1st Edition
Multiplicative Analytic Geometry
This book is devoted to multiplicative analytic geometry. The book reflects recent investigations into the topic. The reader can use the main formulae for investigations of multiplicative differential equations, multiplicative integral equations and multiplicative geometry.
The authors summarize the most recent contributions in this area. The goal of the authors is to bring the most recent research on the topic to capable senior undergraduate students, beginning graduate students of engineering and science and researchers in a form to advance further study. The book contains eight chapters. The chapters in the book are pedagogically organized. Each chapter concludes with a section with practical problems.
Two operations, differentiation and integration, are basic in calculus and analysis. In fact, they are the infinitesimal versions of the subtraction and addition operations on numbers, respectively. In the period from 1967 till 1970, Michael Grossman and Robert Katz gave definitions of a new kind of derivative and integral, moving the roles of subtraction and addition to division and multiplication, and thus established a new calculus, called multiplicative calculus. Multiplicative calculus can especially be useful as a mathematical tool for economics and finance.
Multiplicative Analytic Geometry builds upon multiplicative calculus and advances the theory to the topics of analytic and differential geometry.
1 The Field R*
1.1 Definition
1.2 An Order in R?
1.3 Multiplicative Absolute Value
1.4 The Power Function
1.5 Multiplicative Trigonometric Functions
1.6 Multiplicative Inverse Trigonometric Functions
1.7 Multiplicative Hyperbolic Functions
1.8 Multiplicative Inverse Hyperbolic Functions
1.9 Multiplicative Matrices
1.10 Advanced Practical Problems
2 Multiplicative Plane Euclidean Geometry
2.1 The Multiplicative Vector Space R2 ?
2.2 The Multiplicative Inner Product Space R2 ?
2.3 The Multiplicative Euclidean Plane E?2
2.4 Multiplicative Lines
2.5 Multiplicative Orthonormal Pairs
2.6 Equations of a Multiplicative Line
2.7 Perpendicular Multiplicative Lines
2.8 Multiplicative Parallel and Intersecting Multiplicative Lines
2.9 Multiplicative Reflections
2.10 Multiplicative Congruence and Multiplicative Isometries
2.11 Multiplicative Translations
2.12 Multiplicative Rotations
2.13 Multiplicative Glide Reflections
2.14 Structure of the Multiplicative Isometry Group
2.15 Fixed Points and Fixed Multiplicative Lines
2.16 Advanced Practical Problems
3 Multiplicative Affine Transformations in the Multiplicative
Euclidean Plane
3.1 Multiplicative Affine Transformations
3.2 Fixed Multiplicative Lines
3.3 The Fundamental Theorem
3.4 Multiplicative Affine Reflections
3.5 Multiplicative Shears
3.6 Multiplicative Dilatations
3.7 Multiplicative Similarities
3.8 Multiplicative Affine Symmetries
3.9 Multiplicative Rays and Multiplicative Angles
3.10 Multiplicative Rectilinear Figures
3.11 The Multiplicative Centroid
3.12 Multiplicative Symmetries of a Multiplicative Segment
3.13 Multiplicative Symmetries of a Multiplicative Angle
3.14 Multiplicative Barycentric Coordinates
3.15 Multiplicative Addition of Multiplicative Angles
3.16 Multiplicative Triangles
3.17 Multiplicative Symmetries of a Multiplicative Triangle
3.18 Congruence of Multiplicative Angles
3.19 Congruence Theorems for Multiplicative Triangles
3.20 Multiplicative Angle Sum of Multiplicative Triangles
3.21 Advanced Practical Problems
4 Finite Groups of Multiplicative Isometries of E?2
4.1 Cyclic and Dihedral Groups
4.2 Conjugate Subgroups
4.3 Orbits and Stabilizers
4.4 Regular Multiplicative Polygons
4.5 Similar Regular Multiplicative Polygons
4.6 Advanced Practical Problems
5 Multiplicative Geometry on the Multiplicative Sphere
5.1 The Space E?3
5.2 The Multiplicative Cross Product
5.3 Multiplicative Orthonormal Bases
5.4 Multiplicative Planes
5.5 Incidence Multiplicative Geometry of the Multiplicative Sphere
5.6 The Multiplicative Distance
5.7 Multiplicative Motions on S?2
5.8 Multiplicative Orthogonal Transformations
5.9 The Euler Theorem
5.10 Multiplicative Isometries
5.11 Multiplicative Segments
5.12 Multiplicative Rays, Multiplicative Angles and Multiplicative Triangles
5.13 Multiplicative Spherical Trigonometry
5.14 A Multiplicative Congruence Theorem
5.15 Multiplicative Right Triangles
5.16 Advanced Practical Problems
6 The Projective Multiplicative Plane P?2
6.1 Definition. Incidence Properties of P?2
6.2 Multiplicative Homogeneous Coordinates
6.3 The Desargues Theorem. The Pappus Theorem
6.4 The Projective Multiplicative Group
6.5 The Fundamental Theorem of the Projective Multiplicative Geometry
6.6 Multiplicative Polarities
6.7 Multiplicative Cross Product
6.8 Advanced Practical Problems
7 The Multiplicative Distance Geometry on P?2
7.1 The Multiplicative Distance
7.2 Multiplicative Isometries
7.3 Multiplicative Motions
7.4 Elliptic Multiplicative Geometry
7.5 Advanced Practical Problems
8 The Hyperbolic Multiplicative Plane
8.1 Introduction
8.2 Definition of H2?
8.3 Multiplicative Perpendicular Lines
8.4 Multiplicative Distance of H?2
8.5 Multiplicative Isometries
8.6 Multiplicative Reflections of H?2
8.7 Multiplicative Motions
8.8 Multiplicative Reflections
8.9 Multiplicative Parallel Displacements
8.10 Multiplicative Translations
8.11 Multiplicative Glide Reflections
8.12 Multiplicative Angles, Multiplicative Rays and Multiplicative
Triangles
8.13 Advanced Practical Problems
References
Index
Biography
Svetlin G. Georgiev (born 05 April 1974, Rouse, Bulgaria) is a mathematician who has worked in various areas of mathematics. He currently focuses on harmonic analysis, functional analysis, partial differential equations, ordinary differential equations, Clifford and quaternion analysis, integral equations and dynamic calculus on time scales.
Khaled Zennir was born in Skikda, Algeria, in 1982. He received his PhD in Mathematics in 2013 from Sidi Bel Abbès University, Algeria (Assist. Professor). He obtained his highest diploma in Algeria (Habilitation, Mathematics) from Constantine University, Algeria, in May 2015 (Assoc. Professor). He is now Associate Professor at Qassim University, KSA. His research interests lie in nonlinear hyperbolic partial differential equations: global existence, blow-up and long time behavior.
Aissa Boukarou received his PhD in Mathematics in 2021 from Ghardaia University, Algeria (Assist. Professor). His research interests lie in partial differential equations, harmonic analysis, stochastic PDE and numerical analysis.
