Blancmange Curve - Relation To Simplicial Complexes

Relation To Simplicial Complexes

Let

$N=\binom{n_t}{t}+\binom{n_{t-1}}{t-1}+\ldots+\binom{n_j}{j},\quad n_t > n_{t-1} > \ldots > n_j \geq j\geq 1.$

Define the Kruskal–Katona function

$\kappa_t(N)={n_t \choose t+1} + {n_{t-1} \choose t} + \dots + {n_j \choose j+1}.$

The Kruskal–Katona theorem states that this is the minimum number of (t − 1)-simplexes that are faces of a set of N t-simplexes.

As t and N approach infinity, (suitably normalized) approaches the blancmange curve.

Read more about this topic: Blancmange Curve

Famous quotes containing the words relation to and/or relation:

“The whole point of Camp is to dethrone the serious. Camp is playful, anti-serious. More precisely, Camp involves a new, more complex relation to “the serious.” One can be serious about the frivolous, frivolous about the serious.”
—Susan Sontag (b. 1933)

“Much poetry seems to be aware of its situation in time and of its relation to the metronome, the clock, and the calendar. ... The season or month is there to be felt; the day is there to be seized. Poems beginning “When” are much more numerous than those beginning “Where” of “If.” As the meter is running, the recurrent message tapped out by the passing of measured time is mortality.”
—William Harmon (b. 1938)