Àâòîðèçàöèÿ
Ïîèñê ïî óêàçàòåëÿì
Lasota A., Mackey M.C. — Probabilistic Properties of Deterministic Systems
Îáñóäèòå êíèãó íà íàó÷íîì ôîðóìå Íàøëè îïå÷àòêó?
Íàçâàíèå: Probabilistic Properties of Deterministic SystemsÀâòîðû: Lasota A., Mackey M.C. Àííîòàöèÿ: This book shows how densities arise in simple deterministic systems. Recently there has been explosive growth in interest in physical, biological, and economic systems that can be profitably studied using densities. Due to the inaccessibility of the mathematical literature there has been little diffusion of the applicable mathematics into the study of these 'chaotic' systems. This book will help to bridge that gap. The authors give a unified treatment of a variety of mathematical systems generating densities, ranging from one-dimensional discrete time transformations through continuous time systems described by integro-partial differential equations. They have drawn examples from many scientific fields to illustrate the utility of the techniques presented. The book assumes a knowledge of advanced calculus and differential equations, but basic concepts from measure theory, ergodic theory, the geometry of manifolds, partial differential equations, probability theory and Markov processes, and stochastic integrals and differential equations are introduced as needed
ßçûê: Ðóáðèêà: Ôèçèêà /Äèíàìè÷åñêèå ñèñòåìû /Ñòàòóñ ïðåäìåòíîãî óêàçàòåëÿ: Ãîòîâ óêàçàòåëü ñ íîìåðàìè ñòðàíèö ed2k: ed2k stats Ãîä èçäàíèÿ: 1985Êîëè÷åñòâî ñòðàíèö: 358Äîáàâëåíà â êàòàëîã: 17.09.2005Îïåðàöèè: Ïîëîæèòü íà ïîëêó |
Ñêîïèðîâàòü ññûëêó äëÿ ôîðóìà | Ñêîïèðîâàòü ID
Ïðåäìåòíûé óêàçàòåëü
22 21 21 30 22 13 14 302 305 14 74 14 Abel inequality 117 Abstract ergodic theorem 81 Acoustic feedback system 139 Adjoint operator 43 44 Almost everywhere (a.e.) 15 33 Almost sure convergence 275 Anosov diffeomorphism 51 52 71—72 Arcs, equivalent 153 Asymptotic periodicity 86—90 107 131 Asymptotic stability 95 176 Asymptotic stability of Chandrasekhar — Muench equation 344 Asymptotic stability of convex transformation 129 Asymptotic stability of expanding mappings on manifolds 160 Asymptotic stability of fluid flow 131 Asymptotic stability of integral operators 102 105 Asymptotic stability of integro-differential equations 337—344 Asymptotic stability of Lorenz equations 124 Asymptotic stability of monotonically increasing transformation 119 123 Asymptotic stability of quadratic transformation 142 Asymptotic stability of Renyi transformation 119 Asymptotic stability of stochastically perturbed systems 280 287 Asymptotic stability of strong repellor 129 Asymptotic stability of transformations on R 148 Asymptotic stability, proof via lower-bound function 96 175 Asymptotic stability, relation to conditional entropy 264 Asymptotic stability, relation to exactness 100 Asymptotic stability, relation to Liapunov function 105 329—330 336—337 Asymptotic stability, relation to statistical stability 95 Asymptotic stability, via change of variables 141 Automorphism 74 Baker transformation 48—50 60—61 74—75 260 Baker transformation, relation to dyadic transformation 50 260 Birkhoff ergodic theorem, continuous time 170 Birkhoff ergodic theorem, discrete time 57 Boltzmann equation and entropy 260 Borel 14 Borel measure 14 25—26 Borel measure on manifolds 157 Borel sets 14 Bounded variation, function of 114—119 Brownian motion, d-dimensional 303 Brownian motion, one-dimensional 294 Cartesian product 24 Cauchy condition for convergence 30 Cauchy problem for Fokker — Planck equation 322 Cauchy — Hoelder inequality 23 Cell proliferation 301—302 Cesaro convergence 27 Cesaro convergence of Frobenius — Perron operator 65—66 Cesaro convergence of Koopman operator 68 Chandrasekhar — Muench equation 214 341 344 Change of variables and asymptotic stability 141—147 Change of variables in Lebesgue integral 40—41 Characteristic function 5 Chebyshev inequality 104 273 Chebyshev polynomials 145 Classical solution of Fokker — Planck equation 323 Closed linear subspace 83 Compact support 181 Comparison series 31 Complete measure 27 Complete space 30 Conditional entropy 255—256 263—264 Connected manifold 156 Constant of motion 187 Constrictive Markov operator 87—88 Continuous semigroup 169 Continuous semigroup and ordinary differential equations 187—188 Continuous semigroup of contractions 178 