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Seneta E. — Non-negative matrices: an introduction to theory and application
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Íàçâàíèå: Non-negative matrices: an introduction to theory and applicationÀâòîð: Seneta E. Àííîòàöèÿ: Since its inception by Perron and Frobenius, the theory of non-negative matrices has developed enormously and is now being used and extended in applied fields of study as diverse as probability theory, numerical analysis, demography, mathematical economics, and dynamic programming, while its development is still proceeding rapidly as a branch of pure mathematics in its own right. While there are books which cover this or that aspect of the theory, it is nevertheless not uncommon for workers in one or other branch of its development to be unaware of what is known in other branches, even though there is often formal overlap. One of the purposes of this book, through its aiming at breadth rather than depth, is to relate various aspects of the theory, insofar as this is possible.
ßçûê: Ðóáðèêà: Ìàòåìàòèêà /Ñåðèÿ: Ñäåëàíî â õîëëå Ñòàòóñ ïðåäìåòíîãî óêàçàòåëÿ: Ãîòîâ óêàçàòåëü ñ íîìåðàìè ñòðàíèö ed2k: ed2k stats Ãîä èçäàíèÿ: 1974Êîëè÷åñòâî ñòðàíèö: 256Äîáàâëåíà â êàòàëîã: 23.05.2011Îïåðàöèè: Ïîëîæèòü íà ïîëêó |
Ñêîïèðîâàòü ññûëêó äëÿ ôîðóìà | Ñêîïèðîâàòü ID
Ïðåäìåòíûé óêàçàòåëü
Absorption with probability one 92—93 Absorption, mean time to 94—95 Absorption, probabilities into essential classes 93 Absorption, Probability 158—159 Age structure, evolution of 76 Albert, E. 177 Allen, B. 178 Anderssen, R.S. 178 Backward equations 93 Barriers, absorbing 88 95 158 Barriers, elastic 88 Barriers, reflecting 88 101 179 Bassett, L. 48 Bellman, R. 23 Bernoulli scheme 87 Bernstein, S.N. 99 117 118 119 Birkhoff, G. 25 57 Boolean algebra 50 Boolean algebra, relation matrix 50 107 Borel sets 150—152 Boundary theory 147—152 178 Branching processes 78 Brauer, A. 23 48 55 Brualdi, R.A. 48 57 Buchanan, M.L. 83 Buharaev, R.G. 57 Burger, E. 33 Cantor diagonal selection principle 132 150 Cauchy sequence 148—150 Cherubino, S. 22 Chung, K.L. 23 155 Classes, aperiodic 15 124 Classes, essential 9 91 125 Classes, inessential 9—10 91—99 125 Classes, lead to 11 Classes, period of 14 124—125 Classes, self-communicating 10 124 Classes, subclasses of 16—22 Coale — Lopez Theorem 77 Coding theory 119 Collatz, L. 24 Combinatorial properties 49—55 Contraction properties of P 68 77 104 111 Control theory 66 Control theory, parameter 66 Convergence, geometric in ergodicity of M.C.’s 91 97 Convergence, geometric in weak ergodicity 72 106 Convergence, norm 162 Convergence, parameter 26 61 161—181 Convergence, uniform in strong ergodicity 75 Convergence, uniform in T 78 Convexity in non-negative matrices 82—83 Convolution 126 Cycle of indices 9 26 40 48 Daley, D.J. 178 Darroch, J.N. 47 de Oliveira, G.N. 56 57 205 Debreu, G. 22 26 34 35 38n Decision, linear multistage 66 Decomposition of superregular vector 145 Decomposition, ‘last exit’ 126 Demography 27 67 76—77 Derman, C. 154 168n Determinants, infinite 174—181 Dionisio, J.J. 47 54 Divisor, greatest common 15 182—185 Djokovic, D.i. 57 Dmitriev, N.A. 56 Doeblin, W. 99 100 100n 118 119 Doeblin’s theory of M.C.’s 155 Doob, J.L. 156 Dual approach to countable P 140 157 Dulmage, A.L. 54 Dynamic programming 59 66 Dynkin, E.B. 56 