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Radon Transform for Rotation Estimation, Interaction Picture in Quantum Mechanics, Curvature Tensor in GTR, Gauge Fields
Published in Harish Parthasarathy, Electromagnetics, Control and Robotics, 2023
[ 122] Gauge fields and field equations associated to a matrix Lie group. Let G be a matrix Lie group with ta, a = 1, 2, ...,...N a basis for its Lie algebra g. The commutation relations are [ta,tb]=iC(abc)tc summation over the repeated index c being implied. C(abc) are the structure constants of the Lie algebra associated with the basis {ta} . Let Aaμ (x) be the gauge fields and ψ(x) the matter field. The gauge covariant derivative is given by ∇μ=∂μ+ieAμ where is a constant and Aμ(x)=Aμa(x)ta
Nonassociative Algebras
Published in Leslie Hogben, Richard Brualdi, Anne Greenbaum, Roy Mathias, Handbook of Linear Algebra, 2006
Murray R. Bremner, Lucia I. Murakami, Ivan P. Shestakov
The structure constants of a finite dimensional algebra A over F with basis {x1, …, xn} are the scalars Cijk∈F(i,j,k=1,…,n) defined by: xixj=∑k=1ncijkxk.
Group Theory
Published in Gregory S. Chirikjian, Alexander B. Kyatkin, Engineering Applications of Noncommutative Harmonic Analysis, 2021
Gregory S. Chirikjian, Alexander B. Kyatkin
Recall that the structure constants of a real Lie algebra are defined by [Xi,Xj]=∑k=1NCijkXk.
High-order contact transformations of molecular Hamiltonians: general approach, fast computational algorithm and convergence of ro-vibrational polyad models
Published in Molecular Physics, 2022
Vladimir Tyuterev, Sergey Tashkun, Michael Rey, Andrei Nikitin
The hermicity of rotational and vibrational factors must be the same. As in Section 5.3.2, the vector indices in Equations (68)–(71) u = {u1, u2, u3 … } and b = {b1, b2, b3 … } represent powers of creation and annihilation operators for normal vibration modes and represents powers of rotational components as in Section 6.1.1. In a framework of the conventional perturbation theory for a separation of molecular variables (vibrational zero-order approximation), the rotational factors behave like constant parameters with respect to the operations of CT This means that Equations (64) and (97) and commutator relations are in principle sufficient to conduct computations up to a needed order. Note that the exact formal polynomial algebra for ro-vibrational operator involved in the ro-vibrational Hamiltonian transformations up to arbitrary power M has been implemented [39] accounting for all contributions in commutators and anti-commutators. These calculations are rather involved and are included as a FORTRAN routine of MOL_CT programme or in the form of tables for Lie algebra structure constants readable by an external code.
Instability of planetary flows using Riemann curvature: a numerical study
Published in Geophysical & Astrophysical Fluid Dynamics, 2019
In the spherical harmonics basis, the Riemann tensor is determined by the structure constants (Arakelyan and Savvidy 1989) with and In the Appendix the sectional curvature and the Riemann tensor of Dowker and Mo-Zheng (1990) are quoted.