Geometry and Discrete Mathematics A Selection of Highlights 2nd Edition by Benjamin Fine, Anja Moldenhauer, Gerhard Rosenberger, Annika Schürenberg, Leonard Wienke ISBN 9783110740783 3110740788

Geometry and Discrete Mathematics A Selection of Highlights 2nd Edition by Benjamin Fine, Anja Moldenhauer, Gerhard Rosenberger, Annika Schürenberg, Leonard Wienke – Ebook PDF Instant Download/Delivery: 9783110740783, 3110740788
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Product details:

ISBN 10: 3110740788
ISBN 13: 9783110740783
Author: Benjamin Fine, Anja Moldenhauer, Gerhard Rosenberger, Annika Schürenberg, Leonard Wienke

Geometry and Discrete Mathematics A Selection of Highlights 2nd Edition Table of contents:

1 Geometry and Geometric Ideas

1.1 Geometric Notions, Models and Geometric Spaces

1.1.1 Geometric Notions

1.2 Overview of Euclid’s Method and Approaches to Geometry

1.2.1 Incidence Geometries – Affine Geometries, Finite Geometries, Projective Geometries

1.3 Euclidean Geometry

1.3.1 Birkhoff’s Axioms for Euclidean Geometry

1.4 Neutral or Absolute Geometry

1.5 Euclidean and Hyperbolic Geometry

1.5.1 Consistency of Hyperbolic Geometry

1.6 Elliptic Geometry

1.7 Differential Geometry

1.7.1 Some Special Curves

1.7.2 The Fundamental Existence and Uniqueness Theorem

1.7.3 Computing Formulas for the Curvature, the Torsion and the Components of Acceleration

1.7.4 Integration of Planar Curves

Exercises

2 Isometries in Euclidean Vector Spaces and their Classification in ℝn

2.1 Isometries and Klein’s Erlangen Program

2.2 The Isometries of the Euclidean Plane ℝ2

2.3 The Isometries of the Euclidean Space ℝ3

2.4 The General Case ℝn with n≥2

Exercises

3 The Conic Sections in the Euclidean Plane

3.1 The Conic Sections

3.2 Ellipse

3.3 Hyperbola

3.4 Parabola

3.5 The Principal Axis Transformation

Applications

Exercises

4 Special Groups of Planar Isometries

4.1 Regular Polygons

4.2 Regular Tessellations of the Plane

4.3 Groups of Translations in the Plane ℝ2

4.4 Groups of Isometries of the Plane with Trivial Translation Subgroup

4.5 Frieze Groups

Case 1

Case 2

4.6 Planar Crystallographic Groups

Enumeration of the cases

4.7 A Non-Periodic Tessellation of the Plane ℝ2

Exercises

5 Graph Theory and Graph Theoretical Problems

5.1 Graph Theory

5.2 Coloring of Planar Graphs

5.3 The Marriage Theorem

5.4 Stable Marriage Problem

5.5 Euler Line

5.6 Hamiltonian Line

5.7 The Traveling Salesman Problem

Exercises

6 Spherical Geometry and Platonic Solids

6.1 Stereographic Projection

6.2 Platonic Solids

6.2.1 Cube (C)

6.2.2 Tetrahedron (T)

6.2.3 Octahedron (O)

6.2.4 Icosahedron (I)

6.2.5 Dodecahedron (D)

6.3 The Spherical Geometry of the Sphere S2

6.4 Classification of the Platonic Solids

Exercises

7 Linear Fractional Transformation and Planar Hyperbolic Geometry

7.1 Linear Fractional Transformations

7.2 A Model for a Planar Hyperbolic Geometry

7.3 The (Planar) Hyperbolic Theorem of Pythagoras in ℍ

7.4 The Hyperbolic Area of a Hyperbolic Polygon

Exercises

8 Simplicial Complexes and Topological Data Analysis

8.1 Simplicial Complexes

8.2 Sperner’s Lemma

8.3 Simplicial Homology

8.4 Persistent Homology

Exercises

9 Combinatorics and Combinatorial Problems

9.1 Combinatorics

9.2 Basic Techniques and the Multiplication Principle

9.3 Sizes of Finite Sets and the Sampling Problem

9.3.1 The Binomial Coefficients

9.3.2 The Occupancy Problem

9.3.3 Some Further Comments

9.4 Multinomial Coefficients

9.5 Sizes of Finite Sets and the Inclusion–Exclusion Principle

9.6 Partitions and Recurrence Relations

9.7 Decompositions of Naturals Numbers, Partition Function

9.8 Catalan Numbers

9.9 Generating Functions

9.9.1 Ordinary Generating Functions

9.9.2 Exponential Generating Functions

Exercises

10 Finite Probability Theory and Bayesian Analysis

10.1 Probabilities and Probability Spaces

10.2 Some Examples of Finite Probabilities

10.3 Random Variables, Distribution Functions and Expectation

10.4 The Law of Large Numbers

10.5 Conditional Probabilities

10.6 The Goat or Monty Hall Problem

10.7 Bayes Nets

Exercises

11 Boolean Lattices, Boolean Algebras and Stone’s Theorem

11.1 Boolean Algebras and the Algebra of Sets

11.2 The Algebra of Sets and Partial Orders

11.3 Lattices

11.4 Distributive and Modular Lattices

11.5 Boolean Lattices and Stone’s Theorem

11.6 Construction of Boolean Lattices via 0–1 Sequences

11.7 Boolean Rings

11.8 The General Theorem of Stone

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Tags: Benjamin Fine, Anja Moldenhauer, Gerhard Rosenberger, Annika Schürenberg, Leonard Wienke, Discrete Mathematics