rform.RdRandom \(k\)-form objects and \(k\)-tensors, intended as quick “get you going” examples
rform(terms=9, k=3, n=7, ensure=TRUE, integer=TRUE)
rformm(terms=30, k=7, n=20, ensure=TRUE, integer=TRUE)
rformmm(terms=90, k=15, n=30, ensure=TRUE, integer=TRUE)
rtensor(terms=9, k=3, n=7, integer=TRUE)Number of distinct terms
A \(k\)-form maps \(V^k\) to \(\mathbb{R}\), where \(V=\mathbb{R}^n\)
Boolean with default TRUE meaning to ensure that
the dovs() of the returned value is in fact equal to
n. If FALSE, sometimes the dovs() is strictly
less than n because of random sampling
Boolean specifying whether the coefficients are integers or not
Random \(k\)-form objects and \(k\)-tensors.
By default, function rform() returns a simple \(k\)-form;
rformm() and rformmm() return successively more
complicated objects. Note that argument terms is an upper
bound, as the index matrix might contain repeats which are combined.
Function rtensor() returns a random tensor.
All functions documented here return an object of class kform or
ktensor.
(a <- rform())
#> An alternating linear map from V^3 to R with V=R^7:
#> val
#> 4 5 7 = 9
#> 1 5 7 = 5
#> 1 3 6 = 4
#> 1 6 7 = 4
#> 2 4 6 = 6
#> 1 2 6 = -6
#> 4 6 7 = 1
(b <- rform())
#> An alternating linear map from V^3 to R with V=R^7:
#> val
#> 1 2 4 = 9
#> 1 2 7 = 8
#> 2 4 7 = 6
#> 1 2 6 = -4
#> 1 3 5 = -3
#> 5 6 7 = 2
#> 4 5 7 = -7
#> 1 2 3 = 5
#> 3 5 7 = 1
a ^ b
#> An alternating linear map from V^6 to R with V=R^7:
#> val
#> 1 2 3 5 6 7 = -6
#> 1 3 4 5 6 7 = -31
#> 1 2 4 5 6 7 = 78
#> 1 2 3 4 5 6 = -18
#> 2 3 4 5 6 7 = -6
#> 1 2 3 4 6 7 = -29
#> 1 2 3 4 5 7 = -45
a
#> An alternating linear map from V^3 to R with V=R^7:
#> val
#> 4 5 7 = 9
#> 1 5 7 = 5
#> 1 3 6 = 4
#> 1 6 7 = 4
#> 2 4 6 = 6
#> 1 2 6 = -6
#> 4 6 7 = 1
a ^ dx
#> An alternating linear map from V^4 to R with V=R^7:
#> val
#> 1 4 6 7 = -1
#> 1 2 4 6 = -6
#> 1 4 5 7 = -9
a ^ dx ^ dy
#> An alternating linear map from V^5 to R with V=R^7:
#> val
#> 1 2 4 6 7 = 1
#> 1 2 4 5 7 = 9
(x <- rtensor())
#> A linear map from V^3 to R with V=R^7:
#> val
#> 6 4 7 = 9
#> 2 4 5 = 6
#> 6 2 5 = 7
#> 6 7 5 = 5
#> 6 2 3 = 4
#> 5 7 3 = 8
#> 6 3 2 = 3
#> 3 1 1 = 2
#> 6 3 6 = 1
x %X% x
#> A linear map from V^6 to R with V=R^7:
#> val
#> 6 3 6 6 3 6 = 1
#> 3 1 1 6 3 6 = 2
#> 5 7 3 6 3 6 = 8
#> 6 2 3 6 3 6 = 4
#> 6 2 5 6 3 6 = 7
#> 6 3 2 6 3 6 = 3
#> 6 3 6 3 1 1 = 2
#> 3 1 1 3 1 1 = 4
#> 6 2 3 3 1 1 = 8
#> 6 7 5 3 1 1 = 10
#> 2 4 5 3 1 1 = 12
#> 6 7 5 6 3 6 = 5
#> 6 4 7 3 1 1 = 18
#> 6 3 6 6 3 2 = 3
#> 3 1 1 6 3 2 = 6
#> 6 3 2 6 3 2 = 9
#> 5 7 3 6 3 2 = 24
#> 6 4 7 6 7 5 = 45
#> 5 7 3 2 4 5 = 48
#> 6 7 5 6 2 5 = 35
#> 6 2 3 6 4 7 = 36
#> 2 4 5 6 3 6 = 6
#> 2 4 5 6 7 5 = 30
#> 2 4 5 6 2 5 = 42
#> 6 3 2 2 4 5 = 18
#> 5 7 3 3 1 1 = 16
#> 6 2 3 6 2 5 = 28
#> 3 1 1 2 4 5 = 12
#> 6 2 3 5 7 3 = 32
#> 6 2 5 6 3 2 = 21
#> 6 3 6 2 4 5 = 6
#> 6 4 7 6 4 7 = 81
#> 6 2 5 6 4 7 = 63
#> 6 2 5 6 2 5 = 49
#> 6 2 3 2 4 5 = 24
#> 6 3 2 6 2 5 = 21
#> 3 1 1 6 2 3 = 8
#> 2 4 5 6 4 7 = 54
#> 6 2 5 5 7 3 = 56
#> 6 3 2 6 4 7 = 27
#> 6 7 5 6 4 7 = 45
#> 6 7 5 6 7 5 = 25
#> 3 1 1 6 2 5 = 14
#> 5 7 3 6 4 7 = 72
#> 6 2 5 2 4 5 = 42
#> 6 4 7 2 4 5 = 54
#> 3 1 1 6 4 7 = 18
#> 5 7 3 6 2 5 = 56
#> 6 7 5 2 4 5 = 30
#> 6 3 6 6 4 7 = 9
#> 6 3 6 6 2 5 = 7
#> 6 2 5 6 7 5 = 35
#> 6 4 7 6 2 5 = 63
#> 5 7 3 6 7 5 = 40
#> 3 1 1 6 7 5 = 10
#> 6 2 5 6 2 3 = 28
#> 6 3 6 6 7 5 = 5
#> 6 3 2 6 7 5 = 15
#> 5 7 3 5 7 3 = 64
#> 6 7 5 6 2 3 = 20
#> 6 4 7 6 3 2 = 27
#> 3 1 1 5 7 3 = 16
#> 6 4 7 6 3 6 = 9
#> 5 7 3 6 2 3 = 32
#> 6 3 2 6 2 3 = 12
#> 6 3 6 6 2 3 = 4
#> 6 4 7 5 7 3 = 72
#> 6 7 5 6 3 2 = 15
#> 2 4 5 5 7 3 = 48
#> 2 4 5 2 4 5 = 36
#> 6 4 7 6 2 3 = 36
#> 2 4 5 6 2 3 = 24
#> 6 7 5 5 7 3 = 40
#> 6 2 5 3 1 1 = 14
#> 6 3 2 5 7 3 = 24
#> 6 3 2 3 1 1 = 6
#> 6 3 6 5 7 3 = 8
#> 6 2 3 6 2 3 = 16
#> 2 4 5 6 3 2 = 18
#> 6 2 3 6 7 5 = 20
#> 6 2 3 6 3 2 = 12