equation 3.51 eq:set-indexed

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Supports

• depends_on definition 6.5 Open cover ¶
  • depends_on definition 32.79 Box-counting dimension, restated from Part II ¶
    • depends_on definition 32.81 Correlation dimension ¶
    • depends_on example 32.80 The middle-thirds Cantor set ¶
  • depends_on definition 6.9 Compact set ¶
    • depends_on definition 7.142 Box-counting dimension ¶
      • depends_on definition 32.79 Box-counting dimension, restated from Part II ¶ ↺
      • depends_on proposition 7.143 The Hausdorff dimension never exceeds the box dimension ¶
      • depends_on remark 7.144 What is developed here, and what is not ¶
    • depends_on definition 7.127 Simple regions ¶
      • depends_on lemma A.512 The boundary strip is thin ¶
        • depends_on theorem A.513 Change of variables ¶
      • depends_on remark A.516 The hypotheses of the global form ¶
      • depends_on theorem 7.133 Gauss ¶
        • depends_on lemma A.708 The disturbance flux vanishes ¶
        • depends_on lemma A.712 The force is a far-field integral ¶
        • depends_on lemma A.117 Laplacian in orthogonal coordinates ¶
        • depends_on lemma 44.10 Divergence theorem on $(M,g)$ ¶
        • depends_on lemma 31.11 Transport theorem for a material volume ¶
        • depends_on lemma 106.86 A shift of a divergent integral leaves a surface term ¶
        • depends_on lemma 10.44 Green's identities ¶
        • depends_on lemma 10.58 Darboux's equation for spherical means ¶
        • depends_on proposition 23.54 The geometrical amplitude diverges ¶
        • depends_on proposition 10.57 Energy in a backward cone ¶
        • depends_on theorem 16.36 Euler–Lagrange equations for several independent variables ¶
        • depends_on theorem 32.6 Evolution of phase volume ¶
        • … 1 more
      • depends_on theorem 7.131 Green ¶
        • depends_on proposition 32.33 Bendixson's negative criterion ¶
        • depends_on proposition 9.36 Bendixson–Dulac negative criterion ¶
        • depends_on theorem 7.132 Stokes ¶
        • depends_on theorem 8.12 Cauchy ¶
        • depends_on theorem 13.41 Holonomy equals the enclosed curvature; local Gauss–Bonnet ¶
        • depends_on theorem 10.96 Rankine–Hugoniot condition ¶
    • depends_on definition A.498 Zero content ¶
      • depends_on lemma A.499 What zero content buys ¶
        • depends_on corollary A.514 Degeneracy on a negligible set ¶
        • depends_on lemma A.500 Graphs and $C^{1}$ images have zero content ¶
        • depends_on theorem A.513 Change of variables ¶ ↺
    • depends_on definition A.529 Hausdorff; second countable; locally compact ¶
      • depends_on lemma A.530 Two elementary facts about compactness ¶
        • depends_on proposition A.532 Separation and countability of the quotient ¶
    • depends_on definition 32.8 Attractor and basin ¶
      • depends_on definition 32.29 Limit cycle ¶
        • depends_on phenomenon 32.35 Self-sustained oscillation ¶
      • depends_on definition 32.67 Strange attractor ¶
        • depends_on example 32.70 The Hénon map ¶
        • depends_on phenomenon 32.97 Turbulence sets in after a few transitions ¶
        • depends_on phenomenon 32.68 Strange attractors ¶
      • depends_on theorem 32.74 Oseledets, quoted ¶
        • depends_on phenomenon 32.72 Sensitive dependence on initial conditions ¶
    • depends_on definition 12.41 The operator classes ¶
      • depends_on definition 12.90 Self-adjoint family; commutant; irreducibility ¶
        • depends_on proposition A.589 Cyclic subspaces and the rank of the average ¶
        • depends_on theorem A.579 Stone–von Neumann ¶
        • depends_on theorem 12.91 Schur's lemma, commutant form ¶
        • depends_on theorem 12.114 Stone–von Neumann ¶
      • depends_on definition 12.58 Projection-valued measure ¶
        • depends_on definition 12.60 Functional calculus ¶
        • depends_on lemma A.247 Integration against a projection-valued measure ¶
        • depends_on theorem A.238 Spectral theorem, both forms ¶
