No Arabic abstract
We propose a rational version of the classic Rodrigues rotation formula, which leads to a more accurate and efficient modelling of rotations and their derivatives in finite precision arithmetic. We explain how the rational Rodrigues formula can be used to describe the kinematics of rigid bodies, in a practical example in which we model the rotation of a cell phone using the data obtained from its gyroscope.
In this short paper, we review the Euler-Rodrigues formula for three-dimensional rotation via fractional powers of matrices. We derive the rotations by any angle through the spectral behavior of the fractional powers of the rotation matrix by $frac{pi}{2}$ in $mathbb{R}^3$ about some axis.
We present a simple formula for the generating function for the polynomials in the $d$--dimensional semiclassical wave packets. We then use this formula to prove the associated Rodrigues formula.
In this paper we introduce a family of rational approximations of the reciprocal of a $phi$-function involved in the explicit solutions of certain linear differential equations, as well as in integration schemes evolving on manifolds. The derivation and properties of this family of approximations applied to scalar and matrix arguments are presented. Moreover, we show that the matrix functions computed by these approximations exhibit decaying properties comparable to the best existing theoretical bounds. Numerical examples highlight the benefits of the proposed rational approximations w.r.t.~the classical Taylor polynomials and other rational functions.
We propose a new method for the approximate solution of the Lyapunov equation with rank-$1$ right-hand side, which is based on extended rational Krylov subspace approximation with adaptively computed shifts. The shift selection is obtained from the connection between the Lyapunov equation, solution of systems of linear ODEs and alternating least squares method for low-rank approximation. The numerical experiments confirm the effectiveness of our approach.
We prove a generalization of the Verlinde formula to fermionic rational conformal field theories. The fusion coefficients of the fermionic theory are equal to sums of fusion coefficients of its bosonic projection. In particular, fusion coefficients of the fermionic theory connecting two conjugate Ramond fields with the identity are either one or two. Therefore, one is forced to weaken the axioms of fusion algebras for fermionic theories. We show that in the special case of fermionic W(2,d)-algebras these coefficients are given by the dimensions of the irreducible representations of the horizontal subalgebra on the highest weight. As concrete examples we discuss fusion algebras of rational models of fermionic W(2,d)-algebras including minimal models of the $N=1$ super Virasoro algebra as well as $N=1$ super W-algebras SW(3/2,d).