Power Geometry In Algebraic And Differential Equations
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Power Geometry in Algebraic and Differential Equations by A.D. Bruno Pdf
The geometry of power exponents includes the Newton polyhedron, normal cones of its faces, power and logarithmic transformations. On the basis of the geometry universal algorithms for simplifications of systems of nonlinear equations (algebraic, ordinary differential and partial differential) were developed. The algorithms form a new calculus which allows to make local and asymptotical analysis of solutions to those systems. The efficiency of the calculus is demonstrated with regard to several complicated problems from Robotics, Celestial Mechanics, Hydrodynamics and Thermodynamics. The calculus also gives classical results obtained earlier intuitively and is an alternative to Algebraic Geometry, Differential Algebra, Lie group Analysis and Nonstandard Analysis.
Analytic, Algebraic and Geometric Aspects of Differential Equations by Galina Filipuk,Yoshishige Haraoka,Sławomir Michalik Pdf
This volume consists of invited lecture notes, survey papers and original research papers from the AAGADE school and conference held in Będlewo, Poland in September 2015. The contributions provide an overview of the current level of interaction between algebra, geometry and analysis and demonstrate the manifold aspects of the theory of ordinary and partial differential equations, while also pointing out the highly fruitful interrelations between those aspects. These interactions continue to yield new developments, not only in the theory of differential equations but also in several related areas of mathematics and physics such as differential geometry, representation theory, number theory and mathematical physics. The main goal of the volume is to introduce basic concepts, techniques, detailed and illustrative examples and theorems (in a manner suitable for non-specialists), and to present recent developments in the field, together with open problems for more advanced and experienced readers. It will be of interest to graduate students, early-career researchers and specialists in analysis, geometry, algebra and related areas, as well as anyone interested in learning new methods and techniques.
Linear Differential Equations and Group Theory from Riemann to Poincare by Jeremy Gray Pdf
This book is a study of how a particular vision of the unity of mathematics, often called geometric function theory, was created in the 19th century. The central focus is on the convergence of three mathematical topics: the hypergeometric and related linear differential equations, group theory, and on-Euclidean geometry. The text for this second edition has been greatly expanded and revised, and the existing appendices enriched. The exercises have been retained, making it possible to use the book as a companion to mathematics courses at the graduate level.
Formal and Analytic Solutions of Diff. Equations by Galina Filipuk,Alberto Lastra,Sławomir Michalik Pdf
These proceedings provide methods, techniques, different mathematical tools and recent results in the study of formal and analytic solutions to Diff. (differential, partial differential, difference, q-difference, q-difference-differential.... ) Equations. They consist of selected contributions from the conference "Formal and Analytic Solutions of Diff. Equations", held at Alcalá de Henares, Spain during September 4-8, 2017. Their topics include summability and asymptotic study of both ordinary and partial differential equations. The volume is divided into four parts. The first paper is a survey of the elements of nonlinear analysis. It describes the algorithms to obtain asymptotic expansion of solutions of nonlinear algebraic, ordinary differential, partial differential equations, and of systems of such equations. Five works on formal and analytic solutions of PDEs are followed by five papers on the study of solutions of ODEs. The proceedings conclude with five works on related topics, generalizations and applications. All contributions have been peer reviewed by anonymous referees chosen among the experts on the subject. The volume will be of interest to graduate students and researchers in theoretical and applied mathematics, physics and engineering seeking an overview of the recent trends in the theory of formal and analytic solutions of functional (differential, partial differential, difference, q-difference, q-difference-differential) equations in the complex domain.
Analysis and Applications - ISAAC 2001 by Heinrich G.W. Begehr,R.P. Gilbert,Man-Wah Wong Pdf
This collection of survey articles gives and idea of new methods and results in real and complex analysis and its applications. Besides several chapters on hyperbolic equations and systems and complex analysis, potential theory, dynamical systems and harmonic analysis are also included. Newly developed subjects from power geometry, homogenization, partial differential equations in graph structures are presented and a decomposition of the Hilbert space and Hamiltonian are given. Audience: Advanced students and scientists interested in new methods and results in analysis and applications.
Lecture Notes on O-Minimal Structures and Real Analytic Geometry by Chris Miller,Jean-Philippe Rolin,Patrick Speissegger Pdf
This volume was produced in conjunction with the Thematic Program in o-Minimal Structures and Real Analytic Geometry, held from January to June of 2009 at the Fields Institute. Five of the six contributions consist of notes from graduate courses associated with the program: Felipe Cano on a new proof of resolution of singularities for planar analytic vector fields; Chris Miller on o-minimality and Hardy fields; Jean-Philippe Rolin on the construction of o-minimal structures from quasianalytic classes; Fernando Sanz on non-oscillatory trajectories of vector fields; and Patrick Speissegger on pfaffian sets. The sixth contribution, by Antongiulio Fornasiero and Tamara Servi, is an adaptation to the nonstandard setting of A.J. Wilkie's construction of o-minimal structures from infinitely differentiable functions. Most of this material is either unavailable elsewhere or spread across many different sources such as research papers, conference proceedings and PhD theses. This book will be a useful tool for graduate students or researchers from related fields who want to learn about expansions of o-minimal structures by solutions, or images thereof, of definable systems of differential equations.
Ordinary Differential Equations by Wolfgang Walter Pdf
Based on a translation of the 6th edition of Gewöhnliche Differentialgleichungen by Wolfgang Walter, this edition includes additional treatments of important subjects not found in the German text as well as material that is seldom found in textbooks, such as new proofs for basic theorems. This unique feature of the book calls for a closer look at contents and methods with an emphasis on subjects outside the mainstream. Exercises, which range from routine to demanding, are dispersed throughout the text and some include an outline of the solution. Applications from mechanics to mathematical biology are included and solutions of selected exercises are found at the end of the book. It is suitable for mathematics, physics, and computer science graduate students to be used as collateral reading and as a reference source for mathematicians. Readers should have a sound knowledge of infinitesimal calculus and be familiar with basic notions from linear algebra; functional analysis is developed in the text when needed.
