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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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Adjoint of a matrix 5 Algebra invariants 200 Algebra, defined 5 Algebra, Hopf 120 Algebra, infinite 123 Algebra, Lie 5 Alphabet 5 153 Alphabet of roots 191 Alphabet with one missing letter 208 Alphabet, eigenvalue 196 Alphabet, letters 5 Alphabet, noncommutative 186 Alphabet, reciprocal 193 Alphabet, word 5 Alphabets relations between 191 Alternating function 192 Annihilating polynomial 4 Approximates, Pade 195 Area-preserving, transformation 124 Asymptotic, distribution 29 Asymptotic, equilibrium 29 Attractor of a Markov chain 54 Average on a distribution 29 Basis, matrix 93 106 Bell matrices, eigenvectors 180 226 Bell matrices, group of 167 Bell matrices, introduced 165 Bell matrices, not normal 168 Bell number 159 Bell polynomials and orthogonal polynomials 178 Bell polynomials and partition function 243 Bell polynomials, complete 153 154 177 199 Bell polynomials, definition 151 Bell polynomials, derivatives 181 Bell polynomials, logarithmic 164 Bell polynomials, matrices of 165 Bell polynomials, notation 153 154 Bell polynomials, partial 153 Bell polynomials, power 165 Bell polynomials, properties 155 Bell polynomials, recursion relation 177 Bell polynomials, special cases 160 Bialgebra 120 Boltzmann function 70 Borel transform 155 Bose function 256 Bracket, Moyal 103 114 119 124 133 137 146 Bracket, Poisson 100 103 111 124 132 133 Bracket, Poisson, quantum 129 138 Braid 146 Braid, equation 103 Braid, gas 265 Braid, group 140 236 264 Braid, group, presentation 265 Braid, statistics 264 Braid, statistics and superconductivity 264 Calculus fractional 180 Canonical, ensemble 239 Canonical, partition function 260 Casimir operators 201 Cayley-Hamilton theorem 4 6 84 197 Chapman — Kolmogorov equation 28 Character of a group 123 Character of a group, defined 118 Characteristic, class 203 Characteristic, equation 3 Characteristic, forms 196 Characteristic, matrix 3 Characteristic, polynomial 3 196 Chebyshev polynomials 179 220 Chern, characters 203 Chern, classes 203 Circle theorem 272 Circulant and Fourier transformation 84 Circulant, application of 182 Circulant, basis 88 Circulant, defined 83 Circulant, determinant 84 Circulant, projectors 87 Circulant, semi 169 Circulant, stochastic 53 96 Class, characteristic 203 Classical mechanics 99 Cluster and minor 198 Cluster, cofactor 198 Cluster, expansion 255 Cluster, integral 241 Composition function 151 Condensation 267 Condensation, Lee — Yang theory 270 Condensation, Mayer theory 268 Connected diagram 155 Connected graphs 155 263 Continuous iterate 211 Continuum time in Markov chains 60 Convolution 87 90 Convolution, twisted 102 114 115 123 Coordination number 75 Correlation quantum 251 Correspondence principle 102 112 117 Critical phenomena 267 Cycle decomposition 245 280 Cycle indicator polynomial 234 Cyclic group 85 Darboux theorem 128 Delta Kronecker 85 Derivation 129 Derivative cohomology 131 Detailed balancing 67 68 Detailed balancing and equilibrium 70 Determinant 183 Determinant and characteristic polynomial 197 Determinant and fermions 232 Determinant and traces of powers 185 Determinant, circulant 84 Determinant, Fredholm 276 Determinant, Vandermonde 193 Diagram, connected 155 Differential forms matrix 130 Differintegration 180 Discriminant of a polynomial 195 Displacement operator 89 Dissipation 66 Distribution functions 251 Distribution, equilibrium 29 Distribution, probability 26 Doubly-stochastic matrix 50 96 Duality, Fourier 122 Dynamic quantity, classical 99 Dynamic quantity, classical and quantum 113 117 140 Dynamic quantity, quantum 107 Eigenvalues 4 Eigenvalues of a Fredholm operator 282 Eigenvalues of Bell matrices 219 Eigenvalues of normal matrices 17 Eigenvalues of stochastic matrices 47 Eigenvalues, alphabet 196 Eigenvalues, symmetric functions of 187 Eigenvectors of Bell matrices 226 Eigenvectors of circulants 86 Eigenvectors of stochastic matrices 46 62 Ensemble, Canonical 239 Ensemble, Grand Canonical 253 Ensemble, microcanonical 228 Entire, Chapman — Kolmogorov 28 Entire, characteristic 3 Entire, function 279 Entire, Liouville 127 Entire, master 61 Entire, Maurer — Cartan 129 