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 conceptsExtremes 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.

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.

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.

TheKramers-Kronigrelationsconstitutethemathematicalformulationofthe fundamental connection between the in-phase to the out-of-phase response of a system to a sinusoidal time-varying external perturbation. Such connection exists in both classical and quantum physical systems and derives directly from the principle of causality. Apart from being of great importance in high energy physics, statistical physics, and acoustics, at present the Kramers-Kronig relations are basic and widely-accepted tools for the investigation of the linear optical properties of materials, since they allow performing the so-called inversion of optical data, i.e. acquiring knowledge on dispersive phenomena by measurements of absorptive phenomena over the whole energy spectrum or vice versa. Since the late ’80s, a growing body of theoretical results as well as of experimental evidences has shown that the Kramers-Kronig relations can be adopted for e?ciently acquiring knowledge on nonlinear optical phenomena. These results suggest that the Kramers-Kronig relations may become in a near future standard techniques in the context of nonlinear spectroscopy. Thisbookisthe?rstcomprehensivetreatisedevotedtoprovidingauni- ing picture of the physical backgrounds, of the rigorous mathematical theory, and of the applications of the Kramers-Kronig relations in both ?elds of l- ear and nonlinear optical spectroscopy. Some basic programs written for the 1 MATLAB environment are also included. This book is organized as an argumentative discourse, progressing from the linear to the nonlinear phenomena, from the general to the speci?c s- tems, and from the theoretical to the experimental results.

TheKramers-Kronigrelationsconstitutethemathematicalformulationofthe fundamental connection between the in-phase to the out-of-phase response of a system to a sinusoidal time-varying external perturbation. Such connection exists in both classical and quantum physical systems and derives directly from the principle of causality. Apart from being of great importance in high energy physics, statistical physics, and acoustics, at present the Kramers-Kronig relations are basic and widely-accepted tools for the investigation of the linear optical properties of materials, since they allow performing the so-called inversion of optical data, i.e. acquiring knowledge on dispersive phenomena by measurements of absorptive phenomena over the whole energy spectrum or vice versa. Since the late ’80s, a growing body of theoretical results as well as of experimental evidences has shown that the Kramers-Kronig relations can be adopted for e?ciently acquiring knowledge on nonlinear optical phenomena. These results suggest that the Kramers-Kronig relations may become in a near future standard techniques in the context of nonlinear spectroscopy. Thisbookisthe?rstcomprehensivetreatisedevotedtoprovidingauni- ing picture of the physical backgrounds, of the rigorous mathematical theory, and of the applications of the Kramers-Kronig relations in both ?elds of l- ear and nonlinear optical spectroscopy. Some basic programs written for the 1 MATLAB environment are also included. This book is organized as an argumentative discourse, progressing from the linear to the nonlinear phenomena, from the general to the speci?c s- tems, and from the theoretical to the experimental results.

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 conceptsExtremes 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.

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.

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.

TheKramers-Kronigrelationsconstitutethemathematicalformulationofthe fundamental connection between the in-phase to the out-of-phase response of a system to a sinusoidal time-varying external perturbation. Such connection exists in both classical and quantum physical systems and derives directly from the principle of causality. Apart from being of great importance in high energy physics, statistical physics, and acoustics, at present the Kramers-Kronig relations are basic and widely-accepted tools for the investigation of the linear optical properties of materials, since they allow performing the so-called inversion of optical data, i.e. acquiring knowledge on dispersive phenomena by measurements of absorptive phenomena over the whole energy spectrum or vice versa. Since the late ’80s, a growing body of theoretical results as well as of experimental evidences has shown that the Kramers-Kronig relations can be adopted for e?ciently acquiring knowledge on nonlinear optical phenomena. These results suggest that the Kramers-Kronig relations may become in a near future standard techniques in the context of nonlinear spectroscopy. Thisbookisthe?rstcomprehensivetreatisedevotedtoprovidingauni- ing picture of the physical backgrounds, of the rigorous mathematical theory, and of the applications of the Kramers-Kronig relations in both ?elds of l- ear and nonlinear optical spectroscopy. Some basic programs written for the 1 MATLAB environment are also included. This book is organized as an argumentative discourse, progressing from the linear to the nonlinear phenomena, from the general to the speci?c s- tems, and from the theoretical to the experimental results.

TheKramers-Kronigrelationsconstitutethemathematicalformulationofthe fundamental connection between the in-phase to the out-of-phase response of a system to a sinusoidal time-varying external perturbation. Such connection exists in both classical and quantum physical systems and derives directly from the principle of causality. Apart from being of great importance in high energy physics, statistical physics, and acoustics, at present the Kramers-Kronig relations are basic and widely-accepted tools for the investigation of the linear optical properties of materials, since they allow performing the so-called inversion of optical data, i.e. acquiring knowledge on dispersive phenomena by measurements of absorptive phenomena over the whole energy spectrum or vice versa. Since the late ’80s, a growing body of theoretical results as well as of experimental evidences has shown that the Kramers-Kronig relations can be adopted for e?ciently acquiring knowledge on nonlinear optical phenomena. These results suggest that the Kramers-Kronig relations may become in a near future standard techniques in the context of nonlinear spectroscopy. Thisbookisthe?rstcomprehensivetreatisedevotedtoprovidingauni- ing picture of the physical backgrounds, of the rigorous mathematical theory, and of the applications of the Kramers-Kronig relations in both ?elds of l- ear and nonlinear optical spectroscopy. Some basic programs written for the 1 MATLAB environment are also included. This book is organized as an argumentative discourse, progressing from the linear to the nonlinear phenomena, from the general to the speci?c s- tems, and from the theoretical to the experimental results.