Geometrical Properties of Differential Equations

Applications of the Lie Group Analysis in Financial Mathematics

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Author: Ljudmila A Bordag

Publisher: World Scientific Publishing Company

ISBN: 9814667269

Category: Mathematics

Page: 340

View: 7957

This textbook is a short comprehensive and intuitive introduction to Lie group analysis of ordinary and partial differential equations. This practical-oriented material contains a large number of examples and problems accompanied by detailed solutions and figures. In comparison with the known beginner guides to Lie group analysis, the book is oriented toward students who are interested in financial mathematics, mathematical finance and economics. We provide the results of the Lie group analysis of actual models in Financial Mathematics using recent publications. These models are usually formulated as nonlinear partial differential equations and are rather difficult to make use of. With the help of Lie group analysis it is possible to describe some important properties of these models and to obtain interesting reductions in a clear and understandable algorithmic way. The book can serve as a short introduction for a further study of modern geometrical analysis applied to models in financial mathematics. It can also be used as textbook in a master's program, in an intensive compact course, or for self study. The textbook with a large number of examples will be useful not only for students who are interested in Financial Mathematics but also for people who are working in other areas of research that are not directly connected with Physics (for instance in such areas of Applied Mathematics like mathematical economy, bio systems, coding theory, etc.).

Novel Methods in Computational Finance

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Author: Matthias Ehrhardt,Michael Günther,E. Jan W. ter Maten

Publisher: Springer

ISBN: 3319612824

Category: Mathematics

Page: 606

View: 6254

This book discusses the state-of-the-art and open problems in computational finance. It presents a collection of research outcomes and reviews of the work from the STRIKE project, an FP7 Marie Curie Initial Training Network (ITN) project in which academic partners trained early-stage researchers in close cooperation with a broader range of associated partners, including from the private sector. The aim of the project was to arrive at a deeper understanding of complex (mostly nonlinear) financial models and to develop effective and robust numerical schemes for solving linear and nonlinear problems arising from the mathematical theory of pricing financial derivatives and related financial products. This was accomplished by means of financial modelling, mathematical analysis and numerical simulations, optimal control techniques and validation of models. In recent years the computational complexity of mathematical models employed in financial mathematics has witnessed tremendous growth. Advanced numerical techniques are now essential to the majority of present-day applications in the financial industry. Special attention is devoted to a uniform methodology for both testing the latest achievements and simultaneously educating young PhD students. Most of the mathematical codes are linked into a novel computational finance toolbox, which is provided in MATLAB and PYTHON with an open access license. The book offers a valuable guide for researchers in computational finance and related areas, e.g. energy markets, with an interest in industrial mathematics.

Elementary Lie group analysis and ordinary differential equations

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Author: Nailʹ Khaĭrullovich Ibragimov

Publisher: John Wiley & Sons Inc

ISBN: 9780471974307

Category: Mathematics

Page: 347

View: 3342

Lie group analysis, based on symmetry and invariance principles, is the only systematic method for solving nonlinear differential equations analytically. One of Lie's striking achievements was the discovery that the majority of classical devices for integration of special types of ordinary differential equations could be explained and deduced by his theory. Moreover, this theory provides a universal tool for tackling considerable numbers of differential equations when other means of integration fail. * This is the first modern text on ordinary differential equations where the basic integration methods are derived from Lie group theory * Includes a concise and self contained introduction to differential equations * Easy to follow and comprehensive introduction to Lie group analysis * The methods described in this book have many applications The author provides students and their teachers with a flexible text for undergraduate and postgraduate courses, spanning a variety of topics from the basic theory through to its many applications. The philosophy of Lie groups has become an essential part of the mathematical culture for anyone investigating mathematical models of physical, engineering and natural problems.

The Ricci Flow in Riemannian Geometry

A Complete Proof of the Differentiable 1/4-Pinching Sphere Theorem

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Author: Ben Andrews,Christopher Hopper

Publisher: Springer

ISBN: 364216286X

Category: Mathematics

Page: 302

View: 9181

This book focuses on Hamilton's Ricci flow, beginning with a detailed discussion of the required aspects of differential geometry, progressing through existence and regularity theory, compactness theorems for Riemannian manifolds, and Perelman's noncollapsing results, and culminating in a detailed analysis of the evolution of curvature, where recent breakthroughs of Böhm and Wilking and Brendle and Schoen have led to a proof of the differentiable 1/4-pinching sphere theorem.

