The `Alt()` function in the `stokes` package

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library("stokes")
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Alt

Function Alt() converts a $k$-tensor to alternating form (see kform.Rd). Given a $k$-tensor $T$, we follow Spivak [p78] and define $\operatorname{Alt}(T)$ as follows:

[ \operatorname{Alt}(T)\left(v_1,\ldots,v_k\right)= \frac{1}{k!}\sum_{\sigma\in S_k}\mathrm{sgn}(\sigma)\cdot T\left(v_{\sigma(1)},\ldots,v_{\sigma(k)}\right) ]

where $S_k$ is the set of all permutations of numbers $1$ to $k$ and $\operatorname{sgn}\sigma$ is the sign of permutation $\sigma$ for any $\sigma\in S_k$. Thus for example if $k=3$ we have

[\operatorname{Alt}(T)\left(v_1,v_2,v_3\right)= \frac{1}{6}\left(\begin{array}{cc} +T\left(v_1,v_2,v_3\right)& -T\left(v_1,v_3,v_2\right)\cr -T\left(v_2,v_1,v_3\right)& +T\left(v_2,v_3,v_1\right)\cr +T\left(v_3,v_1,v_2\right)& -T\left(v_3,v_2,v_1\right) \end{array} \right) ]

Package idiom is straightforward:

S <- as.ktensor(rbind(c(1,7,8)))*6  # the "6" is to stop rounding error
S
Alt(S)

and we can see the pattern clearly, observing in passing that the order of the rows is arbitrary. Now, Alt(S) is an alternating form, which is easy to verify, by making it act on an odd permutation of a set of vectors:

V <- matrix(rnorm(24),ncol=3)
c(as.function(Alt(S))(V),as.function(Alt(S))(V[,c(2,1,3)]))

Observing that $\operatorname{Alt}$ is linear, we might apply it to a more complicated tensor, here with two terms:

S <- as.ktensor(rbind(c(1,2,4),c(2,2,3)),c(12,1000))
S
Alt(S)

Note that the 2 2 3 term (with coefficient 1000) disappears in Alt(S): the even permutations (being positive) cancel out the odd permutations (being negative) in pairs. This must be the case for an alternating map by definition. We can see why this cancellation occurs by considering $T$, a linear map corresponding to the second row of S above, that is, [2 2 3]. We have

[ T(v_1,v_2,v_3) = (v_1\cdot e_2)(v_1\cdot e_2)(v_1\cdot e_3)x ]

(where $x\in\mathbb{R}$ is any real number and the dot product), and so

[ T(v_2,v_1,v_3) = (v_1\cdot e_2)(v_1\cdot e_1)(v_1\cdot e_3)x = T(v_1,v_2,v_3). ]

Thus if we require $T$ to be alternating [that is, $T(v_2,v_1,v_3) = -T(v_1,v_2,v_3)$], we must have $x=0$, which is why the [2 2 3] term with coefficient 1000 vanishes (such terms are killed in the function by applying kill_trivial_rows()). We can check that terms with repeated entries are correctly discarded by taking the $\operatorname{Alt}$ of a tensor all of whose entries include at least one repeat:

S <- as.ktensor(matrix(c(
3,2,1,1,
1,4,1,4,
1,1,2,3,
7,7,4,7,
1,2,3,3
),ncol=4,byrow=TRUE),1:5)
S
Alt(S)

We should verify that Alt() does indeed return an alternating tensor for complicated tensors (this is trivial algebraically because $\operatorname{Alt}(\cdot)$ is linear, but it is always good to check):

S <- rtensor(k=5,n=9)
S
AS <- Alt(S)
V <- matrix(rnorm(45),ncol=5) # element of (R^9)^5

Then we swap columns of $V$, using both even and odd permutations, to verify that $\operatorname{Alt}(S)$ is in fact alternating:

V_even <- V[,c(1,2,5,3,4)]  # an even permutation
V_odd  <- V[,c(2,1,5,3,4)]  # an odd permutation
V_rep  <- V[,c(2,1,5,2,4)]  # not a permutation
c(as.function(AS)(V),as.function(AS)(V_even))   # should be identical (even permutation)
c(as.function(AS)(V),as.function(AS)(V_odd))    # should differ in sign only (odd permutation)
as.function(AS)(V_rep)                          # should be zero

