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Isoz. Opera Magistris. Compendium of Elementary Applied Mathematics for Engineers. 3e. 2018.
0 CONTENTS
Contents & Chapter 3 Introduction V1 PDF
1 WARNINGS
1.1 Impressum
1.1.1 Use of content
1.1.2 How to use this book
1.1.3 Data Protection
1.1.4 Use of data
1.1.5 Data transmission
1.1.6 Agreement
1.1.7 Errata
1.2 License
1.2.1 Preamble
1.2.2 Applicability and Definitions
1.2.3 Verbatim Copying
1.2.4 Copying in Quantity
1.2.5 Modifications
1.2.6 Combining Documents
1.2.7 Collections of Documents
1.2.8 Aggregation with independent Works
1.2.9 Generating this document
1.2.10 Translation
1.2.11 Termination
1.2.12 Future revisions of this License
1.3 Roadmap
2 ACKNOWLEDGEMENTS
3 INTRODUCTION
3.1 FOREWORDS
3.1.1 Motivation and goals
3.2 METHODS
3.2.1 Descartes’ Method
3.2.1.1 Blind studies
3.2.2 Research Integrity and Engineering/Scientific Ethics
3.2.2.1 Singapore Statement on Research Integrity
3.2.2.2 European Code of Conduct for Research Integrity
3.2.2.3 Archimedean Oath
3.2.3 Scientific Publication Rules (SPR)
3.2.4 Scientific Mainstream Media communication
3.2.4.1 Social Networks
3.2.4.2 Expert Opinions
3.3 VOCABULARY
3.3.1 On Sciences
3.3.2 Terminology
3.4 SCIENCE & FAITH
3.4.1 Baloney detection kit
3.5 SCIENTIFIC COMMUNICATION BACKFIRE
4 ARITHMETIC
4.1 PROOF THEORY
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4.1.1 Foundations Crisis
4.1.2 Paradoxes
4.1.2.1 Hypothetical-Deductive Reasoning
4.1.3 Propositional Calculus
4.1.3.1 Propositions (premises)
4.1.3.2 Connectors
4.1.3.3 Decision procedures
4.1.3.4 Quantifiers
4.1.4 Predicate Calculus
4.1.4.1 Grammar
4.1.4.2 Languages
4.1.5 Proofs
4.1.5.1 Rules of Proofs
4.2 NUMBERS
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4.2.1 Digital Bases
4.2.2 Type of Numbers
4.2.2.1 Natural Integer Numbers
4.2.2.2 Relative Integer Numbers
4.2.2.3 Rational Numbers
4.2.2.4 Irrational Numbers
4.2.2.5 Real Numbers
4.2.2.6 Transfinite Numbers
4.2.2.7 Complex Numbers
4.2.2.8 Quaternion Numbers
4.2.2.9 Algebraic and Transcendental Numbers
4.2.2.10 Universe Numbers (normal numbers)
4.2.2.11 Abstract Numbers (variables)
4.3 ARITHMETIC OPERATORS
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4.3.1 Binary Relations
4.3.1.1 Equalities
4.3.1.2 Comparators
4.3.2 Fundamental Arithmetic Laws
4.3.2.1 Addition
4.3.2.2 Subtraction
4.3.2.3 Multiplication
4.3.2.4 Division
4.3.3 Arithmetic Polynomials
4.3.4 Absolute Value
4.3.5 Calculation Rules (operators priorities)
4.4 NUMBER THEORY
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4.4.1 Principle of good order
4.4.2 Induction Principle
4.4.3 Divisibility
4.4.3.1 Euclidean Division
4.4.3.2 Euclidean Algorithm
4.4.3.3 Least Common Multiple
4.4.3.4 Fundamental Theorem of Arithmetic
4.4.3.5 Congruences (modular arithmetic)
4.4.3.6 Continued fraction
4.5 SET THEORY
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4.5.1 Zermelo-Fraenkel Axiomatic
