The Element of Volume of the Rotation Group

Murnaghan, F. D.

doi:10.1073/pnas.38.1.69

Cited by 6 publications

(9 citation statements)

References 1 publication

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“…with maximum at cos β n−1 = −(n − 2)/(n − 1). This is in agreement with the result of [23] and other results obtained from different perspectives [32,52] and it is very useful in many instances, e. g., for the parametrization of the families of most probable matrices. The parametrization of Eq.…”

Section: Recursive Haar Measuressupporting

confidence: 91%

“…Permanents are entries of the full 1287 × 1287 matrix; i.e., D-functions for this irreducible representation [46]. Whereas the decomposition of Reck et al [16] (or its primal version by Murnaghan [23]) requires the evaluation of 36 nondiagonal C i j with j > i, their transpose conjugates, and 9 C ii , our scheme requires the evaluation of only 8 C i i+1 matrices, their transpose conjugates, and 9 C ii . As the size of practical interferometers increases, the linear scaling of this scheme thus stands to offer substantial computational savings.…”

Section: Recursive Factorization Of Unitary Transformationsmentioning

confidence: 99%

“…We discuss here a decomposition of any n × n unitary in terms of two (n − 1) × (n − 1) unitaries coupling the same n − 1 modes, and a single 2 × 2 unitary coupling one of those n − 1 to the remaining mode [23]. The scheme is recursive; it can be halted at any dimensionality of subtransformations or performed in its end, resulting in a tidy arrangement of SU (2) gadgets.…”

Section: Introductionmentioning

confidence: 99%

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Simple factorization of unitary transformations

2018

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We demonstrate a method for general linear optical networks that allows one to factorize any SU(n) matrix in terms of two SU(n − 1) blocks coupled by an SU(2) entangling beam splitter. The process can be recursively continued in a straightforward way, ending in a tidy arrangement of SU(2) transformations. The method hinges only on a linear relationship between input and output states, and can thus be applied to a variety of scenarios, such as microwaves, acoustics, and quantum fields.

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Section: Recursive Haar Measuressupporting

confidence: 91%

Section: Recursive Factorization Of Unitary Transformationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Simple factorization of unitary transformations

2018

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“…In we give two choices of H * ( x ). We can parameterize K * by following Murnaghan (1950) and writing it as a product of {( p −1)( p −2)/2} ( p −1)×( p −1) plane rotation matrices which are functions of p −2 longitude or equatorial angles ψ 1 ,…, ψ p −2 ∈ [− π , π ) and ( p −2)( p −3)/2 latitude or meridian angles ν 1 ,…, ν ( p −2)( p −3)/2 ∈ [0, π ]. The are identity matrices with the ( i , i ), ( i , j ), ( j , i ) and ( j , j ) elements replaced by cos ( ψ ), − sin ( ψ ), sin ( ψ ) and cos ( ψ ) respectively.…”

Section: The Modelmentioning

confidence: 99%

Regression for Compositional Data by using Distributions Defined on the Hypersphere

Scealy

Welsh

2011

Journal of the Royal Statistical Society Series B: Statistical Methodology

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Compositional data can be transformed to directional data by the square-root transformation and then modelled by using distributions defined on the hypersphere. One advantage of this approach is that zero components are catered for naturally in the models. The Kent distribution for directional data is a good candidate model because it has a sufficiently general covariance structure. We propose a new regression model which models the mean direction of the Kent distribution as a function of a vector of covariates. Our estimators can be regarded as asymptotic maximum likelihood estimators. We show that these estimators perform well and are suitable for typical compositional data sets, including those with some zero components.

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“…To obtain a uniform distribution of the Euler angles we multiply our integrand by the Jacobian J = sin β/ (8 π 2 ) [22] to obtain…”

Section: Theorymentioning

confidence: 99%

Improvement and analysis of computational methods for prediction of residual dipolar couplings

Berlin

O’Leary

Fushman

2009

Journal of Magnetic Resonance

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We describe a new, computationally effcient method for computing the molecular alignment tensor based on the molecular shape. The increase in speed is achieved by re-expressing the problem as one of numerical integration, rather than a simple uniform sampling (as in the PALES method), and by using a convex hull rather than a detailed representation of the surface of a molecule. This method is applicable to bicelles, PEG/hexanol, and other alignment media that can be modeled by steric restrictions introduced by a planar barrier. This method is used to further explore and compare various representations of protein shape by an equivalent ellipsoid. We also examine the accuracy of the alignment tensor and residual dipolar couplings (RDC) prediction using various ab initio methods. We separately quantify the inaccuracy in RDC prediction caused by the inaccuracy in the orientation and in the magnitude of the alignment tensor, concluding that orientation accuracy is much more important in accurate prediction of RDCs.

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The Element of Volume of the Rotation Group

Cited by 6 publications

References 1 publication

Simple factorization of unitary transformations

Simple factorization of unitary transformations

Regression for Compositional Data by using Distributions Defined on the Hypersphere

Improvement and analysis of computational methods for prediction of residual dipolar couplings

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