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二十面体和5次方程的解的讲义(英文版)
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商品名称:二十面体和5次方程的解的讲义(英文版)
物料号 :51022-00
重量:0.000千克
ISBN:9787040510225
出版社:高等教育出版社
出版年月:2019-05
作者:Felix Klein
定价:168.00
页码:344
装帧:精装
版次:1
字数:420
开本:16开
套装书:否

希腊数学的最高成就是正多面体的分类,即五种所谓的柏拉图体。最复杂的正多面体是二十面体。直到19世纪,数学中最重要的问题是解代数方程。在这本经典著作中,Klein展示了如何将这两个看似无关的主题联系起来,并将它们与另一个新的数学理论联系在一起:超几何函数和单值群。这清楚地表明了克莱因对数学统一性的高瞻远瞩。 本书包括Peter Slodowy的评注和他关于Klein这本经典著作的解释性论文,从而帮助读者理解ADE的分类,以及它们在当前研究中的许多意想不到的联系和应用。 The highest achievement of the Greek mathematics is the classification of regular solids, the five so-called Platonic solids. The most complicated solid is the icosahedron. Up to and through the 19th century, the most important problem in mathematics was to solve algebraic equations. In this classic book, Klein showed how to relate these two seemingly unrelated topics and also tied them together with another new theory of mathematics: hypergeometric functions and monodromy groups. This clearly shows Klein's vision of the unity of mathematics. This book includes Peter Slodowy’s commentaries and his expository paper on Klein's book to help readers to understand the ADE classification, and their many unexpected connections and applications under current study.