4.5.1.1 Cardinals
4.5.1.2 Cartesian Product
4.5.1.3 Intervals
4.5.2 Set Operations
4.5.2.1 Inclusion
4.5.2.2 Intersection
4.5.2.3 Union
4.5.2.4 Difference
4.5.2.5 Symmetric Difference
4.5.2.6 Product
4.5.2.7 Complementarity
4.5.3 Functions and Applications
4.5.3.1 Cantor-Bernstein Theorem
4.5.4 Structures
4.5.4.1 Magma
4.5.4.2 Monoid
4.5.4.3 Groups
4.5.4.4 Ring
4.5.4.5 Field
4.5.4.6 Vector Spaces
4.5.4.7 C-algebra A
4.5.4.8 Summary
4.5.5 Morphisms
4.5.5.1 Ideal
4.6 PROBABILITIES
4.6.1 Event Universe
4.6.1.1 Infinite monkey theorem
4.6.1.2 Do probabilities refute complex structures?
4.6.2 Kolmogorov’s Axioms
4.6.3 Conditional Probabilities
4.6.3.1 Conditional Expectation and Variance
4.6.3.2 Bayesian Networks
4.6.4 Martingales
4.6.4.1 Gambling
4.6.5 Combinatorial Analysis
4.6.5.1 Simple Arrangements with Repetitions
4.6.5.2 Simple Permutations without Repetitions
4.6.5.3 Simple Permutations with Repetitions
4.6.5.4 Simple Arrangements without Repetitions
4.6.5.5 Simple Combinations without Repetitions
4.6.5.6 Simple Combinations with Repetitions
4.6.6 Markov Chains
4.7 STATISTICS
4.7.1 Samples
4.7.2 Averages
4.7.2.1 Laplace Smoothing
4.7.2.2 Means and Averages properties
4.7.3 Type of variables
4.7.3.1 Discrete Variables and Moments
4.7.3.2 Continuous Variables and Moments
4.7.4 Fundamental postulate of statistics
4.7.5 Diversity Index
4.7.6 Distribution Functions (probabilities laws)
4.7.6.1 Discrete Uniform Distribution
4.7.6.2 Bernoulli Distribution
4.7.6.3 Geometric Distribution
4.7.6.4 Binomial Distribution
4.7.6.5 Negative Binomial Distribution
4.7.6.6 Hypergeometric Distribution
4.7.6.7 Multinomial Distribution
4.7.6.8 Poisson Distribution
4.7.6.9 Normal & Gauss-Laplace Distribution
4.7.6.10 Log-Normal Distribution
4.7.6.11 Continuous Uniform Distribution
4.7.6.12 Triangular Distribution
4.7.6.13 Pareto Distribution
4.7.6.14 Exponential Distribution
4.7.6.15 Cauchy Distribution
4.7.6.16 Beta Distribution
4.7.6.17 Gamma Distribution
4.7.6.18 Chi-Square (Pearson) Distribution
4.7.6.19 Student Distribution
4.7.6.20 Fisher Distribution
4.7.6.21 General Folded Normal Distribution
4.7.6.22 Laplace Distribution
4.7.6.23 Benford Distribution
4.7.7 Likelihood Estimators
4.7.7.1 Normal Distribution MLE
4.7.7.2 Poisson Distribution MLE
4.7.7.3 Binomial Distribution MLE
4.7.7.4 Geometric Distribution MLE
4.7.7.5 Weibull Distribution MLE
4.7.7.6 Gamma Distribution MLE
4.7.7.7 Pareto Distribution MLE
4.7.7.8 Censored data MLE
4.7.7.9 Fisher Information Matrix
4.7.8 Finite Population Correction Factor
4.7.9 Confidence Intervals (inference)
4.7.9.1 Frequentist Inference
4.7.9.2 Bayesian Inference
4.7.10 Weak Law of Large Numbers
4.7.11 Characteristic Function
4.7.12 Central Limit Theorem
4.7.13 Univariate Hypothesis and Adequation tests (NHST)
4.7.13.1 Direction of hypothesis test and p-values
4.7.13.2 Fisher’s method for multiple p-values
4.7.13.3 Power of a test
4.7.13.4 Power of the one sample Z-test
4.7.13.5 Power of the one and two samples P-test
4.7.13.6 A/B tests
4.7.13.7 Analysis Of VAriance (ANOVA)
4.7.13.8 Equivalence tests
4.7.13.9 Cochran C test
4.7.13.10 Adequation Tests (Goodness of Fit tests)
4.7.13.11 Kernel density estimation
4.7.13.12 Likelihood-ratio tests
4.7.14 Robustness and Nonparametric Statistics
4.7.14.1 Rank Statistics
4.7.14.2 Range Statistics
4.7.14.3 Extreme Value Theory
4.7.15 Multivariate Statistics
4.7.15.1 Principal Component Analysis
4.7.15.2 Correspondence Factorial Analysis (AFC)
4.7.15.3 Chi-2 Test of Independence
4.7.15.4 Cramér’s V
4.7.15.5 Pearson’s phi coefficient
4.7.15.6 Exact Fisher’s Test
4.7.15.7 Cohen’s kappa agreement
4.7.15.8 McNemar’s test
4.7.16 Survival Statistics
4.7.16.1 Kaplan-Meier Survival Rate
4.7.16.2 Cochran–Mantel–Haenszel tests
4.7.16.3 Cox (Proportional Hazard) Model
4.7.17 Propagation of Errors (experimental uncertainty analysis)
4.7.17.1 Absolute and Relative Uncertainties (direct calculation of bias)
4.7.17.2 Statistical Errors
4.7.17.3 Repeatability
4.7.17.4 Error propagation (linearized approximation)
4.7.17.5 Error propagation (statistical approach)
4.7.17.6 Significant Numbers
4.7.18 Spatial statistics
4.7.18.1 2D Poisson’s spatial distance model
4.7.19 Surveys
4.7.19.1 Survey designs
4.7.19.2 Survey Sampling Error and Non-Sampling Error
4.7.20 A World without statistics
4.7.20.1 Data fallacies .
5 ALGEBRA
5.1 CALCULUS
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5.1.1 Equations and Inequations
5.1.1.1 Equations
5.1.1.2 Inequations
5.1.2 Remarkable Identities
5.1.3 Polynomials
5.1.3.1 Euclidean Division of Polynomials
5.1.3.2 Factorization Theorem of Polynomials
5.1.3.3 Diophantine equation
5.1.3.4 First order univariate Polynomial and Equations
5.1.3.4.1 Solving by circular reference
5.1.3.5 Second order univariate Polynomial and Equations
5.1.3.5.1 Irrational Equations
5.1.3.5.2 Golden Ratio
5.1.3.6 Third order univariate Polynomial and Equations
5.1.3.7 Fourth order univariate Polynomial and Equations
5.1.3.8 Trigonometric Polynomials
5.1.3.9 Cyclotomic Polynomials
5.1.3.10 Legendre Polynomials
5.2. SET ALGEBRA
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5.2.1 Groups Algebra & Geometry
