Geometry Books

Book SynopsisA Course in Topological Combinatorics is the first undergraduate textbook on the field of topological combinatorics, a subject that has become an active and innovative research area in mathematics over the last thirty years with growing applications in math, computer science, and other applied areas.Trade Review“This book is an excellent introduction into the subject. … The book contains a lot of figures and each chapter ends with a group of exercises which help the reader in understanding the hard constructions and proofs. The book may serve for a one- or two-semester undergraduate course depending on the preliminary knowledges of the students.” (János Kincses, Acta Scientiarum Mathematicarum, Vol. 81 (3-4), 2015)“The present book … presents a sequence of combinatorial themes which have shown an affinity for topological methods … . This book is filled with extremely attractive mathematics … and bringing topology into the play of combinatorics and graph theory is a wonderfully elegant manoeuvre. Here it is carried out coherently, and on a pretty grand scale, and we are thus afforded the opportunity to encounter (algebraic) topology in a very seductive uniform context. What a marvelous thing!” (Michael Berg, MAA Reviews, July, 2013)“In the book’s four main chapters, Longueville (Univ. of Applied Sciences, Germany) addresses fair-division problems; graph coloring; graph property evasiveness; and embeddings and mappings. … Basic results of algebraic topology already have powerful consequences for analysis, but the subject’s arcana can look like art for art’s sake. The author’s charting of a novel application domain for a core subject makes this book an essential acquisition. Summing Up: Essential. Upper-division undergraduates and above.” (D. V. Feldman, Choice, Vol. 50 (8), April, 2013)“Topological combinatorics is concerned with the applications of the many powerful techniques of algebraic topology to problems in combinatorics. … The present book aims to give a clear and vivid presentation of some of the most beautiful and accessible results from the area. The text, based upon some courses by the author at Freie Universität Berlin, is designed for an advanced undergraduate student.” (Hirokazu Nishimura, zbMATH, Vol. 1273, 2013)Table of ContentsPreface.- List of Symbols and Typical Notation.- 1 Fair-Division Problems.- 2 Graph-Coloring Problems.- 3 Evasiveness of Graph Properties.- 4 Embedding and Mapping Problems.- A Basic Concepts from Graph Theory.- B Crash Course in Topology.- C Partially Ordered Sets, Order Complexes, and Their Topology.- D Groups and Group Actions.- E Some Results and Applications from Smith Theory.- References.- Index.

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Book Synopsis1 Euclidean Constructions.- 2 The Ruler and Compass.- 3 The Compass and the Mohr-Mascheroni Theorem.- 4 The Ruler.- 5 The Ruler and Dividers.- 6 The Poncelet-Steiner Theorem and Double Rulers.- 7 The Ruler and Rusty Compass.- 8 Sticks.- 9 The Marked Ruler.- 10 Paperfolding.- The Back of the Book.- Suggested Reading and References.Table of Contents1 Euclidean Constructions.- 2 The Ruler and Compass.- 3 The Compass and the Mohr-Mascheroni Theorem.- 4 The Ruler.- 5 The Ruler and Dividers.- 6 The Poncelet-Steiner Theorem and Double Rulers.- 7 The Ruler and Rusty Compass.- 8 Sticks.- 9 The Marked Ruler.- 10 Paperfolding.- The Back of the Book.- Suggested Reading and References.

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Book SynopsisThis book has been written in a frankly partisian spirit-we believe that singularity theory offers an extremely useful approach to bifurcation prob­ lems and we hope to convert the reader to this view.

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Trade Review“This book is an outstanding book on counting integer points of polytopes … . The book contains lots of exercises with very helpful hints. Another essential feature of the book is a vast collection of open problems on different aspects of integer point counting and related areas. … The book is reader-friendly written, self-contained and contains numerous beautiful illustrations. The reader is always accompanied with deep research jokes by famous researchers and valuable historical notes.” (Oleg Karpenkov, zbMATH 1339.52002, 2016)Reviews of the first edition:“You owe it to yourself to pick up a copy of Computing the Continuous Discretely to read about a number of interesting problems in geometry, number theory, and combinatorics.”— MAA Reviews“The book is written as an accessible and engaging textbook, with many examples, historical notes, pithy quotes, commentary integrating the material, exercises, open problems and an extensive bibliography.”— Zentralblatt MATH“This beautiful book presents, at a level suitable for advanced undergraduates, a fairly complete introduction to the problem of counting lattice points inside a convex polyhedron.”— Mathematical Reviews“Many departments recognize the need for capstone courses in which graduating students can see the tools they have acquired come together in some satisfying way. Beck and Robins have written the perfect text for such a course.”— CHOICETable of ContentsPreface.- The Coin-Exchange Problem of Frobenius.- A Gallery of Discrete Volumes.- Counting Lattice Points in Polytopes: The Ehrhart Theory.- Reciprocity.- Face Numbers and the Dehn-Sommerville Relations in Ehrhartian Terms.- Magic Squares.- Finite Fourier Analysis.- Dedekind Sums.- The Decomposition of a Polytope into Its Cones.- Euler-MacLaurin Summation in Rd.- Solid Angles.- A Discrete Version of Green's Theorem Using Elliptic Functions.- Appendix A: Triangulations of Polytopes.- Appendix B: Hints for Selected Exercises.- References.- Index.- List of Symbols.-

