Peter Imkeller's Books

Stochastic Climate Models

By: Peter Imkeller,Jin-Song Von Storch

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A collection of articles written by mathematicians and physicists, designed to describe the state of the art in climate models with stochastic input. Mathematicians will benefit from a survey of simple models, while physicists will encounter mathematically relevant techniques at work.

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Page Count

446

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Springer Science & Business Media

Published Date

ISBN 10

376436520X

ISBN 13

9783764365202

Stochastic Resonance

By: Samuel Herrmann, Peter Imkeller,Ilya Pavlyukevich, Dierk Peithmann

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Stochastic Resonance

By: Samuel Herrmann, Peter Imkeller,Ilya Pavlyukevich, Dierk Peithmann

Stochastic resonance is a phenomenon arising in a wide spectrum of areas in the sciences ranging from physics through neuroscience to chemistry and biology. This book presents a mathematical approach to stochastic resonance which is based on a large deviations principle (LDP) for randomly perturbed dynamical systems with a weak inhomogeneity given by an exogenous periodicity of small frequency. Resonance, the optimal tuning between period length and noise amplitude, is explained by optimizing the LDP's rate function. The authors show that not all physical measures of tuning quality are robust with respect to dimension reduction. They propose measures of tuning quality based on exponential transition rates explained by large deviations techniques and show that these measures are robust. The book sheds some light on the shortcomings and strengths of different concepts used in the theory and applications of stochastic resonance without attempting to give a comprehensive overview of the many facets of stochastic resonance in the various areas of sciences. It is intended for researchers and graduate students in mathematics and the sciences interested in stochastic dynamics who wish to understand the conceptual background of stochastic resonance.

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209

Publisher

American Mathematical Soc.

Published Date

ISBN 10

1470410494

ISBN 13

9781470410490

The Dynamics of Nonlinear Reaction-Diffusion Equations with Small Lévy Noise

By: Arnaud Debussche,Michael Högele,Peter Imkeller

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The Dynamics of Nonlinear Reaction-Diffusion Equations with Small Lévy Noise

By: Arnaud Debussche,Michael Högele,Peter Imkeller

This work considers a small random perturbation of alpha-stable jump type nonlinear reaction-diffusion equations with Dirichlet boundary conditions over an interval. It has two stable points whose domains of attraction meet in a separating manifold with several saddle points. Extending a method developed by Imkeller and Pavlyukevich it proves that in contrast to a Gaussian perturbation, the expected exit and transition times between the domains of attraction depend polynomially on the noise intensity in the small intensity limit. Moreover the solution exhibits metastable behavior: there is a polynomial time scale along which the solution dynamics correspond asymptotically to the dynamic behavior of a finite-state Markov chain switching between the stable states.

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Page Count

175

Publisher

Springer

Published Date

ISBN 10

3319008285

ISBN 13

9783319008288

Two-Parameter Martingales and Their Quadratic Variation

By: Peter Imkeller

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Two-Parameter Martingales and Their Quadratic Variation

By: Peter Imkeller

This book has two-fold aims. In a first part it gives an introductory, thorough and essentially self-contained treatment of the general theory of two-parameter processes that has developed since around 1975. Apart from two survey papers by Merzbach and Meyer it is the first text of this kind. The second part presents the results of recent research by the author on martingale theory and stochastic calculus for two-parameter processes. Both the results and the methods of these two chapters are almost entirely new, and are of particular interest. They provide the fundamentals of a general stochastic analysis of two-parameter processes including, in particular, so far inaccessible jump phenomena. The typical rader is assumed to have some basic knowledge of the general theory of one-parameter martingales. The book should be accessible to probabilistically interested mathematicians who a) wish to become acquainted with or have a complete treatment of the main features of the general theory of two-parameter processes and basics of their stochastic calculus, b) intend to learn about the most recent developments in this area.

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182

Publisher

Springer

Published Date

ISBN 10

3540391487

ISBN 13

9783540391487

Random Ordinary Differential Equations and Their Numerical Solution

By: Xiaoying Han,Peter E. Kloeden

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Random Ordinary Differential Equations and Their Numerical Solution

