Hausdorff Measure - Relation With Hausdorff Dimension

Relation With Hausdorff Dimension

One of several possible equivalent definitions of the Hausdorff dimension is


\operatorname{dim}_{\mathrm{Haus}}(S)=\inf\{d\ge 0:H^d(S)=0\}=\sup\bigl(\{d\ge 0:H^d(S)=\infty\}\cup\{0\}\bigr),

where we take

Read more about this topic:  Hausdorff Measure

Famous quotes containing the words relation with, relation and/or dimension:

    “There is undoubtedly something religious about it: everyone believes that they are special, that they are chosen, that they have a special relation with fate. Here is the test: you turn over card after card to see in which way that is true. If you can defy the odds, you may be saved. And when you are cleaned out, the last penny gone, you are enlightened at last, free perhaps, exhilarated like an ascetic by the falling away of the material world.”
    —Andrei Codrescu (b. 1947)

    “The instincts of the ant are very unimportant, considered as the ant’s; but the moment a ray of relation is seen to extend from it to man, and the little drudge is seen to be a monitor, a little body with a mighty heart, then all its habits, even that said to be recently observed, that it never sleeps, become sublime.”
    —Ralph Waldo Emerson (1803–1882)

    “Authority is the spiritual dimension of power because it depends upon faith in a system of meaning that decrees the necessity of the hierarchical order and so provides for the unity of imperative control.”
    —Shoshana Zuboff (b. 1951)