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Book SynopsisThe Coin-Exchange Problem of Frobenius.- A Gallery of Discrete Volumes.- Counting Lattice Points in Polytopes: The Ehrhart Theory.- Reciprocity.- Face Numbers and the DehnSommerville Relations in Ehrhartian Terms.- Magic Squares.- Finite Fourier Analysis.- Dedekind Sums.- Zonotopes.- h-Polynomials and h*-Polynomials.- The Decomposition of a Polytope Into Its Cones.- EulerMaclaurin Summation in Rd.- Solid Angles.- A Discrete Version of Green's Theorem Using Elliptic Functions.Trade Review“This book is an outstanding book on counting integer points of polytopes … . The book contains lots of exercises with very helpful hints. Another essential feature of the book is a vast collection of open problems on different aspects of integer point counting and related areas. … The book is reader-friendly written, self-contained and contains numerous beautiful illustrations. The reader is always accompanied with deep research jokes by famous researchers and valuable historical notes.” (Oleg Karpenkov, zbMATH 1339.52002, 2016)Reviews of the first edition:“You owe it to yourself to pick up a copy of Computing the Continuous Discretely to read about a number of interesting problems in geometry, number theory, and combinatorics.”— MAA Reviews“The book is written as an accessible and engaging textbook, with many examples, historical notes, pithy quotes, commentary integrating the material, exercises, open problems and an extensive bibliography.”— Zentralblatt MATH“This beautiful book presents, at a level suitable for advanced undergraduates, a fairly complete introduction to the problem of counting lattice points inside a convex polyhedron.”— Mathematical Reviews“Many departments recognize the need for capstone courses in which graduating students can see the tools they have acquired come together in some satisfying way. Beck and Robins have written the perfect text for such a course.”— CHOICETable of ContentsPreface.- The Coin-Exchange Problem of Frobenius.- A Gallery of Discrete Volumes.- Counting Lattice Points in Polytopes: The Ehrhart Theory.- Reciprocity.- Face Numbers and the Dehn-Sommerville Relations in Ehrhartian Terms.- Magic Squares.- Finite Fourier Analysis.- Dedekind Sums.- The Decomposition of a Polytope into Its Cones.- Euler-MacLaurin Summation in Rd.- Solid Angles.- A Discrete Version of Green's Theorem Using Elliptic Functions.- Appendix A: Triangulations of Polytopes.- Appendix B: Hints for Selected Exercises.- References.- Index.- List of Symbols.-

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Book SynopsisThoroughly updated, featuring new material on important topics such as hyperbolic geometry in higher dimensions and generalizations of hyperbolicity Includes full solutions for all exercises Successful first edition sold over 800 copies in North America Table of ContentsThe Basic Spaces.- The General Möbius Group.- Length and Distance in ?.- Planar Models of the Hyperbolic Plane.- Convexity, Area, and Trigonometry.- Nonplanar models.

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Book SynopsisThis book teaches algebra and geometry. The authors dedicate chapters to the key issues of matrices, linear equations, matrix algorithms, vector spaces, lines, planes, second-order curves, and elliptic curves. The text is supported throughout with problems, and the authors have included source code in Python in the book. The book is suitable for advanced undergraduate and graduate students in computer science. Trade Review“It is most interesting to combine a classical mathematical topic with a new evolving programming language and exactly this is obtained by this book. … This material is used as a case study for their implementation for solving problems in theoretical and practical cryptography. The ‘roadmap’ of the content of this also quite interesting.” (Panayiotis Vlamos, zbMATH 1480.00002, 2022)Table of ContentsMatrices and Matrix Algorithms.- Matrix Algebra.- Systems of Linear Equations.- Complex Numbers and Matrices.- Vector Spaces.- Vectors in a Three-Dimensional Space.- Equation of a Straight Line on a Plane.- Equation of a Plane in Space.- Equation of a Line in Space.- Bilinear and Quadratic Forms.- Curves of the Second-Order.- Elliptic Curves.- Appendix A, Basic Operators in Python and C.- Appendix B, Trigonometric Formulae.- Appendix C, The Greek Alphabet.- References.- Name Index.- Subject Index.

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Book SynopsisThis book provides a detailed exposition of a wide range of topics in geometric group theory, inspired by Gromov’s pivotal work in the 1980s. It includes classical theorems on nilpotent groups and solvable groups, a fundamental study of the growth of groups, a detailed look at asymptotic cones, and a discussion of related subjects including filters and ultrafilters, dimension theory, hyperbolic geometry, amenability, the Burnside problem, and random walks on groups. The results are unified under the common theme of Gromov’s theorem, namely that finitely generated groups of polynomial growth are virtually nilpotent. This beautiful result gave birth to a fascinating new area of research which is still active today.The purpose of the book is to collect these naturally related results together in one place, most of which are scattered throughout the literature, some of them appearing here in book form for the first time. In this way, the connections between these topics are revealed, providing a pleasant introduction to geometric group theory based on ideas surrounding Gromov's theorem. The book will be of interest to mature undergraduate and graduate students in mathematics who are familiar with basic group theory and topology, and who wish to learn more about geometric, analytic, and probabilistic aspects of infinite groups.Table of Contents- Foreword.- Preface.- Part I Algebraic Theory: 1. Free Groups.- 2. Nilpotent Groups.- 3. Residual Finiteness and the Zassenhaus Filtration.- 4. Solvable Groups.- 5. Polycyclic Groups.- 6. The Burnside Problem.- Part II Geometric Theory: 7. Finitely Generated Groups and Their Growth Functions.- 8. Hyperbolic Plane Geometry and the Tits Alternative.- 9. Topological Groups, Lie Groups, and Hilbert Fifth Problem.- 10. Dimension Theory.- 11. Ultrafilters, Ultraproducts, Ultrapowers, and Asymptotic Cones.- 12. Gromov’s Theorem.- Part III Analytic and Probabilistic Theory: 13. The Theorems of Polya and Varopoulos.- 14. Amenability, Isoperimetric Profile, and Følner Functions.- 15. Solutions or Hints to Selected Exercises.- References.- Subject Index.- Index of Authors.

