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Register a new userEuler-Rodrigues formula for three-dimensional rotation via fractional powers of matrices
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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.
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Entrywise powers of matrices have been well-studied in the literature, and have recently received renewed attention in the regularization of high-dimensional correlation matrices. In this paper, we study powers of positive semidefinite block matrices $(H_{st})_{s,t=1}^n$ with complex entries. We first characterize the powers $alphainmathbb{R}$ such that the blockwise power map $(H_{st}) mapsto (H_{st}^alpha)$ preserves Loewner positivity. The characterization is obtained by exploiting connections with the theory of matrix monotone functions developed by Loewner. Second, we revisit previous work by Choudhury [Proc. AMS 108] who had provided a lower bound on $alpha$ for preserving positivity when the blocks $H_{st}$ pairwise commute. We completely settle this problem by characterizing the full set of powers preserving positivity in this setting. Our characterizations generalize previous work by FitzGerald-Horn, Bhatia-Elsner, and Hiai from scalars to arbitrary block size, and in particular, generalize the Schur Product Theorem. Finally, a natural and unifying framework for studying the case of diagonalizable blocks consists of replacing real powers by general characters of the complex plane. We thus classify such characters, and generalize our results to this more general setting. In the course of our work, given $betainmathbb{Z}$, we provide lower and upper bounds for the threshold power $alpha >0$ above which the complex characters $re^{itheta}mapsto r^alpha e^{ibetatheta}$ preserve positivity when applied entrywise to positive semidefinite matrices. In particular, we completely resolve the $n=3$ case of a question raised in 2001 by Xingzhi Zhan. As an application, we extend previous work by de Pillis [Duke Math. J. 36] by classifying the characters $K$ of the complex plane for which the map $(H_{st})_{s,t=1}^n mapsto (K({rm tr}(H_{st})))_{s,t=1}^n$ preserves positivity.
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.
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.
We give a local Euler-Maclaurin formula for rational convex polytopes in a rational euclidean space . For every affine rational polyhedral cone C in a rational euclidean space W, we construct a differential operator of infinite order D(C) on W with constant rational coefficients, which is unchanged when C is translated by an integral vector. Then for every convex rational polytope P in a rational euclidean space V and every polynomial function f (x) on V, the sum of the values of f(x) at the integral points of P is equal to the sum, for all faces F of P, of the integral over F of the function D(N(F)).f, where we denote by N(F) the normal cone to P along F.
We propose a new acquisition geometry for electron density reconstruction in three dimensional X-ray Compton imaging using a monochromatic source. This leads us to a new three dimensional inverse problem where we aim to reconstruct a real valued function $f$ (the electron density) from its integrals over spindle tori. We prove injectivity of a generalized spindle torus transform on the set of smooth functions compactly supported on a hollow ball. This is obtained through the explicit inversion of a class of Volterra integral operators, whose solutions give us an expression for the harmonic coefficients of $f$. The polychromatic source case is later considered, and we prove injectivity of a new spindle interior transform, apple transform and apple interior transform on the set of smooth functions compactly supported on a hollow ball. A possible physical model is suggested for both source types. We also provide simulated density reconstructions with varying levels of added pseudo random noise and model the systematic error due to the attenuation of the incoming and scattered rays in our simulation.
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