Arithmetic Geometry

Lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, September 10-15, 2007

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Author: Jean-Louis Colliot-Thélène,Peter Swinnerton-Dyer,Paul Vojta

Publisher: Springer

ISBN: 3642159451

Category: Mathematics

Page: 232

View: 1426

Arithmetic Geometry can be defined as the part of Algebraic Geometry connected with the study of algebraic varieties through arbitrary rings, in particular through non-algebraically closed fields. It lies at the intersection between classical algebraic geometry and number theory. A C.I.M.E. Summer School devoted to arithmetic geometry was held in Cetraro, Italy in September 2007, and presented some of the most interesting new developments in arithmetic geometry. This book collects the lecture notes which were written up by the speakers. The main topics concern diophantine equations, local-global principles, diophantine approximation and its relations to Nevanlinna theory, and rationally connected varieties. The book is divided into three parts, corresponding to the courses given by J-L Colliot-Thelene, Peter Swinnerton Dyer and Paul Vojta.

Geometrical Methods of Mathematical Physics

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Author: Bernard F. Schutz

Publisher: Cambridge University Press

ISBN: 9780521298872

Category: Mathematics

Page: 250

View: 1529

For physicists and applied mathematicians working in the fields of relativity and cosmology, high-energy physics and field theory, thermodynamics, fluid dynamics and mechanics. This book provides an introduction to the concepts and techniques of modern differential theory, particularly Lie groups, Lie forms and differential forms.

Fundamentals of Differential Geometry

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Author: Serge Lang

Publisher: Springer Science & Business Media

ISBN: 9780387985930

Category: Mathematics

Page: 540

View: 3486

This book provides an introduction to the basic concepts in differential topology, differential geometry, and differential equations, and some of the main basic theorems in all three areas. This new edition includes new chapters, sections, examples, and exercises. From the reviews: "There are many books on the fundamentals of differential geometry, but this one is quite exceptional; this is not surprising for those who know Serge Lang's books." --EMS NEWSLETTER

Distributions

Theory and Applications

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Author: J.J. Duistermaat,Johan A.C. Kolk

Publisher: Springer Science & Business Media

ISBN: 9780817646752

Category: Mathematics

Page: 445

View: 6114

This textbook is an application-oriented introduction to the theory of distributions, a powerful tool used in mathematical analysis. The treatment emphasizes applications that relate distributions to linear partial differential equations and Fourier analysis problems found in mechanics, optics, quantum mechanics, quantum field theory, and signal analysis. The book is motivated by many exercises, hints, and solutions that guide the reader along a path requiring only a minimal mathematical background.

UC Santa Cruz

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Author: University of California, Santa Cruz

Publisher: N.A

ISBN: N.A

Category:

Page: N.A

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Introduction to Smooth Manifolds

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Author: John M. Lee

Publisher: Springer Science & Business Media

ISBN: 9780387954486

Category: Mathematics

Page: 628

View: 9115

Author has written several excellent Springer books.; This book is a sequel to Introduction to Topological Manifolds; Careful and illuminating explanations, excellent diagrams and exemplary motivation; Includes short preliminary sections before each section explaining what is ahead and why

The Statistical Analysis of Time Series

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Author: Theodore W. Anderson

