RankedMin

RankedMin[list,n]

gives the n smallest element in list.

RankedMin[list,-n]

gives the n largest element in list.

Details

RankedMin yields a definite result if all its arguments are real numbers.
RankedMin[{x₁,…,x_m},n] is often indicated using or .
RankedMin[{x₁,…,x_m},1] is equivalent to Min[{x₁,…,x_m}]. »
RankedMin[{x₁,…,x_m},-1] is equivalent to Max[{x₁,…,x_m}].
RankedMin[{x₁,…,x_m},m] is equivalent to Max[{x₁,…,x_m}]. »
RankedMin[{x₁,…,x_m},n] is equivalent to Quantile[{x₁,…,x_m},n/m]. »

Examples

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Basic Examples (3)

The second smallest of three numbers:

The third smallest of four numbers:

Plot the second-largest function over a subset of the reals:

Scope (22)

Numerical Evaluation (4)

Evaluate the second smallest of three numbers:

The fourth smallest—i.e. the largest—of four numbers:

The second largest of five numbers:

The fourth largest of five numbers:

The fifth largest—i.e. the smallest—of five numbers:

Evaluate to high precision:

Evaluate efficiently at high precision:

Specific Values (4)

Values at infinity:

Evaluate symbolically:

Solve equations and inequalities:

Find a value of for which the RankedMin[{Sin[x],Cos[x],Exp[x]},2]1:

Visualization (3)

Plot RankedMin of several functions:

Plot RankedMin in three dimensions:

Plot RankedMin of three functions in three dimensions:

Function Properties (8)

RankedMin is only defined for real-valued inputs:

The range of RankedMin is real numbers:

Basic symbolic simplification is done automatically:

Multi-argument ranked RankedMin is generally not an analytic function:

It will have singularities where the functions cross, but it will be continuous:

is neither nondecreasing nor nonincreasing:

is not injective:

is not surjective:

is non-negative:

Integration (3)

Series expansion of the second-smallest function at the origin:

Asymptotic expansion at Infinity:

Integrate expressions involving RankedMin:

Applications (8)

Plot the bivariate RankedMin functions:

Plot the contours of bivariate and trivariate RankedMin functions:

RankedMin[{y₁,…,y_n,x},k] as a function of x:

Compute the expectation of the second smallest (median) variable:

Alternatively, use OrderDistribution:

Compute the probability of the second smallest variable being less than 1:

Show that OrderDistribution is a special case of TransformedDistribution:

Find the height of the third shortest child in a class:

Find the second-shortest Length of borders in kilometers:

It is Luxembourg:

Properties & Relations (6)

RankedMin[{x₁,…,x_m},1] is equivalent to Min[x₁,…,x_m]:

RankedMin[{x₁,…,x_m},m] is equivalent to Max[x₁,…,x_m]:

RankedMin[{x₁,…,x_m},n] is equivalent to RankedMax[{x₁,…,x_m},m-n+1]:

RankedMin[{x₁,…,x_m},n] is equivalent to Quantile[{x₁,…,x_m},n/m]:

RankedMin[{x₁,…,x_m},n] is equivalent to Sort[{x₁,…,x_m}]〚n〛:

The equivalent Piecewise function has disjoint piecewise case domains:

Algebraically prove the piecewise case domains are disjoint:

Visually show it:

Algebraically prove the piecewise case domains are pairwise disjoint:

Visually show it:

Neat Examples (2)

Two-dimensional sublevel sets:

Three-dimensional sublevel sets:

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RankedMin

Details

Examples

Basic Examples (3)