Continuous semigroup of contractions and infinitesimal operator 200 Continuous stochastic processes 294 Continuous stochastic semigroup 175 Continuous time stochastic process 221—222 Continuous time system and discrete time systems 172 219—220 Continuous time system, ergodic 171 172 Continuous time system, exact 173 297—302 Continuous time system, mixing 172 194—200 Contracting operator 34 177 Convergence, almost sure 275 Convergence, Cauchy condition for 30 Convergence, Cesaro 27 Convergence, comparison series for 31 Convergence, in different 29 Convergence, in mean 274 Convergence, stochastic 274 Convergence, strong 27 Convergence, to set of functions 87 Convergence, weak 27 Convergence, weak Cesaro 27 Convex transformation 128—130 Counterimage 5 Counting measure 14 Counting process 222 Curvature 199—200 CYCLE 92 Cyclical permutation 92 Cylinder 195 298 Dense subset of densities 99 density 5 10 36 Density of absolutely continuous measure 36 Density of random variable 221 Density, evolution of by Frobenius — Perron operator 38 215—217 Derivative, right lower 111 Derivative, strong 181 Determinant of differential on manifold 158 Diffeomorphism 52 Differential delay equation as semidynamical system 164—165 Differential equation as dynamical system 164 Differential of transformation on manifold 154 Differential, determinant of, on manifold 158 Discrete time stochastic process 221—222 Discrete time system and Poisson processes 226—229 Discrete time system as sampled semidynamical system 219—220 Discrete time system, embedded in continuous time system 220 Distance, between function and set of functions 86—87 Distance, in 22 Distance, on manifold 156 Dyadic transformation 9 61—62 260 Dyadic transformation, related to baker transformation 50 Dynamical system 165 Dynamical system and exactness 173 Dynamical system, ergodic 171 Dynamical system, mixing 172 Elementary events 220 Endomorphism 74 entropy 249 Entropy and exact transformations 258 260 262 Entropy and Frobenius — Perron operator 257—260 262 Entropy and Hamiltonian systems 257—258 Entropy and invertible transformations 257 Entropy and Liouville equation 260 Entropy and Markov operators 254—257 Entropy of reversible and irreversible systems 260 Entropy, conditional 255 Equivalent arcs on manifold 153 Ergodic Birkhoff theorem 57 170 Ergodic dynamical system 171 Ergodic Markov operator 72 92 Ergodic semidynamical system 171 Ergodic transformation 53 Ergodicity and Hamiltonian systems 205 Ergodicity and linear Boltzmann equation 240—241 Ergodicity and rotational transformation 56—57 69—70 171—172 Ergodicity of motion on torus 190—192 Ergodicity, conditions for via Frobenius — Perron operator 55 65—66 86 194 Ergodicity, conditions for via Koopman operator 54 68 189 194 204 Ergodicity, illustrated 63 Ergodicity, necessary and sufficient conditions for 53 Ergodicity, relation to mixing, exactness, and K-automorphisms 73 Essential supremum 23 Euler — Bernstein equations 315 Events, elementary 220 Events, independent 221 Events, mutually disjoint 221 Exact Markov operator 72 93 Exact semidynamical system 173 Exact semidynamical system with continuous time 297—302 Exact semigroup of linear Boltzmann equation 241 Exact transformation 62 Exactness and entropy 258 262 Exactness, illustrated 65 Exactness, necessary and sufficient conditions for via Frobenius — Perron operator 65—66 194 Exactness, of r-adic transformation 70 Exactness, of transformations on torus 162 Exactness, relation to dynamical systems 173 Exactness, relation to ergodicity, mixing, and K-automorphisms 62 73 76 Exactness, relation to statistical stability 100—101 143 Expanding mappings 158—162 Factor of transformation 75 Finite measure space 15 First return map 219 Fixed point of Markov operator 35 Fluid flow 131 Fokker — Planck equation, and Cauchy problem 322 Fokker — Planck equation, and Liouville equation 329 Fokker — Planck equation, and stochastic semigroups 327 Fokker — Planck equation, asymptotic stability of solutions 330 Fokker — Planck equation, classical solution 323 Fokker — Planck equation, derivation of 318—322 Fokker — Planck equation, existence and uniqueness of solutions 323—324 Fokker — Planck equation, for Langevin equation 325 332 Fokker — Planck equation, for second-order system 335 Fokker — Planck equation, for stochastic differential