Publisher: John Wiley & Sons

ISBN: 1118150392

Category: Mathematics

Page: 704

View: 6348

The Wiley Classics Library consists of selected books that havebecome recognized classics in their respective fields. With thesenew unabridged and inexpensive editions, Wiley hopes to extend thelife of these important works by making them available to futuregenerations of mathematicians and scientists. Currently availablein the Series: T. W. Anderson Statistical Analysis of Time SeriesT. S. Arthanari & Yadolah Dodge Mathematical Programming inStatistics Emil Artin Geometric Algebra Norman T. J. Bailey TheElements of Stochastic Processes with Applications to the NaturalSciences George E. P. Box & George C. Tiao Bayesian Inferencein Statistical Analysis R. W. Carter Simple Groups of Lie TypeWilliam G. Cochran & Gertrude M. Cox Experimental Designs,Second Edition Richard Courant Differential and Integral Calculus,Volume I Richard Courant Differential and Integral Calculus, VolumeII Richard Courant & D. Hilbert Methods of MathematicalPhysics, Volume I Richard Courant & D. Hilbert Methods ofMathematical Physics, Volume II D. R. Cox Planning of ExperimentsHarold M. S. Coxeter Introduction to Modern Geometry, SecondEdition Charles W. Curtis & Irving Reiner Representation Theoryof Finite Groups and Associative Algebras Charles W. Curtis &Irving Reiner Methods of Representation Theory with Applications toFinite Groups and Orders, Volume I Charles W. Curtis & IrvingReiner Methods of Representation Theory with Applications to FiniteGroups and Orders, Volume II Bruno de Finetti Theory ofProbability, Volume 1 Bruno de Finetti Theory of Probability,Volume 2 W. Edwards Deming Sample Design in Business Research Amosde Shalit & Herman Feshbach Theoretical Nuclear Physics, Volume1 --Nuclear Structure J. L. Doob Stochastic Processes NelsonDunford & Jacob T. Schwartz Linear Operators, Part One, GeneralTheory Nelson Dunford & Jacob T. Schwartz Linear Operators,Part Two, Spectral Theory--Self Adjoint Operators in Hilbert SpaceNelson Dunford & Jacob T. Schwartz Linear Operators, PartThree, Spectral Operators Herman Fsehbach Theoretical NuclearPhysics: Nuclear Reactions Bernard Friedman Lectures onApplications-Oriented Mathematics Gerald d. Hahn & Samuel S.Shapiro Statistical Models in Engineering Morris H. Hansen, WilliamN. Hurwitz & William G. Madow Sample Survey Methods and Theory,Volume I--Methods and Applications Morris H. Hansen, William N.Hurwitz & William G. Madow Sample Survey Methods and Theory,Volume II--Theory Peter Henrici Applied and Computational ComplexAnalysis, Volume 1--Power Series--lntegration--ConformalMapping--Location of Zeros Peter Henrici Applied and ComputationalComplex Analysis, Volume 2--Special Functions--IntegralTransforms--Asymptotics--Continued Fractions Peter Henrici Appliedand Computational Complex Analysis, Volume 3--Discrete FourierAnalysis--Cauchy Integrals--Construction of ConformalMaps--Univalent Functions Peter Hilton & Yel-Chiang Wu A Coursein Modern Algebra Harry Hochetadt Integral Equations Erwin O.Kreyezig Introductory Functional Analysis with Applications WilliamH. Louisell Quantum Statistical Properties of Radiation All HasanNayfeh Introduction to Perturbation Techniques Emanuel ParzenModern Probability Theory and Its Applications P.M. Prenter Splinesand Variational Methods Walter Rudin Fourier Analysis on Groups C.L. Siegel Topics in Complex Function Theory, Volume I--EllipticFunctions and Uniformization Theory C. L. Siegel Topics in ComplexFunction Theory, Volume II--Automorphic and Abelian integrals C. LSiegel Topics in Complex Function Theory, Volume III--AbelianFunctions & Modular Functions of Several Variables J. J. StokerDifferential Geometry J. J. Stoker Water Waves: The MathematicalTheory with Applications J. J. Stoker Nonlinear Vibrations inMechanical and Electrical Systems

Introduction to Differential Geometry for Engineers

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Author: Brian F. Doolin,Clyde F. Martin

Publisher: Courier Corporation

ISBN: 0486488160

Category: Mathematics

Page: 163

View: 1831

This outstanding guide supplies important mathematical tools for diverse engineering applications, offering engineers the basic concepts and terminology of modern global differential geometry. Suitable for independent study as well as a supplementary text for advanced undergraduate and graduate courses, this volume also constitutes a valuable reference for control, systems, aeronautical, electrical, and mechanical engineers. The treatment's ideas are applied mainly as an introduction to the Lie theory of differential equations and to examine the role of Grassmannians in control systems analysis. Additional topics include the fundamental notions of manifolds, tangent spaces, vector fields, exterior algebra, and Lie algebras. An appendix reviews concepts related to vector calculus, including open and closed sets, compactness, continuity, and derivative.