Entire, Schroeder 222 Entire, secular 3 197 Entire, Yang — Baxter 103 120 143 Equation braid 103 Equation of state, Boyle Mariotte 268 Equation of state, Mayer 268 Equation of state, virial 256 Equilibrium 46 54 Equilibrium and detailed balancing 70 Equilibrium, asymptotic 29 Equilibrium, distribution 29 53 Equilibrium, unstable 66 Ergodic, theorem 67 74 Eternel retour 66 Evanescent root 53 Faa di Bruno formula 153 161 181 Fermi function 256 Feynman diagrams number of 242 Fibonacci numbers 164 Formula, Faa di Bruno 153 161 181 Formula, Lagrange 173 Formula, Leibniz 172 Formula, Matsubara 263 Formula, Mayer 256 Formula, Newton 191 Formula, Prakash- Sudarshan 237 Formula, Viete 194 Formula, Wronski 169 Fourier transformation and circulants 84 Fourier transformation and cyclic groups 85 Fourier transformation and quantum groups 122 Fourier transformation, double 95 Fourier transformation, operator 113 Fourier, duality 122 Fractional calculus 180 Fredholm, alternative 275 Fredholm, determinant 276 Fredholm, first 278 Fredholm, minor 185 Fredholm, theory 185 245 274 Free energy, Helmholtz 247 Frobenius, spectral theorem 47 49 Fugacity 254 Function of a Bell matrix 207 Function of a circulant 91 Function of a matrix 8 Function, Bose 256 Function, composition 151 Function, entire 279 Function, Fermi 256 Function, Heaviside 195 Function, inverse 157 Fundamental group 265 Fundamental theorem of symmetric functions 187 Gas, braid 265 Gas, ideal 245 Gas, quantum 248 256 260 Gas, real 243 256 267 Gas, relativistic 248 260 Gas, ultrarelativistic 262 Gegenbauer, polynomials 178 Generating function for Bell numbers 160 Generating function for Bell polynomials 153 177 Generating function for conditional partition numbers 211 Generating function for Hermite polynomials 178 Generating function for Legendre polynomials 178 Generating function for number of partitions 153 Generating function for Stirling numbers 158 Generating function for symmetric functions, elementary 187 208 Generating function for symmetric functions, homogeneous 187 Generating function for symmetric functions, power-sum 188 Generating function, Chebyshev polynomials 179 Generating function, cycle indicator as 234 Generating function, Gegenbauer polynomials 178 Generating function, manipulations 162 Generators of the braid group 141 265 Generators of the symmetric group 235 Gibbs potential 253 Gibbs — Di Marzio law 34 39 78 Glass, binary 72 77 Golden ratio 164 grand canonical ensemble 253 Grand canonical ensemble, graphs connected 155 245 263 Grand canonical ensemble, partition function 188 253 260 Group 26 Group of permutations 26 234 Group, braid 140 264 265 Group, compact 201 Group, complex linear 203 Group, cyclic 85 Group, Heisenberg 104 108 111 116 123 Group, Lorentz 202 Group, orthogonal 202 Group, Poincare 202 Group, rotation 201 Group, semisimple 201 Group, symmetric 26 234 Group, unitary 203 Hamiltonian, approach to Markov chains 61 Harmonic analysis 122 harmonic oscillator 221 Heaviside function 195 Heisenberg, group 104 108 111 116 123 Helmholtz free energy 247 Hermite polynomials 178 Hermitian conjugate of a matrix 5 Hermitian matrix 5 Homotopy, integration by 256 Hopf algebra 120 Hypermatrix 49 136 Hypersensibility to initial conditions 66 Ideal gas 245 Idempotent number 163 Imprimitivity 48 INDEX 71 Insensibility to 56 Interpolating polynomial Lagrange 11 12 Invariants of a Lie algebra 200 Invariants of a matrix 199 Invariants, Casimir 201 Invariants, Killing 201 Inverse of a function 157 Inverse of a matrix 9 198 Inverse of a series 173 Involution 193 Ising model and circulants 87 Ising model and detailed balancing 70 Ising model, stochastic approach 58 Iterate, continuous 211 215 Jacobi identity 5 Kerner model 23 Killing — Cartan metric 201 Kronecker delta 85 Lagrange, interpolation 11 12 Lagrange, series inversion 171 Lah number 163 Law, Gibbs — Di Marzio 34 39 78 Lee-Yang, circle theorem 272 Lee-Yang, matrix 274 Lee-Yang, phase transition theory 270 Legendre polynomials 178 Leibniz, formula 172 Lie algebra 5 Lie algebra invariants 200 Lie derivative 128 Liouville, canonical form 128 138 Liouville, equation 127 Lissajous curve 221 Logarithmic Bell polynomials 164 Logistic map 166 219
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