, Mathematics and Physical Sciences at the University of Exeter, UK. TOBIAS KUNA, PhD, is Associate Professor in the Department of Mathematics and Statistics at the University of Reading, UK. MATTHEW NICOL, PhD, is Professor of Mathematics at the University of Houston, USA.
MIKE TODD, PhD, is Lecturer in the School of Mathematics and Statistics at the University of St. Andrews, Scotland.
SANDRO VAIENTI, PhD, is Professor of Mathematics at the University of Toulon and Researcher at the Centre de Physique Théorique, France.
eBook
Mathematical analysis : a very short introduction
Richard Earl describes the nascent evolution of mathematical analysis, its development as a subject in its own right, and its wide-ranging applications in mathematics and science, modelling reality from acoustics to fluid dynamics, from biological systems to quantum theory.
Book
Multilinear Singular Integral Forms of Christ-Journé Type
We introduce a class of multilinear singular integral forms
eBook
Advanced Numerical and Semi-Analytical Methods for Differential Equations
<p><b>Examines numerical and semi-analytical methods for differential equations that can be used for solving practical ODEs and PDEs</b> <p>This student-friendly book deals with various approaches for solving differential equations numerically or semi-analytically depending on the type of equations and offers simple example problems to help readers along. <p>Featuring both traditional and recent methods, <i>Advanced Numerical and Semi-Analytical Methods for Differential Equations</i> begins with a review of basic numerical methods. It then looks at Laplace, Fourier, and weighted residual methods for solving differential equations. A new challenging method of Boundary Characteristics Orthogonal Polynomials (BCOPs) is introduced next. The book then discusses Finite Difference Method (FDM), Finite Element Method (FEM), Finite Volume Method (FVM), and Boundary Element Method (BEM). Following that, analytical/semi<i>-</i>analytic methods like Akbari Ganji's Method (AGM) and Exp-function are used to solve nonlinear differential equations. Nonlinear differential equations using semi-analytical methods are also addressed, namely Adomian Decomposition Method (ADM), Homotopy Perturbation Method (HPM), Variational Iteration Method (VIM), and Homotopy Analysis Method (HAM). Other topics covered include: emerging areas of research related to the solution of differential equations based on differential quadrature and wavelet approach; combined and hybrid methods for solving differential equations; as well as an overview of fractal differential equations. Further, uncertainty in term of intervals and fuzzy numbers have also been included, along with the interval finite element method. This book: <ul> <li>Discusses various methods for solving linear and nonlinear ODEs and PDEs</li> <li>Covers basic numerical techniques for solving differential equations along with various discretization methods</li> <li>Investigates nonlinear differential equations using semi-analytical methods</li> <li>Examines differential equations in an uncertain environment</li> <li>Includes a new scenario in which uncertainty (in term of intervals and fuzzy numbers) has been included in differential equations</li> <li>Contains solved example problems, as well as some unsolved problems for self-validation of the topics covered</li> </ul> <p><i>Advanced Numerical and Semi-Analytical Methods for Differential Equations</i> is an excellent textbook for graduate as well as post graduate students and researchers studying various methods for solving differential equations, numerically and semi-analytically.
eBook