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Book SynopsisThis textbook teaches the transformations of plane Euclidean geometry through problems, offering a transformation-based perspective on problems that have appeared in recent years at mathematics competitions around the globe, as well as on some classical examples and theorems. It is based on the combined teaching experience of the authors (coaches of several Mathematical Olympiad teams in Brazil, Romania and the USA) and presents comprehensive theoretical discussions of isometries, homotheties and spiral similarities, and inversions, all illustrated by examples and followed by myriad problems left for the reader to solve. These problems were carefully selected and arranged to introduce students to the topics by gradually moving from basic to expert level. Most of them have appeared in competitions such as Mathematical Olympiads or in mathematical journals aimed at an audience interested in mathematics competitions, while some are fundamental facts of mathematics discussed in the framework of geometric transformations. The book offers a global view of the geometric content of today's mathematics competitions, bringing many new methods and ideas to the attention of the public.Talented high school and middle school students seeking to improve their problem-solving skills can benefit from this book, as well as high school and college instructors who want to add nonstandard questions to their courses. People who enjoy solving elementary math problems as a hobby will also enjoy this work.Trade Review“This book … is a nice addition to the literature. … for instructors teaching geometry courses in which these are a topic, this book should provide an excellent source of interesting examples and problems. The large number of solved problems should also make useful reading for people preparing for mathematical contests and Olympiads.” (Mark Hunacek, MAA Reviews, October 4, 2022)Table of ContentsIntroduction.- Part I: Problems - 1. Isometries.- 2. Homotheties and Spiral Similarities.- 3. Inversions.- 4. A Synthesis.- Part II: Hints - 5. Isometries.- 6. Homotheties and Spiral Similarities.- 7. Inversions.- 8. A Synthesis.- Part III: Solutions - 9. Isometries.- 10. Homotheties and Spiral Similarities.- 11. Inversions.- 12. A Synthesis.

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Book Synopsis- 1. Introduction.- 2. Interview with Athanase Papadopoulos.- 3. A glance at S. Novikov's theory of multivalued Morse functions.- 4. A twisted invariant of a compact Riemann surface.- 5. Directional moduli and pseudoconvexity.- 6. Angle Defect for Super Triangles.- 7. Lipschitz and quasiconformal mappings in cartography.- 8. Spherical representations of the group of isometries of semi-homogeneous trees.- 9. Trees of fractions.- 10. Binary quadratic forms: modern developments.- 11. A Note on Reversibility of Unipotent Matrices.- 12. Le complément supérieur: On the poetics of mathematics.- 13. Pythagorean Book II of the Elements restored and Pythagorean Incommensurabilities reconstructed.

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Book SynopsisChapter 1. Introduction.- Chapter 2. Background on classical real hedgehogs.- Chapter 3. Volumes and mixed volumes.- Chapter 4. Special convex bodies, hedgehogs or multihedgehog.- Chapter 5. The Minkowski problem for hedgehogs.- Chapter 6. Complex hedgehogs in Cn+1 or Pn+1 (C).- Chapter 7. Hedgehogs in non-Euclidean spaces.- Chapter 8. Marginally trapped hedgehogs.- Chapter 9. Focal of hedgehogs in Rn+1 and concurrent normals conjecture.- Chapter 10. Miscellaneous questions regarding hedgehogs.-Chapter 11. List of selected problems.

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Book SynopsisIn the two-volume set ‘A Selection of Highlights’ we present basics of mathematics in an exciting and pedagogically sound way. This volume examines many fundamental results in Geometry and Discrete Mathematics along with their proofs and their history. In the second edition we include a new chapter on Topological Data Analysis and enhanced the chapter on Graph Theory for solving further classical problems such as the Traveling Salesman Problem.

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Book SynopsisThis work covers quantum mechanics by answering questions such as where did the Planck constant and Heisenberg algebra come from, what motivated Feynman to introduce his path integral and why does one distinguish two types of particles, the bosons and fermions. The author addresses all these topics with utter mathematical rigor. The high number of instructive Appendices and numerous Remark sections supply the necessary background knowledge.

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Book SynopsisThis textbook treats two important and related matters in convex geometry: the quantification of symmetry of a convex set—measures of symmetry—and the degree to which convex sets that nearly minimize such measures of symmetry are themselves nearly symmetric—the phenomenon of stability. By gathering the subject’s core ideas and highlights around Grünbaum’s general notion of measure of symmetry, it paints a coherent picture of the subject, and guides the reader from the basics to the state-of-the-art. The exposition takes various paths to results in order to develop the reader’s grasp of the unity of ideas, while interspersed remarks enrich the material with a behind-the-scenes view of corollaries and logical connections, alternative proofs, and allied results from the literature. Numerous illustrations elucidate definitions and key constructions, and over 70 exercises—with hints and references for the more difficult ones—test and sharpen the reader’s comprehension.The presentation includes: a basic course covering foundational notions in convex geometry, the three pillars of the combinatorial theory (the theorems of Carathéodory, Radon, and Helly), critical sets and Minkowski measure, the Minkowski–Radon inequality, and, to illustrate the general theory, a study of convex bodies of constant width; two proofs of F. John’s ellipsoid theorem; a treatment of the stability of Minkowski measure, the Banach–Mazur metric, and Groemer’s stability estimate for the Brunn–Minkowski inequality; important specializations of Grünbaum’s abstract measure of symmetry, such as Winternitz measure, the Rogers–Shepard volume ratio, and Guo’s Lp -Minkowski measure; a construction by the author of a new sequence of measures of symmetry, the kth mean Minkowski measure; and lastly, an intriguing application to the moduli space of certain distinguished maps from a Riemannian homogeneous space to spheres—illustrating the broad mathematical relevance of the book’s subject.Trade Review“The book under review is a graduate-level textbook on convexity, which presents the topic from a new and interesting point of view. … The book offers the reader a new approach to the study of convexity, focusing on the important topics of measures of symmetry and stability. It moves from the very beginning background to recent research, and therefore both students and researchers can benefit from it.” (María A. Hernández Cifre, Mathematical Reviews, December, 2016) “This is a graduate-level textbook on convex geometry in finite-dimensional Euclidean spaces, which has some interesting special features. … Each chapter has illustrating figures and concludes with exercises … . The book has a surprising appendix, where certain of the symmetry measures are applied to convex bodies … . This book is an unconventional introduction to convexity, full of appealing intuitive geometry; it may equally well serve the beginner and the experienced researcher in the field.” (Rolf Schneider, zbMATH 1335.52002, 2016)Table of ContentsFirst Things First on Convex Sets.- Affine Diameters and the Critical Set.- Measures of Stability and Symmetry.- Mean Minkowski Measures.

