X-Nico

3 unusual facts about monotonic function


Erdős–Rényi model

If P is any graph property which is monotone with respect to the subgraph ordering (meaning that if A is a subgraph of B and A satisfies P, then B will satisfy P as well), then the statements "P holds for almost all graphs in G(n, p)" and "P holds for almost all graphs in G(n, \tbinom{n}{2} p) " are equivalent (provided pn2 → ∞).

Strausberg

After an increase of population since the Second World War, Strausberg has stopped its "growth" remaining at almost 26,000 inhabitants in last one, with a little decrease in early years of 2000.

YAFFS

YAFFS2 marks every newly written block with a sequence number that is monotonically increasing.


Holographic algorithm

Soon after, Valiant found holographic algorithms with reductions to matchgates for #Pl-Rtw-Mon-3CNF and #7Pl-3/2Bip-VC.

Order theory

A simple example of an order theoretic property for functions comes from analysis where monotone functions are frequently found.

Roll-off

For some filter classes, such as the Butterworth filter, the insertion loss is still monotonically increasing with frequency and quickly asymptotically converges to a roll-off of 6n dB/8ve, but in others, such as the Chebyshev or elliptic filter the roll-off near the cut-off frequency is much faster and elsewhere the response is anything but monotonic.


see also