Wedgie/Archived version: Difference between revisions

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[math] \displaystyle H=\begin{pmatrix} 1&0&-4&-13\\0&1&4&10\end{pmatrix}[/math]

Let us take another [[Dave Keenan & Douglas Blumeyer's guide to EA for RTT|example]]. Suppose we have [math] \alpha_0=dx_1\wedge dx_2\wedge dx_3+2dx_1\wedge dx_2\wedge dx_4-2dx_1\wedge dx_3\wedge dx_4 -5dx_2\wedge dx_3\wedge dx_4[/math]. Now we have [math] L_0 (b)=b\wedge \alpha_0 [/math] and the basis for nullspace is

[math] \displaystyle b_1=dx_1+5dx_4 \quad, \quad b_2=dx_2+2dx_4 \quad, \quad b_3=dx_3+2dx_4 [/math].

and one can check that [math] \alpha=b_1\wedge b_2 \wedge b_3[/math]. Note by the way that n-1 forms are always decomposable (here n=4 and we computed the decomposition of 3 form).

==Truncation of wedgies==

A useful operation to perform on any multivector, including wedgies, is truncation of the wedgie to a lower prime limit. This in effect sets all the basis vectors of a p-limit wedgie which are greater than q, the prime limit being truncated to, to zero. An algorithm to produce the truncation is to list the r-subsets of the primes to p in alphabetical order, and add the corresponding coefficient to the list of the q-limit truncation if and only if the maximum prime in the r-subet is less than or equal to q. Truncating a wedgie can lead to a non-wedgie if the GCD of the coefficients is greater than one; this means that in the lower limit, [[Wedgies_and_Multivals|contortion]] has appeared.

@@ Line 124: / Line 124: @@
 [math] \displaystyle  H=\begin{pmatrix} 1&0&-4&-13\\0&1&4&10\end{pmatrix}[/math]
+Let us take another [[Dave Keenan & Douglas Blumeyer's guide to EA for RTT|example]]. Suppose we have  [math] \alpha_0=dx_1\wedge dx_2\wedge dx_3+2dx_1\wedge dx_2\wedge dx_4-2dx_1\wedge dx_3\wedge dx_4 -5dx_2\wedge dx_3\wedge dx_4[/math]. Now we have  [math] L_0 (b)=b\wedge \alpha_0 [/math] and the basis for nullspace is
+[math] \displaystyle   b_1=dx_1+5dx_4 \quad, \quad b_2=dx_2+2dx_4  \quad, \quad b_3=dx_3+2dx_4 [/math].
+and one can check that  [math] \alpha=b_1\wedge b_2 \wedge b_3[/math]. Note by the way that n-1 forms are always decomposable (here n=4 and we computed the decomposition of 3 form).
 ==Truncation of wedgies==
 A useful operation to perform on any multivector, including wedgies, is truncation of the wedgie to a lower prime limit. This in effect sets all the basis vectors of a p-limit wedgie which are greater than q, the prime limit being truncated to, to zero. An algorithm to produce the truncation is to list the r-subsets of the primes to p in alphabetical order, and add the corresponding coefficient to the list of the q-limit truncation if and only if the maximum prime in the r-subet is less than or equal to q. Truncating a wedgie can lead to a non-wedgie if the GCD of the coefficients is greater than one; this means that in the lower limit, [[Wedgies_and_Multivals|contortion]] has appeared.