zonoseg {zonohedra}

zonoseg construction, querying, and printing

Description

Construct a zonoseg from a numeric matrix with one row.

A zonoseg ("zonotope" + "segment") is my own personal term for a 1-dimensional zonotope. I could not find an alternative term. It is a linear image of the unit cube [0,1]^n in the real numbers, and a compact segment of reals. The order of the generators has no effect on the zonoseg.

The image of the 2-transition subcomplex of [0,1]^n is a compact subsegment of the zonoseg. The order of the generators affects this subsegment in a major way.

Usage

zonoseg( mat, e0=0, ground=NULL )

## S3 method for class 'zonoseg'
getsegment( x )

## S3 method for class 'zonoseg'
getsegment2trans( x )

## S3 method for class 'zonoseg'
print( x, ... )

Arguments

mat

a numeric matrix with 1 row whose entries determine the zonoseg. One or more entries must be non-zero. It is OK to have both positive and negative entries.
mat can also be a numeric vector which is then converted to a matrix with 1 row.

e0

threshold for an entry of mat to be considered 0. Since the default is e0=0, by default an entry must be exactly 0 to become a loop in the associated matroid.

ground

The ground set of the associated matroid of rank 1 - an integer vector in strictly increasing order, or NULL.
When ground is NULL, it is set to 1:ncol(mat). If ground is not NULL, length(ground) must be equal to ncol(mat). The point ground[i] corresponds to the i'th column of mat.

x

a zonoseg object as returned by zonoseg()

...

not used

Details

A zonoseg object is a list with only 3 items: the associated matroid, the endpoints of the segment, and endpoints of the 2-transition subsegment.

print.zonoseg() prints some information about the generators, and the endpoints of the segment plus the 2 vertices of the unit cube that map to these endpoints. It prints similar data for the 2-transition subsegment. Finally, it prints data on the associated matroid.

Value

zonoseg() returns a list with S3 class 'zonoseg'. In case of error, e.g. invalid mat, the function prints an error message and returns NULL.

getsegment() and getsegment2trans() return numeric 2-vectors - the min and max endpoints of the corresponding segments.

print.zonoseg() returns TRUE or FALSE.

Note

The ground set of positive integers should not be too sparse; otherwise performance may suffer.

References

Matroid - Wikipedia.
https://en.wikipedia.org/w/index.php?title=Matroid&oldid=1086234057

See Also

rank()

Examples

zono1 = zonoseg( c(1,-2,3,0,-3,-4) )
zono1

# generators:        6 -- 3 negative, 2 positive, and 1 loops.
#
# segment:                   [-9,4]
#      value pcube.1 pcube.2 pcube.3 pcube.4 pcube.5 pcube.6
# zmin    -9       0       1       0       0       1       1
# zmax     4       1       0       1       0       0       0
#
# 2-transition subsegment:   [-8,3]
#             value source.1 source.2 source.3 source.4 source.5 source.6
# tmin-2trans    -8        1        1        0        0        1        1
# tmax-2trans     3        0        0        1        1        0        0
# 
# matroid:
# ground set:           6 points   {1 2 3 4 5 6}
# hyperplanes:          1     {4}
# rank:                 1
# loops:                1   {4}
# multiple groups:      1     {1 2 3 5 6}
# uniform:              FALSE
# paving:               TRUE
# simple:               FALSE
# This matroid is constructed from a 1x6 real matrix.
#      1  2 3 4  5  6
# [1,] 1 -2 3 0 -3 -4
# 
# The summary of the simplified matroid is:
#     ground set:           1 points   {1}
#                       Point 1 corresponds to the multiple group {1 2 3 5 6} in the original ...
#      hyperplanes:          1     {}
#      rank:                 1
#      loops:                0   {}
#      multiple groups:      0   {}
#      uniform:              TRUE
#      paving:               TRUE
#      simple:               TRUE
#      This matroid is constructed from a 1x1 real matrix.
#           1+...+6
#      [1,]     -13


## so the 2-transition subsegment is a proper subset of the zonoseg