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Book SynopsisThis unique collection of new and classical problems provides full coverage of geometric inequalities. Many of the 1,000 exercises are presented with detailed author-prepared-solutions, developing creativity and an arsenal of new approaches for solving mathematical problems. This book can serve teachers, high-school students, and mathematical competitors. It may also be used as supplemental reading, providing readers with new and classical methods for proving geometric inequalities. Trade Review“‘The goal of the book is to teach the reader new and classical methods for proving geometric inequalities.’ ... The book contains more than 1000 problems. ... intended for mathematics competitions and Olympiads. Every chapter contains problems for self-study and solutions.” (Sándor Nagydobai Kiss, zbMATH 1375.51001, 2018)Table of ContentsTheorem on the Length of the Broken Line.- Application of Projection Method.- Areas.- Application of Trigonometric Inequalities.- Inequalities for Radiuses.- Miscellaneous Inequalities.- Some Applications of Geometric Inequalities.

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Book SynopsisThis book presents in a systematic and almost self-contained way the striking analogy between classical function theory, in particular the value distribution theory of holomorphic curves in projective space, on the one hand, and important and beautiful properties of the Gauss map of minimal surfaces on the other hand. Both theories are developed in the text, including many results of recent research. The relations and analogies between them become completely clear. The book is written for interested graduate students and mathematicians, who want to become more familiar with this modern development in the two classical areas of mathematics, but also for those, who intend to do further research on minimal surfaces.Table of ContentsContents: The Gauss map of minimal surfaces in R3 - The derived curves of a holomorphic curve - The classical defect relations for holomorphic curves - Modified defect relation for holomorphic curves - The Gauss Map of complete minimal surfaces in Rm.

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Table of ContentsThe etale topology of schemes.- Algebraic spaces.- Quasicoherent sheaves on noetherian locally separated algebraic spaces.- The Finiteness Theorem.- Formal algebraic spaces.

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Book Synopsis Diese Einführung besticht durch zwei ungewöhnliche Aspekte: Sie gibt einen Einblick in die Mathematik als Bestandteil unserer Kultur, und sie vermittelt die Hintergründe der Mathematik vom Schulstoff ausgehend bis zum Niveau von Mathematikvorlesungen im ersten Studienjahr. Die Stoffdarstellung geht vom Aufbau der natürlichen Zahlen aus; der Schwerpunkt liegt aber in den exakten Begründungen der Zahlenbegriffe, der Geometrie der Ebene und der Funktionen einer Veränderlichen. Dabei werden alle Sätze bis hin zum Hauptsatz der Algebra vollständig bewiesen. Der klare Aufbau des Buches mit Stichwortregister wichtiger Begriffe erleichtert das systematische Lernen und Nachschlagen. Die zweite Auflage enthält teilweise ausführliche Darstellungen für die Lösungen der zahlreichen Übungsaufgaben.Da viele Aspekte zur Sprache kommen, die so weder im Unterricht noch im Studium behandelt werden, ergänzt die Einführung ideal den Vorlesungsstoff für Lehramtskandidaten und Diplomstudenten.Trade Review"...dies ist eine Art "Brückenkurs"', der Aspekte der Schulmathematik von höherer Warte aus diskutiert... Der Autor steckt sich im Vorwort selbst das ehrgeizige Ziel, einen ‚Einblick in die Mathematik als einen Bestandteil unserer Kultur‘ zu geben, indem er sich ‚am Schulstoff (zwar) orientiert, aber über diesen hinausgeht und ihn hinterfragt.‘ Die Erreichbarkeit dieses Zieles stellt er mit diesem schönen Buch sehr überzeugend unter Beweis. Dabei wird beileibe nicht der Schulstoff ‚formalisiert‘, und noch weniger der Universitätsstoff ‚trivialisiert‘, sondern es kommen Aspekte zur Sprache, die im Mathematikunterricht wegen ihrer Schwierigkeit und im Mathematikstudium aus Zeitgründen kaum zur Sprache kommen. Dies ist ebenso verdienstvoll wie ungewöhnlich; als Ergebnis ist ein Buch herausgekommen, welches im ausufernden Markt tatsächlich eine Lücke füllt. Man kann grob drei Stoffgebiete unterscheiden, die behandelt werden, nämlich Zahlen (Kapitel 1-4 und 9), Geometrie (Kapitel 5 und 10) und Reelle Analysis (Kapitel 6-8). Wie ernst der Autor seine Aufgabe genommen hat, zeigt die sehr lesenswerte Einleitung, die auch den formalen Aufbau und inhaltliche Einzelheiten erklärt. Man kann allen Erstsemesterstudenten der Mathematik und Physik wärmstens empfehlen, dieses Buch als Ergänzung zu der von ihrem Dozenten empfohlenen Literatur zu kaufen und regelmäßig zu konsultieren." Jürgen Appell, Würzburg, in Zentralblatt MATH Table of ContentsNatürliche Zahlen.- Die 0 und die ganzen Zahlen.- Rationale Zahlen.- Reelle Zahlen.- Euklidische Geometrie der Ebene.- Reelle Funktionen einer Veränderlichen.- Maß und Integral.- Trigonometrie.- Die komplexen Zahlen.- Nicht-euklidische Geometrie.- Lösungen der Aufgaben.

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Book SynopsisIn the early 70's and 80's the field of integrable systems was in its prime youth: results and ideas were mushrooming all over the world. It was during the roaring 70's and 80's that a first version of the book was born, based on our research and on lectures which each of us had given. We owe many ideas to our colleagues Teruhisa Matsusaka and David Mumford, and to our inspiring graduate students (Constantin Bechlivanidis, Luc Haine, Ahmed Lesfari, Andrew McDaniel, Luis Piovan and Pol Vanhaecke). As it stood, our first version lacked rigor and precision, was rough, dis- connected and incomplete...In the early 90's new problems appeared on the horizon and the project came to a complete standstill, ultimately con- fined to a floppy. A few years ago, under the impulse of Pol Vanhaecke, the project was revived and gained real momentum due to his insight, vision and determination. The leap from the old to the new version is gigantic. The book is designed as a teaching textbook and is aimed at a wide read- ership of mathematicians and physicists, graduate students and professionals.Trade ReviewFrom the reviews of the first edition: "The aim of this book is to explain ‘how algebraic geometry, Lie theory and Painlevé analysis can be used to explicitly solve integrable differential equations’. … One of the main advantages of this book is that the authors … succeeded to present the material in a self-contained manner with numerous examples. As a result it can be also used as a reference book for many subjects in mathematics. In summary … a very good book which covers many interesting subjects in modern mathematical physics." (Vladimir Mangazeev, The Australian Mathematical Society Gazette, Vol. 33 (4), 2006) "This is an extensive volume devoted to the integrability of nonlinear Hamiltonian differential equations. The book is designed as a teaching textbook and aims at a wide readership of mathematicians and physicists, graduate students and professionals. … The book provides many useful tools and techniques in the field of completely integrable systems. It is a valuable source for graduate students and researchers who like to enter the integrability theory or to learn fascinating aspects of integrable geometry of nonlinear differential equations." (Ma Wen-Xiu, Zentralblatt MATH, Vol. 1083, 2006)Table of Contents1 Introduction.- 2 Lie Algebras.- 3 Poisson Manifolds.- 4 Integrable Systems on Poisson Manifolds.- 5 The Geometry of Abelian Varieties.- 6 A.c.i. Systems.- 7 Weight Homogeneous A.c.i. Systems.- 8 Integrable Geodesic Flow on SO(4).- 9 Periodic Toda Lattices Associated to Cartan Matrices.- 10 Integrable Spinning Tops.- References.