前辅文
Part I Theory of the Icosahedron Itself
  Chapter I The Regular Solids and the Theory of Groups
   1.Statement of the Question.
   2.Preliminary Notions of the Group-Theory
   3.The Cyclic Rotation Groups
   4.The Group of theDihedral Rotations.
   5.The Quadratic Group.
   6.The Group of the Tetrahedral Rotations.
   7.The Group of the Octahedral Rotations
   8.The Group of the Icosahedral Rotations
   9.On the Planes of Symmetry inOur Configurations
   10.General Groups of Points—Fundamental Domains
   11.The Extended Groups
   12.Generation of the IcosahedralGroup.
   13.Generation of the Other Groups of Rotations.
  Chapter II Introduction of (x +i y)
   1.First Presentation and Survey of the Developments of This Chapter
   2.On Those Linear Transformations of (x +i y) WhichCorrespond to Rotations Round the Centre
   3.Homogeneous Linear Substitutions—Their Composition.
   4.Return to the Groups of Substitutions—the Cyclic and DihedralGroups
   5.The Groups of the Tetrahedron andOctahedron
   6.The IcosahedralGroup.
   7.Non-Homogeneous Substitutions—Consideration of the Extended Groups.
   8.Simple Isomorphism in the Case of Homogeneous Groups of Substitutions
   9.Invariant Forms Belonging to a Group—The Set of Forms for the Cyclic andDihedralGroups
   10.Preparation for the Tetrahedral andOctahedral Forms.
   11.The Set of Forms for the Tetrahedron.
   12.The Set of Forms for the Octahedron
   13.The Set of Forms for the Icosahedron.
   14.The Fundamental Rational Functions
   15.Remarks on the Extended Groups
  Chapter III Statement and Discussion of the Fundamental Problem, According to the Theory of Functions
   1.Definition of the Fundamental Problem.
   2.Reduction of the Form-Problem.
   3.Plan of the Following Investigations
   4.On the Conformable Representation byMeans of the Function z(Z)
   5.March of the z1, z2 Function in General—Development in Series
   6.Transition to theDifferential Equations of the ThirdOrder.
   7.Connection with Linear Differential Equations of the Second Order
   8.Actual Establishment of the Differential Equation of the Third Order for z[Z].
   9.Linear Differential Equations of the Second Order for z1 and z2
   10.Relations to Riemann’s P-Function.
  Chapter IV On the Algebraical Character of Our Fundamental Problem
   1.Problemof the Present Chapter
   2.On the Group of an Algebraical Equation.
   3.General Remarks on Resolvents.
   4.The Galois Resolvent in Particular
   5.Marshalling of our Fundamental Equations
   6.Consideration of the Form-Problems
   7.The Solution of the Equations of the Dihedron,Tetrahedron,andOctahedron.
   8.The Resolvents of the Fifth Degree for the Icosahedral Equation
   9.The Resolvent of the r ’s
   10.Computation of the Forms t andW
   11.The Resolvent of the u’s.
   12.The Canonical Resolvent of the Y ’s
   13.Connection of the New Resolvent with the Resolvent of the r ’s
   14.On the Products of Differences for the u’s and the Y ’s.
   15.The Simplest Resolvent of the SixthDegree.
   16.Concluding Remarks.
  Chapter V General Theorems and Survey of the Subject.
   1.Estimation of our Process of Thought so far, and Generalisations Thereof
   2.Determination of all Finite Groups of Linear Substitutions of a Variable
   3.Algebraically Integrable Linear Homogeneous Differential Equations of the Second Order
   4.Finite Groups of Linear Substitutions for a Greater Number of Variables.
   5.Preliminary Glance at the Theory of Equations of the Fifth Degree,and Formulation of a General Algebraical Problem.
   6.InfiniteGroups of Linear Substitutions of a Variable
   7.Solution of the Tetrahedral, Octahedral, and Icosahedral Equations by EllipticModular Functions.
   8.Formulae for the Direct Solution of the Simplest Resolvent of the SixthDegree for the Icosahedron
   9.Significance of the Transcendental Solutions.
Part II Theory of Equations of the Fifth Degree
  Chapter I The HistoricalDevelopment of the Theory of Equations of the Fifth Degree
   1.Definition of Our First Problem.
   2.Elementary Remarks on the Tschirnhausian Transformation—Bring’s Form.
   3.Data Concerning Elliptic Functions
   4.On Hermite’sWork of 1858
   5.The Jacobian Equations of the SixthDegree
   6.Kronecker’sMethod for the Solution of Equations of the Fifth Degree
   7.On Kronecker’sWork of 1861.
   8.Object of our Further Developments
  Chapter II Introduction of GeometricalMaterial
   1.Foundation of the Geometrical Interpretation
   2.Classification of the Curves and Surfaces.
   3.The Simplest Special Cases of Equations of the Fifth Degree.
   4.Equations of the Fifth DegreeWhich Appertain to the Icosahedron.
   5.Geometrical Conception of the Tschirnhausian Transformation.
   6.Special Applications of the Tschirnhausian Transformation
   7.Geometrical Aspect of the Formation of Resolvents
   8.On Line Co-ordinates in Space.
   9.A Resolvent of the Twentieth Degree of Equations of the Fifth Degree
   10.Theory of the Surface of the Second Degree.
  Chapter III The Canonical Equations of the Fifth Degree
   1.Notation–The Fundamental Lemma.
   2.Determination of the Appropriate Parameter λ
   3.Determination of the Parameter μ.
   4.The Canonical Resolvent of the Icosahedral Equation.
   5.Solution of the Canonical Equations of the Fifth Degree
   6.Gordan’s Process
   7.Substitutions of the λ,μ’s—Invariant Forms.
   8.General Remarks on the Calculations WhichWe Have to Perform.
   9.Fresh Calculation of theMagnitude m1
   10.Geometrical Interpretation ofGordan’s Theory
   11.Algebraical Aspects (After Gordan)
   12.The Normal Equation of The rν’s
   13.Bring’s Transformation
   14.TheNormal Equation ofHermite.
  Chapter IV The Problem of the A’s and the Jacobian Equations of the Sixth Degree
   1.The Object of the Following Developments
   2.The Substitutions of the A’s—Invariant Forms.
   3.Geometrical Interpretation—Regulation of the Invariant Expressions
   4.The Problem of the A’s and Its Reduction.
   5.On the Simplest Resolvents of the Problem of the A’s
   6.The General Jacobian Equation of the SixthDegree
   7.Brioschi’s Resolvent
   8.Preliminary Remarks on the Rational Transformation of Our Problem.
   9.Accomplishment of the Rational Transformation
   10.Group-Theory Significance of Cogredience and Contragredience
   11.Introductory to the Solution of Our Problem.
   12.Corresponding Formulae.
  Chapter V The General Equation of the Fifth Degree
   1.Formulation of TwoMethods of Solution.
   2.Accomplishment of Our FirstMethod
   3.Criticismof theMethods of Bring andHermite.
   4.Preparation for Our SecondMethod of Solution.
   5.Of the Substitutions of the A,A’s—Definite Formulation
   6.The Formulae of Inversion ofOur SecondMethod
   7.Relations to Kronecker and Brioschi
   8.Comparison of Our TwoMethods.
   9.On theNecessity of the Accessory Square Root
   10.Special Equations of the Fifth DegreeWhich Can Be Rationally Reduced to an Icosahedral Equation.
   11.Kronecker’s Theorem.
Appendix

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