Asymptotic Differential Algebra And Model Theory Of Transseries

Author : Matthias Aschenbrenner,Lou van den Dries,Joris van der Hoeven
Publisher : Princeton University Press
Page : 880 pages
File Size : 48,8 Mb
Release : 2017-06-06
Category : Mathematics
ISBN : 9781400885411

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Asymptotic Differential Algebra and Model Theory of Transseries by Matthias Aschenbrenner,Lou van den Dries,Joris van der Hoeven Pdf

Asymptotic differential algebra seeks to understand the solutions of differential equations and their asymptotics from an algebraic point of view. The differential field of transseries plays a central role in the subject. Besides powers of the variable, these series may contain exponential and logarithmic terms. Over the last thirty years, transseries emerged variously as super-exact asymptotic expansions of return maps of analytic vector fields, in connection with Tarski's problem on the field of reals with exponentiation, and in mathematical physics. Their formal nature also makes them suitable for machine computations in computer algebra systems. This self-contained book validates the intuition that the differential field of transseries is a universal domain for asymptotic differential algebra. It does so by establishing in the realm of transseries a complete elimination theory for systems of algebraic differential equations with asymptotic side conditions. Beginning with background chapters on valuations and differential algebra, the book goes on to develop the basic theory of valued differential fields, including a notion of differential-henselianity. Next, H-fields are singled out among ordered valued differential fields to provide an algebraic setting for the common properties of Hardy fields and the differential field of transseries. The study of their extensions culminates in an analogue of the algebraic closure of a field: the Newton-Liouville closure of an H-field. This paves the way to a quantifier elimination with interesting consequences.

Author : Matthias Aschenbrenner,Lou Van den Dries,Joris Hoeven
Publisher : Unknown
Page : 849 pages
File Size : 45,9 Mb
Release : 2017
Category : Asymptotic expansions
ISBN : 0691175438

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Asymptotic Differential Algebra and Model Theory of Transseries by Matthias Aschenbrenner,Lou Van den Dries,Joris Hoeven Pdf

Author : Joris van der Hoeven
Publisher : Springer
Page : 260 pages
File Size : 49,6 Mb
Release : 2006-10-31
Category : Mathematics
ISBN : 9783540355915

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Transseries and Real Differential Algebra by Joris van der 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.

Author : Joris van der Hoeven
Publisher : Unknown
Page : 0 pages
File Size : 50,5 Mb
Release : 2006
Category : Difference equations
ISBN : 835403559X

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Transseries and Real Differential Algebra by Joris van der 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.

Author : Joris Hoeven
Publisher : Unknown
Page : 255 pages
File Size : 50,5 Mb
Release : 2006
Category : Differential algebra
ISBN : 661070029X

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Transseries and Real Differential Algebra by 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 A0/00calle'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.

Author : Henri Cartan,Samuel Eilenberg
Publisher : Princeton University Press
Page : 410 pages
File Size : 50,8 Mb
Release : 1999-12-19
Category : Mathematics
ISBN : 0691049912

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Homological Algebra by Henri Cartan,Samuel Eilenberg Pdf

When this book was written, methods of algebraic topology had caused revolutions in the world of pure algebra. To clarify the advances that had been made, Cartan and Eilenberg tried to unify the fields and to construct the framework of a fully fledged theory. The invasion of algebra had occurred on three fronts through the construction of cohomology theories for groups, Lie algebras, and associative algebras. This book presents a single homology (and also cohomology) theory that embodies all three; a large number of results is thus established in a general framework. Subsequently, each of the three theories is singled out by a suitable specialization, and its specific properties are studied. The starting point is the notion of a module over a ring. The primary operations are the tensor product of two modules and the groups of all homomorphisms of one module into another. From these, "higher order" derived of operations are obtained, which enjoy all the properties usually attributed to homology theories. This leads in a natural way to the study of "functors" and of their "derived functors." This mathematical masterpiece will appeal to all mathematicians working in algebraic topology.