Geometry of PDEs and Mechanics by Agostino Prastaro Pdf
This volume presents the theory of partial differential equations (PDEs) from a modern geometric point of view so that PDEs can be characterized by using either technique of differential geometry or algebraic geometry. This allows us to recognize the richness of the structure of PDEs. It presents, for the first time, a geometric theory of non-commutative (quantum) PDEs and gives a general application of this theory to quantum field theory and quantum supergravity.
Interactions of Classical and Numerical Algebraic Geometry by Daniel James Bates Pdf
This volume contains the proceedings of the conference on Interactions of Classical and Numerical Algebraic Geometry, held May 22-24, 2008, at the University of Notre Dame, in honor of the achievements of Professor Andrew J. Sommese. While classical algebraic geometry has been studied for hundreds of years, numerical algebraic geometry has only recently been developed. Due in large part to the work of Andrew Sommese and his collaborators, the intersection of these two fields is now ripe for rapid advancement. The primary goal of both the conference and this volume is to foster the interaction between researchers interested in classical algebraic geometry and those interested in numerical methods. The topics in this book include (but are not limited to) various new results in complex algebraic geometry, a primer on Seshadri constants, analyses and presentations of existing and novel numerical homotopy methods for solving polynomial systems, a numerical method for computing the dimensions of the cohomology of twists of ideal sheaves, and the application of algebraic methods in kinematics and phylogenetics.
Linear Algebra and Differential Equations by Alexander Givental Pdf
The material presented in this book corresponds to a semester-long course, ``Linear Algebra and Differential Equations'', taught to sophomore students at UC Berkeley. In contrast with typical undergraduate texts, the book offers a unifying point of view on the subject, namely that linear algebra solves several clearly-posed classification problems about such geometric objects as quadratic forms and linear transformations. This attractive viewpoint on the classical theory agrees well with modern tendencies in advanced mathematics and is shared by many research mathematicians. However, the idea of classification seldom finds its way to basic programs in mathematics, and is usually unfamiliar to undergraduates. To meet the challenge, the book first guides the reader through the entire agenda of linear algebra in the elementary environment of two-dimensional geometry, and prior to spelling out the general idea and employing it in higher dimensions, shows how it works in applications such as linear ODE systems or stability of equilibria. Appropriate as a text for regular junior and honors sophomore level college classes, the book is accessible to high school students familiar with basic calculus, and can also be useful to engineering graduate students.
Transseries and Real Differential Algebra by Joris van der Hoeven,Joris Hoeven Pdf
Transseries are formal objects constructed from an infinitely large variable x and the reals using infinite summation, exponentiation and logarithm. They are suitable for modeling "strongly monotonic" or "tame" asymptotic solutions to differential equations and find their origin in at least three different areas of mathematics: analysis, model theory and computer algebra. They play a crucial role in Écalle's proof of Dulac's conjecture, which is closely related to Hilbert's 16th problem. The aim of the present book is to give a detailed and self-contained exposition of the theory of transseries, in the hope of making it more accessible to non-specialists.
Applying Power Series to Differential Equations by James Sochacki,Anthony Tongen Pdf
This book is aimed to undergraduate STEM majors and to researchers using ordinary differential equations. It covers a wide range of STEM-oriented differential equation problems that can be solved using computational power series methods. Many examples are illustrated with figures and each chapter ends with discovery/research questions most of which are accessible to undergraduate students, and almost all of which may be extended to graduate level research. Methodologies implemented may also be useful for researchers to solve their differential equations analytically or numerically. The textbook can be used as supplementary for undergraduate coursework, graduate research, and for independent study.
This text presents a graduate-level introduction to differential geometry for mathematics and physics students. The exposition follows the historical development of the concepts of connection and curvature with the goal of explaining the Chern–Weil theory of characteristic classes on a principal bundle. Along the way we encounter some of the high points in the history of differential geometry, for example, Gauss' Theorema Egregium and the Gauss–Bonnet theorem. Exercises throughout the book test the reader’s understanding of the material and sometimes illustrate extensions of the theory. Initially, the prerequisites for the reader include a passing familiarity with manifolds. After the first chapter, it becomes necessary to understand and manipulate differential forms. A knowledge of de Rham cohomology is required for the last third of the text. Prerequisite material is contained in author's text An Introduction to Manifolds, and can be learned in one semester. For the benefit of the reader and to establish common notations, Appendix A recalls the basics of manifold theory. Additionally, in an attempt to make the exposition more self-contained, sections on algebraic constructions such as the tensor product and the exterior power are included. Differential geometry, as its name implies, is the study of geometry using differential calculus. It dates back to Newton and Leibniz in the seventeenth century, but it was not until the nineteenth century, with the work of Gauss on surfaces and Riemann on the curvature tensor, that differential geometry flourished and its modern foundation was laid. Over the past one hundred years, differential geometry has proven indispensable to an understanding of the physical world, in Einstein's general theory of relativity, in the theory of gravitation, in gauge theory, and now in string theory. Differential geometry is also useful in topology, several complex variables, algebraic geometry, complex manifolds, and dynamical systems, among other fields. The field has even found applications to group theory as in Gromov's work and to probability theory as in Diaconis's work. It is not too far-fetched to argue that differential geometry should be in every mathematician's arsenal.