Hodge dual: Difference between revisions

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== Computing the dual ==

Again with a basis of dimension n, suppose we have a k-multival V and wish to find its dual multimonzo M. The elements of V are associated with k-combinations, and of M with (n-k)-combinations, of the basis elements. Because of the symmetry of binomial coefficients, V and M will have the same length. To find M we adjust the signs of V with the following procedure

Again with a basis of dimension ''n'', suppose we have a ''k''-multival '''V''' and wish to find its dual multimonzo '''M'''. The elements of '''V''' are associated with ''k''-combinations, and of '''M''' with {{nowrap|(''n'' − ''k'')}}-combinations, of the basis elements. Because of the symmetry of binomial coefficients, '''V''' and '''M''' will have the same length. To find '''M''' we adjust the signs of '''V''' with the following procedure

1. Let C be the k-combinations of the numbers 1..n in lexicographic order

1. Let '''C''' be the ''k''-combinations of the numbers 1 through ''n'' in lexicographic order

2. C will have the same length as V and M

2. '''C''' will have the same length as '''V''' and '''M'''

3. Sum the numbers in each combination Ci with ceil(k/2) to find Si

3. Sum the numbers in each combination '''C'''''i'' with {{ceil|{{frac|''k''|2}}}} to find '''S'''''i''

4. Multiply the ~~ith~~ element of V by −1~~^(Si)~~

4. Multiply the ''i''th element of ''V'' by −1'''S'''''i''

and then reverse the elements of V.

and then reverse the elements of '''V'''.

To find an unknown V from a known M, first reverse M and then adjust the signs.

To find an unknown '''V''' from a known '''M''', first reverse '''M''' and then adjust the signs.

== Using the dual ==

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 == Computing the dual ==
-Again with a basis of dimension n, suppose we have a k-multival V and wish to find its dual multimonzo M. The elements of V are associated with k-combinations, and of M with (n-k)-combinations, of the basis elements. Because of the symmetry of binomial coefficients, V and M will have the same length. To find M we adjust the signs of V with the following procedure
+Again with a basis of dimension ''n'', suppose we have a ''k''-multival '''V''' and wish to find its dual multimonzo '''M'''. The elements of '''V''' are associated with ''k''-combinations, and of '''M''' with {{nowrap|(''n'' − ''k'')}}-combinations, of the basis elements. Because of the symmetry of binomial coefficients, '''V''' and '''M''' will have the same length. To find '''M''' we adjust the signs of '''V''' with the following procedure
-. Let C be the k-combinations of the numbers 1..n in lexicographic order
+. Let '''C''' be the ''k''-combinations of the numbers 1 through ''n'' in lexicographic order
-. C will have the same length as V and M
+. '''C''' will have the same length as '''V''' and '''M'''
-. Sum the numbers in each combination Ci with ceil(k/2) to find Si
+. Sum the numbers in each combination '''C'''<sub>''i''</sub> with {{ceil|{{frac|''k''|2}}}} to find '''S'''<sub>''i''</sub>
-. Multiply the ith element of V by −1^(Si)
+. Multiply the ''i''th element of ''V'' by −1<sup>'''S'''<sub>''i''</sub></sup>
-and then reverse the elements of V.
+and then reverse the elements of '''V'''.
-To find an unknown V from a known M, first reverse M and then adjust the signs.
+To find an unknown '''V''' from a known '''M''', first reverse '''M''' and then adjust the signs.
 == Using the dual ==