equations 318 Fokker — Planck equation, fundamental solution 323 Fokker — Planck equation, generalized solution 327 Fokker — Planck equation, stationary solutions 332 333 334 336 Forced oscillator 136—138 Frobenius — Perron operator 37 173—174 Frobenius — Perron operator and evolution of densities 4—12 38 Frobenius — Perron operator and invariant measure 46 189 203 Frobenius — Perron operator and Koopman operator 177 Frobenius — Perron operator and Liouville equation 203 Frobenius — Perron operator and ordinary differential equations 185—187 Frobenius — Perron operator and semidynamical systems 173—174 189 Frobenius — Perron operator as Markov operator 38 Frobenius — Perron operator for Anosov diffeomorphism 51—52 Frobenius — Perron operator for baker transformation 48—50 Frobenius — Perron operator for dyadic transformation 9 Frobenius — Perron operator for Hamiltonian system 187—188 Frobenius — Perron operator for invertible transformations 42 Frobenius — Perron operator for quadratic map 7 47 Frobenius — Perron operator for r-adic transformation 9 46 Frobenius — Perron operator for transformations on 40 Frobenius — Perron operator for transformations on R 10 38 148 Frobenius — Perron operator, relation to entropy 257—260 262 Frobenius — Perron operator, relation to ergodicity 55 65—66 86 194 Frobenius — Perron operator, relation to infinitesimal operator 185—187 Frobenius — Perron operator, relation to Koopman operator 43 177 215—217 Frobenius — Perron operator, relation to mixing 65—66 194 Frobenius — Perron operator, semigroups of 173—174 Frobenius — Perron operator, support of 39 Frobenius — Perron operator, weak continuity of 43—44 Fubini's theorem 25 Function of bounded variation 115 Function, left lower semicontinuous 111 Function, lower semicontinuous 111 Function, support of 34 Fundamental solution of Fokker — Planck equation 323—324 Gas dynamics 194—198 244—247 Gaussian density 251 286 294 303 Gaussian kernel 176 208 214 218 324 Generalized solution of Fokker — Planck equation 327 Geodesic 199 Geodesic, flow 199 Geodesic, motion on 198—200 Gibbs canonical distribution function 253 Gibbs inequality 249 Gradient of function 152 Gradient, length of 156 Hahn — Banach theorem 83 Hamiltonian 187 Hamiltonian, system 187 192 198 204 205 257 Hat map 142 Hausdorff space 152 Heat equation 176 208 Henon map 50 Hille — Yosida theorem 200 Homeomorphism 152 Ideal gas 194—198 244—247 Independent 302 Independent events 221 Independent increments 222 Independent random variables 221 266 Indicator function 5 Inequality, Cauchy — Hoelder 23 Inequality, triangle 22 Infinitesimal operator 180 Infinitesimal operator and differential equations 181 Infinitesimal operator and ergodicity 189 Infinitesimal operator and Frobenius — Perron operator 185—187 203 Infinitesimal operator and Hamiltonian systems 187 Infinitesimal operator and Hille — Yosida theorem 200 Infinitesimal operator and invariant measure 203 Infinitesimal operator and Koopman operator 184—185 204 Infinitesimal operator and ordinary differential equations 183—185 Infinitesimal operator and partial differential equation 182—183 Infinitesimal operator as strong derivative 181 Infinitesimal operator of continuous semigroup of contractions 200 Infinitesimal operator, illustrated by heat equation 208 Infinitesimal operator, illustrated by parabolic differential equations 208—209 Integrable function 17 Integral, Ito 304—309 Integral, Lebesgue 16—18 Integral, Stochastic 305 311 Integro-differential equations 213 214 337 341 Intermittency 131 Invariant measure 45 169 Invariant measure and differential equations 203 Invariant measure and Frobenius — Perron operator 46 189 Invariant measure and Hamiltonian systems 204 Invariant measure and infinitesimal operators 203 Invariant measure and Liouville's theorem 203—204 Invariant measure for monotonic transformations 132 Invariant set 53 171 Invertibility 50 62 165 257 260 Ito integral 304—309 Ito sum 305 Jacobian matrix 41 Jensen inequality 253 Joint density function 267 K-automorphism 73—76 K-automorphism and exactness 76 K-automorphism and geodesic flows 200 K-automorphism and mixing 76 Kolmogorov automorphism 73 Kolmogorov equation see "Fokker — Planck equation" Koopman operator 42—44 177 Koopman operator and Anosov diffeomorphism 71—72
Ðåêëàìà
distance 
-algebra 
|
|
Î ïðîåêòå