Economics, mathematical 30—33 40 59 66 Eigenvalues of cyclic matrix 21—25 Eigenvalues of Perron matrix 43 47 Eigenvalues, analogue of Perron — Frobenius 160 Eigenvalues, bounds for 6 23 28 49 55—57 Eigenvalues, generalized problem of 48 Eigenvalues, invariance under rotation 22 Eigenvalues, Perron — Frobenius 20; see also "Perron — Frobenius" Erdoes — Feller — Pollard Theorem 156 Erdoes, P. 156 Ergodic theorems 69 73 91 97 138 156 Ergodicity of primitive M.C.’s 91 Ergodicity of regular M.C.’s 97 Ergodicity, characterization of 102 Ergodicity, coefficient of 118 Ergodicity, geometric 170—171 178 Ergodicity, pseudo- 98—99 Ergodicity, strong 73 104 111—121 Ergodicity, uniform weak 108 Ergodicity, weak 69 103—121 Fan, K. 23 33 47 Feller, W. 155 156 Fiedler, M. 47 Forward equations 93 Frechet, M. 99 Frobenius, G. 22 25 33 34 55 Functions, convex 194—195 Functions, generating 125—129 141 162—165 Functions, subadditive 184 Functions, superconvex 194—195 Functions, supermultiplicative 184 Functions, upper semicontinuous 1 193—194 Gale, D. 34 Gantmacher, F.R. 22 33 47 56 Gauss — Seidel iteration 36—39 102 Gauss — Seidel iteration, matrix 36 Generators, extreme 48 Georgescu-Roegen, N. 33 Golub, G. 178 Gordon, P. 23 Green’s function 143 Hadamard, J. 100 Hajnal, J. 118 Harris — Veech condition 156 Harris, T.E. 156 Hawkins — Simon condition 32 35 Hawkins, D. 33 Haynsworth, E.V. 56n Heap, B.R. 54 Heathcote, C.R. 8 Helly selection principle 151—152 Helly — Bray lemma 152 Herstein, I.N. 22 24 26 34 35 38n Hille, E. 184n Hoelder’s inequality 83 194—195 Holladay, J.C. 54 Holmes, P.T. 156 Homogeneity, asymptotic 111—118 Hostinsky, B. 99 Householder, A.S. 23 39 Hypergeometric probabilities 101 Index of primitivity 50—52 Index, aperiodic 15 90 130—134 141 Index, consequent 10 Index, essential 9 90 102 123 155 Index, inessential 9 90 102 123 155 157 Index, leads to 9 Index, null-recurrent 127 Index, period of 14 90 123 160 Index, periodic 156 Index, positive-recurrent 127 157 Index, R-null 163—181 Index, R-positive 163—181 Index, R-recurrent 163—181 Index, R-transient 163—181 Index, recurrent 127 130—134 141 Index, transient 127 157—159 Indices, chain of 9 Indices, Chung’s exposition on 23 Indices, classification of 9 49 90 124—127 161—165 Indices, communicating 9 90 Indices, cycle of 9 26 40 48 Indices, Kolmogorov’s classification of 23 Indices, relabelling of 12 Indices, residue class of 16—22 Indices, simultaneous permutation of 12 Indices, subclass of 16—22 Inequality, fundamental for minimal vector 143 147 Information theory 119 Initial probability distribution 86 91 97—99 Iterative methods 35—39 Jacobi iteration 36—39 Jacobi iteration, matrix 36 Joffe, A. 77 Jordan canonical form 38 Karlin, S. 23 34 Karpelivich, F.I. 56 Kaucky, J. 102 Keilson, J.H. 205 Kemeny, J.G. 34 77 97n 99 155 156 180 Kendall, D.G. 156 177 178 179 Kernel 143 Khintchine, A.Y. 184n Kingman, J.F.C. 83 178 184n 195n Klimko, L.A. 205 Knapp, A.W. 155 156 Knopp, P. 57 Kolmogorov, A.N. 23 77 100 117 155 156 Kolmogorov’s theory of M.C.’s 155 Konecny, M. 102 Kotelyanskii — Gantmacher assertion 33 Kotelyanskii, D.M. 33 35 49 Kozniewska, I. 118 Kuich, W. 178 179 180 Larisse, J. 119 Law of iterated logarithm 119 Ledermann, W. 49 55 Leontief model 30—38 Leontief model, dynamic version 32 38 Leontief model, matrix 40 Leontief model, open 30 32 34—35 Leontief model, static version 