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Aldrovandi R. — Special matrices of mathematical physics (stochastic, circulant and bell matrices)
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Название: Special matrices of mathematical physics (stochastic, circulant and bell matrices)Автор: Aldrovandi R. Аннотация: 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.
Язык: Рубрика: Физика /Статус предметного указателя: Готов указатель с номерами страниц ed2k: ed2k stats Год издания: 2001Количество страниц: 323Добавлена в каталог: 28.03.2010Операции: Положить на полку |
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Logistic map, extreme 222 Lorentz group 202 Lyapunov exponent 218 Lyapunov exponent, main root 53 56 Markov chain 28 Markov chain, cyclic 51 Markov chain, ergodic 67 Markov chain, irreducible 51 67 Markov chain, non-cyclic 51 Markov chain, positive-recurrent 75 Markov chain, recurrent 66 Markov chain, reducible 51 Markov chain, regular 52 Markov chain, reversible 70 Markov chain, totally regular 51 Markov, condition 28 Markov, process 28 Master equation 61 Matrix, adjoint 5 Matrix, basis 106 Matrix, Bell 165 Matrix, characteristic 3 Matrix, circulant 83 182 Matrix, completely 26 Matrix, degenerate or not 4 Matrix, differential geometry 127 Matrix, doubly 50 96 Matrix, eigenvalues 3 196 Matrix, functions of a 6 207 Matrix, hermitian 5 Matrix, Hermitian conjugate 5 Matrix, imprimitive 48 Matrix, invariants 199 Matrix, inversion 9 172 198 Matrix, irreducible 47 49 Matrix, Lee — Yang 274 Matrix, linear basis 93 Matrix, non-negative 25 Matrix, normal 5 17 Matrix, positive 25 47 Matrix, primitive 48 Matrix, projectors of a 6 Matrix, projectors of a normal 17 Matrix, reducible 26 27 50 Matrix, reducible stochastic 29 Matrix, regular 52 Matrix, Schwinger basis 105 Matrix, spectrum 3 Matrix, stochastic 27 Matrix, totally regular stochastic 51 Matrix, unitary 5 Matrix, unitary basis 93 Maurer — Cartan equation 129 Mayer, equation of state 268 Mayer, function 240 Mayer, theory of condensation 268 Mayer, virial formula 256 262 Metric 134 microcanonical ensemble 228 Minor and cofactor 198 Minor, Fredholm 185 278 Model, Ising 58 Model, Kerner stochastic 35 Model, Lee — Yang 270 Model, Mayer 268 Model, Moyal bracket 103 114 119 124 133 137 146 Model, Potts 272 Multi-phase system 62 Multinomial theorem 152 Navier — Stokes equation 219 Newton formula 191 No-fluctuation condition and detailed balancing 72 Noncommutative geometry 100 Normal, matrix 5 17 Number, Bell 159 Number, Fibonacci 164 Number, idempotent 163 Number, Lah 163 Operator, displacement, or shift 89 Optical theorem 60 Orbit of a distribution 55 Orthogonal polynomials 178 Pade approximants 195 Partition 152 Partition function canonical 260 Partition function canonical for a braid gas 267 Partition function canonical for a real gas 243 Partition function, grand, canonical 188 253 254 260 Partition function, recursion formula 246 Partition function, Widom 247 Partition, conditional 210 Pauli matrices 109 periodicity 63 Permanents and bosons 232 Permutation 27 234 Perron theorem 47 49 Phase multiple 62 Phase quantum 109 Phase space 99 Phase transitions 267 Phase transitions, Mayer theory 268 Phase transitions, theory 270 Phase volume 228 Poincare group 202 Poincare map 218 Poisson bracket 100 103 111 124 132 133 Poisson quantum 129 138 Polynomial in general 193 Polynomial, annihilating 4 Polynomial, characteristic 3 196 Polynomial, Chebyshev 179 220 Polynomial, cycle indicator 234 Polynomial, Gegenbauer 178 Polynomial, Hermite 178 Polynomial, Legendre 178 Polynomial, minimal 5 Polynomial, orthogonal 178 Pontryagin 122 Pontryagin duality 122 Posit iverecurrent, chain 75 Posit iverecurrent, state 75 Potential, Gibbs 253 Potential, Helmholtz 247 Potts model 272 Power Bell polynomials 165 Prakash-Sudarshan, recursion formula 237 Prescription, Weyl 102 112 119 146 Presentation, braid group 265 Presentation, braid group, defined 235 