5.2.1.1 Cyclic Groups
5.2.1.2 Transformations Groups
5.2.1.3 Group of Symetries
5.2.1.4 Permutations Groups
5.2.2 Galois Theory
5.2.2.1 Elementary symmetric and Invariant Polynomials
5.2.2.2 General Vieta’s formulas
5.3. DIFFERENTIAL & INTEGRAL CALCULUS
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5.3.1 Differential Calculus
5.3.1.1 Differentials
5.3.1.2 Usual Derivatives
5.3.1.3 Implicit Differentiation
5.3.1.4 Smoothness
5.3.2 Integral Calculus
5.3.2.1 Definite Integral
5.3.2.2 Indefinite Integral
5.3.2.3 Double Integral
5.3.2.4 Integration by Substitution
5.3.2.5 Integration by Parts
5.3.2.6 Usual Primitives
5.3.2.7 Integral representation of first kind Bessel’s function
5.3.2.8 Dirac Function
5.3.2.9 Gamma Euler Function
5.3.2.10 Curvilinear Integrals
5.3.2.11 Integrals involving parametric equations
5.3.2.12 Improper Integrals
5.3.2.13 Elliptic Integrals
5.3.3 Differential Equations
5.3.3.1 First order differential equations
5.3.3.2 Linear differential equations
5.3.3.3 Resolution methods of differential equations
5.3.3.4 Classification of partial differential equations
5.3.4 Systems of Differential Equations
5.3.5 Regular Methods of Perturbations
5.3.5.1 Perturbation theory for algebraic equations
5.3.5.2 Perturbation theory of differential equations
5.4. SEQUENCES & SERIES
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5.4.1 Sequences
5.4.1.1 Arithmetic Sequences
5.4.1.2 Harmonic Sequences
5.4.1.3 Geometric Sequences
5.4.1.4 Cauchy Sequence
5.4.1.5 Fibonacci Sequence
5.4.1.6 Logic Sequences/Psychologist Sequences
5.4.2 Series
5.4.2.1 Gauss Series
5.4.2.2 Arithmetic Series
5.4.2.3 Geometric Series
5.4.2.4 Telescoping Series
5.4.2.5 Grandi’s Series
5.4.2.6 Taylor and Maclaurin Series
5.4.2.7 Fourier Series (trigonometric series)
5.4.2.8 Bessel Series
5.4.3 Convergence Criteria
5.4.3.1 Integral Test
5.4.3.2 D’Alembert Rule
5.4.3.3 Alternating Series Test
5.4.3.4 Fixed Point Theorem
5.4.4 Generating Functions (transformation of a sequence into a series)
5.4.4.1 Ordinary Generating Functions (transformation of a sequence into a series)
5.4.4.2 Multivariate Generating Functions
5.4.4.3 Functional Generating Functions
5.5. VECTOR CALCULUS
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5.5.1 Concept of Arrow
5.5.2 Set of Vectors
5.5.2.1 Pseudo-Vectors
5.5.2.2 Normal vector
5.5.2.3 Multiplication by a scalar .
5.5.3 Vector Spaces
5.5.3.1 Linear Combinations
5.5.3.2 Sub-vector spaces
5.5.3.3 Generating families
5.5.3.4 Linear Dependence or Independence
5.5.3.5 Base of a vectorial space
5.5.3.6 Direction Angles
5.5.3.7 Dimensions of a vector space
5.5.3.8 Extension of a free family
5.5.3.9 Rank of a finite family
5.5.3.10 Direct Sums
5.5.3.11 Affine spaces
5.5.4 Euclidean Vector Spaces
5.5.4.1 Scalar Product (Dot Product)
5.5.4.2 Cross Product
5.5.4.3 Mixed Product (triple product)
5.5.5 Vectorial Functional Space
5.5.6 Hermitian Vector Space
5.5.6.1 Hermitian Inner Product
5.5.6.2 Types of Vectors Spaces
5.5.7 System of Coordinates
5.5.7.1 Cartesian (rectangular) Coordinate System
5.5.7.2 Spherical Coordinate System
5.5.7.3 Cylindrical Coordinate System
5.5.7.4 Polar Coordinate System
5.5.8 Differential Operators
5.5.8.1 Gradients of Scalar Field
5.5.8.2 Gradients of Vector Field
5.5.8.3 Divergences of a Vector Field
5.5.8.4 Rotationals of a Vector Field (Curl)
5.5.8.5 Laplacians of Scalar Fields (Laplace Operator)
5.5.8.6 Laplacians of Vector Fields
5.5.8.7 Remarkable Identities
5.5.8.8 Summary
5.6. LINEAR ALGEBRA
5.6.1 Linear Systems
5.6.2 Linear Transformations
5.6.3 Matrices
5.6.3.1 Rank of a matrix
5.6.3.2 Matrix Algebra
5.6.3.3 Type of Matrices
5.6.3.4 Determinant
5.6.4 Change of basis (frames)
5.6.5 Eigenvalues and Eigenvectors
5.6.5.1 Rotation Matrices and Eigenvalues
5.6.6 Spectral Theorem
5.6.7 Matrix Decompositions
5.6.7.1 Singular Value Decomposition (SVD)
5.6.7.2 LU Decomposition
5.6.7.3 Cholesky’s Decomposition
5.6.7.4 QR Decomposition
5.7. TENSOR CALCULUS
5.7.1 Tensor
5.7.2 Indicial Notation
5.7.2.1 Summation on multiple index
5.7.2.2 Kronecker Symbol
5.7.2.3 Antisymmetric Symbol (Levi-Civita symbol)
5.7.3 Metric and Signature
5.7.4 Gram’s Determinant
5.7.5 Contravariant and Covariant Components
5.7.6 Operation in Basis
5.7.6.1 Gram-Schmidt Orthogonalization Method
5.7.6.2 Change of Basis
5.7.6.3 Reciprocal Basis (Dual Basis)
5.7.7 Euclidean Tensors (cartesian tensor)
5.7.7.1 Fundamental Tensor
5.7.7.2 Tensor product (dyadic) of two vectors and matrices
5.7.7.3 Tensor Spaces
5.7.7.4 Linear combination of tensors
5.7.7.5 Contraction of indices
5.7.8 Special Tensors
5.7.8.1 Symmetric Tensor
5.7.8.2 Antisymmetric Tensor
5.7.8.3 Fundamental Tensor
5.7.9 Curvilinear Coordinates
5.7.9.1 Natural basis in spherical coordinates (curvilinear basis in spherical coordinates)
5.7.9.2 Natural basis in polar coordinates (curvilinear basis in polar coordinates)
5.7.9.3 Natural basis in cylindrical coordinates (cylindrical basis in polar coordinates)
5.7.10 Christoffel symbols
5.7.11 Ricci Theorem
5.7.12 Riemann-Christoffel symbols
5.7.13 Ricci curvature (Ricci tensor)
5.7.14 Einstein Tensor
5.8. SPINOR CALCULUS
5.8.1 Unit Spinor
5.8.2 Geometric Properties
5.8.2.1 Plane Symmetries
5.8.2.2 Rotations
5.8.2.3 Properties of Pauli Matrices
6 ANALYSIS
6.1 FUNCTIONAL ANALYSIS
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6.1.1 Representations
6.1.1.1 Tabular Representation
6.1.1.2 Graphical Representation
6.1.1.3 Analytical Representation
6.1.2 Functions
6.1.2.1 Limits and Continuity of Functions
6.1.2.2 Asymptotes
6.1.2.3 Concavity/Convexity of a function
6.1.3 Logarithms
6.1.4 Convolutions
6.1.4.1 Continuous and Discrete Linear Convolution Product
6.1.4.2 Matrix Convolution
6.1.5 Integral Transforms
6.1.5.1 Fourier Transform
6.1.5.2 Laplace Transform
6.1.5.3 Z-Transform
6.1.5.4 Hilbert Transform
6.1.6 Functional dot product (inner product)
6.1.6.1 Cauchy-Schwarz inequality for integrals
6.2 COMPLEX ANALYSIS
6.2.1 Linear Applications
6.2.2 Holomorphic Functions
6.2.2.1 Orthogonality of real and imaginary iso-curves
6.2.3 Complex Logarithm
6.2.4 Complex Integral Calculus
6.2.4.1 Convergence of a complex series
6.2.5 Path Decomposition
6.2.5.1 Inverse Path
6.2.6 Laurent Series
6.2.7 Singularities
6.2.8 Residue Theorem
6.2.8.1 Pole at infinity
6.3 TOPOLOGY
6.3.1 General Topology
6.3.1.1 Topological Spaces
6.3.2 Metric Space and Distance
6.3.2.1 Equivalent Distances
6.3.2.2 Lipschitz Functions
6.3.2.3 Continuity and Uniform Continuity