By: Xiaoying Han,Peter E. Kloeden

This book is intended to make recent results on the derivation of higher order numerical schemes for random ordinary differential equations (RODEs) available to a broader readership, and to familiarize readers with RODEs themselves as well as the closely associated theory of random dynamical systems. In addition, it demonstrates how RODEs are being used in the biological sciences, where non-Gaussian and bounded noise are often more realistic than the Gaussian white noise in stochastic differential equations (SODEs). RODEs are used in many important applications and play a fundamental role in the theory of random dynamical systems. They can be analyzed pathwise with deterministic calculus, but require further treatment beyond that of classical ODE theory due to the lack of smoothness in their time variable. Although classical numerical schemes for ODEs can be used pathwise for RODEs, they rarely attain their traditional order since the solutions of RODEs do not have sufficient smoothness to have Taylor expansions in the usual sense. However, Taylor-like expansions can be derived for RODEs using an iterated application of the appropriate chain rule in integral form, and represent the starting point for the systematic derivation of consistent higher order numerical schemes for RODEs. The book is directed at a wide range of readers in applied and computational mathematics and related areas as well as readers who are interested in the applications of mathematical models involving random effects, in particular in the biological sciences.The level of this book is suitable for graduate students in applied mathematics and related areas, computational sciences and systems biology. A basic knowledge of ordinary differential equations and numerical analysis is required.

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Page Count

252

Publisher

Springer

Published Date

ISBN 10

981106265X

ISBN 13

9789811062650

Paris-Princeton Lectures on Mathematical Finance 2002

By: Peter Bank,Fabrice Baudoin,Hans Föllmer,L. C. G. Rogers,Halil Mete Soner,Nizar Touzi

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Paris-Princeton Lectures on Mathematical Finance 2002

By: Peter Bank,Fabrice Baudoin,Hans Föllmer,L. C. G. Rogers,Halil Mete Soner,Nizar Touzi

The Paris-Princeton Lectures in Financial Mathematics, of which this is the first volume, will, on an annual basis, publish cutting-edge research in self-contained, expository articles from outstanding - established or upcoming! - specialists. The aim is to produce a series of articles that can serve as an introductory reference for research in the field. It arises as a result of frequent exchanges between the finance and financial mathematics groups in Paris and Princeton. The present volume sets standards with articles by P. Bank/H. Föllmer, F. Baudoin, L.C.G. Rogers, and M. Soner/N. Touzi.

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Page Count

185

Publisher

Springer

Published Date

ISBN 10

3540448594

ISBN 13

9783540448594

A Course on Rough Paths

With an Introduction to Regularity Structures

By: Peter K. Friz,Martin Hairer

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A Course on Rough Paths

With an Introduction to Regularity Structures

By: Peter K. Friz,Martin Hairer

With many updates and additional exercises, the second edition of this book continues to provide readers with a gentle introduction to rough path analysis and regularity structures, theories that have yielded many new insights into the analysis of stochastic differential equations, and, most recently, stochastic partial differential equations. Rough path analysis provides the means for constructing a pathwise solution theory for stochastic differential equations which, in many respects, behaves like the theory of deterministic differential equations and permits a clean break between analytical and probabilistic arguments. Together with the theory of regularity structures, it forms a robust toolbox, allowing the recovery of many classical results without having to rely on specific probabilistic properties such as adaptedness or the martingale property. Essentially self-contained, this textbook puts the emphasis on ideas and short arguments, rather than aiming for the strongest possible statements. A typical reader will have been exposed to upper undergraduate analysis and probability courses, with little more than Itô-integration against Brownian motion required for most of the text. From the reviews of the first edition: "Can easily be used as a support for a graduate course ... Presents in an accessible way the unique point of view of two experts who themselves have largely contributed to the theory" - Fabrice Baudouin in the Mathematical Reviews "It is easy to base a graduate course on rough paths on this ... A researcher who carefully works her way through all of the exercises will have a very good impression of the current state of the art" - Nicolas Perkowski in Zentralblatt MATH

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Page Count

346

Publisher

Springer Nature

Published Date

ISBN 10

3030415562

ISBN 13

9783030415563

Nonautonomous Dynamical Systems

By: Peter E. Kloeden,Martin Rasmussen

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Nonautonomous Dynamical Systems

By: Peter E. Kloeden,Martin Rasmussen

The theory of nonautonomous dynamical systems in both of its formulations as processes and skew product flows is developed systematically in this book. The focus is on dissipative systems and nonautonomous attractors, in particular the recently introduced concept of pullback attractors. Linearization theory, invariant manifolds, Lyapunov functions, Morse decompositions and bifurcations for nonautonomous systems and set-valued generalizations are also considered as well as applications to numerical approximations, switching systems and synchronization. Parallels with corresponding theories of control and random dynamical systems are briefly sketched. With its clear and systematic exposition, many examples and exercises, as well as its interesting applications, this book can serve as a text at the beginning graduate level. It is also useful for those who wish to begin their own independent research in this rapidly developing area.

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274

Publisher

American Mathematical Soc.