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Book SynopsisThere has recently been a renewal of interest in Fokker-Planck operators, motivated by problems in statistical physics, in kinetic equations, and differential geometry. Compared to more standard problems in the spectral theory of partial differential operators, those operators are not self-adjoint and only hypoelliptic. The aim of the analysis is to give, as generally as possible, an accurate qualitative and quantitative description of the exponential return to the thermodynamical equilibrium. While exploring and improving recent results in this direction, this volume proposes a review of known techniques on: the hypoellipticity of polynomial of vector fields and its global counterpart, the global Weyl-Hörmander pseudo-differential calculus, the spectral theory of non-self-adjoint operators, the semi-classical analysis of Schrödinger-type operators, the Witten complexes, and the Morse inequalities.Trade ReviewFrom the reviews of the first edition: "The aim of this text is to give an account of how the known techniques from partial differential equations and spectral theory can be applied for the analysis of Fokker-Plank operators or Witten Laplacians … . This synthetic text is very challenging and useful for researchers in partial differential equations, probability theory and mathematical physics." (Viorel Iftimie, Zentralblatt MATH, Vol. 1072, 2005)Table of Contents1. Introduction.- 2. Kohn's Proof of the Hypoellipticity of the Hörmander Operators.- 3. Compactness Criteria for the Resolvent of Schrödinger Operators.- 4. Global Pseudo-differential Calculus.- 5. Analysis of some Fokker-Planck Operator.- 6. Return to Equillibrium for the Fokker-Planck Operator.- 7. Hypoellipticity and nilpotent groups.- 8. Maximal Hypoellipticity for Polynomial of Vector Fields and Spectral Byproducts.- 9. On Fokker-Planck Operators and Nilpotent Techniques.- 10. Maximal Microhypoellipticity for Systems and Applications to Witten Laplacians.- 11. Spectral Properties of the Witten-Laplacians in Connection with Poincaré inequalities for Laplace Integrals.- 12. Semi-classical Analysis for the Schrödinger Operator: Harmonic Approximation.- 13. Decay of Eigenfunctions and Application to the Splitting.- 14. Semi-classical Analysis and Witten Laplacians: Morse Inequalities.- 15. Semi-classical Analysis and Witten Laplacians: Tunneling Effects.- 16. Accurate Asymptotics for the Exponentially Small Eigenvalues of the Witten Laplacian.- 17. Application to the Fokker-Planck Equation.- 18. Epilogue.- References.- Index.

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Book SynopsisComputational synthetic geometry deals with methods for realizing abstract geometric objects in concrete vector spaces. This research monograph considers a large class of problems from convexity and discrete geometry including constructing convex polytopes from simplicial complexes, vector geometries from incidence structures and hyperplane arrangements from oriented matroids. It turns out that algorithms for these constructions exist if and only if arbitrary polynomial equations are decidable with respect to the underlying field. Besides such complexity theorems a variety of symbolic algorithms are discussed, and the methods are applied to obtain new mathematical results on convex polytopes, projective configurations and the combinatorics of Grassmann varieties. Finally algebraic varieties characterizing matroids and oriented matroids are introduced providing a new basis for applying computer algebra methods in this field. The necessary background knowledge is reviewed briefly. The text is accessible to students with graduate level background in mathematics, and will serve professional geometers and computer scientists as an introduction and motivation for further research.Table of ContentsPreliminaries.- On the existence of algorithms.- Combinatorial and algebraic methods.- Algebraic criteria for geometric realizability.- Geometric methods.- Recent topological results.- Preprocessing methods.- On the finding of polyheadral manifolds.- Matroids and chirotopes as algebraic varieties.

