Geometry Books

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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 Synopsis

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Book Synopsis

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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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Book SynopsisBoth classical geometry and modern differential geometry have been active subjects of research throughout the 20th century and lie at the heart of many recent advances in mathematics and physics. The underlying motivating concept for the present book is that it offers readers the elements of a modern geometric culture by means of a whole series of visually appealing unsolved (or recently solved) problems that require the creation of concepts and tools of varying abstraction. Starting with such natural, classical objects as lines, planes, circles, spheres, polygons, polyhedra, curves, surfaces, convex sets, etc., crucial ideas and above all abstract concepts needed for attaining the results are elucidated. These are conceptual notions, each built "above" the preceding and permitting an increase in abstraction, represented metaphorically by Jacob's ladder with its rungs: the 'ladder' in the Old Testament, that angels ascended and descended... In all this, the aim of the book is to demonstrate to readers the unceasingly renewed spirit of geometry and that even so-called "elementary" geometry is very much alive and at the very heart of the work of numerous contemporary mathematicians. It is also shown that there are innumerable paths yet to be explored and concepts to be created. The book is visually rich and inviting, so that readers may open it at random places and find much pleasure throughout according their own intuitions and inclinations. Marcel Berger is the author of numerous successful books on geometry, this book once again is addressed to all students and teachers of mathematics with an affinity for geometry.Trade Review“It is a must own book for anyone serious about developing a conceptual understanding of the interconnected web of modern geometry and the ever-growing intertwining of geometry with practically all other branches of mathematics. … It is remarkable for a book to provide such a detailed glimpse of contemporary geometry via well developed discussions of so many questions of current interest. It provides the most extensive exposition of geometric thinking I’ve ever seen in a book at this level.” (William H. Barker, MAA Reviews, August, 2017)“Geometry Revealed is to give the reader a feel for the conceptual frameworks of modern geometry, attempting to reach as far as possible with a minimum of assumed knowledge and formal scaffolding. … Geometry Revealed being useful for research mathematicians as a still reasonably up-to-date survey. … Geometry Revealed offered an ascent into the wonders of a new world.” (Danny Yee, Danny Yee’s Book Reviews, dannyreviews.com, July, 2015)“By considering a hierarchy of ‘natural’ geometrical objects … it sets out to investigate significant geometrical problems which are either unsolved or were solved only recently. … it is undoubtedly a major tour de force, and if you really want to gain an idea of where geometry is going in the 21st century, you will find plenty of exquisite material here.” (Gerry Leversha, The Mathematical Gazette, Vol. 96 (356), July, 2012)“The book contains twelve chapters, each of them is a collection of such problems about geometric objects with more and more complexity … . The chapters are independent from each other, any of them can serve as a course. Researchers in geometry can use it as a source for further research. … the book is accessible to a wide audience of people who are interested in geometry.” (János Kincses, Acta Scientiarum Mathematicarum (Szeged), Vol. 78 (1-2), 2012)“‘Geometry Revealed’ is a massive text of 831 pages which is organized in twelve chapters and which additionally provides indices for names, subjects and symbols … throughout the author quite carefully lays out the historical perspective. … a typical chapter starts with an observation or a problem in elementary geometry. Large parts of the text are very accessible, and a reader who likes (mathematical) physics will often get something extra.” (Michael Joswig, Zentralblatt MATH, Vol. 1232, 2012)“The author provides the reader with an enormous amount of detailed information and thus yields deep insight into the various topics. … All in all an overwhelming book which is a must … for everyone having sufficient mathematical knowledge.” (G. Kowol, Monatshefte für Mathematik, Vol. 164 (2), October, 2011)“The book is a very readable account of several branches of geometry, classical and modern, elementary and advanced. … Every chapter is extremely interesting and alive. … The book is rich in ideas, written in an informal style, with no formulae and no unnecessary technical details. … Every part of this book is interesting and should be accessible to a wide audience of mathematicians. … Every mathematician will experience great pleasure in reading this book.” (Athanase Papadopoulos, Mathematical Reviews, Issue 2011 m)Table of ContentsPoints and lines in the plane.- Circles and spheres.- The sphere by itself: can we distribute points on it evenly?.- Conics and quadrics.- Plane curves.- Smooth surfaces.- Convexity and convex sets.- Polygons, polyhedra, polytopes.- Lattices, packings and tilings in the plane.- Lattices and packings in higher dimensions.- Geometry and dynamics I: billiards.- Geometry and dynamics II: geodesic flow on a surface.

