Symmetry Studies: An Introduction to the Analysis of Structured Data in Applications

Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 7
Preface......Page 11
1.1 Introduction......Page 13
1.2 Symmetry and Classification......Page 14
1.3 Data Indexed by Symmetries......Page 15
1.4 Symmetry and Data Reduction......Page 17
1.5 Statistical Aspects......Page 20
1.6 Algebraic Aspects......Page 21
1.7 Structured Data......Page 23
1.8 Partitions......Page 27
A view from mechanics......Page 28
Canonical projections......Page 33
Further Reading......Page 35
Exercises......Page 37
Appendix A......Page 41
2.2 Permutations......Page 42
Parity of a permutation......Page 43
Conjugacy classes......Page 44
Integer partitions and Young frames......Page 45
2.3 Groups and Homomorphisms......Page 46
Group homomorphisms......Page 47
Cyclic groups......Page 48
Cosets......Page 49
Semidirect and direct products of groups......Page 52
Dihedral groups in the plane......Page 54
Matrix groups......Page 55
Isometry groups......Page 56
The group of the quaternions......Page 57
Orbits, stabilizers, and transitive actions......Page 59
Group actions and permutations......Page 60
Binary sequences in length of 2......Page 61
Cyclic orbits for binary sequences in length of 4......Page 62
Dihedral orbits for binary sequences in length of 4......Page 63
Ternary sequences in length of 4......Page 64
2.6 Genotypic Classification......Page 65
Data indexed by short branches or transitions......Page 67
2.8 Counting Orbits in Linkage Analysis......Page 68
Binary sequences in length of 4......Page 70
Exercises......Page 71
Permutation representations......Page 75
Equivalent representations......Page 76
The regular representation......Page 77
3.3 Unitary Representations......Page 78
3.4 Regular Representations and Group Algebras......Page 80
3.5 Tensor Representations......Page 81
3.6 Matrices with Group Symmetry......Page 82
Matrices with dihedral structure......Page 83
Matrices with complex structure......Page 84
Matrices with quaternionic structure......Page 85
Stable subspaces......Page 86
Irreducible representations......Page 87
3.8 Schur’s Lemma......Page 91
3.9 Constructing the Irreducible Representations of Sn......Page 94
An irreducible representation of S4......Page 98
Further Reading......Page 99
Exercises......Page 100
4.2 Characters of a Linear Representation......Page 103
4.3 Orthogonality Relations for Characters......Page 104
Irreducible characters......Page 105
The characters of the regular representation......Page 107
Class functions......Page 110
Reducing the conjugacy action on S3......Page 112
The canonical projections for the Sloan Fonts study......Page 113
The canonical projections for the binary sequences study......Page 115
The canonical projections for the regular representation of S3......Page 116
The canonical projections for the regular representation of D4......Page 119
Invariant plots......Page 123
4.5 The Standard Decomposition......Page 124
Sampling considerations......Page 126
Matrices with the symmetry of Sn......Page 127
Linear representations of order statistics and ranks......Page 128
4.6 Harmonic Analysis......Page 129
A decomposition for x ∈ F (G)......Page 131
A decomposition for x ∈ F (S3)......Page 132
A Poisson summation formula......Page 133
Exercises......Page 134
5.2 Analysis of Variance......Page 140
Analysis of variance for a simple triangular array......Page 141
One-way analysis of variance......Page 142
Two-way analysis of variance......Page 143
Latin squares......Page 144
5.5 Cyclic Reduction of Binary Sequences......Page 146
5.6 Dihedral Reduction of Binary Sequences......Page 148
Dihedral stratifications for voting preferences......Page 149
5.7 Projections in the Space of Scalar-Valued Functions......Page 150
5.8 Decompositions in the Dual Space......Page 151
Planar rotations......Page 152
Coinvariants of C4......Page 154
The Standard Decomposition of Entropy......Page 160
Invariant plots in the H1 × H2 space......Page 161
The standard decomposition of the entropy of the Sloan fonts......Page 162
Geological compositions......Page 163
The regular decomposition of entropy......Page 164
5.10 A Two-Way Cyclic Decomposition......Page 165
Data structures induced by C3v......Page 166
5.12 Data Indexed by Homogeneous Polynomials......Page 168
5.13 Likelihood Decompositions......Page 170
Further Reading......Page 172
Exercises......Page 173
6.2 Symmetry Studies of Four Sequences in Length of 3......Page 177
6.3 Reductions by Position Symmetry......Page 178
Determining the classes of transitivity......Page 179
Canonical decompositions in the partition λ = 210(2)......Page 180
Canonical decompositions in the partition λ = 1(3)0......Page 181
6.4 Reductions by Symbol Symmetry......Page 183
6.5 Dihedral Studies......Page 185
Exercises......Page 187
Appendix A......Page 189
Appendix B......Page 190
7.1 Introduction......Page 192
7.3 Astigmatic and Stigmatic Constraints......Page 193
7.4 Ranking Permutations......Page 194
Two-color topography......Page 195
Three-level gray scale......Page 196
7.5 Classification of Astigmatic Mappings......Page 197
The distribution of the y\'P y components......Page 199
The likelihood of an astigmatic mapping......Page 200
7.7 Dihedral Fourier Analysis......Page 202
The dihedral Fourier coefficients......Page 203
Dihedral Fourier analysis of refractive profiles......Page 204
Dihedral Fourier analysis-related applications......Page 205
Further Reading......Page 206
Appendix A......Page 210
Appendix B......Page 211
Appendix C......Page 212
8.1 Introduction......Page 213
8.2 Characterizing Rotations and Reversals......Page 215
8.3 Canonical Classification of Handedness in Elementary Images......Page 217
D4 symmetry......Page 218
Line (y = x) symmetry......Page 219
C4 symmetry......Page 220
Analytically generated images......Page 221
Sampling the mapping space......Page 223
Exercises......Page 225
Appendix A......Page 229
Appendix B......Page 230
Appendix C......Page 232
Appendix A: Computing Algorithms......Page 233
Appendix B: Glossary of Selected Symbols, Notations, and Terms......Page 237
Bibliography......Page 239
Index......Page 245