Author : Ovidiu Costin
Publisher : CRC Press
Page : 256 pages
File Size : 52,9 Mb
Release : 2008-12-04
Category : Mathematics
ISBN : 1420070320

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Asymptotics and Borel Summability by Ovidiu Costin Pdf

Incorporating substantial developments from the last thirty years into one resource, Asymptotics and Borel Summability provides a self-contained introduction to asymptotic analysis with special emphasis on topics not covered in traditional asymptotics books. The author explains basic ideas, concepts, and methods of generalized Borel summability, transseries, and exponential asymptotics. He provides complete mathematical rigor while supplementing it with heuristic material and examples, so that some proofs may be omitted by applications-oriented readers. To give a sense of how new methods are used in a systematic way, the book analyzes in detail general nonlinear ordinary differential equations (ODEs) near a generic irregular singular point. It enables readers to master basic techniques, supplying a firm foundation for further study at more advanced levels. The book also examines difference equations, partial differential equations (PDEs), and other types of problems. Chronicling the progress made in recent decades, this book shows how Borel summability can recover exact solutions from formal expansions, analyze singular behavior, and vastly improve accuracy in asymptotic approximations.

Author : Joris van der Hoeven,Joris Hoeven
Publisher : Springer Science & Business Media
Page : 265 pages
File Size : 53,9 Mb
Release : 2006-09-15
Category : Mathematics
ISBN : 9783540355908

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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.

Author : Ahmed Abbes,Michel Gros,Takeshi Tsuji
Publisher : Princeton University Press
Page : 618 pages
File Size : 54,6 Mb
Release : 2016-02-09
Category : Mathematics
ISBN : 9781400881239

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The p-adic Simpson Correspondence (AM-193) by Ahmed Abbes,Michel Gros,Takeshi Tsuji Pdf

The p-adic Simpson correspondence, recently initiated by Gerd Faltings, aims at describing all p-adic representations of the fundamental group of a proper smooth variety over a p-adic field in terms of linear algebra—namely Higgs bundles. This book undertakes a systematic development of the theory following two new approaches, one by Ahmed Abbes and Michel Gros, the other by Takeshi Tsuji. The authors mainly focus on generalized representations of the fundamental group that are p-adically close to the trivial representation. The first approach relies on a new family of period rings built from the torsor of deformations of the variety over a universal p-adic thickening defined by J. M. Fontaine. The second approach introduces a crystalline-type topos and replaces the notion of Higgs bundles with that of Higgs isocrystals. The authors show the compatibility of the two constructions and the compatibility of the correspondence with the natural cohomologies. The last part of the volume contains results of wider interest in p-adic Hodge theory. The reader will find a concise introduction to Faltings' theory of almost étale extensions and a chapter devoted to the Faltings topos. Though this topos is the general framework for Faltings' approach in p-adic Hodge theory, it remains relatively unexplored. The authors present a new approach based on a generalization of P. Deligne's covanishing topos.

Author : Daniel T. Wise
Publisher : Princeton University Press
Page : 374 pages
File Size : 42,7 Mb
Release : 2021-05-04
Category : Mathematics
ISBN : 9780691170459

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The Structure of Groups with a Quasiconvex Hierarchy by Daniel T. Wise Pdf

"This monograph weaves together fundamentals of Mikhail Leonidovich Gromov's hyperbolic groups with the theory of cube complexes dual to spaces with walls. Many fundamental new ideas and methodologies are presented here for the first time: A cubical small-cancellation theory generalizing ideas from the 1960's, a version of "Dehn Filling" that works in the category of special cube complexes, and a variety of new results about right-angled Artin groups. The book culminates by providing an unexpected new theorem about the nature of hyperbolic groups that are constructible as amalgams. Among the stunning applications, are the virtual fibering of cusped hyperbolic 3-manifolds and the resolution of R.J. Baumslag's conjecture on the residual finiteness of one-relator groups with torsion. Most importantly, the book outlines the author's program towards the resolution of the most important remaining conjectures of William Thurston, and achieves substantial progress in this direction. This monograph, which is richly illustrated with over 100 drawings, will be of interest to graduate students and scholars working in geometry, algebra, and topology. This groundbreaking monograph, intended for the Annals of Math series, lays the mathematical groundwork for the solution of the Thurston-Haken Conjecture, a significant result in geometric group theory. It outlines one of the deepest and most surprising pieces of this result, which also has a variety of other implications for geometric group theory. This work also has applications to low-dimensional topology, and the results in this book have since been used by other mathematicians to provide other important results"--

Author : Fabrizio Broglia
Publisher : American Mathematical Soc.
Page : 366 pages
File Size : 41,8 Mb
Release : 2017
Category : Forms, Quadratic
ISBN : 9781470429669

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Ordered Algebraic Structures and Related Topics by Fabrizio Broglia Pdf

This volume contains the proceedings of the international conference ""Ordered Algebraic Structures and Related Topics'', held from October 12-16, 2015, at CIRM, Luminy, Marseilles, France. Papers contained in this volume cover topics in real analytic geometry, real algebra, and real algebraic geometry including complexity issues, model theory of various algebraic and differential structures, Witt equivalence of fields, and the moment problem.

