Wedgie/Archived version: Difference between revisions

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=====Computing the previous example in Maple=====

In fact one can directly do many computations in Maple. Let us associate to the i'th prime the variable [math] x_i [/math]. So for example 7 corresponds to [math] x_4 [/math]. Then we introduce a basis vector [math] dx_i [/math] associated to the variable [math] x_i [/math]. Then to a pair of primes, for example [math] (3,7) [/math], we associate a basis vector [math] dx_2 \wedge dx_4 [/math]. Similarly if we have 3 or more primes.

In fact one can directly do many computations in Maple. Let us associate to the i'th prime the variable [math] x_i [/math]. So for example 7 corresponds to [math] x_4 [/math]. Then we introduce a basis vector [math] dx_i [/math] associated to the variable [math] x_i [/math]. Then to a pair of primes, for example [math] (3,7) [/math], we associate a basis vector [math] dx_2 \wedge dx_4 [/math]. Similarly if we have 3 or more primes. Expressions where there are [math] dx_i [/math] can be called 1 forms, [math] dx_i \wedge dx_j [/math] 2 forms etc.

In this way let's write E19 = {{val| 19 30 44 53 }} and E31 = {{val| 31 49 72 87 }} as [math] e_{19}=19dx_1+30dx_2+44dx_3+53dx_4 [/math] and [math] e_{31}=31dx_1+49dx_2+72dx_3+87dx_4 [/math]. Then we simply compute the exterior product

[math] \displaystyle e_{19} \wedge e_{31}=dx_1\wedge dx_2+4dx_1\wedge dx_3+10dx_1\wedge dx_4+4dx_2\wedge dx_3 +13dx_2\wedge dx_4+12dx_3\wedge dx_4[/math].

[math] \displaystyle \alpha =e_{19} \wedge e_{31}=dx_1\wedge dx_2+4dx_1\wedge dx_3+10dx_1\wedge dx_4+4dx_2\wedge dx_3 +13dx_2\wedge dx_4+12dx_3\wedge dx_4[/math].

A form is said to be decomposable if it can be written as an exterior product of 1 forms. So given above [math] \alpha [/math] how do we know if it is decomposable or not? Let us introduce a linear map [math] L (b)=b\wedge \alpha [/math]. This is a map from 1 forms to 3 forms. Now a kernel or nullspace of this map are all 1 forms such that [math] L(b)=0 [/math]. A basis for this nullspace in the present case is

[math] \displaystyle b_1=dx_1-4dx_3-13dx_4 \quad, \quad b_2=dx_2+4dx_3+10dx_4 [/math].

Now one can check that [math] \alpha=b_1\wedge b_2 [/math]. All these computations can be done easily in Maple when the things are properly set up. But is this useful to anyone?

==Truncation of wedgies==

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 =====Computing the previous example in Maple=====
-In fact one can directly do many computations in Maple. Let us associate to the i'th prime the variable [math] x_i [/math]. So for example 7 corresponds to [math] x_4 [/math]. Then we introduce a basis vector [math] dx_i [/math] associated to the variable [math] x_i [/math]. Then to a pair of primes, for example [math] (3,7) [/math], we associate a basis vector [math] dx_2 \wedge dx_4 [/math]. Similarly if we have 3 or more primes.
+In fact one can directly do many computations in Maple. Let us associate to the i'th prime the variable [math] x_i [/math]. So for example 7 corresponds to [math] x_4 [/math]. Then we introduce a basis vector [math] dx_i [/math] associated to the variable [math] x_i [/math]. Then to a pair of primes, for example [math] (3,7) [/math], we associate a basis vector [math] dx_2 \wedge dx_4 [/math]. Similarly if we have 3 or more primes. Expressions where there are [math] dx_i [/math] can be called 1 forms,  [math] dx_i \wedge dx_j [/math] 2 forms etc.
 In this way let's write E19 = {{val| 19 30 44 53 }} and  E31 = {{val| 31 49 72 87 }}  as [math] e_{19}=19dx_1+30dx_2+44dx_3+53dx_4 [/math] and  [math] e_{31}=31dx_1+49dx_2+72dx_3+87dx_4 [/math].  Then we simply compute the exterior product
-[math] \displaystyle e_{19}  \wedge e_{31}=dx_1\wedge dx_2+4dx_1\wedge dx_3+10dx_1\wedge dx_4+4dx_2\wedge dx_3 +13dx_2\wedge dx_4+12dx_3\wedge dx_4[/math].
+[math] \displaystyle  \alpha =e_{19}  \wedge e_{31}=dx_1\wedge dx_2+4dx_1\wedge dx_3+10dx_1\wedge dx_4+4dx_2\wedge dx_3 +13dx_2\wedge dx_4+12dx_3\wedge dx_4[/math].
+A form is said to be decomposable if it can be written as an exterior product of 1 forms. So given above [math] \alpha [/math] how do we know if it is decomposable or not? Let us introduce a linear map [math] L (b)=b\wedge \alpha [/math]. This is a map from 1 forms to 3 forms. Now a kernel or nullspace of this map are all 1 forms such that [math] L(b)=0 [/math]. A basis for this nullspace in the present case is
+[math] \displaystyle   b_1=dx_1-4dx_3-13dx_4 \quad, \quad b_2=dx_2+4dx_3+10dx_4 [/math].
+Now one can check that [math] \alpha=b_1\wedge b_2 [/math]. All these computations can be done easily in Maple when the things are properly set up. But is this useful to anyone?
 ==Truncation of wedgies==