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Measurement Technique: it’s → its; then → than

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Fix CS1 deprecated date parameter errors

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In [[mathematics]] — specifically, in [[fractal geometry]] — the '''Assouad dimension''' is a definition of [[fractal dimension]] for subsets of a [[metric space]]. It was introduced by [[Patrice Assouad]] in his 1977 [[PhD]] [[thesis]] and later published in 1979. As well as being used to study fractals, the Assouad dimension has also been used to study [[quasiconformal mapping]]s and [[embedding|embeddability problems]].

==Definition==

Let (''X'', ''d'') be a [[metric space]], and let ''E'' be a non-empty subset of ''X''. For ''r'' > 0, let ''N''''r''(''E'') denote the least number of metric [[open ball]]s of radius less than or equal to ''r'' with which it is possible to [[open cover]] the set ''E''. The '''Assouad dimension''' of ''E'' is defined to be the [[infimum|infimal]] ''α'' ≥ 0 for which there exist positive constants ''C'' and ''ρ'' so that, whenever
:<math>0 < r < R \leq \rho,</math>
the following bound holds:
:<math>\sup_{x \in E} N_{r}(B_{R}(x) \cap E) \leq C \left( \frac{R}{r} \right)^{\alpha}.</math>

The intuition underlying this definition is that, for a set ''E'' with “ordinary” integer dimension ''n'', the number of small balls of radius ''r'' needed to cover the intersection of a larger ball of radius ''R'' with ''E'' will scale like (''R'' &frasl; ''r'')''n''.

==References==

* {{cite journal
| last = Assouad
| first = Patrice
| title = Étude d'une dimension métrique liée à la possibilité de plongements dans '''R'''''n''
| journal = C. R. Acad. Sci. Paris Sér. A-B
| volume = 288
| year = 1979
| issue = 15
| pages = A731–A734
| issn = 0151-0509}} {{MathSciNet|id=532401}}

[[Category:Dimension theory]]
[[Category:Fractals]]
[[Category:Metric geometry]]

Satellite surface salinity - Revision history

en>Mild Bill Hiccup: /* Measurement Technique */ it’s → its; then → than

en>Monkbot: Fix CS1 deprecated date parameter errors