pdimStats -- statistics on projective dimension of a list of objects

Synopsis

Usage:

pdimStats(List)
Inputs:
- ideals, a list, of objects of type MonomialIdeal or any type to which pdim can be applied.
Optional inputs:
- ShowTally => ..., default value false, optional input to choose if the tally is to be returned
- Verbose => ..., default value false, optional input to request verbose feedback
Outputs:
- a sequence, whose first entry is the mean projective dimension, the second entry is the standard deviation of the projective dimension, and third entry (if option turned on) is the projective dimension tally for quotient rings of ideals in the list ideals.

Description

The function pdimStats computes the mean and standard deviation of the projective dimension of elements in the list:

i1 : R=ZZ/101[a,b,c];

i2 : ideals = {monomialIdeal(a^3,b,c^2), monomialIdeal(a^3,b,a*c)}

                      3      2                   3
o2 = {monomialIdeal (a , b, c ), monomialIdeal (a , b, a*c)}

o2 : List

i3 : pdimStats(ideals)

o3 = (3, 0)

o3 : Sequence

The function can also output the projective dimension tally as follows:

i4 : R=ZZ/101[a,b,c];

i5 : ideals = {monomialIdeal(a,c),monomialIdeal(b),monomialIdeal(a^2*b,b^2)}

                                                             2    2
o5 = {monomialIdeal (a, c), monomialIdeal b, monomialIdeal (a b, b )}

o5 : List

i6 : pdimStats(ideals, ShowTally=>true)

o6 = (1.66667, .471405, Tally{1 => 1})
                              2 => 2

o6 : Sequence

The following examples use the existing functions randomMonomialIdeals to automatically generate a list of ideals, rather than creating the list manually:

i7 : ideals = randomMonomialIdeals(4,3,1.0,3)

o7 = {monomialIdeal (x , x , x , x ), monomialIdeal (x , x , x , x ),
                      1   2   3   4                   1   2   3   4  
     ------------------------------------------------------------------------
     monomialIdeal (x , x , x , x )}
                     1   2   3   4

o7 : List

i8 : pdimStats(ideals)

o8 = (4, 0)

o8 : Sequence

i9 : ideals = randomMonomialIdeals(4,6,0.01,10)

                     2 3                       2     3                    
o9 = {monomialIdeal(x x x ), monomialIdeal (x x x , x x x ), monomialIdeal
                     2 3 4                   1 2 4   1 3 4                
     ------------------------------------------------------------------------
                         5       3   2                    2 2  
     (), monomialIdeal (x x , x x , x x ), monomialIdeal(x x ),
                         2 3   2 3   1 4                  2 3  
     ------------------------------------------------------------------------
                    2 2                   5       4       2   2  
     monomialIdeal(x x ), monomialIdeal (x x , x x x , x x x x ),
                    1 4                   1 3   2 3 4   1 2 3 4  
     ------------------------------------------------------------------------
                      2                   3       2 2     2  
     monomialIdeal(x x ), monomialIdeal (x x x , x x x , x ),
                    2 3                   1 3 4   1 3 4   4  
     ------------------------------------------------------------------------
                      3
     monomialIdeal(x x x )}
                    2 3 4

o9 : List

i10 : pdimStats(ideals)

o10 = (1.6, 1.0198)

o10 : Sequence

Ways to use pdimStats :

pdimStats(List)

For the programmer

The object pdimStats is a method function with options.

pdimStats -- statistics on projective dimension of a list of objects

Synopsis

Description

See also

Ways to use pdimStats :

For the programmer