Author : Charles F. Miller
Publisher : Princeton University Press
Page : 124 pages
File Size : 55,6 Mb
Release : 1971-11-21
Category : Mathematics
ISBN : 0691080917

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On Group-theoretic Decision Problems and Their Classification by Charles F. Miller Pdf

Part exposition and part presentation of new results, this monograph deals with that area of mathematics which has both combinatorial group theory and mathematical logic in common. Its main topics are the word problem for groups, the conjugacy problem for groups, and the isomorphism problem for groups. The presentation depends on previous results of J. L. Britton, which, with other factual background, are treated in detail.

Author : Joan C. Artés,Jaume Llibre,Dana Schlomiuk,Nicolae Vulpe
Publisher : Springer Nature
Page : 699 pages
File Size : 54,9 Mb
Release : 2021-07-19
Category : Mathematics
ISBN : 9783030505707

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Geometric Configurations of Singularities of Planar Polynomial Differential Systems by Joan C. Artés,Jaume Llibre,Dana Schlomiuk,Nicolae Vulpe Pdf

This book addresses the global study of finite and infinite singularities of planar polynomial differential systems, with special emphasis on quadratic systems. While results covering the degenerate cases of singularities of quadratic systems have been published elsewhere, the proofs for the remaining harder cases were lengthier. This book covers all cases, with half of the content focusing on the last non-degenerate ones. The book contains the complete bifurcation diagram, in the 12-parameter space, of global geometrical configurations of singularities of quadratic systems. The authors’ results provide - for the first time - global information on all singularities of quadratic systems in invariant form and their bifurcations. In addition, a link to a very helpful software package is included. With the help of this software, the study of the algebraic bifurcations becomes much more efficient and less time-consuming. Given its scope, the book will appeal to specialists on polynomial differential systems, pure and applied mathematicians who need to study bifurcation diagrams of families of such systems, Ph.D. students, and postdoctoral fellows.

Author : Ichiro Satake
Publisher : Princeton University Press
Page : 340 pages
File Size : 45,5 Mb
Release : 2014-07-14
Category : Mathematics
ISBN : 9781400856800

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Algebraic Structures of Symmetric Domains by Ichiro Satake Pdf

This book is a comprehensive treatment of the general (algebraic) theory of symmetric domains. Originally published in 1981. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Author : Predrag Cvitanović
Publisher : Princeton University Press
Page : 278 pages
File Size : 52,5 Mb
Release : 2020-05-26
Category : Mathematics
ISBN : 9780691202983

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Group Theory by Predrag Cvitanović Pdf

If classical Lie groups preserve bilinear vector norms, what Lie groups preserve trilinear, quadrilinear, and higher order invariants? Answering this question from a fresh and original perspective, Predrag Cvitanovic takes the reader on the amazing, four-thousand-diagram journey through the theory of Lie groups. This book is the first to systematically develop, explain, and apply diagrammatic projection operators to construct all semi-simple Lie algebras, both classical and exceptional. The invariant tensors are presented in a somewhat unconventional, but in recent years widely used, "birdtracks" notation inspired by the Feynman diagrams of quantum field theory. Notably, invariant tensor diagrams replace algebraic reasoning in carrying out all group-theoretic computations. The diagrammatic approach is particularly effective in evaluating complicated coefficients and group weights, and revealing symmetries hidden by conventional algebraic or index notations. The book covers most topics needed in applications from this new perspective: permutations, Young projection operators, spinorial representations, Casimir operators, and Dynkin indices. Beyond this well-traveled territory, more exotic vistas open up, such as "negative dimensional" relations between various groups and their representations. The most intriguing result of classifying primitive invariants is the emergence of all exceptional Lie groups in a single family, and the attendant pattern of exceptional and classical Lie groups, the so-called Magic Triangle. Written in a lively and personable style, the book is aimed at researchers and graduate students in theoretical physics and mathematics.