6.3.3 Opened and Closed Set
6.3.3.1 Balls
6.3.3.2 Partititions
6.3.3.3 Formal Ball
6.3.3.4 Diameter
6.3.4 Varieties
6.3.4.1 Subvariety
6.3.4.2 Surfaces Homeomorphism
6.3.4.3 Differential Varieties
6.4 MEASURE THEORY
6.4.1 Measurable Spaces
6.4.1.1 Monotone Classes
7 GEOMETRY
7.1 TRIGONOMETRY
7.1.1 Radian
7.1.2 Circle Trigonometry
7.1.2.1 Remarkable trigonometric triangle identities
7.1.3 Hyperbolic Trigonometry
7.1.3.1 Remarkable hyperbolic identities
7.1.4 Spherical Trigonometry
7.1.5 Solid Angle
7.2 EUCLIDEAN GEOMETRY
7.2.1 Objects of Euclidean Geometry
7.2.1.1 Dimensions
7.2.2 Euclid’s Constructions
7.2.2.1 Segments and Lines
7.2.3 Plane Geometry
7.2.3.1 Displacements and Turnarounds
7.2.3.2 Plane angles
7.2.3.3 Triangles
7.2.3.4 Parallelism
7.2.3.5 Circle
7.2.4 Hilbert’s Axioms
7.2.4.1 Incidence Axioms (axioms of association)
7.2.4.2 Order Axioms
7.2.4.3 Congruence Axioms
7.2.4.4 Continuity Axioms
7.2.4.5 Parellels Axioms
7.2.5 Barycenter (centroid)
7.2.6 Geometric Transformations
7.2.6.1 Translation
7.2.6.2 Homothety (scaling)
7.2.6.3 Shear (skew) transformation
7.2.6.4 Rotation
7.2.6.5 Reflection
7.3 NON-EUCLIDEAN GEOMETRY
7.3.1 Curvature(s)
7.3.2 Axioms of non-euclidean geometry
7.3.3 Geodesic and Metric Equation
7.3.4 Riemann Spaces
7.4 PROJECTIVE GEOMETRY
7.4.1 Conical Perspective (Central Perspective)
7.4.1.1 Images of Points
7.4.1.2 Images of Straight Lines
7.4.2 Affine projections
7.4.2.1 Isometric perspective
7.4.2.2 Oblique perspective
7.4.2.3 Orthogonal projection
7.4.3 Spherical projections
7.4.3.1 Stereographic projection
7.4.3.2 Cylindrical projection
7.4.3.3 Mercator projection
7.4.3.4 Lambert’s equivalent projection (Peters projection)
7.4.4 Other perspectives
7.4.5 Homogeneous Coordinates (projection coordinates)
7.4.5.1 P2 Projective Space
7.4.5.2 P3 Projective Space
7.5 ANALYTICAL GEOMETRY
7.5.1 Conics
7.5.1.1 Algebraic approach
7.5.1.2 Geometric Approach
7.5.1.3 Dudelin Theorem (Dudelin Spheres)
7.5.1.4 Classification of conical by the determinant
7.5.2 Parametrizations
7.5.2.1 Equation of the Plane
7.5.2.2 Equation of the Straight line
7.5.2.3 Equation of a Square
7.5.2.4 Equation of a Cycloid
7.5.2.5 Equation of an Epicycloid
7.5.2.6 Equation of an Hypocycloid
7.5.2.7 Surface of revolution
7.6 DIFFERENTIAL GEOMETRY
7.6.1 Parametric Curves
7.6.2 Isolines
7.6.3 Frenet Frame
7.6.4 Surface Patchs
7.6.4.1 Metric of a Surface Patch
7.7 GEOMETRIC SHAPES
7.7.1 Usual Surfaces (Areas)
7.7.1.1 Polygons
7.7.1.2 Rectangle
7.7.1.3 Square
7.7.1.4 Unspecified Triangle
7.7.1.5 Isosceles Triangle
7.7.1.6 Equilateral Triangle
7.7.1.7 Right Triangle
7.7.1.8 Trapezoid
7.7.1.9 Parallelogram
7.7.1.10 Hexagon
7.7.1.11 Rhombus
7.7.1.12 Circle
7.7.1.13 Ellipse
7.7.2 Usual Volumes
7.7.2.1 Polyhedron
7.7.2.2 Usual Solids of Revolution
7.8 GRAPH THEORY
7.8.1 Type of Graphs and Structures
7.8.2 Graph Adjacency Matrix
7.8.3 Categories
7.9 KNOT THEORY
7.9.1 Braids Representation
7.9.1.1 Braids Group
7.9.2 Knot Representation
7.9.2.1 Knots Group
7.9.3 Tait’s Knot
7.9.4 Mathematical Formalisation
7.9.4.1 Planar Representation
8 MECHANICS
8.1 PRINCIPIA
8.1.1 System of Units
8.1.1.1 Dimensional Analysis
8.1.1.2 Scientific Notation and Metric Prefixes
8.1.1.3 Scales of Measurements
8.1.2 Distributions
8.1.3 Constants
8.1.3.1 Mathematical Constants
8.1.3.2 Universal Constants (fundamental constants)
8.1.3.3 Astronomical/Astrophysical parameters and constants
8.1.3.4 Chemical parameters
8.1.3.5 Material parameters
8.1.3.6 Planck’s constants
8.1.4 Principles of Physics
8.1.4.1 Principle of Causality
8.1.4.2 Principle of Conservation of Energy
8.1.4.3 Principle of Least Action
8.1.4.4 Noether’s Principle (Noether’s theorem)
8.1.4.5 Curie’s Principle
8.1.5 Point Spaces
8.2 ANALYTICAL/LAGRANGIAN MECHANICS
8.2.1 Lagrangian formalism
8.2.1.1 Generalized coordinates and frames
8.2.1.2 Variational Principle
8.2.1.3 Euler-Lagrange Equation
8.2.2 Canonical Formalism
8.2.2.1 Legendre Transform
8.2.2.2 Hamiltonian
8.2.2.3 Poisson bracket
8.2.2.4 Canonical transformations
8.3 CLASSICAL MECHANICS
8.3.1 Newton’s Laws
8.3.1.1 Newton’s First Law (Inertia Law)
8.3.1.2 Newton’s Second Law (Fundamental Principle of Dynamics)
8.3.1.3 Newton’s Third Law (Law of Action and Reaction)
8.3.2 Center of Mass and Reduced Weight
8.3.2.1 Center of Mass Theorem
8.3.2.2 Guldin’s Theorem
8.3.3 Kinematics of Rectilinear Motion
8.3.3.1 Position
8.3.3.2 Velocity
8.3.3.3 Acceleration
8.3.3.4 Galilean Relativity Principle
8.3.4 Angular Momentum
8.3.4.1 Moments
8.3.4.2 Static Forces
8.3.5 Ballistics
8.3.5.1 Ballistic equation without friction
8.3.5.2 Ballistic equation with friction
8.3.6 Kinematics of Circular Motion
8.3.7 Energy, Work and Power
8.3.7.1 Conservative vector field
8.3.7.2 Kinetic Energy
8.3.7.3 Gravitational Potential Energy
8.3.7.4 Conservation of Total Mechanical Energy
8.3.7.5 Conservation of Linear Momentum
8.3.7.6 Power
8.3.8 Relative Movements and Inertial Forces
8.3.8.1 Coriolis force and deflection magnitude
8.3.9 Oscillating Movements
8.3.9.1 Newton’s cradle
8.3.9.2 Simple Pendulum
8.3.9.3 Physical Pendulum
8.3.9.4 Elastic Pendulum (spring pendulum)
8.3.9.5 Conical Pendulum
8.3.9.6 Torsion Pendulum
8.3.9.7 Foucault’s Pendulum
8.3.9.8 Huygens’ Pendulum (and brachistochrone curve)
8.3.9.9 Double Pendulum
8.3.9.10 Inverted Pendulum
8.3.10 Tribology
8.3.10.1 Exponential Friction
8.3.10.2 Horizontal Viscous Friction
8.3.10.3 Vertical Viscous Friction
8.3.10.4 Stokes’ Vertical Viscous Friction
8.3.10.5 Stokes’ Horizontal Viscous Friction
8.3.10.6 Friction’s Heat Factor
8.4 WAVE MECHANICS
8.4.1 Wave Function
8.4.2 Wave Equation
8.4.3 Type of Waves
8.4.3.1 Periodic Waves
8.4.3.2 Harmonic Waves
8.4.3.3 Stationnary Waves
8.4.3.4 Vibration Modes in a Stretch String
8.4.4 Non-relativistic Lagrangian of a String
8.4.5 Vibrational modes of a circular membrane
8.4.6 Phasors
8.4.7 Superposition of periodic waves
8.4.7.1 AM/FM Broadcasting
8.5 STATISTICAL MECHANICS
8.5.1 Statistical Information Theory
8.5.1.1 Kullback–Leibler divergence
8.5.2 Boltzmann Law
8.5.3 Statistical Physics Distributions
8.5.3.1 Maxwell Distribution (velocity distribution)
8.5.3.2 Maxwell-Boltzmann Distribution
8.5.4 Brownian Motion (random walk)
8.6 THERMODYNAMICS
8.6.1 Thermodynamic Variables
8.6.2 Thermodynamics Systems
8.6.3 Thermodynamic Transformations
8.6.4 State Variables
8.6.4.1 Phases
8.6.5 Equation of State
8.6.5.1 Ideal Gaz Law
8.6.5.2 State equation of a Liquid