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Book SynopsisFrom the reviews: "A well-written, very thorough account ... Among the topics are lattices, reduction, Minkowskis Theorem, distance functions, packings, and automorphs; some applications to number theory; excellent bibliographical references." The American Mathematical MonthlyTrade ReviewFrom the reviews:"The work is carefully written. It is well motivated, and interesting to read, even if it is not always easy... historical material is included... the author has written excellent account of an interesting subject." -Mathematical Gazette"A well-written, very thorough account ... Among the topi are lattices, reduction, Minkowskis Theorem, distance functions, packings, and automorphs; some applications to number theory; excellent bibliographical references." -The American Mathematical Monthly“It is very clearly written, and assumes little in the way of prerequisites. In particular, it is accessible to an undergraduate who is willing to work a bit, and I speak from experience as I first read the book the summer before I started graduate school. At the same time, it is a serious work giving an exhaustive (and not at all watered down) account of Minkowski’s theory. … This book certainly earns its place in a series on the ‘Classics in Mathematics.’” (Darren Glass, The Mathematical Association of America, January, 2011)Table of ContentsNotation Prologue Chapter I. Lattices 1. Introduction 2. Bases and sublattices 3. Lattices under linear transformation 4. Forms and lattices 5. The polar lattice Chapter II. Reduction 1. Introduction 2. The basic process 3. Definite quadratic forms 4. Indefinite quadratic forms 5. Binary cubic forms 6. Other forms Chapter III. Theorems of Blichfeldt and Minkowski 1. Introduction 2. Blichfeldt's and Mnowski's theorems 3. Generalisations to non-negative functions 4. Characterisation of lattices 5. Lattice constants 6. A method of Mordell 7. Representation of integers by quadratic forms Chapter IV. Distance functions 1. Introduction 2. General distance-functions 3. Convex sets 4. Distance functions and lattices Chapter V. Mahler's compactness theorem 1. Introduction 2. Linear transformations 3. Convergence of lattices 4. Compactness for lattices 5. Critical lattices 6. Bounded star-bodies 7. Reducibility 8. Convex bodies 9. Speres 10. Applications to diophantine approximation Chapter VI. The theorem of Minkowski-Hlawka 1. Introduction 2. Sublattices of prime index 3. The Minkowski-Hlawka theorem 4. Schmidt's theorems 5. A conjecture of Rogers 6. Unbounded star-bodies Chapter VII. The quotient space 1. Introduction 2. General properties 3. The sum theorem Chapter VIII. Successive minima 1. Introduction 2. Spheres 3. General distance-functions Chapter IX. Packings 1. Introduction 2. Sets with V(/varphi) =n^2/Delta(/varphi) 3. Voronoi's results 4. Preparatory lemmas 5. Fejes Tóth's theorem 6. Cylinders 7. Packing of spheres 8. The proudctio of n linear forms Chapter X. Automorphs 1. Introduction 2. Special forms 3. A method of Mordell 4. Existence of automorphs 5. Isolation theorems 6. Applications of isolation 7. An infinity of solutions 8. Local methods Chapter XI. Ihomogeneous problems 1. Introduction 2. Convex sets 3. Transference theorems for convex sets 4. The producti of n linear forms Appendix References Index quotient space. successive minima. Packings. Automorphs. Inhomogeneous problems.

