Kigami J. — Analysis on Fractals :: Электронная библиотека попечительского совета мехмата МГУ

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Kigami J. — Analysis on Fractals

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Название: Analysis on Fractals

Автор: Kigami J.

Аннотация:

This graduate textbook introduces the analysis of fractals, particularly dynamical aspects such as heat diffusion on fractals and the vibration of a material with fractal structure. Kigami (Kyoto University) explains how to construct Dirichlet forms, harmonic functions, Green's functions and Laplacians on post critically finite self-similar sets, and examines eigenvalues and eigenfunctions of Laplacians and heat kernels associated with Laplacians.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2001

Количество страниц: 226

Добавлена в каталог: 17.01.2007

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Предметный указатель

$L^{\infty}$ -solution of the heat equation on $[0, \infty)$       164
$\Delta-Y$ transform      48 67 129
$\gamma$ -elliptic measure      89
$\Lambda$ -harmonic function      139
Affine nested fractal      116 117 150
Arcwise connected      33
Arithmetic case      207
bernoulli      25
Borel, $\sigma$ -algebra      25
Borel, measure      25
Borel, regular measure      25 26
Borel, set      25
Boundary condition      92 93
Boundary condition, Dirichlet      (see Dirichlet boundary condition)
Boundary condition, Neumann      (see Neumann boundary condition)
Boundary of self-similar set      19
Cantor set      15 39 66
Cantor set, topological      13 19
Characteristic function      42 203
Closable      197
Closed form      197
Compact operator      92 94 198
Compact resolvent      64 92 93 133 198
Compatible sequence      51 65 69
Complete measure      25
Connected      33
Connected, arcwise      (see Arcwise connected)
Connected, locally      (see Locally connected)
Contraction      9
Contraction principle      9 10
Contraction property      199
Contraction ratio      9
Core      202
Critical set      19
D-eigenfunction      (see Dirichlet eigenfunction)
D-eigenvalue      (see Dirichlet eigenvalue)
Diameter      14
Diffusion process      66 193
Directly Riemann integrable      207
Dirichlet boundary condition      93 158
Dirichlet eigenfunction      132
Dirichlet eigenvalue      132
Dirichlet form      65 88 92 185 202
Dirichlet form on a finite set      42
Dirichlet form, local      (see local Dirichlet form)
Dirichlet form, regular      see regular Dirichlet form)
Dirichlet Laplacian      93 95 111
Dirichlet problem for Poisson’s equation      114
Effective resistance      46 51 65
Effective resistance metric      48 66 83 138
Eigenfunction, Dirichlet      (see Dirichlet eigenfunction)
Eigenfunction, localized      (see Localized eigenfunction)
Eigenfunction, Neumann      (see Neumann eigenfunction)
Eigenfunction, pre-localized      (see Pre-localized eigenfunction)
Eigenvalue, counting function      133 190
Eigenvalue, decimation method      135 156
Eigenvalue, Dirichlet      (see Dirichlet eigenvalue)
Eigenvalue, Neumann      (see Neumann eigenvalue)
Equilibrium point      9
Extension of a form      197
Extension, Friedrichs      (see Friedrichs extension)
Finite rank operator      92 198
Finitely ramified      23
Fixed point      9 14 70
Focal point      180
Focal point, Rriedrichs extension      111 188 198
Frostman’s lemma      29
Fundamental solution      162 7
Gauss — Green’s formula      110 145 187
Generator      199
Green’s function      95 98 102 188 189
Green’s operator      94 95 102 103 145
Green’s operator, extended      102
Harmonic function      45 52 73 75 143
Harmonic structure      69 73 138
Harmonic structure, regular      (see Regular harmonic structure)
Harnack inequality      46 78 100
Hata’s tree-like set      16 24 71 129
Hausdorff, dimension      28 29
Hausdorff, measure      28 138
Hausdorff, metric      10
Heat kernel      152
Heat, equation      162 164 190
Heat, kernel      158 171 190
Heat, semigroup      190
Hitting time      194
Inequality, Harnack      (see Harnack inequality)
Inequality, Nash      (see Nash inequality)
Integrated density of states      135 155
Isomorphism between self-similar structures      18
Koch curve      15 17
Laplacian      65 66 92 107 108 186
Laplacian, Dirichlet      (see Dirichlet Laplacian)
Laplacian, Neumann      (see Neumann Laplacian)
Laplacian, on a finite set      42
Laplacian, standard      (see Standard Laplacian)
Lattice case      134 142 207
Lipschitz      9
Lipschitz constant      9
Local Dirichlet form      65 88 202
Local property      93 202
Localized eigenfunction      141
Locally connected      33