Published Date

ISBN 10

0821868713

ISBN 13

9780821868713

Lernpunkt Deutsch

Copymasters

By: Peter Morris,Alan Wesson

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Lernpunkt Deutsch

Copymasters

By: Peter Morris,Alan Wesson

Lernpunkt Deutsch provides a challenging and thorough approach to grammar, and encourages vocabulary acquisition, practice and retention. It is a three stage course which is rigorous and motivating.

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Page Count

180

Publisher

Nelson Thornes

Published Date

ISBN 10

0174402643

ISBN 13

9780174402640

Lernpunkt Deutsch

With New German Spellings

By: Peter Morris,Alan Wesson

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Lernpunkt Deutsch

With New German Spellings

By: Peter Morris,Alan Wesson

Lernpunkt Deutsch provides a challenging and thorough approach to grammar, and encourages vocabulary acquisition, practice and retention. It is a three stage course which is rigorous and motivating.

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180

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Nelson Thornes

Published Date

ISBN 10

017440266X

ISBN 13

9780174402664

Taylor Approximations for Stochastic Partial Differential Equations

By: Arnulf Jentzen,Peter E. Kloeden

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Taylor Approximations for Stochastic Partial Differential Equations

By: Arnulf Jentzen,Peter E. Kloeden

This book presents a systematic theory of Taylor expansions of evolutionary-type stochastic partial differential equations (SPDEs). The authors show how Taylor expansions can be used to derive higher order numerical methods for SPDEs, with a focus on pathwise and strong convergence. In the case of multiplicative noise, the driving noise process is assumed to be a cylindrical Wiener process, while in the case of additive noise the SPDE is assumed to be driven by an arbitrary stochastic process with Hl̲der continuous sample paths. Recent developments on numerical methods for random and stochastic ordinary differential equations are also included since these are relevant for solving spatially discretised SPDEs as well as of interest in their own right. The authors include the proof of an existence and uniqueness theorem under general assumptions on the coefficients as well as regularity estimates in an appendix.

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234

Publisher

SIAM

Published Date

ISBN 10

1611972019

ISBN 13

9781611972016

Dissipative Lattice Dynamical Systems

By: Xiaoying Han,Peter Kloeden

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Dissipative Lattice Dynamical Systems

By: Xiaoying Han,Peter Kloeden

There is an extensive literature in the form of papers (but no books) on lattice dynamical systems. The book focuses on dissipative lattice dynamical systems and their attractors of various forms such as autonomous, nonautonomous and random. The existence of such attractors is established by showing that the corresponding dynamical system has an appropriate kind of absorbing set and is asymptotically compact in some way.There is now a very large literature on lattice dynamical systems, especially on attractors of all kinds in such systems. We cannot hope to do justice to all of them here. Instead, we have focused on key areas of representative types of lattice systems and various types of attractors. Our selection is biased by our own interests, in particular to those dealing with biological applications. One of the important results is the approximation of Heaviside switching functions in LDS by sigmoidal functions.Nevertheless, we believe that this book will provide the reader with a solid introduction to the field, its main results and the methods that are used to obtain them.

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Page Count

381

Publisher

World Scientific

Published Date

ISBN 10

9811267774

ISBN 13

9789811267772

Grid Homology for Knots and Links

By: Peter S. Ozsváth,András I. Stipsicz,Zoltán Szabó

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Grid Homology for Knots and Links

By: Peter S. Ozsváth,András I. Stipsicz,Zoltán Szabó

Knot theory is a classical area of low-dimensional topology, directly connected with the theory of three-manifolds and smooth four-manifold topology. In recent years, the subject has undergone transformative changes thanks to its connections with a number of other mathematical disciplines, including gauge theory; representation theory and categorification; contact geometry; and the theory of pseudo-holomorphic curves. Starting from the combinatorial point of view on knots using their grid diagrams, this book serves as an introduction to knot theory, specifically as it relates to some of the above developments. After a brief overview of the background material in the subject, the book gives a self-contained treatment of knot Floer homology from the point of view of grid diagrams. Applications include computations of the unknotting number and slice genus of torus knots (asked first in the 1960s and settled in the 1990s), and tools to study variants of knot theory in the presence of a contact structure. Additional topics are presented to prepare readers for further study in holomorphic methods in low-dimensional topology, especially Heegaard Floer homology. The book could serve as a textbook for an advanced undergraduate or part of a graduate course in knot theory. Standard background material is sketched in the text and the appendices.

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423

Publisher

American Mathematical Soc.