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Book SynopsisA. Lineare Algebra I.- 1. Vektorräume.- 2. Matrizen.- 3. Determinant en.- B. Analytische Geometrie.- 4. Elementar-Geometrie in der Ebene.- 5. Euklidische Vektorräume.- 6. Der ?aun als Euklidischer Vektorraum.- 7. Geometrie im dreidimensionalen Raum.- C. Lineare Algebra II.- 8. Polynome und Matrizen.- 9. Homomorphismen von Vektorräumen.- Literatur.- Namenverzeichnis.Table of ContentsA. Lineare Algebra I.- 1. Vektorräume.- § 1. Der Begriff eines Vektorraumes.- 1. Vorbemerkung.- 2. Vektorräume.- 3. Unterräume.- 4. Geraden.- 5. Das Standardbeispiel Kn.- 6. Geometrische Deutung.- 7. Anfänge einer Geometrie im ?2.- § 2*. Über den Ursprung der Vektorräume.- 1. Die Grassmannsche Ausdehnungslehre.- 2. Grassmann: Übersicht über die allgemeine Formenlehre.- 3. Extensive Größen als Elemente eines Vektorraumes.- 4. Reaktion der Mathematiker.- 5. Der moderne Vektorraumbegriff.- § 3. Beispiele von Vektorräumen.- 1. Einleitung.- 2. Reelle Folgen.- 3. Vektorräume von Abbildungen.- 4. Stetige Funktionen.- 5. Reelle Polynome.- 6*. Reell-analytische Funktionen.- 7* Lineare Differentialgleichungen n-ter Ordnung mit konstanten Koeffizienten.- 8. Die Vektorräume Abb[M, K].- § 4. Elementare Theorie der Vektorräume.- 1. Vorbemerkung.- 2. Homogene Gleichungen.- 3. Erzeugung von Unterräumen.- 4. Lineare Abhängigkeit.- 5. Der Begriff einer Basis.- 6. Die Dimension eines Vektorraums.- 7. Der Dimensions-Satz.- 8*. Der Basis-Satz für beliebige Vektorraume.- 9*. Ein Glasperlen-Spiel.- § 5. Anwendungen.- 1. Die reellen Zahlen als Vektorraum über Q.- 2. Beispiele.- 3. Der Rang einer Teilmenge.- 4. Anwendung auf lineare Gleichungssysteme.- § 6. Homomorphismen von Vektorräumen.- 1. Einleitung.- 2. Definition und einfachste Eigenschaften.- 3. Kern und Bild.- 4. Die Dimensionsformel für Homomorphismen.- 5. Äquivalenz-Satz fÄr Homomorphismen.- 6. Der Rang eines Homomorphismus.- 7. Anwendung auf homogene lineare Gleichungen.- 8. Beispiele.- 9*. Die Funktionalgleichung f(x + y) = f(x) + f(y).- § 7*. Linearformen und der duale Raum.- 1. Vorbemerkungen.- 2. Definition und Beispiele.- 3. Existenz von Linearformen.- 4. Der Dual-Raum.- 5. Linearformen des Vektorraums der stetigen Funktionen.- § 8*. Direkte Summen und Komplemente.- 1. Summe und direkte Summe.- 2. Komplemente.- 3. Die Dimensionsformel für Summen.- 4. Die Bild-Kern-Zerlegung.- 2. Matrizen.- § 1. Erste Eigenschaften.- 1. Der Begriff einer Matrix.- 2. Über den Vorteil von Doppelindizes.- 3. Mat(m, n; K) als K-Vektorraum.- 4. Das Transponierte einer Matrix.- 5. Spalten- und Zeilenrang.- 6. Elementare Umformungen.- 7. Die Ranggleichung.- 8. Kästchenschreibweise und Rangberechnung.- 9. Zur Geschichte des Rang-Begriffes.- § 2. Matrizenrechnung.- 1. Arthur Cayley oder die Erfindung der Matrizenrechnung.- 2. Produkte von Matrizen.- 3. Produkte von Vektoren.- 4. Homomorphismen zwischen Standard-Raumen.- 5. Erntezeit.- 6. Das Skalarprodukt.- 7*. Rang A ? r.- 8. Kästchenrechnung.- § 3. Algebren.- 1. Einleitung.- 2. Der Begriff einer Algebra.- 3. Invertierbare Elemente.- 4. Ringe.- 5. Beispiele.- § 4. Der Begriff einer Gruppe.- 1. Halbgruppen.- 2. Gruppen.- 3. Untergruppen.- 4. Kommutative Gruppen.- 5. Homomorphismen.- 6. Normalteiler.- 7. Historische Bemerkungen.- § 5. Matrix-Algebren.- 1. Mat(n; K) und GL(n; K).- 2. Der Äquivalenz-Satz für invertierbare Matrizen.- 3. Die Invarianz des Ranges.- 4. Spezielle invertierbare Matrizen.- 5*. Zentralisator und Zentrum.- 6. Die Spur einer Matrix.- 7. Die Algebra Mat(2; K).- § 6. Der Normalformen-Satz.- 1. Elementar-Matrizen.- 2. Zusammenhang mit elementaren Umformungen.- 3. Anwendungen.- 4*. Die Weyr-Frobenius-Ungleichungen.- 5. Aufgaben zum Normalformen-Satz.- 6. Zur Geschichte des Normalformen-Satzes.- § 7. Gleichungssysteme.- 1. Erinnerung an lineare Gleichungen.- 2. Wiederholung von Problemen und Ergebnissen.- 3. Der Fall m = n.- 4. Anwendung des Normalformen-Satzes.- 5. Lösungsverfahren.- 6. Basiswechsel in Vektorräumen.- § 8*. Pseudo-Inverse.- 1. Motivation.- 2. Der Begriff des Pseudo-Inversen.- 3. Ein Kriterium für Gleichungssysteme.- 4. Zerlegung in eine direkte Summe.- 3. Determinant en.- § 1. Erste Ergebnisse über Determinanten.- 1. Eine Motivation.- 2. Determinanten-Funktionen.- 3. Existenz.- 4. Eigenschaften.- 5. Anwendungen auf die Gruppe GL(n; K).- 6. Die Cramerche Regel.- § 2. Das Inverse einer Matrix.- 1. Vorbemerkung.- 2. Die Entwicklungs-Sätze.- 3. Die komplementäre Matrix.- 4. Beschreibung des Inversen.- § 3. Existenzbeweise.- 1. Durch Induktion.- 2. Permutationen.- 3. Die Leibnizsche Formel.- 4. Permutationsmatrizen.- 5. Ein weiterer Existenzbeweis.- § 4. Erste Anwendungen.- 1. Lineare Gleichungssysteme.- 2. Zweidimensionale Geometrie.- 3. Lineare Abhängigkeit.- 4. Rangberechnung.- 5. Die Determinanten-Rekursionsformel.- 6. Das charakteristische Polynom.- 7*. Mehrfache Nullstellen von Polynomen.- 8*. Eine Funktionalgleichung.- 9. Orientierung von Vektorräumen.- § 5. Symmetrische Matrizen.- 1. Einleitung.- 2. Der Vektorraum der symmetrischen Matrizen.- 3. Quadratische Ergänzung.- 4. Die Jacobische Normalform.- 5. Normalformen-Satz.- 6*. Trägheits-Satz.- § 6. Spezielle Matrizen.- 1. Schiefsymmetrische Matrizen.- 2. Die Vandermondesche Determinante.- 3. Bandmatrizen.- 4. Aufgaben.- § 7. Zur Geschichte der Determinanten.- 1. Gottfried Wilhelm LEIBNIZ.- 2. BALTZER’S Lehrbuch.- 3. Die weitere Entwicklung.- B. Analytische Geometrie.- 4. Elementar-Geometrie in der Ebene.- § 1. Grundlagen.- 1. Skalarprodukt, Abstand und Winkel.- 2. Die Abbildung x ? x? 3..- 3. Geraden.- 4. Schnittpunkt zwischen zwei Geraden.- 5. Abstand zwischen Punkt und Gerade.- 6. Fläche eines Dreiecks.- 7. Der Höhenschnittpunkt.- § 2. Die Gruppe O(2).- 1. Drehungen und Spiegelungen.