8.6.5.3 State equation of Solids
8.6.6 Laws of Thermodynamics
8.6.7 Calorific Capacities (heat capacity)
8.6.8 Internal Energy
8.6.8.1 Work (energy) of Mechanical Forces
8.6.8.2 Enthalpy
8.6.8.3 Laplace’s Law
8.6.8.4 Saint-Venant Thermodynamic Equation
8.6.8.5 Thermoelastics coefficients
8.6.9 Heat
8.6.9.1 Entropy
8.6.9.2 Carnot Cycle
8.6.10 Maxwell relations
8.6.11 Continuity Equation
8.6.11.1 Heat Equation
8.6.12 Thermal radiation
8.6.12.1 Black Body radiation
8.7 CONTINUUM MECHANICS
8.7.1 Rigid Bodies
8.7.1.1 Pressures
8.7.1.2 Elasticity of Solids
8.7.2 Liquids (fluid mechanics)
8.7.2.1 Pascal’s Fluid Theorem
8.7.2.2 Viscosity
8.7.2.3 Bernouilli’s Theorem
8.7.2.4 Navier-Stokes Equations
8.7.2.5 Hydrostatic Pressure
8.7.2.6 Archimedes’ principle
8.7.2.7 Speed of sound in a liquid
8.7.3 Gas
8.7.3.1 Types of Gas
8.7.3.2 Virial Theorem
8.7.3.3 Kinetic pressures (kinetic theory of gases)
8.7.3.4 Kinetic Temperature
8.7.3.5 Amagat and Dalton’s law
8.7.3.6 Mean free path (in kinetic theory)
8.7.3.7 Generalized Bernoulli equation
8.7.4 Plasmas
8.7.4.1 Plasma Frequency
9 ELECTROMAGNETISM
9.1 ELECTROSTATICS
9.1.1 Electric Force
9.1.2 Electric Potential
9.1.2.1 Path Independence
9.1.3 Equipotential and Field lines
9.1.3.1 Infinite straight wire
9.1.3.2 Electric Rigid Dipole
9.1.4 Electric Flux
9.1.4.1 Capacitor
9.1.5 Electrostatic potential energy
9.2 MAGNETOSTATICS
9.2.1 Ampere’s theorem
9.2.1.1 Infinitely long solenoid
9.2.1.2 Toroidal coils
9.2.1.3 Electromagnet
9.2.2 Maxwell-Ampere Relation
9.2.3 Biot-Savart law
9.2.3.1 Magnetic field for a current loop
9.2.3.2 Magnetic field for an infinite wire
9.2.4 Magnetic dipole
9.2.4.1 Magnetic torque
9.2.5 Lorentz law (Lorentz force)
9.2.5.1 Magnetic Vector Potential
9.2.5.2 Work of Magnetic Field
9.2.5.3 Classical Hall effect
9.2.5.4 Larmor radius
9.2.5.5 Energy of a magnetic dipole
9.2.6 Langevin treatment of Diamagnetism and Paramagnetism
9.2.6.1 Langevin model of diamagnetism
9.2.6.2 Langevin model of paramagnetism
9.3 ELECTRODYNAMICS
9.3.1 Maxwell Equations
9.3.1.1 First Maxwell Equation (constant electric flow)
9.3.1.2 Second Maxwell Equation (non-existence of magnetic monopole)
9.3.1.3 Third Maxwell Equation
9.3.1.4 Fourth Maxwell Equation
9.3.1.5 Magnetic Monopoles
9.3.2 Charge conservation equation
9.3.3 Gauge Theory
9.3.3.1 Electromagnetic field tensor
9.3.4 Electromagnetic wave equation
9.3.4.1 Helmholtz equation
9.3.4.2 Energy flow transportation (Poynting vector)
9.3.4.3 Emissions
9.3.5 Synchrotron radiation (bremsstrahlung)
9.3.5.1 Liénard-Wiechert potentials
9.3.5.2 Retarded Electric and Magnetic fields
9.3.6 Dipole antenna radiation pattern derivation (Hertz Electric Dipole)
9.3.6.1 The L = λ/2 (half-wave) far-field resonant antenna
9.4 ELECTROKINETICS
9.4.1 Kirchoff’s laws
9.4.1.1 Mesh law (Kirchhoff’s Loop Law)
9.4.1.2 Nodes law (Kirchhoff’s Point Law)
9.4.2 Drude model
9.4.3 Ohm’s law
9.4.3.1 Complex Conductivity
9.4.3.2 Equivalent Resistance
9.4.3.3 Equivalent Capacities
9.4.4 Electromotive Force
9.4.4.1 Faraday’s law of induction
9.4.4.2 Inductance
9.4.5 Skin effect
9.4.6 Semiconductors
9.4.6.1 Non-degenerated statistic density of negative electric charge carriers
9.4.6.2 Non-degenerated statistic density of positive electric charge carriers
9.4.6.3 Energy bands (conduction band, band gap, valence band)
9.4.6.4 Ohm’s law of Semiconductors
9.4.7 Superconductivity
9.4.7.1 Meissner effect (second London’s equation)
9.5 OPTICS (RAY OPTICS)
9.5.1 Sources and Shadows
9.5.2 Colors
9.5.3 Radiometry/Photometry
9.5.3.1 Energy flow
9.5.3.2 Light Intensity (Radiant Intensity)
9.5.3.3 Energy Emittance (Radiant Emittance)
9.5.3.4 Radiance and Luminance
9.5.3.5 Kirchhoff’s law of Radiation
9.5.3.6 Spectral Decomposition
9.5.4 Law of Refraction
9.5.4.1 Refractive index
9.5.4.2 Snell’s law
9.5.4.3 Cherenkov radiation
9.5.5 Descartes’ Formulas
9.5.5.1 Stigmatism
9.5.5.2 Lenses
9.5.5.3 Triangular Prism
9.5.5.4 Pentaprism
9.5.6 Rainbow
9.6 WAVE OPTICS
9.6.1 Huygens’ principle
9.6.2 Fraunhofer Diffraction
9.6.2.1 Case of a rectangular aperture
9.6.2.2 Case of a network of rectangular apertures
9.6.2.3 Young’s interference experiment
9.6.3 Light polarization
9.6.3.1 Linear polarization
9.6.3.2 Elliptical polarization
9.6.3.3 Circular polarization
9.6.3.4 Natural polarization
9.6.3.5 Malus’ law
9.6.4 Coherence and interference
9.6.5 LASER
10 ATOMISTIC
10.1 CORPUSCULAR QUANTUM PHYSICS
10.1.1 Dalton’s model
10.1.2 Thomson’s model
10.1.3 Rhuterfords’s model
10.1.4 Bohr’s Model
10.1.4.1 Bohr’s Postulates
10.1.4.2 Quantification
10.1.4.3 Hydrogen Type Atoms Model without dragging
10.1.4.4 Hydrogen Type Atoms Model with dragging
10.1.4.5 Neutron Assumption
10.1.5 Wilson and Sommerfeld’s Model
10.1.6 Relativistic Sommerfeld Model
10.1.6.1 Magnetic dipole moment
10.1.6.2 Spin
10.1.6.3 Pauli exclusion principle
10.1.7 Electron configuration (atomic orbital)
10.2 WAVE QUANTUM PHYSICS
10.2.1 Postulates
10.2.1.1 1st Postulate: Quantum State
10.2.1.2 2nd postulate: Time evolution of a quantum state
10.2.1.3 3rd postulate: Observables and operators
10.2.1.4 4th postulate: Measure of a property
10.2.1.5 5th postulate: Average of a property
10.2.2 Classical principles of uncertainty
10.2.2.1 First classical uncertainty relation
10.2.2.2 Second classical uncertainty relation
10.2.2.3 Third classical uncertainty relation
10.2.3 Quantum algebra
10.2.3.1 Linear functional operators
10.2.3.2 Robertson uncertainty relations
10.2.4 Schrödinger Model
10.2.4.1 de Broglie associated wave
10.2.4.2 Classical Schrödinger Wave Equation
10.2.4.3 Classical Shrödinger equation of evolution
10.2.4.4 Implications and Applications
10.2.4.5 Superposition principle
10.2.4.6 Ehrenfest theorem
10.2.5 Angular momentum and Spin
10.2.5.1 Spin–orbit interaction (LS coupling)
10.2.5.2 Spin operator for spin 1/2 particles
10.2.6 Planck dimensions
10.2.7 Wave Quantum Physics Interpretation
10.2.7.1 Copenhagen interpretation
10.2.7.2 De Broglie-Bohm (pilot-wave) interpretation
10.3 RELATIVISTIC QUANTUM PHYSICS
10.3.1 Relativistic Schrödinger evolution equation
10.3.1.1 Antimatter
10.3.2 Generalized Klein-Gordon Equation
10.3.2.1 Generalized Klein-Gordon Equation continuity equation
10.3.3 Classical free Dirac equation
10.3.3.1 Gamma, alpha, beta matrices
10.3.3.2 Free particle solution
10.3.3.3 Free particle solution at rest
10.3.3.4 Majorana equation
10.3.4 Linearized Dirac Equation
10.3.4.1 Linearized Dirac Equation continuity equation