        • depends_on theorem 12.59 Spectral theorem for a bounded self-adjoint operator ¶
      • depends_on definition 12.64 Strongly continuous one-parameter unitary group ¶
        • depends_on definition A.580 Weyl operator ¶
        • depends_on definition 12.109 Weyl system ¶
        • depends_on lemma A.255 Smoothed vectors lie in the domain ¶
        • depends_on lemma A.581 Composition law ¶
        • depends_on lemma A.582 Joint strong continuity ¶
        • depends_on proposition A.256 Density ¶
        • depends_on proposition 12.65 Exponential of a bounded self-adjoint operator ¶
        • depends_on proposition 12.67 The generator is symmetric, and generates the motion ¶
        • depends_on theorem A.253 Stone ¶
        • depends_on theorem 12.66 Stone ¶
        • depends_on theorem 25.34 Stone–von Neumann ¶
      • depends_on lemma A.231 Restriction to an invariant closed subspace ¶
        • depends_on lemma A.233 Construction of the system ¶
      • depends_on lemma A.230 Sequential characterisation ¶
        • depends_on lemma A.232 Attainment ¶
        • depends_on lemma A.231 Restriction to an invariant closed subspace ¶ ↺
        • depends_on lemma A.234 The eigenvalues tend to zero, with finite multiplicity ¶
      • depends_on proposition 12.42 Elementary consequences ¶
        • depends_on lemma A.232 Attainment ¶ ↺
        • depends_on proposition 12.43 Norm of a self-adjoint operator ¶
      • depends_on theorem A.229 Hilbert–Schmidt ¶
      • depends_on theorem 12.44 Hilbert–Schmidt: compact self-adjoint operators ¶
        • depends_on theorem A.461 Completeness in the weighted and in the energy norm ¶
        • depends_on theorem A.471 Spectral decomposition and completeness in $L^{2}_{r}$ ¶
      • depends_on theorem 12.55 The spectrum of a self-adjoint operator is real ¶
        • depends_on example 12.56 Multiplication by the coordinate: spectrum without eigenvectors ¶
        • depends_on theorem A.238 Spectral theorem, both forms ¶ ↺
        • depends_on theorem 12.59 Spectral theorem for a bounded self-adjoint operator ¶ ↺
    • depends_on definition 13.66 Smooth action; free; proper; orbit ¶
      • depends_on example 13.68 Why each hypothesis is there ¶
        • depends_on remark A.542 Where each hypothesis is spent ¶
      • depends_on lemma A.531 The projection is open ¶
        • depends_on lemma A.536 A slice is a chart domain downstairs ¶
        • depends_on proposition A.532 Separation and countability of the quotient ¶ ↺
      • depends_on lemma A.533 The orbit map has constant rank $d$ ¶
        • depends_on proposition A.534 Every orbit is an embedded copy of $G$ ¶
        • depends_on proposition A.535 Existence of a slice ¶
      • depends_on proposition A.535 Existence of a slice ¶ ↺
      • depends_on proposition A.532 Separation and countability of the quotient ¶ ↺
      • depends_on theorem 13.67 Quotient manifold theorem ¶
        • depends_on example 13.68 Why each hypothesis is there ¶ ↺
        • depends_on proposition A.566 The isotropy group acts, and the quotient is smooth ¶
        • depends_on theorem A.563 Marsden–Weinstein reduction ¶
    • depends_on lemma A.506 Locality ¶
      • depends_on proposition A.511 The substitution property is universal ¶
        • depends_on remark A.516 The hypotheses of the global form ¶ ↺
        • depends_on theorem A.513 Change of variables ¶ ↺
    • depends_on lemma A.505 A continuous partition of unity ¶
      • depends_on lemma A.506 Locality ¶ ↺
    • depends_on lemma A.530 Two elementary facts about compactness ¶ ↺
    • depends_on lemma A.307 Partition of unity on a compact manifold ¶
      • depends_on definition A.308 Integral over the manifold ¶
        • depends_on theorem A.311 General Stokes theorem ¶
    • depends_on lemma 13.136 Discrete subgroups of $\R^{f}$ ¶
    • … 11 more
  • depends_on lemma 6.30 Lebesgue number ¶
    • depends_on theorem 6.31 Compactness and sequential compactness ¶
      • depends_on lemma A.230 Sequential characterisation ¶ ↺
