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15
Symmetry Studies and Decompositions of Entropy
, 2006
"... This paper describes a grouptheoretic method for decomposing the entropy of a finite ensemble when symmetry considerations are of interest. The cases in which the elements in the ensemble are indexed by {1, 2,...,n} and by the permutations of a finite set are considered in detail and interpreted ..."
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Cited by 7 (6 self)
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This paper describes a grouptheoretic method for decomposing the entropy of a finite ensemble when symmetry considerations are of interest. The cases in which the elements in the ensemble are indexed by {1, 2,...,n} and by the permutations of a finite set are considered in detail and interpreted as particular cases of ensembles with elements indexed by a set subject to the actions of a finite group. Decompositions for the entropy in binary ensembles and in ensembles indexed by short DNA words are discussed. Graphical descriptions of the decompositions of the entropy in geological samples are illustrated. The decompositions derived in the present cases follow from a systematic data analytic tool to study entropy data in the presence of symmetry considerations.
CANONICAL INVARIANTS FOR THREECANDIDATE PREFERENCE RANKINGS
"... ABSTRACT. It is shown that the data space for the threecandidate Condorcet Rule can be decomposed as the sum of two onedimensional and one twodimensional permutation invariant subspaces. The nontrivial onedimensional invariant describes the variation in total number of votes between fully distinc ..."
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Cited by 3 (3 self)
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ABSTRACT. It is shown that the data space for the threecandidate Condorcet Rule can be decomposed as the sum of two onedimensional and one twodimensional permutation invariant subspaces. The nontrivial onedimensional invariant describes the variation in total number of votes between fully distinct preferences and preferences that agree on the ranking of exactly one candidate. The twodimensional invariant describes the voting difference between the extreme (win vs. show) rankings for any two candidates. In contrast, the data space for the original voting data has one additional twodimensional invariant subspace corresponding to win vs. place (or place vs. show) data for any two candidates. Canonical bases for these subspaces are constructed, interpreted, and graphically displayed as invariant plots. Permutation invariant distances among data points in the invariant subspaces are obtained. The presented
Dihedral Fourier Analysis
 CONTEMPORARY MATHEMATICS
, 2010
"... Dataanalytic aspects of dihedral Fourier analysis are presented within the context of the canonical decomposition theorem for finite groups. ..."
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Cited by 3 (1 self)
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Dataanalytic aspects of dihedral Fourier analysis are presented within the context of the canonical decomposition theorem for finite groups.
Symmetry Studies for Data Analysis
 METHODOL COMPUT APPL PROBAB (2007) 9:325–341
, 2007
"... This paper describes some of the basic applications of the algebraic theory of canonical decomposition to the analysis of data. The notions of structured data and symmetry studies are discussed and applied to demonstrate their role in well known principles of analysis of variance and their applicab ..."
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Cited by 2 (2 self)
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This paper describes some of the basic applications of the algebraic theory of canonical decomposition to the analysis of data. The notions of structured data and symmetry studies are discussed and applied to demonstrate their role in well known principles of analysis of variance and their applicability in more general experimental settings.
DATA ANALYTIC ASPECTS OF CHIRALITY
, 2005
"... This paper interprets chirality in the context of dihedral (finite) group algebras. It is argued that the presence and quantification of chirality or handedness in certain objects is dependent on how these objects respond when subject to (dihedral) rotations and reversals. The absence of response ..."
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Cited by 1 (1 self)
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This paper interprets chirality in the context of dihedral (finite) group algebras. It is argued that the presence and quantification of chirality or handedness in certain objects is dependent on how these objects respond when subject to (dihedral) rotations and reversals. The absence of response to reversals is an indication of absence of chirality. Rotationreversal displays are introduced in order to describe an object’s responds to dihedral actions. A formal quantification of deviations from chirality in elementary numerical matrices, and methods of statistical inference for testing those deviations via canonical decompositions are obtained.
A SYMMETRY STUDY OF SLOAN FONTS
"... Abstract. This preliminary report defines the group of symmetries for the Sloan fonts and the canonical invariants for data indexed by those symmetries. Further, the association between line symmetries in the ETDRS Charts and the corresponding line difficulty is discussed. 1. ..."
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Abstract. This preliminary report defines the group of symmetries for the Sloan fonts and the canonical invariants for data indexed by those symmetries. Further, the association between line symmetries in the ETDRS Charts and the corresponding line difficulty is discussed. 1.
Symmetry and Data Analysis
"... A brief account of symmetry in its Greek origins is presented and its role in the analysis of data indexed by permutations is discussed, with examples of ranked preference data and short biological sequences. 1 2 1. ..."
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A brief account of symmetry in its Greek origins is presented and its role in the analysis of data indexed by permutations is discussed, with examples of ranked preference data and short biological sequences. 1 2 1.
A geometric algebra reformulation of 2x2 matrices: the dihedral group D4 in braket notation
, 811
"... Abstract. We represent vector rotation operators in terms of bras or kets of halfangle exponentials in Clifford (geometric) algebra Cl3,0. We show that SO3 is a rotation group and we define the dihedral group D4 as its finite subgroup. We use the EulerRodrigues formulas to compute the multiplicati ..."
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Abstract. We represent vector rotation operators in terms of bras or kets of halfangle exponentials in Clifford (geometric) algebra Cl3,0. We show that SO3 is a rotation group and we define the dihedral group D4 as its finite subgroup. We use the EulerRodrigues formulas to compute the multiplication table of D4 and derive its group algebra identities. We take the linear combination of rotation operators in D4 to represent the four Fermion matrices in Sakurai, which in turn we use to decompose any 2 × 2 matrix. We show that bra and ket operators generate left and rightacting matrices, respectively. We also show that the Pauli spin matrices are not vectors but vector rotation operators, except for ˆσ2 which requires a subsequent multiplication by the imaginary number i geometrically interpreted as the unit oriented volume. 1