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Book SynopsisEinstein's equations stem from General Relativity. In the context of Riemannian manifolds, an independent mathematical theory has developed around them. This is the first book which presents an overview of several striking results ensuing from the examination of Einstein’s equations in the context of Riemannian manifolds. Parts of the text can be used as an introduction to modern Riemannian geometry through topics like homogeneous spaces, submersions, or Riemannian functionals.Trade ReviewFrom the reviews: "[...] an efficient reference book for many fundamental techniques of Riemannian geometry. [...] despite its length, the reader will have no difficulty in getting the feel of its contents and discovering excellent examples of all interaction of geometry with partial differential equeations, topology, and Lie groups. Above all, the book provides a clear insight into the scope and diversity of problems posed by its title."S.M. Salamon in MathSciNet 1988 "It seemed likely to anyone who read the previous book by the same author, namely Manifolds all of whose geodesic are closed, that the present book would be one of the most important ever published on Riemannian geometry. This prophecy is indeed fulfilled."T.J. Wilmore in Bulletin of the London Mathematical Society 1987 "Einstein Manifolds is accordingly described as Besse’s second book … . there is no doubt that Einstein Manifolds is a magnificient work of mathematical scholarship. … It is truly a seminal work on an incomparably fascinating and important subject." (Michael Berg, MathDL, March, 2008) "The present book is intended to be a complete reference book. … The book under review serves several purposes. It is an efficient reference for many fundamental techniques of Riemannian geometry as well as excellent examples of the interaction of geometry with partial differential equations, topology and Lie groups. Certainly the monograph provides a clear insight into the scope and diversity of problems posed by its title." (Adela-Gabriela Mihai, Zentralblatt MATH, Vol. 1147, 2008)Table of ContentsBasic Material.- Basic Material (Continued): Kähler Manifolds.- Relativity.- Riemannian Functionals.- Ricci Curvature as a Partial Differential Equation.- Einstein Manifolds and Topology.- Homogeneous Riemannian Manifolds.- Compact Homogeneous Kähler Manifolds.- Riemannian Submersions.- Holonomy Groups.- Kähler-Einstein Metrics and the Calabi Conjecture.- The Moduli Space of Einstein Structures.- Self-Duality.- Quaternion-Kähler Manifolds.- A Report on the Non-Compact Case.- Generalizations of the Einstein Condition.

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Book SynopsisThis volume deals with the theory of finite topological spaces and its relationship with the homotopy and simple homotopy theory of polyhedra. The interaction between their intrinsic combinatorial and topological structures makes finite spaces a useful tool for studying problems in Topology, Algebra and Geometry from a new perspective. In particular, the methods developed in this manuscript are used to study Quillen's conjecture on the poset of p-subgroups of a finite group and the Andrews-Curtis conjecture on the 3-deformability of contractible two-dimensional complexes. This self-contained work constitutes the first detailed exposition on the algebraic topology of finite spaces. It is intended for topologists and combinatorialists, but it is also recommended for advanced undergraduate students and graduate students with a modest knowledge of Algebraic Topology.Trade ReviewFrom the reviews:“This book deals with the algebraic topology of finite topological spaces and its applications, and includes well-known results on finite spaces and original results developed by the author. The book is self-contained and well written. It is understandable and enjoyable to read. It contains a lot of examples and figures which help the readers to understand the theory.” (Fumihiro Ushitaki, Mathematical Reviews, March, 2014)“This book illustrates convincingly the idea that the study of finite non-Hausdorff spaces from a homotopical point of view is useful in many areas and can even be used to study well-known problems in classical algebraic topology. … This book is a revised version of the PhD Thesis of the author. … All the concepts introduced with the chapters are usefully illustrated by examples and the recollection of all these results gives a very nice introduction to a domain of growing interest.” (Etienne Fieux, Zentralblatt MATH, Vol. 1235, 2012)Table of Contents1 Preliminaries.- 2 Basic topological properties of finite spaces.- 3 Minimal finite models.- 4 Simple homotopy types and finite spaces.- 5 Strong homotopy types.- 6 Methods of reduction.- 7 h-regular complexes and quotients.- 8 Group actions and a conjecture of Quillen.- 9 Reduced lattices.- 10 Fixed points and the Lefschetz number.- 11 The Andrews-Curtis conjecture.

