Conformal Geometric Algebra in Stochastic Optimization Problems ...

248 APPENDIX A. SELECTED ASPECTS UNDERLYING THIS WORK
Next it is analyzed how w changes under an elementary transposition of an element
from v with a neighboring element not in v. It is ultimately intended to learn how w
changes under a couple of elementary transpositions so that those summands with
identical remainder can be grouped together. In this
(A) (B)
b1 ∧ b2 ∧ b3 ∧ b4 ∧ b5 ∧ b6 −→ ( b2 ∧ b4 ∧ b6 ) ∧ b1 ∧ b3 ∧ b5
↑ ↑ ↑ ↑ ↑ ↑
v1 v2 v3 −→ v ′ 1 v ′ 2 v ′ 3
(A.17)
example, several elementary transpositions lead to v = v ′ = (2, 4, 6) but w = (2,3, 4)
−→ w ′ = (1, 1, 1). It is easy to see that moving a vector from v by one position
to the left, e.g. b3 ∧ b4 � b4 ∧ b3 (the right, e.g. b4 ∧ b5 � b5 ∧ b4) causes the
respective index in w to decrease (increase) by one. In the v = (2, 4, 6) example one
obtains ∆w = (−1, −2, −3).
Another thing to reflect about is what happens to w if two neighboring elements
in v are exchanged, so if for example v = (2, 4, 6) −→ v ′ = (4,2,6). First of all,
since the elements are neighboring, their composite action regarding the effects on
w is atomic, i.e. the uninvolved indices of w stay unchanged. The situation may be
subsumed as follows
vi, vi+1 −→ v ′ i := vi+1, v ′ i+1 := vi
wi, wi+1 −→ w ′ i = ?, w′ i+1 = ?
1 ≤ i ≤ k − 1.
Two cases must be taken into account when exchanging vi and vi+1: first, assume
that vi < vi+1. Then, bvi precedes bvi+1 in B 〈l〉 so that the effect of taking out bvi
from B 〈l〉 prior to bvi+1 must be that the index wi+1 is already decreased by one.
Note that, as a consequence, wi = vi − (i − 1) if the elements in v are in ascending
order. Conversely, if vi > vi+1 then the removal of bvi does not affect the index
wi+1. After exchanging vi and vi+1 the update for w is
vi < vi+1 =⇒ w ′ i = wi+1 + 1 and w ′ i+1 = wi
vi > vi+1 =⇒ w ′ i = wi+1 and w ′ i+1 = wi − 1.
(A.18)
For example, v = (1, 2,...,6,9, 7, 8) ←→ v ′ = (1, 2,...,9,6,7,8) corresponds
to w = (1, 1, ...,1,3, 1,1) ←→ w ′ = (1, 1,...,4,1, 1,1).
The sum in equation (A.15) consists of � � l l (l−1) ...(l−k+1)
k = k! terms whereas the sum
in the k-fold application of equation (A.16) is to be taken over l (l−1) ... (l−k+1)
summands, where k! of them can each be grouped together as they belong to the
same remainder. This becomes clear by observing that there are k! permutations of
a sequence v = (v1,v2,...,vk). It is thus planned to rearrange the k-fold application
of equation (A.16) such that it resembles equation (A.15) in proposition 2.8.
The next equation shows a first generalization towards the k-fold application of
proposition 2.7. The sum �
w in the formula is intended to be taken over all valid