M-harmonic      81
Markov property      42 56 93 162 202
Markov property of a semigroup      202
Mass distribution principle      29
Maximum principle      45 52 76 77 81
Maximum principle, parabolic      (see Parabolic maximum principle)
Maximum principle, strong version      81
Maximum principle, weak version      77
Measure, Bernoulli      25
Measure, Borel      (see Borel measure)
Measure, Borel regular      (see Borel regular measure)
Measure, complete      (see Complete measure)
Measure, Hausdorff      (see Hausdorff measure)
Measure, probability      (see Probability measure)
Measure, self-similar      (see Self-similar measure)
Minimal      21
Modified Sierpinski gasket      129 129
Moran’s theorem      32
Multi-harmonic function      143
N-eigenfunction      (see Neumann eigenfunction)
N-eigenvalue      (see Neumann eigenvalue)
Nash inequality      153 172 202 204
Nested fractal      117 146
Nested fractal, affine      (see Affine nested fractal)
Nesting condition      117
Net      10
Neumann boundary condition      92 158
Neumann derivative      110
Neumann eigenfunction      132
Neumann eigenvalue      132
Neumann Laplacian      92 111
Neumann problem for Poissons equation      129
Non-arithmetic case      208
Non-lattice case      134 208
Open set condition      30 32
Operator, compact      (see Compact operator)
Operator, finite rank      (see Finite rank operator)
Operator, Green’s      (see Green’s operator)
Operator, renormalization      (see Renormalization operator)
Operator, self-adjoint      (see Self-adjoint operator)
Operator, symmetric      (see Symmetric operator)
Parabolic maximum principle      164
Partition      22 23 83 139 173
Pentakun      119
Piecewise harmonic function      81
Poisson’s equation      114 129
Post critical set      19 72
Post critically finite      23 34
Pre-localized eigenfunction      141 146 153
Probability measure      25
Quadratic form, associated with a self-adjoint operator      197
Quadratic form, non-negative      197
R-network      (see Resistance network)
Random walk      54
Refinement of partition      22
Regular Dirichlet form      55 65 88 202
Regular harmonic structure      69 75 167
Renewal, equation      137 207
Renewal, theorem      135 137 207
Renormalization operator      70
Resistance, form      55 61
Resistance, metric      56 61 70 75
Resistance, network      43 69
Self-adjoint operator      196
Self-affine set      10
Self-similar, measure      25 26 105
Self-similar, set      10 13
Self-similar, structure      18 25 26
Self-similarity of a form      83
Semigroup      160 199
Semigroup of symmetric operators      199
Semigroup, property      199
Semigroup, strong continuous      (see Strong continuous semigroup)
Shift, map      13
Shift, space      12 18
Sierpinski, carpet      24
Sierpinski, gasket      15 24 71 108 128 135 137 149 155
Similarity dimension of harmonic, structure      138
Similitude      9 13
Snowflake      120
Solution of the heat equation on $(0, \infty)$       162
Solution of the heat equation, $L^{\infty}-$       (see $L^{\infty}$ -solution of the heat equation)
Spectral, dimension      135 137 138 155
Spectral, exponent      134 137
Standard harmonic structure      71 77
Standard Laplacian      109 135 149
Strong continuous semigroup      199
Strongly symmetric      116
Symbol      12 18 20
Symmetric, operator      196
Symmetric, strongly      (see Strongly symmetric)
Symmetric, weakly      (see Weakly symmetric)
Symmetry of a harmonic structre      115
Symmetry, group      146
Theorem, renewal      (see Renewal theorem)
Theorem, Weyl’s      (see Weyl’s theorem)
Totally bounded      10
Unit contraction      202
Variational formula      136 140 198
Weakly symmetric      37
Weyl’s theorem      133
Weyl’s theorem, empty      12
Weyl’s theorem, length of      12
Weyl’s theorem, word      12

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