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Book SynopsisIn this textbook the authors present first-year geometry roughly in the order in which it was discovered. The first five chapters show how the ancient Greeks established geometry, together with its numerous practical applications, while more recent findings on Euclidian geometry are discussed as well. The following three chapters explain the revolution in geometry due to the progress made in the field of algebra by Descartes, Euler and Gauss. Spatial geometry, vector algebra and matrices are treated in chapters 9 and 10. The last chapter offers an introduction to projective geometry, which emerged in the 19thcentury.Complemented by numerous examples, exercises, figures and pictures, the book offers both motivation and insightful explanations, and provides stimulating and enjoyable reading for students and teachers alike.Trade ReviewFrom the book reviews:Choice - Outstanding Academic Title in 2012“This is an excellent, challenging textbook as well as a valuable resource for historical information, problems, and student projects. The historical content is broad based, comprehensive, and reliable. Each chapter has extensive exercises, many taken directly from or based on historical sources … . Hints and solutions for all problems are given in an appendix. Thorough bibliography. Summing Up: Highly recommended. Lower-division undergraduates and above.” (C. A. Gorini, Choice, Vol. 50 (3), November, 2012)“The book under review is a treasure chest of interesting theorems and problems in geometry together with their illuminating histories. … This is the kind of book that one would enjoy browsing through and reading while sitting relaxedly in an armchair without any paper or pencil and starting at almost any page or paragraph. It should be on the shelf of every lover of geometry.” (Mowaffaq Hajja, zbMATH, Vol. 1288, 2014)“This book belongs on the bookshelf of every geometer. … The authors have penned their book with students of geometry as well as science in mind. In fact, the book would serve well as a second year mathematics course in a classical liberal arts setting. … the book treats many interesting and beautiful problems, introducing powerful concepts along the way, and yet is written at a level suitable for an introductory course of geometry or even advanced mathematics.” (Alan S. McRae, Mathematical Reviews, February, 2013)“There is a lot of interesting material in this book, supplemented by a lot of very nice artwork and many interesting exercises … . I would think that any other college instructor … with an interest in geometry would also want a copy on his or her shelf.” (Mark Hunacek, The Mathematical Association of America, June, 2012)Table of ContentsPreface.- Part I: Classical Geometry.- Thales and Pythagoras.- The Elements of Euclid.- Conic Sections.- Further Results on Euclidean Geometry.- Trigonometry.- Part II: Analytic Geometry.- Descartes' Geometry.- Cartesian Coordinates.- To be Constructible, or not to be.- Spatial Geometry and Vector Algebra.- Matrices and Linear Mappings.- Projective Geometry.- Solutions to Exercises.- References.- Figure Source and Copyright.- Index.

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Book SynopsisDieser Buchtitel ist Teil des Digitalisierungsprojekts Springer Book Archives mit Publikationen, die seit den Anfängen des Verlags von 1842 erschienen sind. Der Verlag stellt mit diesem Archiv Quellen für die historische wie auch die disziplingeschichtliche Forschung zur Verfügung, die jeweils im historischen Kontext betrachtet werden müssen. Dieser Titel erschien in der Zeit vor 1945 und wird daher in seiner zeittypischen politisch-ideologischen Ausrichtung vom Verlag nicht beworben.