- 2. Orthogonale Matrizen.- 3. Bewegungen.- 4. Ein Beispiel.- 5. Die Hauptachsentransformation fur 2 Matrizen.- 6. Fix-Geraden.- 7. Die beiden Orientierungen der Ebene.- § 3. Geometrische Sätze.- 1. Der Kreis.- 2. Tangente.- 3. Die beiden Sehnensätze.- 4. Der Umkreis eines Dreiecks.- 5. Die Euler-Gerade.- 6. Der Feuerbach-Kreis.- 7. Das Mittendreieck.- 5. Euklidische Vektorräume.- § 1. Positiv definite Bilinearformen.- 1. Symmetrische Bilinearformen.- 2. Beispiele.- 3. Positiv definite Bilinearformen.- 4. Positiv definite Matrizen.- 5. Die Cauchy-Schwrzsche Ungleichung.- 6. Normierte Vektorraume.- § 2. Das Skalarprodukt.- 1. Der Begriff eines euklidischen Vektorraumes.- 2. Winkelmessung.- 3. Orthonormalbasen.- 4. Basisdarstellung.- 5. Orthogonales Komplement und orthogonale Summe.- 6. Linearformen.- § 3. Erste Anwendungen.- 1. Positiv definite Matrizen.- 2. Die adjungierte Abbildung.- 3. Systeme linearer Gleichungen.- 4. Ein Kriterium für gleiche Orientierung.- 5*. Legendre-Polynome.- §4. Geometrie in euklidischen Vektorräumen.- 1. Geraden.- 2. Hyperebenen.- 3. Schnittpunkt von Gerade und Hyperebene.- 4. Abstand von einer Hyperebene.- 5*. Orthogonale Projektion.- 6*. Abstand zweier Unterräume.- 7*. Volumenberechnung.- 8*. Duale Basen.- § 5. Die orthogonale Gruppe.- 1. Bewegungen.- 2. Spiegelungen.- 3. Die Transitivitat von O(V,?) auf Sphären.- 4*. Die Erzeugung von O(V,?) durch Spiegelungen.- 5*. Winkeltreue Abbildungen.- 6. Der ?aun als Euklidischer Vektorraum.- § 1. Der ?n und die orthogonale Gruppe O(n).- 1. Der euklidische Vektorraum ?n.- 2. Orthogonale Matrizen.- 3. Die Gruppe O(n).- 4. Spiegelungen.- 5. Erzeugung von O(n) durch Spiegelungen.- 6*. Drehungen.- 7. Anwendung der Determinanten-Theorie.- 8*. Eine Parameterdarstellung.- 9. Euler, Cauchy, Jacobi Und Cayley.- § 2. Die Hauptachsentransformation.- 1. Problemstellung.- 2. Der Vektorraum der symmetrischen Matrizen.- 3. Positiv semi-definite Matrizen.- 4. Das Minimum einer quadratischen Form.- 5. Satz uber die Hauptachsentransformation.- 6. Eigenwerte.- 7. Eigenräume.- § 3. Anwendungen.- 1. Vorbemerkung.- 2. Positiv definite Matrizen.- 3. Hyperflächen.- 2. Grades.- 4*. Der Quadratwurzel-Satz.- 5*. Polar-Zerlegung.- 6*. Orthogonale Normalform.- 7*. Das Moorw-Penrose-Inverse.- § 4*. Topologische Eigenschaften.- 1. Zusammenhang.- 2. Kompaktheit.- 3. Hauptachsentransformation.- 7. Geometrie im dreidimensionalen Raum.- § 1. Das Vektorprodukt.- 1. Definition und erste Eigenschaften.- 2. Zusammenhang mit Determinanten.- 3. Geometrische Deutung.- 4. Ebenen.- 5. Parallelotope.- 6. Vektorrechnung im Anschauungsraum.- § 2*. Sphärische Geometrie.- 1. Über den Ursprung der Sphärik.- 2. Das sphärische Dreieck.- 3. Das Polardreieck.- 4. Entfernung auf der Erde.- § 3. Die Gruppe O(3).- 1. Beschreibung durch das Vektorprodukt.- 2. Erzeugung durch Drehungen.- 3. Spiegelungen.- 4. Fix-Geraden.- 5. Die Normalform.- 6. Die Drehachse.- 7*. Die Eulersche Formel.- 8*. Drehungen um eine Achse.- § 4. Bewegungen.- 1. Fixpunkte.- 2. Bewegungen mit Fixpunkt.- 3. Schraubungen.- C. Lineare Algebra II.- 8. Polynome und Matrizen.- § 1. Polynome.- 1. Der Vektorraum Pol K.- 2. Pol K als Ring.- 3. Zerfallende Polynome.- 4. Pol K als Hauptidealring.- 5*. Unbestimmte.- § 2. Die komplexen Zahlen.- 1. Der Körper C der komplexen Zahlen.- 2. Konjugation und Betrag.- 3. Der Fundamentalsatz der Algebra.- § 3. Struktursatz für zerfallende Matrizen.- 1. Der Begriff der Diagonalisierbarkeit.- 2. Das charakteristische Polynom.- 3. Äquivalenz-Satz für Eigenwerte.- 4. Nilpotente Matrizen.- 5. Idempotente Matrizen.- 6. Zerfallende Matrizen.- 7. Diagonalisierbarkeits-Kriterium.- 8*. Ein Beispiel zum Struktur-Satz.- 9*. Elementarsymmetrische Funktionen und Potenzsummen.- §4. Die Algebra K[A].- 1. Eine Warnung.- 2. Matrix-Polynome.- 3. Das Minimalpolynom.- 4. Eigenwerte.- 5. Das Rechnen mit Kästchen-Diagonalmatrizen.- 6. Satz von Cayley.- 7. Äquivalenz-Satz für Diagonalisierbarkeit.- 8. Spektralscharen.- 9. Eigenräume.- §5. Die Jordan-Chevalley-Zerlegung.- 1. Existenz-Satz.- 2. Summen von diagonalisierbaren Matrizen.- 3. Die Eindeutigkeit.- 4. Anwendungen.- § 6. Normalformen reeller und komplexer Matrizen.- 1. Normalformen komplexer Matrizen.- 2. Reelle und komplexe Matrizen.- 3*. Hermitesche Matrizen.- 4. Invariante Unterräume.- 5. Die Stufenform.- 6. Der Satz über die Stufenform.- 7. Orthogonale Matrizen.- 8. Schiefsymmetrische Matrizen.- 9*. Normale Matrizen.- § 7*. Der höhere Standpunkt.- 1. Einfache und halbeinfache Algebren.- 2. Kommutative Algebren.- 3. Die Struktursätze.- 4. Die weitere Entwicklung.- 5. Der generische Standpunkt.- 9. Homomorphismen von Vektorräumen.- § 1. Der Vektorraum Hom(V, V?).- 1. Der Vektorraum Abb(M, V?).- 2. Hom(V, V?) als Unterraum von Abb(V, V?).- 3. Mat(m, n; K) als Beispiel.- 4. Verknüpfungen von Hom(V, V?) und Hom(V?, V?).- § 2. Beschreibung der Homomorphismen im endlich-dimensionalen Fall.- 1. Isomorphic mit Standard-Räumen.- 2. Darstellung der Homomorphismen.- 3. Basiswechsel.- 4. Die Algebra End V.- 5. Diagonalisierbarkeit.- 6. Die Linksmultiplikation in Mat(n; K).- 7. Polynome.- § 3. Euklische Vektorräume.- 1. Der Satz über die Hauptachsentransformation.- 2. Spiegelungen.- 3*. Unitäre Vektorräume.- § 4. Der Quotientenraum.- 1. Einleitung.- 2. Nebenklassen.- 3. Der Satz über den Quotientenraum.- 4. Der Satz über den kanonischen Epimorphismus.- 5. Kanonische Faktorisierung.- 6. Anwendungen.- 7. Beispiele.- § 5*. Nilpotente Endomorphismen.- 1. Problemstellung.- 2. Zyklische Unterräume.- 3. Der Struktur-Satz.- 4. Nilzyklische Matrizen.- 5. Die Normalform.- 6. Satz von der JoRDANSchen Normalform.- 7. Anwendungen auf Differentialgleichungen.- Literatur.- Namenverzeichnis.

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