10.3.4.2 Linearized Dirac Equation Angular Momentum + Helicity Conservation
10.3.5 Generalized Dirac Equation
10.3.6 Pauli Equation
10.3.6.1 Landé g factor
10.3.6.2 Lamb Shift
10.4 NUCLEAR PHYSICS
10.4.1 Nuclear Weapon
10.4.2 Radioactivity
10.4.2.1 Disintegration
10.4.2.2 Activity
10.4.2.3 Two level radioactive cascade
10.4.2.4 Radioactive phenomena
10.4.3 Radiation protection
10.4.3.1 Bether formula
10.4.3.2 Compton scattering
10.4.3.3 Photoelectric effect
10.4.3.4 Rutherford scattering
10.4.3.5 X-rays and Gamma rays
10.4.4 Liquid Drop Model of Nucleus
10.4.4.1 Volumic binding energy
10.4.4.2 Superficial binding energy
10.4.4.3 Coulomb electrostatic repulsive energy
10.4.4.4 Energy of asymmetry (Pauli energy)
10.4.4.5 Pairing Energy
10.5 QUANTUM FIELD THEORY
10.5.1 Yukawa potential
10.5.1.1 Mass fields
10.5.1.2 Non-mass fields
10.5.2 Euler-Lagrange equation for Fields
10.5.3 Gauge Theories
10.5.3.1 Global Gauge invariance
10.5.3.2 Local Gauge invariance
10.6 ELEMENTARY PARTICLE PHYSICS
10.6.1 Coupling Constants
10.6.2 Spin magnetic resonance
11 COSMOLOGY
11.1 ASTRONOMY (CELESTIAL MECHANICS)
11.1.1 Drake Equation
11.1.2 Kepler’s Laws
11.1.2.1 First Kepler’s Law (conicity law)
11.1.2.2 Second Kepler’s Law (area law)
11.1.2.3 Third Kepler’s Law (periods’ law)
11.1.3 Newton’s Gravitational Law
11.1.3.1 Gaussian Formulation of Newtonian Gravity
11.1.3.2 Shell Theorem
11.1.3.3 Orbital speed
11.1.3.4 Asteroids/Meteors impact velocity
11.1.3.5 Spherisation of Celestial Bodies
11.1.3.6 Stability of Atmospheres
11.1.4 Roche’s Limit
11.1.5 Keplerian Orbitals
11.1.5.1 First Binet Formula
11.1.5.2 Second Binet Formula
11.1.5.3 Keplerian orbital period
11.1.5.4 Classical deflection of light (light bending)
11.1.5.5 Classical precession of perihelion
11.1.6 Duration of the diurnal arc
11.1.7 Trigonometric parallax
11.1.8 Planets’ Motion
11.1.8.1 Synodic and Sidereal period
11.1.8.2 Planet’s apparent retrograde motion
11.1.9 Lagrange Points
11.1.9.1 Equilibrium points of the first type
11.1.9.2 Equilibrium points of the second type
11.1.10 Relativistic Doppler-Fizeau Effect
11.1.10.1 Apparent speed
11.2 ASTROPHYSICS
11.2.1 Stars
11.2.1.1 Stellar Physics
11.2.1.2 Pulsative Variable Stars
11.2.1.3 Neutron Stars (magnetars)
11.2.2 Galaxies
11.2.2.1 Probability of collisions in merging galaxies
11.2.2.2 Radial Speed Anomaly
11.3 SPECIAL RELATIVITY
11.3.1 Assumptions and Principles
11.3.1.1 Postulate of Invariance
11.3.1.2 Cosmological Principle
11.3.1.3 Special Relativity Principle
11.3.2 Lorentz Transformations/Boost
11.3.2.1 Displacement four-vector
11.3.2.2 Velocity four-vector
11.3.2.3 Current four-vector
11.3.2.4 Acceleration four-vector
11.3.2.5 Relativistic sum of velocities
11.3.2.6 Relativistic lengths variation (length contraction)
11.3.2.7 Relativistic time variation (time dilatation)
11.3.2.8 Apparent relativistic mass
11.3.2.9 Relativistic electrodynamics
11.3.3 Minkowski space-time
11.3.3.1 Four-vectors
11.3.3.2 Universe light cone
11.4 GENERAL RELATIVITY
11.4.1 Assumptions and Principles
11.4.1.1 Equivalence Postulates
11.4.1.2 Mach Principle
11.4.2 Metrics
11.4.2.1 Schild Criteria (Einstein redshift effect Newtonian approach)
11.4.3 Equations of movement
11.4.3.1 Geodesic equations
11.4.3.2 Newtonian Limit
11.4.4 Stress-Energy Tensor
11.4.4.1 Stress-Energy Tensor for a perfect fluid
11.4.4.2 Electromagnetic stress–energy tensor
11.4.5 Einstein’s Field Equations
11.4.5.1 Cosmological Constant
11.4.5.2 Weak field approximation with cosmological constant
11.4.5.3 Einstein-Maxwell equations
11.4.5.4 Schwarzschild Solution
11.4.6 Experimental Tests
11.4.6.1 Gravitational Redshift
11.4.6.2 Precession of Mercury’s Perihelion
11.4.6.3 Deflection of Light (light bending)
11.4.6.4 Shapiro Effect (delay)
11.4.6.5 Hafele–Keating experiment (general relativity version)
11.4.6.6 Black Holes
11.4.6.7 Gravitational waves
11.4.7 Einstein-Hilbert action
11.4.7.1 Einstein-Hilbert action with cosmological constant
11.4.7.2 Einstein-Hilbert action with matter
11.5 COSMOGONY
11.5.1 Newtonian (ie non-relativistic) Cosmological Models
11.5.1.1 Hubble’s Law
11.5.2 Friedmann Equations (Newtonian derivation)
11.5.2.1 Critical Density .
11.5.3 Friedmann-Lemaitre Cosmological Models
11.5.3.1 Flat spaces (k = 0)
11.5.3.2 Spherical spaces (k > 0)
11.5.3.3 Hyperbolic spaces (k < 0)
11.5.4 Observable Universe
11.5.5 Cosmic Microwave Background (CMB)
11.5.5.1 Decoupling time
11.5.5.2 Background radiation temperature
11.5.5.3 Decoupling temperature
11.5.6 Friedmann-Lemaître-Robertson-Walker Cosmological Models
11.5.6.1 Robertson-Walker metric
11.5.6.2 Cosmological Redshift
11.5.6.3 Comobile Universe Diameter
11.5.6.4 General Friedmann equations
11.5.6.5 Einstein (static) Universe model
11.5.6.6 Friedmann Universe models
11.5.7 The Black Hole Universe
11.6 QUANTUM GRAVITY
11.6.1 Newton Quantum Gravity
11.7 STRING THEORY
11.7.1 Wave equation of a transervsal string
11.7.2 Non-relativistic Wave equation of a transversal string
11.7.2.1 Nambu-Goto Action
11.7.3 Lagrangian of a String
12 CHEMISTRY
12.1 QUANTUM CHEMISTRY
12.1.1 Infinite three-dimensional rectangular potential
12.1.2 Molecular Vibrations
12.1.3 Hydrogenoid Atom
12.1.4 Rigid Rotator
12.1.4.1 Potential Profile
12.2 MOLECULAR CHEMISTRY
12.2.1 Orbital Approximations
12.2.2 LCAO Method
12.2.3 Molecular Rotational Energy Levels
12.2.4 Molecular Vibrational Energy Levels
12.3 ANALYTICAL CHEMISTRY
12.3.1 Analytical chemistry process
12.3.2 Simple Mixtures
12.3.3 Reactions
12.4 THERMOCHEMISTRY
12.4.1 Chemical transformations
12.4.2 Molar Quantities
2.4.2.1 Standard enthalpy of reaction
13 THEORETICAL COMPUTING
13.1 NUMERICAL METHODS/ANALYSIS
13.1.1 Computer Representation of Numbers
13.1.1.1 Decimal System
13.1.1.2 Binary system
13.1.1.3 Hexadecimal System
13.1.1.4 Octal System
13.1.1.5 Conversion of decimal system to non-decimal system:
13.1.2 Algorithm Complexity
13.1.2.1 NP-Completude
13.1.3 Integer Part
13.1.4 Heron’s Square Root Algorithm
13.1.5 Archimedes Algorithm
13.1.6 Euler’s Number e
13.1.7 Stirling’s factorial approximation
13.1.8 Linear Systems of Equations
13.1.8.1 One equation with one unknown
13.1.8.2 Two equations with two unknowns
13.1.8.3 Three equations with three unknowns
13.1.8.4 n equations with n unknowns
13.1.9 Polynomials
13.1.10 Regression Techniques
13.1.10.1 Univariate linear regression model
13.1.10.2 Univariate linear regression Gaussian Model
13.1.10.3 Linear univariate regression forced through the origin(RTO)
