Lovegrove Mathematicals

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"Dedicated to making Likelinesses the entity of prime interest"

Step distributions

Step-down distributions

Let f∈S(N) and c∈XN be such that  [j≤c<k]⇒f(j)>f(k). Then f is a step-down distribution with step at c.

SD concept

The set of all such distributions is denoted by SD(c,N).

Ranked step-down distributions

Let f∈S(N) and c∈XN be such that [i<j≤c<k]⇒f(i)>f(j)>f(k). Then f is a ranked step-down distribution with step at c.

RSD concept

The set of all such distributions is denoted by RSD(c,N).

Every ranked distribution is a ranked step-down distribution with a step anywhere, ie. (∀c∈XN) R(N)⊂RSD(c,N)

Tailed step-down distributions

Let f∈S(N) and c∈XN be such that [i≤c<j<k]⇒f(i)>f(j)>f(k). Then f is a tailed step-down distribution with step at c.

TSD concept

The set of all such distributions is denoted by TSD(c,N).

Step-up distributions

Let f∈S(N) and c∈XN be such that  [j≤c<k]⇒f(j)<f(k). Then f is a Step-up distribution with step at c.

Heaviside concept

The set of all such distributions is denoted by SU(c,N).

A step-up distribution with step at c is not the mirror image of a step-down distribution with step at c. This is because of the placement of c relative to the step.

Algorithm to find injective f∈SD(c,N)

Select f∈R(N) and randomise f(1), ..., f(c) and also f(c+1),...,f(N)

Algorithm to find injective f∈RSD(c,N)

Select f∈R(N) and randomise f(c+1), ..., f(N)

Algorithm to find injective f∈TSD(c,N)

Select f∈R(N) and randomise f(1), ..., f(c)

It is a matter of some practical usefulness that if c=N-1 and P is any of the above sets of step-down distributions then, because f(N) is not moved by any of the randomisations, LP(N)=LR(N)(N) [=1/N2]

Algorithm to find injective f∈SU(c,N)

Select f∈RR(c,N) and randomise f(1), ..., f(c) and also f(c+1), ..., f(N)

element of SD(6,29) Likelinesses for step-down
element of RSD(6,29) Likelinesses for RSD(6,29)

The step in a ranked step-down distribution can sometimes be interpreted as a 'limit of discrimination' for a ranked distribution: a limit beyond which it is not possible to tell the various outputs apart.

element of TSD(6,29) Likeliness of TSD(6,29)
element of SU(23,29) Likeliness of SU(23,29)