      • depends_on lemma A.488 Small chords cut off small arcs ¶
        • depends_on theorem A.490 Equicontinuity of the normalised class ¶
• depends_on definition 6.1 Topological space ¶
  • depends_on definition A.529 Hausdorff; second countable; locally compact ¶ ↺
  • depends_on definition A.298 Half-space; smoothness on it ¶
    • depends_on definition A.299 Manifold with boundary ¶
      • depends_on lemma A.300 The boundary is well defined, and is a manifold ¶
        • depends_on definition A.309 The induced orientation of the boundary ¶
      • depends_on lemma A.307 Partition of unity on a compact manifold ¶ ↺
  • depends_on definition 11.33 Sigma-algebra ¶
    • depends_on definition 11.34 Measure; measure space ¶
      • depends_on definition 11.38 Measure-preserving map; invariant measure; ergodicity ¶
        • depends_on theorem 11.39 Birkhoff's pointwise ergodic theorem ¶
      • depends_on definition 11.35 Null set; almost everywhere ¶
        • depends_on definition 11.38 Measure-preserving map; invariant measure; ergodicity ¶ ↺
        • depends_on proposition 11.36 Monotonicity, countable subadditivity, and null sets ¶
      • depends_on proposition 11.36 Monotonicity, countable subadditivity, and null sets ¶ ↺
  • depends_on definition 6.6 Continuous map ¶
    • depends_on definition A.361 Covering map ¶
      • depends_on lemma A.366 Unique path lifting ¶
        • depends_on lemma A.367 Homotopy lifting ¶
      • depends_on lemma A.369 $\Phi$ is a two-sheeted covering ¶
      • depends_on theorem A.364 Covering homomorphisms ¶
      • depends_on theorem A.363 Monodromy ¶
        • depends_on theorem A.364 Covering homomorphisms ¶ ↺
    • depends_on definition 12.64 Strongly continuous one-parameter unitary group ¶ ↺
    • depends_on definition 6.7 Homeomorphism ¶
      • depends_on definition 13.50 Diffeomorphism ¶
      • depends_on definition 13.53 Immersion, submersion, embedding ¶
        • depends_on definition 13.54 Embedded submanifold ¶
        • depends_on example 13.55 An injective immersion that is not an embedding ¶
        • depends_on lemma A.533 The orbit map has constant rank $d$ ¶ ↺
        • depends_on lemma A.538 Submersions have smooth local sections ¶
        • depends_on proposition A.534 Every orbit is an embedded copy of $G$ ¶ ↺
        • depends_on theorem A.537 The smooth structure on the orbit space ¶
      • depends_on lemma A.536 A slice is a chart domain downstairs ¶ ↺
      • depends_on theorem 32.15 Hartman–Grobman, restated from Part II ¶
        • depends_on definition 32.22 Bifurcation ¶
        • depends_on example 32.18 A linear centre that is really a stable focus ¶
      • depends_on theorem 9.32 Hartman–Grobman; quoted ¶
        • depends_on theorem 32.15 Hartman–Grobman, restated from Part II ¶ ↺
    • depends_on definition 6.15 Path-connected space ¶
      • depends_on definition A.362 Path homotopy and the fundamental group ¶
        • depends_on lemma A.368 The concatenation rules ¶
        • depends_on theorem A.363 Monodromy ¶ ↺
      • depends_on definition 6.17 Simply connected space ¶
        • depends_on corollary 8.15 Deformation of contours ¶
        • depends_on definition A.362 Path homotopy and the fundamental group ¶ ↺
        • depends_on definition 16.62 Field of extremals; slope function ¶
        • depends_on example 6.18 ex:top-simply-connected ¶
        • depends_on lemma 14.34 The $n$-sphere is simply connected for $n \ge 2$ ¶
        • depends_on proposition 32.33 Bendixson's negative criterion ¶ ↺
        • depends_on proposition 14.31 $\SO(3,\R)$ is not simply connected ¶
        • depends_on proposition 9.36 Bendixson–Dulac negative criterion ¶ ↺
        • depends_on proposition 6.23 The punctured plane is not simply connected ¶
      • depends_on proposition 6.16 prop:top-path-implies-connected ¶
    • depends_on definition 6.17 Simply connected space ¶ ↺
    • depends_on lemma A.505 A continuous partition of unity ¶ ↺
    • depends_on lemma A.531 The projection is open ¶ ↺