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Book SynopsisIn July 2004, a conference on graph theory was held in Paris in memory of Claude Berge, one of the pioneers of the field. The event brought together many prominent specialists on topics such as perfect graphs and matching theory, upon which Claude Berge's work has had a major impact. This volume includes contributions to these and other topics from many of the participants.Table of ContentsClaude Berge — Sculptor of Graph Theory.- ?-path-connectivity and mk-generation: an Upper Bound on m.- Automated Results and Conjectures on Average Distance in Graphs.- Brambles, Prisms and Grids.- Dead Cell Analysis in Hex and the Shannon Game.- Ratios of Some Domination Parameters in Graphs and Claw-free Graphs.- Excessive Factorizations of Regular Graphs.- Odd Pairs of Cliques.- Recognition of Perfect Circular-arc Graphs.- On Edge-maps whose Inverse Preserves Flows or Tensions.- On the Extremal Number of Edges in 2-Factor Hamiltonian Graphs.- Generalized Colourings (Matrix Partitions) of Cographs.- A Note on [k, l]-sparse Graphs.- Even Pairs in Bull-reducible Graphs.- Kernels in Orientations of Pretransitive Orientable Graphs.- Nonrepetitive Graph Coloring.- A Characterization of the 1-well-covered Graphs with no 4-cycles.- A Graph-theoretical Generalization of Berge’s Analogue of the Erd?s-Ko-Rado Theorem.- Independence Polynomials and the Unimodality Conjecture for Very Well-covered, Quasi-regularizable, and Perfect Graphs.- Precoloring Extension on Chordal Graphs.- On the Enumeration of Bipartite Minimum Edge Colorings.- Kempe Equivalence of Colorings.- Acyclic 4-choosability of Planar Graphs with Girth at Least 5.- Automorphism Groups of Circulant Graphs — a Survey.- Hypo-matchings in Directed Graphs.- On Reed’s Conjecture about ?,? and ?.- On the Generalization of the Matroid Parity Problem.- Reconstruction of a Rank 3 Oriented Matroids from its Rank 2 Signed Circuits.- The Normal Graph Conjecture is True for Circulants.- Two-arc Transitive Near-polygonal Graphs.- Open Problems.

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Book SynopsisThis book develops the geometric intuition of the reader by examining the symmetries (or rigid motions) of the space in question. This approach introduces in turn all the classical geometries: Euclidean, affine, elliptic, projective and hyperbolic. The main focus is on the mathematically rich two-dimensional case, although some aspects of 3- or $n$-dimensional geometries are included. Basic notions of algebra and analysis are used to convey better understanding of various concepts and results. Concepts of geometry are presented in a very simple way, so that they become easily accessible: the only pre-requisites are calculus, linear algebra and basic analytic geometry.Trade ReviewFrom the reviews:“This book offers an introduction to classical geometry. … The book contains many figures, which help the reader to develop geometric intuition, and lots of exercises. To follow the presentation, only a basic background in analysis and linear algebra is required, some necessary facts are collected in an appendix.” (A. Cap, Monatshefte für Mathematik, Vol. 158 (2), October, 2009)Table of ContentsEuclidean geometry.- Affine geometry.- Projective geometry.- Hyperbolic geometry.- Appendices.