Presentation, braid group, symmetric group 235 probability distribution 26 Projector 6 8 Projector for degenerate matrices 14 Projector for nondegenerate matrices 10 12 Projector of a Bell matrix 213 224 Projector of a normal matrix 17 Projector, circulant 90 Projector, closed expression 16 208—210 Quantization 120 125 Quantum degeneracy 262 Quantum group 120 quantum mechanics 17 Quantum Mechanics of a Lie algebra 200 Quantum Mechanics of a matrix 200 Quantum Mechanics, discrete 92 99 Quantum Mechanics, general 96 Quantum Mechanics, Markov chain 66 Quantum Mechanics, random walk 78 Quantum Mechanics, reciprocal series 161 169 Quantum Mechanics, simple 97 Quantum Mechanics, Weyl — Wigner picture 101 REAL 243 Reducible matrix 27 Regular stochastic matrix 52 Relativistic gases ideal 260 Resolvent of a matrix 4 9 Resolvent, other sense 275 Retour, eternel 66 Reversibility 67 70 Reversible, Maxkov chain 70 Root, evanescent 53 Root, main 53 Rotations, group of 13 201 Rule, Leibniz 172 Schrorder equation 222 Schwinger on Quantum Mechanics 108 Schwinger, basis 105 106 109 115 119 144 Secular equation 3 197 Semi-circulant matrix 169 Series, generating functions 162 Series, group of 157 Series, invertible 157 Series, Lagrange inversion 173 Series, non-invertible 168 Series, reciprocal 161 169 Series, truncated 154 Series, unit 219 Series, virial 256 Shift operator 89 Spectrum of a matrix 3 Spectrum of a stochastic matrix 45 State, aperiodic 67 State, independence of initial 56 State, positive-recurrent 75 State, recurrent 66 State, transient 51 66 Statistical mechanics 227 Statistical Mechanics, canonical ensemble 239 Statistical Mechanics, grand canonical ensemble 253 Statistical Mechanics, microcanonical ensemble 228 Stax, or twisted product 101 115 116 118 119 138 Stirling numbers 158 171 225 Stirling numbers, meaning 159 Stochastic, circulant 96 Stochastic, matrix, asymptotic 53 Stochastic, matrix, defined 27 Superconductivity and braid statistics 264 Symmetric functions 186 Symmetric functions, complete homogeneous 187 Symmetric functions, elementary 186 Symmetric functions, fundamental theorem 187 192 Symmetric functions, power-sum 188 273 Symmetric group 26 234 247 Symmetric group, generators 235 Symmetric group, presentation 235 Symplectic, form 127 132 138 Symplectic, structure 132 Tanaka — Krein 122 Tanaka — Krein duality 122 Theorem, Cayley — Hamilton 4 6 197 Theorem, Darboux 128 Theorem, ergodic 67 74 Theorem, Frobenius 47 49 Theorem, Lee — Yang circle 272 Theorem, Multinomial 152 Theorem, optical 60 Theorem, Perron 47 49 Theorem, van Hove 272 Thermal wavelength, non-relativistic 238 Thermal wavelength, relativistic 248 Time reversal 70 Totally regular, stochastic matrix 51 53 Traces 183 Traces, relations to determinants 185 Transformation, area-preserving 124 Transformation, braid 264 Transformation, canonical 124 128 Transformation, cyclic 62 Transformation, double 95 Transformation, Fourier 81 86 109 Transformation, general 122 Transformation, group 121 200 Transformation, operator 113 Transformation, permutation 264 Transformation, rotation 13 Transformation, similarity 17 87 199 239 Transformation, Weyl — Wigner 102 109 112 138 Transience 63 Transient state in a Markov chain 66 Transposition elementary 26 Turbulence, application to 219 Twisted convolution 102 115 Twisted, or star product 101 115 116 118 119 138 Unitary, basis 93 Unitary, matrix 5 Unitary, Schwinger basis 105 Vacuum, multiple 62 van Hove theorem 272 Vandermonde determinant 193 Vector field 128 Vector space dual to 18 Viete’s formulae 194 Wavelength thermal 238 Weierstrass expression for a polynomial 194 Weyl 108 Weyl — Heisenberg group 103 116 Weyl, prescription 102 112 119 146 Weyl-Wigner, transformations 102 109 112 125 138 139 Widom, partition function 247 Wigner, density 113 115 116 Wigner, function 101 113 116 118 Wigner, word 5 Wronski formula 169 Yang-Baxter, equation 103 120 143
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