32 Lopez, A. 77 Lynn, M.S. 54 M matrix 47 Malecot, G. 101 Mandl, P. 47 66 67 178 Mangasarian, O.L. 48 Mapping, continuous 44 Mappingá upper semicontinuous 193—194 Marcus, M. 55 56 57 Markov chains 23 34—35 77 84—102 123—126 140 147 157 Markov matrices 105 117—121 178 Markov processes 36 40 47 Markov property 85 Markov regular chains 97 Markov, A.A. 77 99 117 Markov’s theorem 117 Martin entrance boundary 158—159 Martin exit boundary 140 147—150 178 Matrices of Laplace — Stieltjes transforms 83 Matrices of moduli 25 Matrices, acyclic (aperiodic) 15 Matrices, arbitrary non-negative 1 25 54 123 Matrices, block sub- 13 21 Matrices, Boolean relation 50 107 Matrices, canonical form 10—11 17 27 124 Matrices, characterization of irreducible ML 42 Matrices, column finite 179—180 Matrices, combinational properties of 49—55 Matrices, convergence radius of irreducible 26 Matrices, cyclic (periodic) 9 15 20—25 179 180 Matrices, determinantal and cofactor properties ofinfinite 171—181 Matrices, deterministic 52 Matrices, diagonalization of 57—58 Matrices, doubly stochastic 56—57 101 157 Matrices, essentially positive 40 Matrices, fully indecomposable 57—58 Matrices, fundamental for absorbing M.C. 94 Matrices, Gauss — Seidel 36 Matrices, graph theory of 23 50 54 Matrices, incidence 1 50 106—107 111 Matrices, infinite 23 66 123—181 Matrices, inhomogeneous products of 67—78 Matrices, irreducible 7 9 15—27 43—51 90 124 134—140 Matrices, irreducible ML 40 Matrices, Jacobi 36 Matrices, Leontief 40 Matrices, M 47 Matrices, Markov 105 108 117—121 178 Matrices, Metzler 40 Matrices, Minkowski 40 Matrices, ML 40 48—49 Matrices, Morishima 48 Matrices, normed 111 120 Matrices, null-recurrent 135—140 156—157 170 Matrices, path diagrams of 10 23 124 Matrices, periodic (cyclic) 9 15 20—25 179 180 Matrices, permanents of 57 Matrices, permutable structure of 22 Matrices, permutation of 57 Matrices, Perron 43—49 Matrices, positive-recurrent 135—140 156—157 Matrices, power-positive 43—48 Matrices, quasi-Markov 121 Matrices, R-null 164—181 Matrices, R-positive 164—181 Matrices, R-recurrent 164—181 Matrices, R-transient 164—181 Matrices, recurrent stochastic 135—140 156—157 Matrices, regular stochastic 97 105—108 120 Matrices, representation of cyclic 25 Matrices, row finite 179—180 Matrices, scrambling stochastic 108—110 118—120 Matrices, sets of irreducible 59—83 Matrices, sets of positive 67 Matrices, similarity transformation of 12 Matrices, slowly spreading stochastic 180 Matrices, spectral decomposition of 7 42 Matrices, stable stochastic 104 Matrices, stochastic 23 56 68 77 84—121 123—160 163—165 168—171 176—181 Matrices, sub-stochastic 31 34 98—99 142 171 Matrices, Tambs — Lyche 40 Matrices, transient stochastic 135—140 156—157 Matrices, transition 86 Matrices, transition intensity 40 47 Matrices, truncations of infinite 171—181 Matrices, unitary 80 190—192 Maybee, J.S. 48 McFarland, D.D. 77 Measure, 166—168 Measure, 166—168 Measure, bounds for R-invariant 175 Measure, invariant 135—140 155—156 168 181 Measure, mean recurrence 127 Measure, minimal subinvariant 137 Measure, probability 150—152 Measure, R-invariant 163—181 Measure, R-subinvariant 163—181 Measure, subinvariant 134—140 155 Medlin, G.W. 205 Mendelsohn, N.S. 54 Metric d on R 148—150 Metric space 148—152 155 Metzler matrix 40
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