Lectures on the Geometry of Poisson Manifolds

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Author: Izu Vaisman

Publisher: Springer Science & Business Media

ISBN: 9783764350161

Category: Mathematics

Page: 206

View: 3521

Everybody having even the slightest interest in analytical mechanics remembers having met there the Poisson bracket of two functions of 2n variables (pi, qi) f g ~(8f8g 8 8 ) (0.1) {f,g} = L... ~[ji - [ji~ , ;=1 p, q q p, and the fundamental role it plays in that field. In modern works, this bracket is derived from a symplectic structure, and it appears as one of the main in­ gredients of symplectic manifolds. In fact, it can even be taken as the defining clement of the structure (e.g., [TIl]). But, the study of some mechanical sys­ tems, particularly systems with symmetry groups or constraints, may lead to more general Poisson brackets. Therefore, it was natural to define a mathematical structure where the notion of a Poisson bracket would be the primary notion of the theory, and, from this viewpoint, such a theory has been developed since the early 19708, by A. Lichnerowicz, A. Weinstein, and many other authors (see the references at the end of the book). But, it has been remarked by Weinstein [We3] that, in fact, the theory can be traced back to S. Lie himself [Lie].

Stochastic Models, Information Theory, and Lie Groups, Volume 1

Classical Results and Geometric Methods

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Author: Gregory S. Chirikjian

Publisher: Springer Science & Business Media

ISBN: 0817648038

Category: Mathematics

Page: 383

View: 7363

This unique two-volume set presents the subjects of stochastic processes, information theory, and Lie groups in a unified setting, thereby building bridges between fields that are rarely studied by the same people. Unlike the many excellent formal treatments available for each of these subjects individually, the emphasis in both of these volumes is on the use of stochastic, geometric, and group-theoretic concepts in the modeling of physical phenomena. Stochastic Models, Information Theory, and Lie Groups will be of interest to advanced undergraduate and graduate students, researchers, and practitioners working in applied mathematics, the physical sciences, and engineering. Extensive exercises and motivating examples make the work suitable as a textbook for use in courses that emphasize applied stochastic processes or differential geometry.

Geometric Methods and Applications

For Computer Science and Engineering

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Author: Jean Gallier

Publisher: Springer Science & Business Media

ISBN: 9781441999610

Category: Mathematics

Page: 680

View: 3769

This book is an introduction to the fundamental concepts and tools needed for solving problems of a geometric nature using a computer. It attempts to fill the gap between standard geometry books, which are primarily theoretical, and applied books on computer graphics, computer vision, robotics, or machine learning. This book covers the following topics: affine geometry, projective geometry, Euclidean geometry, convex sets, SVD and principal component analysis, manifolds and Lie groups, quadratic optimization, basics of differential geometry, and a glimpse of computational geometry (Voronoi diagrams and Delaunay triangulations). Some practical applications of the concepts presented in this book include computer vision, more specifically contour grouping, motion interpolation, and robot kinematics. In this extensively updated second edition, more material on convex sets, Farkas’s lemma, quadratic optimization and the Schur complement have been added. The chapter on SVD has been greatly expanded and now includes a presentation of PCA. The book is well illustrated and has chapter summaries and a large number of exercises throughout. It will be of interest to a wide audience including computer scientists, mathematicians, and engineers. Reviews of first edition: "Gallier's book will be a useful source for anyone interested in applications of geometrical methods to solve problems that arise in various branches of engineering. It may help to develop the sophisticated concepts from the more advanced parts of geometry into useful tools for applications." (Mathematical Reviews, 2001) "...it will be useful as a reference book for postgraduates wishing to find the connection between their current problem and the underlying geometry." (The Australian Mathematical Society, 2001)