Special Matrices Of Mathematical Physics: Stochastic, Circulant And Bell Matrices Stochastic, Circulant and Bell Matrices

this Book Expounds Three Special Kinds Of Matrices That Are Of Physical Interest, Centering On Physical Examples. Stochastic Matrices Describe Dynamical Systems Of Many Different Types, Involving (or Not) Phenomena Like Transience, Dissipation, Ergodicity, Nonequilibrium, And Hypersensitivity To Initial Conditions. The Main Characteristic Is Growth By Agglomeration, As In Glass Formation. Circulants Are The Building Blocks Of Elementary Fourier Analysis And Provide A Natural Gateway To Quantum Mechanics And Noncommutative Geometry. Bell Polynomials Offer Closed Expressions For Many Formulas Concerning Lie Algebra Invariants, Differential Geometry And Real Gases, And Their Matrices Are Instrumental In The Study Of Chaotic Mappings. This work expounds three special kinds of matrices that are of physical interest, centring on physical examples. Stochastic matrices describe dynamical systems of many different types, involving (or not) phenomena like transience, dissipation, ergodicity, non-equilibrium, and hypersensitivity to initial conditions. The main characteristic is growth by agglomeration, as in glass formation. Circulants are the building blocks of elementary Fourier analysis and provide a natural gateway to quantum mechanics and non-commutative geometry. Bell polynomials offer closed expressions for many formulas concerning Lie algebra invariants, differential geometry and real gases, and their matrices are instrumental in the study of chaotic mappings Chapter 12 An organizing toolChapter 13 Bell polynomials; 13.1 Definition and elementary properties; 13.2 The matrix representation; 13.3 The Lagrange inversion formula; 13.4 Developments; Chapter 14 Determinants and traces; 14.1 Introduction; 14.2 Symmetric functions; 14.3 Polynomials; 14.4 Characteristic polynomials; 14.5 Lie algebras invariants; Chapter 15 Projectors and iterates; 15.1 Projectors revisited; 15.2 Continuous iterates; Chapter 16 Gases: real and ideal; 16.1 Microcanonical ensemble; 16.2 The canonical ensemble; 16.3 The grand canonical ensemble; 16.4 Braid statistics Chapter 7 Equilibrium dissipation and ergodicity7.1 Recurrence transience and periodicity; 7.2 Detailed balancing and reversibility; 7.3 Ergodicity; CIRCULANT MATRICES; Chapter 8 Prelude; Chapter 9 Definition and main properties; 9.1 Bases; 9.2 Double Fourier transform; 9.3 Random walks; Chapter 10 Discrete quantum mechanics; 10.1 Introduction; 10.2 Weyl-Heisenberg groups; 10.3 Weyl-Wigner transformations; 10.4 Braiding and quantum groups; Chapter 11 Quantum symplectic structure; 11.1 Matrix differential geometry; 11.2 The symplectic form; 11.3 The quantum fabric; BELL MATRICES 16.5 Condensation theories16.6 The Fredholm formalism; Appendix A Formulary; A.1 General formulas; A.2 Algebra; A.3 Stochastic matrices; A.4 Circulant matrices; A.5 Bell polynomials; A.5.1 Orthogonal polynomials; A.5.2 Differintegration derivatives of Bell polynomials; A.6 Determinants minors and traces; A.6.1 Symmetric functions; A.6.2 Polynomials; A.6.3 Characteristic polynomials and classes; A.7 Bell matrices; A.7.1 Schroder equation; A.7.2 Fredholm theory; A.8 Statistical mechanics; A.8.1 Microcanonical ensemble; A.8.2 Canonical ensemble; A.8.3 Grand canonical ensemble Preface; Contents; BASICS; Chapter 1 Some fundamental notions; 1.1 Definitions; 1.2 Components of a matrix; 1.3 Matrix functions; 1.3.1 Nondegenerate matrices; 1.3.2 Degenerate matrices; 1.4 Normal matrices; STOCHASTIC MATRICES; Chapter 2 Evolving systems; Chapter 3 Markov chains; 3.1 Non-negative matrices; 3.2 General properties; Chapter 4 Glass transition; Chapter 5 The Kerner model; 5.1 A simple example: Se-As glass; Chapter 6 Formal developments; 6.1 Spectral aspects; 6.2 Reducibility and regularity; 6.3 Projectors and asymptotics; 6.4 Continuum time (1) Given an N x N matrix M, its characteristic matrix is I - M, a function of M, of the N x N identity matrix I and of the complex variable. A.8.4 Ideal relativistic quantum gasesBibliography; Index