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Book SynopsisThis book arose out of original research on the extension of well-established applications of complex numbers related to Euclidean geometry and to the space-time symmetry of two-dimensional Special Relativity. The system of hyperbolic numbers is extensively studied, and a plain exposition of space-time geometry and trigonometry is given. Commutative hypercomplex systems with four unities are studied and attention is drawn to their interesting properties.Trade ReviewFrom the reviews: “It is worth pointing out that the book is mainly a text about commutative hypercomplex numbers and some of their applications to a 2-dimensional Minkowski spacetime. … This book should be interesting to anybody who is interested in applications of hypercomplex numbers … . In conclusion, I recommend this book to anyone who wants to learn about hypercomplex numbers.” (Emanuel Gallo, Mathematical Reviews, Issue 2010 d)Table of ContentsThe Mathematics of Minkowski Space-Time: 1 N-Dimensional Hypercomplex Numbers and the associated Geometries.- Commutative Hypercomplex Number Systems.- The General Two-Dimensional System.- Linear Transformations and Geometries.- The Geometries Associated with Hypercomplex Numbers.- Conclusions.- 2 Trigonometry in the Minkowski Plane.- Geometrical Representation of Hyperbolic Numbers.- Basics of Hyperbolic Trigonometry.- Geometry in Pseudo-Euclidean Cartesian Plane.- Trigonometry in the Pseudo-Euclidean Plane.- Theorems on Equilateral Hyperbolas in the Pseudo-Euclidean Plane.- Some Examples of Triangle Solutions in the Minkowski Plane.- Conclusions.- 3 Uniform and Accelerated Motions in the Minkowski Space-Time (Twin Paradox).- Inertial Motions.- Inertial and Uniformly Accelerated Motions.- Non-uniformly Accelerated Motions.- Conclusions.- 4 General Two-Dimensional Hypercomplex Numbers.-Geometrical Representation.- Geometry and Trigonometry in Two-Dimensional Algebras.- Some Properties of Fundamental Conic Section.- Numerical Examples.- 5 Functions of a Hyperbolic Variable.- Some Remarks on Functions of a Complex Variable.- Functions of Hypercomplex Variables.- The Functions of a Hyperbolic Variable.- The Elementary Functions of a Canonical Hyperbolic Variable.- H-Conformal Mappings.- Commutative Hypercomplex Systems with Three Unities.- 6 Hyperbolic Variables on Lorentz Surfaces.- Introduction.- Gauss: Conformal Mapping of Surfaces.- Extension of Gauss Theorem: Conformal Mapping of Lorentz Surfaces.- Beltrami: Complex Variables on a Surface.- Beltrami’s Integration of Geodesic Equations.- Extension of Beltrami’s Equation to Non-Definite Differential Forms.- 7 Constant Curvature Lorentz Surfaces.- Introduction.- Constant Curvature RiemannSurfaces.- Constant Curvature Lorentz Surfaces.- Geodesics and Geodesic Distances on Riemann and Lorentz Surfaces.- Conclusions.- 8 Generalization of Two-Dimensional Special Relativity (Hyperbolic Transformations and the Equivalence Principle).- Physical Meaning of Transformations by Hyperbolic Functions.- Physical Interpretation of Geodesics on Riemann and Lorentz Surfaces with Positive Constant Curvature.- Einstein’s Way to General Relativity.- Conclusions.- II An Introduction to Commutative Hypercomplex Numbers.- 9 Commutative Segre’s Quaternions.- Introduction.- Hypercomplex Systems with Four Units.- Historical Introduction of Segre’s Quaternion.- Algebraic Properties of Commutative Quaternions.- Functions of a Quaternion Variable.- Mapping by Means of Quaternion Functions.- Elementary Functions of the Quaternions.- Elliptic-Hyperbolic Quaternions.- Elliptic-Parabolic Generalized Segre’s Quaternions.- 10 Constant Curvature Segre’s Quaternion Spaces.- Introduction.- Quaternion differential geometry and geodesic equations.- Orthogonality in Segre’s Quaternion Space.- Constant Curvature Quaternion Spaces.- Geodesic Equations in Quaternion Space.- Beltrami’s Integration Method for Quaternion Spaces.- Beltrami’s Integration Method for Quaternion Spaces.- Conclusions.- 11 A Matrix Formalization for Commutative Hypercomplex Systems.- Mathematical Operations.- Properties of the Characteristic Matrix M.- Functions of Hypercomplex Variable.- Functions of a Two-Dimensional Hypercomplex Variable.- Derivatives of a Hypercomplex Function.- Characteristic Differential Equation.- A Equivalence Between the Formalizations of Hypercomplex Numbers.

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