13.1.10.4 Deming regression (orthogonal regression)
13.1.10.5 Multiple linear regression Gaussian Model
13.1.10.6 Ridge, LASSO and Elastic Net regularization
13.1.10.7 Polynomial regression
13.1.10.8 Kernel regression
13.1.10.9 Logistic Regressions (LOGIT)
13.1.10.10 General Linear Models (GLM)
13.1.10.11Robust, M-estimators and W-estimators
13.1.11 Interpolation Techniques
13.1.11.1 Bezier Curves (B-Splines)
13.1.11.2 Linear ordering isotonic regression
13.1.11.3 Euler Method
13.1.11.4 Polynomial of collocation
13.1.11.5 Lagrange polynomials
13.1.11.6 Newton polynomials (divided differences)
13.1.11.7 Errors in Polynomial Interpolation
13.1.12 Roots search
13.1.12.1 Proportional parts methods
13.1.12.2 Bisection method
13.1.12.3 Secant method (Regula Falsi or False Position)
13.1.12.4 Newton’s method
13.1.13 Numerical Differentiation
13.1.14 Numerical Integration
13.1.14.1 Rectangles method
13.1.14.2 Trapezoidal method
13.1.14.3 Newton–Cotes formulas
13.1.14.4 Multidimensional integrals
13.1.14.5 Numerical solution of ordinary differential equations
13.1.15 Optimization
13.1.15.1 Linear programming (Linear Optimization)
13.1.15.2 Nonlinear programming (Nonlinear optimization)
13.1.16 Resampling statistics
13.1.16.1 Monte Carlo Simulations
13.1.16.2 Bootstrapping
13.1.16.3 Jackknife Resampling
13.1.16.4 Permutation Tests
13.1.17 Finite difference method (F.D.M.)
13.1.17.1 One space dimension F.D.M
13.1.17.2 Space-time F.D.M (finite-volume method)
13.1.18 Data Mining
13.1.18.1 Training VS Test Data
13.1.18.2 Hyperparameters vs Parameters
13.1.18.3 Association Rules
13.1.18.4 Clustering and Classification
13.1.18.5 Gradient Boosting
13.1.18.6 Neural networks
13.1.18.7 Genetic Algorithms
13.2 FRACTALS
13.2.1 IFS Fractals
13.2.1.1 Fractals Metric Space
13.2.2 Fractals Visualization
13.2.2.1 Cantor’s Fractal (Cantor Set)
13.2.2.2 Triangle Sierpinski Fractal
13.2.2.3 Sierpinski carpet fractal
13.2.2.4 Fractal spirals
13.2.2.5 Von Koch fractal (Koch snowflake)
13.2.2.6 Natural fractals
13.2.3 Escape Time Algorithm Fractals
13.2.3.1 Mandelbrot set
13.2.3.2 Julia set
13.2.3.3 Newton set
13.3 LOGICAL SYSTEMS
13.3.1 Strict Logic
13.3.1.1 Boolean Algebra
13.3.1.2 Logical Functions (Boolean operators)
13.3.1.3 Karnaugh maps
13.3.1.4 Arithmetic Boolean (binary) operations
13.3.2 Fuzzy logic
13.3.2.1 Fuzzy set
13.4 ERROR-CORRECTING CODES
13.4.1 CheckSum
13.4.1.1 Luhn algorithm
13.4.2 Check Digit
13.4.2.1 European Article Numbering (EAN-13)
13.4.2.2 Swiss Post Payment slip
13.4.2.3 International Bank Account Number (IBAN)
13.4.2.4 UIC wagon numbers
13.4.3 Permutations
13.4.4 Encoders
13.4.4.1 Block code
13.4.4.2 Systematic codes
13.5 AUTOMATA THEORY
13.5.1 Von Neumann machine
13.5.2 Turing machine
13.5.3 Chomsky hierarchy
13.5.3.1 Formal language
13.5.3.2 Syntax
13.5.3.3 Grammar
13.5.3.4 Associated automata
13.5.4 Terminology
13.5.4.1 Words
13.5.4.2 Languages
13.5.4.3 Equations
13.5.4.4 Codes
13.5.5 Linguistic algorithms
13.5.5.1 Huffmann algorithm
13.5.5.2 Sardinas and Patterson algorithm
13.6 CRYPTOGRAPHY
13.6.1 Cryptographic systems
13.6.1.1 Kerckhoffs’ principle
13.6.2 Traps
13.6.3 Secret-key encryption system
13.6.3.1 Feistel Schemes
13.6.4 Public key encryption
13.6.4.1 Diffie-Hellman protocol
13.6.4.2 R.S.A system
13.6.5 Hash functions
13.6.5.1 MD5 message digest condensation function
13.6.5.2 SHA-1 Secure Hash Algorithm condensation function
13.6.6 Certificate based authentication
13.6.7 Quantum cryptography
13.6.8 Alternative cryptography
13.7 QUANTUM COMPUTING
13.7.1 Schrödinger’s Cat superposition
13.7.2 Photon polarization
13.7.3 Qubit
13.7.3.1 Bloch sphere
13.7.4 Entangled qubit system
13.7.5 Quantum logic gates
14 SOCIAL SCIENCES
14.1 POPULATION DYNAMICS
14.1.1 Birth rate and mortality tables (biometric features)
14.1.1.1 Population Renevewal
14.1.2 Population Models
14.1.2.1 Exponential model
14.1.2.2 Deterministic Logistic Model (Verlhust)
14.1.2.3 Chaotic Logistic Model
14.1.2.4 Malthusian Growth Law
14.1.2.5 Leslie model
14.1.2.6 SIR Model (Susceptible-Infected-Recovered) for spread of disease
14.1.2.7 Lotka–Volterra predator–prey model
14.1.3 Schaefer’s Optimal capture model
14.1.4 Hardy-Weinberg model
14.1.5 Mendel’s law
14.1.6 Growth rate with temperature
14.2 GAME AND DECISION THEORY
14.2.1 Behavorial decision bias (cognitive bias)
14.2.1.1 Sunk Cost
14.2.1.2 Anchoring Bias
14.2.1.3 Wisdom of Crowds
14.2.2 Utility
14.2.2.1 Pareto Optimum
14.2.2.2 Nash Equilibrium
14.2.3 Games Representations
14.2.3.1 Extensive representation of a Game
14.2.3.2 Extensive representation of a Decision
14.2.3.3 Normal representation of a Game
14.2.3.4 Set representation of a Game
14.2.3.5 Graphical representation of a Game
14.2.4 Expected Utility
14.2.4.1 Hurwitz Criteria
14.2.4.2 Laplace Criteria
14.2.5 Evolutionary Game
14.2.5.1 Dove & Hawk game in pure strategy (without probabilities)5619
14.2.5.2 Dove & Hawk game in stable evolutionary strategy (with probabilities)
14.2.6 Cournot Competition
14.2.7 Markov Decision Processes (MDPs)
14.2.8 Multi-Criteria Decision Making (MCDM)
14.2.8.1 Analytic Hierarchy Process (AHP)
14.3 ECONOMY
14.3.1 Concepts
14.3.1.1 Microeconomics
14.3.1.2 Macroeconomics
14.3.2 Monetary Model
14.3.2.1 Monetary base conservation
14.3.2.2 Walras’ law
14.3.3 Price Index and GDP
14.3.3.1 Paasche and Laspeyres price indices
14.3.3.2 Fisher index and Marshall–Edgeworth index
14.3.3.3 Gross domestic product (GDP)
14.3.4 Supply and Demand Theory
14.3.4.1 Expected utility theory
14.3.5 Net gain/loss opposite feedback model
14.3.6 Capitalization and Actuarial
14.3.6.1 Dates Intervals
14.3.6.2 Rates Equivalence
14.3.6.3 Simple Interest
14.3.6.4 Compound Interest
14.3.6.5 Continuous Interest
14.3.7 Progressive interest (annuities)
14.3.7.1 Postnumerando annuities
14.3.7.2 Praenumerando annuities
14.3.8 Rounding
14.3.9 Loans Amortization/Repayments
14.3.9.1 Fixed-Term Loan
14.3.9.2 Loan with constant amortization
14.3.9.3 Loan with constant annuity
14.3.10 Modern Portfolio Theory
14.3.10.1 No Arbitrage Opportunity (N.A.O.)
14.3.10.2 Liquidity and Illiquidity
14.3.10.3 Portfolios
14.3.11 Exchange rate
14.3.12 Credit default Risk
14.3.12.1 Credit spread risk
14.3.12.2 Value At Risk equity coverage rating
14.3.12.3 Merton’s Distance to Default and KMV Expected Default Frequency
14.3.13 Time Series Analysis
14.3.13.1 Type of Errors
14.3.13.2 Decompositions
14.3.13.3 Types of Forecasting Models
14.3.13.4 Autoregressive Models
