definition 7.142 Box-counting dimension

open in the book · parts/02-mathematical-methods/05-real-analysis.tex:4853 · p. 262

ground object -- no derivation owed

Rests on

depends_on definition 7.140 Hausdorff measure ¶
- depends_on axiom 7.1 Completeness of $\R$ ¶
- depends_on definition 7.138 Set of measure zero ¶
  - depends_on definition 7.125 Multiple integral ¶
    - depends_on axiom 7.1 Completeness of $\R$ ¶ ↺
    - depends_on definition 7.39 Darboux sums and the definite integral ¶
      - depends_on axiom 7.1 Completeness of $\R$ ¶ ↺
  - depends_on definition 3.65 Countable set ¶
    - depends_on axiom 3.33 Peano axioms ¶
    - depends_on definition 3.62 Equipotence ¶
      - depends_on definition 3.47 Bijective map ¶
      - depends_on definition 3.40 Cardinal ¶
- depends_on definition 6.24 Metric ¶
  - depends_on definition 3.43 Map ¶
    - depends_on definition 3.24 Quantifiers ¶
      - depends_on definition 3.23 Predicate ¶
    - depends_on definition 3.28 Set ¶
  - depends_on definition 3.38 Ordered pair and Cartesian product ¶
depends_on definition 6.9 Compact set ¶
- depends_on definition 6.5 Open cover ¶
  - depends_on definition 6.2 Open set ¶
    - depends_on definition 6.1 Topological space ¶
      - depends_on definition 3.31 Empty set ¶
      - depends_on definition 3.35 Union, intersection, difference ¶
      - depends_on definition 3.30 Subset ¶
      - depends_on equation 3.51 eq:set-indexed ¶
  - depends_on equation 3.51 eq:set-indexed ¶ ↺

Supports

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 proposition 7.143 The Hausdorff dimension never exceeds the box dimension ¶
depends_on remark 7.144 What is developed here, and what is not ¶

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Hausdorff measure	declared	parts/02-mathematical-methods/05-real-analysis.tex:4871
`depends_on`	→	Compact set	declared	parts/02-mathematical-methods/05-real-analysis.tex:4871
`depends_on`	←	Box-counting dimension, restated from Part II	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:2364
`depends_on`	←	The Hausdorff dimension never exceeds the box dimension	declared	parts/02-mathematical-methods/05-real-analysis.tex:4883
`depends_on`	←	What is developed here, and what is not	declared	parts/02-mathematical-methods/05-real-analysis.tex:4939