    • depends_on proposition 6.10 Continuous images of compact sets ¶
      • depends_on lemma A.530 Two elementary facts about compactness ¶ ↺
    • depends_on proposition 6.28 $\varepsilon$–$\delta$ characterization ¶
      • depends_on proposition 12.36 Boundedness is continuity ¶
        • depends_on theorem 12.74 Hellinger–Toeplitz ¶
  • depends_on definition 6.26 Open ball; metric topology ¶
    • depends_on definition A.76 Star-shaped set ¶
      • depends_on lemma A.77 Poincaré lemma, converse form ¶
    • depends_on lemma 6.30 Lebesgue number ¶ ↺
    • depends_on proposition 6.28 $\varepsilon$–$\delta$ characterization ¶ ↺
  • depends_on definition 6.2 Open set ¶
    • depends_on definition A.361 Covering map ¶ ↺
    • depends_on definition A.141 Mollification ¶
      • depends_on proposition A.143 Reproduction identity ¶
    • depends_on definition A.134 Mean-value property ¶
      • depends_on corollary A.146 Locally uniform limits of harmonic functions ¶
      • depends_on proposition A.143 Reproduction identity ¶ ↺
      • depends_on theorem A.135 Converse of the mean-value property ¶
        • depends_on corollary A.146 Locally uniform limits of harmonic functions ¶ ↺
        • depends_on corollary A.145 Harmonic functions are smooth ¶
    • depends_on definition A.529 Hausdorff; second countable; locally compact ¶ ↺
    • depends_on definition 13.108 Star-shaped domain ¶
      • depends_on theorem 13.109 Converse of the Poincaré lemma on a star-shaped domain ¶
        • depends_on example 13.110 Closed but not exact: the angle form ¶
        • depends_on theorem 13.112 Symmetric analogue of the converse Poincaré lemma ¶
    • depends_on definition 10.1 Partial differential equation; order ¶
      • depends_on definition A.102 Normal Cauchy problem of order $k$ ¶
        • depends_on definition A.103 First-order quasilinear system with zero data ¶
        • depends_on lemma A.105 Formal reduction of a normal problem ¶
      • depends_on definition 10.2 Linear, semilinear, quasilinear ¶
        • depends_on definition A.103 First-order quasilinear system with zero data ¶ ↺
        • depends_on definition 10.93 Scalar conservation law ¶
        • depends_on definition 10.3 Principal part and principal symbol ¶
        • depends_on lemma 10.28 The separation constant ¶
        • depends_on proposition 28.44 Superposition ¶
        • depends_on proposition 10.15 Method of characteristics, first order ¶
        • depends_on proposition 10.43 Representation of the solution ¶
        • depends_on theorem 10.22 Cauchy–Kovalevskaya ¶
        • depends_on theorem 10.52 Duhamel's principle ¶
      • depends_on definition 10.24 Well-posed problem ¶
        • depends_on corollary A.152 Uniqueness ¶
        • depends_on corollary 10.76 Uniqueness and stability for the Dirichlet problem ¶
        • depends_on corollary 10.81 Uniqueness for the heat equation ¶
        • depends_on definition 30.22 Boundary conditions of elastostatics ¶
        • depends_on proposition 10.25 The backward heat problem is ill posed ¶
        • depends_on theorem 44.45 Local existence and uniqueness; Choquet-Bruhat; imported ¶
    • depends_on definition 6.3 Closed set ¶
      • depends_on definition 12.70 Graph; closed and closable operators ¶
        • depends_on proposition 12.73 The adjoint is always closed ¶
      • depends_on definition 10.79 Parabolic boundary ¶
        • depends_on theorem 10.80 Parabolic maximum principle ¶
      • depends_on theorem 6.12 Heine–Borel in $\R^{N}$ ¶
    • depends_on definition 6.13 Connected space ¶
      • depends_on proposition 6.16 prop:top-path-implies-connected ¶ ↺
      • depends_on theorem 10.77 Strong maximum principle for harmonic functions ¶
      • depends_on theorem 6.14 Intervals are connected ¶
        • depends_on proposition 6.16 prop:top-path-implies-connected ¶ ↺
    • depends_on definition 6.6 Continuous map ¶ ↺
    • depends_on definition 6.4 Neighbourhood ¶
    • depends_on definition 6.26 Open ball; metric topology ¶ ↺
    • depends_on definition 6.5 Open cover ¶ ↺
    • … 2 more