14.3.13.5 Forecasting Stationary Processes
14.3.13.6 TBATS
14.3.13.7 Recipes for modeling time series data
14.4 QUANTITATIVE MANAGEMENT
14.4.1 Corporate and Government Finance Management
14.4.1.1 Basic Accounting Equation
14.4.1.2 Ratio Analysis
14.4.1.3 Weighted Average Cost of Capital (WACC)
14.4.1.4 Break-even Point Analysis (BEPA)
14.4.1.5 Investment Strategies
14.4.1.6 Company Valuation Methods
14.4.1.7 Capital Goods
14.4.1.8 Wages model
14.4.2 Project Management
14.4.2.1 Probabilistic PERT (Beta-PERT Distribution)
14.4.2.2 Project planning variance reduction
14.4.2.3 Process Reliability
14.4.2.4 Divide and rule?
14.4.3 Lean Management (Six Sigma Process)
14.4.3.1 Pareto Analysis
14.4.3.2 Weighted Ishikawa Diagram
14.4.4 Supply Chain Management
14.4.4.1 Supply Chain Management in uncertain future
14.4.4.2 Optimal initial stock management with zero rotation
14.4.4.3 Wilson’s Models
14.4.5 Queueing Theory
14.4.5.1 M/M/ . . . arrival times modelisation
14.4.5.2 M/M/ . . . service times modelisation
14.4.5.3 Kendall queues notation
14.4.5.4 Modeling of arrivals and departures M/M/1
14.4.5.5 Probability of standby in a M/M/k/k queue (Erlang-B formula)
14.4.5.6 Probability M/M/K/ + ∞ of standby (Erlang-C formula)
14.4.6 Insurance
14.4.6.1 Premium pricing
14.4.6.2 Taking account of the return on experience
14.4.6.3 Discounting factor of a retirement insurance
14.4.6.4 Pension insurance (life annuities)
14.4.7 Sensitivity Analysis
14.4.7.1 Direct Bias Method
14.4.7.2 Correlation Method
14.5 MUSIC MATHEMATICS (PHYSICS OF HEARING)
14.5.1 Longitudinal Sound Waves
14.5.1.1 Power carried by a sound wave
14.5.1.2 Measuring the intensity of a sound
14.5.2 Spherical Sound Waves
14.5.3 Doppler effect
14.5.3.1 Fixed source-Moving observer
14.5.3.2 Moving source-Fixed observer
14.5.3.3 Moving source and observer
14.5.4 Shock waves
14.5.5 Music Scales
14.5.6 Harmonic Oscillator
14.5.6.1 Damping and Resonance
15 ENGINEERING
15.1 MARINE & WEATHER ENGINEERING
15.1.1 Visual horizon
15.1.2 Wind direction
15.1.3 Atmospheric Profile Models
15.1.3.1 Atmospheric Exponential Profile Model
15.1.3.2 Adiabatic Atmosphere Model
15.1.4 Planetary equilibrium temperature
15.1.4.1 Greenhouse effect
15.1.4.2 Milankovitch cycles
15.1.5 Weather (sounding) balloon
15.1.6 Cyclogenesis and Anticyclogenesis
15.1.7 Tides
15.1.7.1 First approach
15.1.7.2 Second approach
15.1.8 Lorenz equation
15.1.8.1 Rayleigh-Bénard convection cells (Benard-Marangoni instability)
15.1.8.2 Lorenz attractor and chaos
15.1.9 Waves
15.1.9.1 Depth of a wave
15.1.9.2 Wave’s amplitude
15.2 MECHANICAL ENGINEERING
15.2.1 Mechanical Efficiency
15.2.2 Gears
15.2.2.1 Transmission ratios
15.2.2.2 Gears association
15.2.2.3 Type of Gears
15.2.3 Strength of materials
15.2.3.1 Quadratic moments
15.2.3.2 Equation of the elastic line
15.2.3.3 Torsion
15.2.3.4 Buckling
15.2.3.5 Traction
15.3 ELECTRICAL ENGINEERING
15.3.1 Elementary Primitive Electrical Symbols
15.3.2 Alternative current VS Direct current
15.3.2.1 Average power
15.3.3 Transformers
15.3.3.1 Transformer universal EMF equation
15.3.4 Steady State Linear Circuits
15.3.4.1 RC series circuit
15.3.4.2 RL series circuit
15.3.4.3 RLC circuit
15.3.5 Linear Circuit in Forced Regime
15.3.5.1 Low-pass filter
15.3.5.2 High-pass filter
15.3.5.3 Integrator and differentiator
15.3.6 Signal Processing
15.3.6.1 Nyquist-Shannon Theorem (Nyquist sampling theorem)
15.3.7 Amplifiers
15.4 CIVIL ENGINEERING
15.4.1 Static
15.4.2 Pulleys
15.4.2.1 Windlass
15.4.3 Cornu spiral
15.4.4 Overhead cable
15.4.4.1 Free overhead cable (catenary)
15.4.4.2 Charges overhead cable (suspended bridge)
15.4.4.3 Very tense cable
15.4.5 Falling chimney (naive approach)
15.4.6 Dams
15.5 AEROSPACE ENGINEERING
15.5.1 Airfoil Lift
15.5.1.1 Newton’s lift argument (skipping stone argument)
15.5.1.2 Bernoulli’s lift argument (equal time argument)
15.5.1.3 Euler’s lift argument
15.5.1.4 Coand˘a lift argument
15.5.1.5 Kutta-Joukowski lift argument
15.5.2 Cosmic speeds
15.5.3 Fundamental Equation of Propulsion (Tsiolkovsky rocket equation)
15.5.4 Geostationary orbit
15.5.5 Vis-Viva Equation
15.5.6 Hohmann Transfer orbit
15.6 SOFTWARE ENGINEERING
15.6.1 Algorithm
15.6.2 Dichotomic Search algorithm
15.6.2.1 Bisection algorithm
15.6.2.2 Binary search algorithm
15.6.3 Tower of Hanoi algorithm
15.6.4 Sorting Algorithms
15.6.4.1 Bubble sort
15.6.4.2 QuickSort algorithm
15.6.5 Dijkstra’s algorithm
15.6.6 Google PageRank algorithm
15.6.6.1 Weighted Count
15.6.6.2 Recursive counting
15.6.6.3 Absorbing states
15.7 INDUSTRIAL ENGINEERING
15.7.1 Six Sigma
15.7.1.1 Quality Control
15.7.1.2 Defaults/Errors
15.7.1.3 Capability Indices
15.7.1.4 Quality Levels
15.7.2 Taguchi Model
15.7.3 Preventive Maintenance
15.7.3.1 Planned Obsolescence
15.7.3.2 Reliability Empirical Estimators
15.7.3.3 Weibull Distribution
15.7.3.4 Topology of Systems
15.7.3.5 Maximum Likelihood for failure rate determination of samples
15.7.3.6 Kaplan-Meier Survival Rate
15.7.3.7 ABC Method
15.7.3.8 XY Z Method
15.7.4 Design of Experiments (DoE)
15.7.4.1 Two levels factorial Designs
15.7.4.2 General factorial Designs
15.7.4.3 Taguchi Designs and Nomenclature (robust designs)
15.7.4.4 Response Surface Methodology (Box Domains)
15.7.4.5 Optimal Designs
15.7.4.6 Mixture Design
15.7.4.7 General DoE diagnostic tools
15.7.5 Quality Control on Reception (Lot Acceptance Sampling Plans)
15.7.5.1 Simple acceptance sampling plan by measurement for a unique tolerance with known standard deviation
15.7.5.2 Simple acceptance sampling plan by attribute
15.7.5.3 Double acceptance sampling plan by attribute
15.7.5.4 Operating characteristic curve (OC)
15.7.5.5 Average outgoing quality (AOQ)
15.7.6 Quality Control Charts (CC)
15.7.6.1 WECO’s empirical rules
15.7.6.2 Sample size and Sampling frequency for Control Charts
15.7.6.3 Attributes Control Charts (qualitative CC)
15.7.6.4 Measurement Control Charts (quantitative CC)
15.7.6.5 Autocorrelated Measurement Control Charts (time weighted control charts)
15.7.6.6 Rare events control charts
15.7.6.7 Control Charts Operating Characteristic (OC) Curves
15.7.7 Design of reliability tests
15.7.7.1 Chi-squared time of test
15.7.7.2 Binomial sampling size
15.7.7.3 Test to demonstrate MTBF (or MTTF)
15.7.7.4 Beta-binomial sampling size
16 EPILOGUE
17 BIOGRAPHIES
18 CHRONOLOGY
19 HUMOR
20 LINKS
21 QUOTES
22 CHANGE LOG
23 NOMENCLATURE
LIST OF FIGURES
LIST OF TABLES
LIST OF ALGORITHMS
BIBLIOGRAPHY
INDEX
24 DONATE