Contents -- Preface -- 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 -- Chapter 7 Equilibrium dissipation and ergodicity -- 7.1 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 -- Chapter 12 An organizing tool -- Chapter 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 -- 16.5 Condensation theories -- 16.6 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 -- A.8.4 Ideal relativistic quantum gases -- Bibliography -- Index.