Published Date

ISBN 10

1470417375

ISBN 13

9781470417376

Stability and Bifurcation Theory for Non-Autonomous Differential Equations

Cetraro, Italy 2011, Editors: Russell Johnson, Maria Patrizia Pera

By: Anna Capietto,Peter Kloeden,Jean Mawhin,Sylvia Novo,Miguel Ortega

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Stability and Bifurcation Theory for Non-Autonomous Differential Equations

Cetraro, Italy 2011, Editors: Russell Johnson, Maria Patrizia Pera

By: Anna Capietto,Peter Kloeden,Jean Mawhin,Sylvia Novo,Miguel Ortega

This volume contains the notes from five lecture courses devoted to nonautonomous differential systems, in which appropriate topological and dynamical techniques were described and applied to a variety of problems. The courses took place during the C.I.M.E. Session "Stability and Bifurcation Problems for Non-Autonomous Differential Equations," held in Cetraro, Italy, June 19-25 2011. Anna Capietto and Jean Mawhin lectured on nonlinear boundary value problems; they applied the Maslov index and degree-theoretic methods in this context. Rafael Ortega discussed the theory of twist maps with nonperiodic phase and presented applications. Peter Kloeden and Sylvia Novo showed how dynamical methods can be used to study the stability/bifurcation properties of bounded solutions and of attracting sets for nonautonomous differential and functional-differential equations. The volume will be of interest to all researchers working in these and related fields.

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314

Publisher

Springer

Published Date

ISBN 10

3642329063

ISBN 13

9783642329067

Recent Developments in Computational Finance

Foundations, Algorithms and Applications

By: Thomas Gerstner,Peter E. Kloeden

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Recent Developments in Computational Finance

Foundations, Algorithms and Applications

By: Thomas Gerstner,Peter E. Kloeden

Computational finance is an interdisciplinary field which joins financial mathematics, stochastics, numerics and scientific computing. Its task is to estimate as accurately and efficiently as possible the risks that financial instruments generate. This volume consists of a series of cutting-edge surveys of recent developments in the field written by leading international experts. These make the subject accessible to a wide readership in academia and financial businesses. The book consists of 13 chapters divided into 3 parts: foundations, algorithms and applications. Besides surveys of existing results, the book contains many new previously unpublished results.

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481

Publisher

World Scientific

Published Date

ISBN 10

9814436437

ISBN 13

9789814436434

Wie, bitte?

Introductory German for Proficiency

By: William B. Fischer,Peter N. Richardson

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Wie, bitte?

Introductory German for Proficiency

By: William B. Fischer,Peter N. Richardson

A brief, proficiency-based introductory program stressing communicative abilities, and organized according to ACTFL (American Council on the Teaching of Foreign Languages). Use of authentic listening and reading materials (e.g., radio broadcasts, transportation schedules, ads, magazine excerpts) promotes practical fluency. Cultural material is integrated into every aspect of the program. Class text has situational dialogues, brief grammar notes, cultural sections, class activities, and realia. Workbook contains chapter vocabularies, grammar and conversational exercises, lab manual, and reference grammar. Text includes many illustrations.

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Page Count

408

Publisher

Wiley

Published Date

ISBN 10

0471848441

ISBN 13

9780471848448

Book's by Peter Imkeller

Stochastic Climate Models

Stochastic Climate Models

Stochastic Resonance

Stochastic Resonance

The Dynamics of Nonlinear Reaction-Diffusion Equations with Small Lévy Noise

The Dynamics of Nonlinear Reaction-Diffusion Equations with Small Lévy Noise

Two-Parameter Martingales and Their Quadratic Variation

Two-Parameter Martingales and Their Quadratic Variation

Random Ordinary Differential Equations and Their Numerical Solution

Random Ordinary Differential Equations and Their Numerical Solution

Paris-Princeton Lectures on Mathematical Finance 2002

Paris-Princeton Lectures on Mathematical Finance 2002

A Course on Rough Paths

A Course on Rough Paths

Nonautonomous Dynamical Systems

Nonautonomous Dynamical Systems

Lernpunkt Deutsch

Lernpunkt Deutsch

Lernpunkt Deutsch

Lernpunkt Deutsch

Taylor Approximations for Stochastic Partial Differential Equations

Taylor Approximations for Stochastic Partial Differential Equations

Dissipative Lattice Dynamical Systems

Dissipative Lattice Dynamical Systems

Grid Homology for Knots and Links

Grid Homology for Knots and Links

Stability and Bifurcation Theory for Non-Autonomous Differential Equations

Stability and Bifurcation Theory for Non-Autonomous Differential Equations

Recent Developments in Computational Finance

Recent Developments in Computational Finance

Wie, bitte?

Wie, bitte?

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