Matrix Methods and Fractional Calculus

The following sections are included:

• Preface
• Acknowledgments
• Contents
• List of Symbols

The following sections are included:

• Introduction
  • Exercise 1.1
• A Vector Differential Operator
  • Linear forms
  • Quadratic forms
  • Quadratic form, hermitian form and definiteness
    • Definiteness of matrices and quadratic forms
    • Definiteness of hermitian forms
  • Exercise 1.2
• Expansion of a Multivariate Function
  • Maxima/minima of functions of many variables
    • Some special cases
  • Exercise 1.3
  • Maxima/minima subject to constraints
  • Optimization of a quadratic form subject to general quadratic constraint
    • Principal components analysis
    • Maximizing a quadratic form subject to linear constraint
  • Maximizing a linear function subject to quadratic constraint
  • Maximizing a bilinear form subject to quadratic constraint
    • Canonical correlation analysis
  • Exercise 1.3 (Continued)
• Derivative of a Scalar Function with Respect to a Matrix
  • Exercise 1.4
• Acknowledgments
• Bibliography

The following sections are included:

• Introduction
• Linear Transformations
  • Exercise 2.2
• Nonlinear Transformations Involving Scalar Variables
  • Exercise 2.3
• Some Nonlinear Matrix Transformations
  • Exercise 2.4
• Transformations Involving Orthonormal and Semiorthonormal Matrices
  • Exercise 2.5
• Some Matrix Transformations in the Complex Domain
  • Exercise 2.6
• Acknowledgments
• Bibliography

The following sections are included:

• Introduction
• Riemann–Liouville Fractional Calculus
• Liouville–Weyl Fractional Calculus
• Riesz–Feller Fractional Calculus
  • The Riesz fractional integrals and derivatives
  • The Feller fractional integrals and derivatives
• Riemann–Liouville and Caputo Fractional Calculus
  • The Riemann–Liouville fractional integrals and derivatives
  • The Caputo fractional derivative
• Mittag-Leffler Functions
  • Reference-historical background
  • The Mittag-Leffler functions E_α(z), E_α,β(z)
  • The Mittag-Leffler integral representation and asymptotic expansions
  • The Laplace transform pairs related to the Mittag-Leffler functions
  • Fractional relaxation and fractional oscillation
  • Other formulas: summation and integration
  • The Mittag-Leffler functions of rational order
  • Some plots of the Mittag-Leffler functions
• Wright Functions
  • The series representation of the Wright function
  • The Wright integral representation and asymptotic expansions
  • The Wright functions as generalization of the Bessel functions
  • The auxiliary functions F_v(z) and M_v(z) of the Wright function
  • Laplace transform pairs related to the Wright function
  • Some plots of the M-Wright functions
• Bibliography

The following sections are included:

• Introduction
• Fractional Derivatives/Integrals
  • Differ–integral
  • Riemann–Liouville fractional integral/derivative
  • Caputo derivative
  • Existence, uniqueness and stability of solutions
  • Dependence of solution on initial conditions
• Decomposition Methods for Solving FDEs
  • Adomian decomposition method
  • New iterative method
  • Numerical methods for solving FDEs
• Fractional Boundary Value Problems
• Systems of FDEs
  • Linear case
  • Chaos in nonlinear dynamics
  • Phase portraits (Lorenz system)
  • Fractional-ordered systems and chaos
• Bibliography

The following sections are included:

• Introduction
• Matrix-Variate Statistical Distributions
• Right-Sided or Second Kind Fractional Integrals
  • Second kind fractional integrals in the matrix-variate case, general definition
  • Special cases
• Left-Sided or First Kind Fractional Integrals in the Real Matrix-Variate Case
• Fractional Derivatives in the Matrix-Variate Case
  • Exercise 5.5
• Bibliography

The following sections are included:

• Introduction
• Matrix Groups
  • Exercise 6.2
• Manifold
• Lie Groups
  • Local Lie groups
  • Examples of Lie groups
  • Special orthogonal groups SO(n, ℝ)
• Lie Bracket and Lie Algebra
  • Examples of Lie algebra
• Lie Algebra of the Endomorphism of the Vector Space V
  • Application of Theorem 6.6.4
• Monomiality Principle
  • Multiplicative and derivative operators
  • Monomiality of Appell polynomials
  • Exercise 6.7

The following sections are included:

• Introduction
• Multivariate Distributions
  • Definitions and properties
  • The transformation of variables technique
  • The multivariate normal distribution
  • Quadratic forms in normal vector random variables
  • Exercise 7.2
• Maximum Likelihood Estimation
  • The maximum likelihood estimates of μ and 𝕍
  • The distribution of the $\mathcal{\text{MLE}}$s
  • The Wishart distribution
  • Exercise 7.3
• Certain Test Statistics
  • Introduction
  • A test of independence
  • The sphericity test
  • Testing that μ equals a given vector
  • Multivariate analysis of variance
  • Exercise 7.4
• Acknowledgments
• Bibliography

The following sections are included:

• Author Index
• Subject Index