Conformal Geometric Algebra in Stochastic Optimization Problems ...

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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
A.3. PROOFS AND DERIVATIONS 249 index combinations w = (i1,i2,...,ik) ∈ � k . Besides, v is well-defined 3 in terms of w, and vice versa. This property is indicated by writing v(w) or w(v), respectively. A 〈k〉 · B 〈l〉 = � (−1) �k w i=1 wi±k k� j=1 � �� ak−(j−1) · bv(w)j B 〈l〉 \ k� � bv(w)r This formula may be rewritten in terms of a sum extending over all valid sequences v ∈ � k . More precisely, the sum is split into two sums such that the outer one captures all � � l k k-combinations for which a common remainder exists. The inner sum captures all k! permutations of the indices v1 < v2 < ... < vk in v that belong to the actual remainder. A 〈k〉 · B 〈l〉 = � 1≤v1
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INSTITUT FÜR INFORMATIK Conformal
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Conformal Geometric Algebra in Stoc
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Dedicated to my beloved wife, Steph
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ACKNOWLEDGMENT This thesis as is ca
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3 The Conformal Geometric Algebra 8
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8 Applications in Omnidirectional V
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2 CHAPTER 1. INTRODUCTION with a si
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4 CHAPTER 1. INTRODUCTION three poi
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6 CHAPTER 1. INTRODUCTION Spherical
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8 CHAPTER 1. INTRODUCTION Fig. 1.3:
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10 CHAPTER 1. INTRODUCTION incidenc
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12 CHAPTER 1. INTRODUCTION tabular
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14 CHAPTER 1. INTRODUCTION
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16 CHAPTER 2. GEOMETRIC ALGEBRA Gra
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18 CHAPTER 2. GEOMETRIC ALGEBRA and
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20 CHAPTER 2. GEOMETRIC ALGEBRA 2.1
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22 CHAPTER 2. GEOMETRIC ALGEBRA fur
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24 CHAPTER 2. GEOMETRIC ALGEBRA Fig
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26 CHAPTER 2. GEOMETRIC ALGEBRA Fig
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28 CHAPTER 2. GEOMETRIC ALGEBRA Ret
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30 CHAPTER 2. GEOMETRIC ALGEBRA dou
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32 CHAPTER 2. GEOMETRIC ALGEBRA whe
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34 CHAPTER 2. GEOMETRIC ALGEBRA In
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36 CHAPTER 2. GEOMETRIC ALGEBRA 2.2
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38 CHAPTER 2. GEOMETRIC ALGEBRA geo
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40 CHAPTER 2. GEOMETRIC ALGEBRA fol
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42 CHAPTER 2. GEOMETRIC ALGEBRA Def
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44 CHAPTER 2. GEOMETRIC ALGEBRA sug
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46 CHAPTER 2. GEOMETRIC ALGEBRA In
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48 CHAPTER 2. GEOMETRIC ALGEBRA due
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50 CHAPTER 2. GEOMETRIC ALGEBRA the
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52 CHAPTER 2. GEOMETRIC ALGEBRA Cor
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54 CHAPTER 2. GEOMETRIC ALGEBRA Pur
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56 CHAPTER 2. GEOMETRIC ALGEBRA Pro
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58 CHAPTER 2. GEOMETRIC ALGEBRA Acc
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60 CHAPTER 2. GEOMETRIC ALGEBRA Pro
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62 CHAPTER 2. GEOMETRIC ALGEBRA it
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64 CHAPTER 2. GEOMETRIC ALGEBRA whe
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66 CHAPTER 2. GEOMETRIC ALGEBRA Let
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68 CHAPTER 2. GEOMETRIC ALGEBRA Sin
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70 CHAPTER 2. GEOMETRIC ALGEBRA Def
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72 CHAPTER 2. GEOMETRIC ALGEBRA The
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74 CHAPTER 2. GEOMETRIC ALGEBRA dua
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76 CHAPTER 2. GEOMETRIC ALGEBRA It
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78 CHAPTER 2. GEOMETRIC ALGEBRA If
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80 CHAPTER 2. GEOMETRIC ALGEBRA
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82 CHAPTER 3. THE CONFORMAL GEOMETR
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100 CHAPTER 3. THE CONFORMAL GEOMET
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136 CHAPTER 4. A PRIMER ON POSE EST
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146 CHAPTER 5. PARAMETER ESTIMATION
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180 CHAPTER 6. PRACTICAL ASPECTS OF
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190 CHAPTER 7. APPLICATIONS IN COMP
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