Further properties of Alt()

In his theorem 4.3, Spivak proves the following statements:

We have demonstrated the first point above. For the second, we need to construct a tensor that is alternating, and then show that Alt() does not change it:

P <- as.ktensor(1+diag(2),c(-7,7))
P
P == Alt(P)

The third point, idempotence is also easy:

P <- rtensor()*6 # the "6" avoids numerical round-off issues
Alt(Alt(P))==Alt(P)   # should be TRUE

Wedge product

Spivak defines the wedge product as follows. Given alternating forms $\omega\in\Lambda^k(V)$ and $\eta\in\Lambda^l(V)$ we have

[ \omega\wedge\eta=\frac{(k+l)!}{k!l!}\operatorname{Alt}(\omega\otimes\eta) ]

So for example:

omega <- as.ktensor(2+diag(2),c(-7,7))
eta   <- Alt(rtensor(2))*30
omega
eta
f <- as.function(Alt(omega %X% eta))

(the tensor itself is quite long, having 60 nonzero components). We may verify that f() is in fact alternating:

V <-  matrix(rnorm(35),ncol=5)
c(f(V),f(V[,c(2:1,3:5)]))

Further further properties of Alt()

Spivak goes on to prove the following three statements. If $S,T$ are tensors and $\omega,\eta,\theta$ are alternating tensors of arity $k,l,m$ respectively, then

Taking the points in turn. Firstly $\operatorname{Alt}(S\otimes T)=\operatorname{Alt}(T\otimes S)=0$:

(S <- as.ktensor(rbind(c(1,2,3,3),c(1,1,2,3)),1000:1001)) 
Alt(S)  # each row of S includes repeats
T <- rtensor()
c(is.zero(Alt(S %X% T)), is.zero(Alt(T %X% S)))

secondly, $\operatorname{Alt}(\operatorname{Alt}(\omega\otimes\eta)\otimes\theta) =\operatorname{Alt}(\omega\otimes\eta\otimes\theta)= \operatorname{Alt}(\omega\otimes\operatorname{Alt}(\eta\otimes\theta))$:

omega <- Alt(as.ktensor(rbind(1:3),6))
eta <- Alt(as.ktensor(rbind(4:5),60))
theta <- Alt(as.ktensor(rbind(6:7),14))

omega
eta
theta

f1 <- as.function(Alt(Alt(omega %X% eta) %X% theta))
f2 <- as.function(Alt(omega %X% eta %X% theta))
f3 <- as.function(Alt(omega %X% Alt(eta %X% theta)))
V <- matrix(rnorm(9*14),ncol=9)
c(f1(V),f2(V),f3(V))

Verifying the third identity $(\omega\wedge\eta)\wedge\theta = \omega\wedge(\eta\wedge\theta) = \frac{(k+l+m)!}{k!l!m!}\operatorname{Alt}(\omega\otimes\eta\otimes\theta)$ needs us to coerce from a $k$-form to a $k$-tensor:

omega <- rform(2,2,19)
eta <- rform(3,2,19)
theta <- rform(2,2,19)

a1 <- as.ktensor(omega ^ (eta ^ theta))
a2 <- as.ktensor((omega ^ eta) ^ theta)
a3 <- Alt(as.ktensor(omega) %X% as.ktensor(eta) %X% as.ktensor(theta))*90

c(is.zero(a1-a2),is.zero(a1-a3),is.zero(a2-a3))

Argument give_kform

Function Alt() takes a Boolean argument give_kform. We have been using Alt() with give_kform taking its default value of FALSE, which means that it returns an object of class ktensor. However, an alternating form can be much more efficiently represented as an object of class kform, and this is returned if give_kform is TRUE:

(rand_tensor <- rtensor(k=5,n=9)*120)
S1 <- Alt(rand_tensor)  # 720 terms, too long to print
(SA1 <- Alt(rand_tensor,give_kform=TRUE))

Verification:

V <- matrix(rnorm(45),ncol=5)
c(as.function(S1)(V),as.function(SA1)(V)) # should match


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stokes documentation built on Jan. 18, 2022, 1:11 a.m.