Rotations about nonorthogonal axes.

Davenport, P. B.

doi:10.2514/3.6842

Cited by 32 publications

(14 citation statements)

References 4 publications

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“…Knowing how to diagonalize, say, 35 s , how do we then find the corresponding action on the 35 c (and vice-versa)? Logarithmizing the group element to obtain the algebra element is out of the question, but we can employ a higher-dimensional generalization of Davenport chained rotations [130,131] to write G d as a product of a sequence of up to 28 rotations in coordinate-aligned planes, Once the 35 s has been diagonalized by any such method that also gives us the effect of the rotation which was employed on the 35 c , we can proceed to determine the residual subalgebra of so(8) that keeps the diagonalized 35 s unchanged. We can employ this subalgebra to reduce the number of off-diagonal entries for the matrix carrying the 35 c representation, but in general not completely.…”

Section: Coordinate-aligning Rotationsmentioning

confidence: 99%

SO(8) supergravity and the magic of machine learning

Comşa

Firsching

Fischbacher

2019

J. High Energ. Phys.

106

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Using de Wit-Nicolai D = 4 N = 8 SO(8) supergravity as an example, we show how modern Machine Learning software libraries such as Google's TensorFlow can be employed to greatly simplify the analysis of high-dimensional scalar sectors of some M-Theory compactifications. We provide detailed information on the location, symmetries, and particle spectra and charges of 192 critical points on the scalar manifold of SO(8) supergravity, including one newly discovered N = 1 vacuum with SO(3) residual symmetry, one new potentially stabilizable non-supersymmetric solution, and examples for "Galois conjugate pairs" of solutions, i.e. solution-pairs that share the same gauge group embedding into SO(8) and minimal polynomials for the cosmological constant. Where feasible, we give analytic expressions for solution coordinates and cosmological constants.As the authors' aspiration is to present the discussion in a form that is accessible to both the Machine Learning and String Theory communities and allows adopting our methods towards the study of other models, we provide an introductory overview over the relevant Physics as well as Machine Learning concepts. This includes short pedagogical code examples. In particular, we show how to formulate a requirement for residual Supersymmetry as a Machine Learning loss function and effectively guide the numerical search towards supersymmetric critical points. Numerical investigations suggest that there are no further supersymmetric vacua beyond this newly discovered fifth solution.At the moment, the N = 8 Supergravity Theory is the only candidate in sight. There are likely to be a number of crucial calculations within the next few years that have the possibility of showing that the theory is no good. If the theory survives these tests, it will probably be some years more before we develop computational methods that will enable us to make predictions and before we can account for the initial conditions of the universe as well as the local physical laws. These will be the outstanding problems for theoretical physics in the next twenty years or so.But to end on a slightly alarmist note, they may not have much more time than that.At present, computers are a useful aid in research, but they have to be directed by human minds. If one extrapolates their recent rapid rate of development, however, it would seem quite possible that they will take over altogether in theoretical physics. So, maybe the end is in sight for theoretical physicists, if not for theoretical physics. S. Hawking, Conclusion of his 1981 Inaugural lecture [1]"Is the end in sight for theoretical physics?"

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Section: Coordinate-aligning Rotationsmentioning

confidence: 99%

SO(8) supergravity and the magic of machine learning

Comşa

Firsching

Fischbacher

2019

J. High Energ. Phys.

106

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“…(9), we need to find the composition rule as well as the Euler angle increment depending on the incremental rotation vector . Since there are 12 possible sets of Euler angles [13,19,44], we present a general procedure to determine the Euler angle increment for an arbitrary set of Euler angles. In Appx.…”

Section: Proposed Approach For the Time Integration Of Euler Anglesmentioning

confidence: 99%

“…(56)-(57) with Eqs. (53)-(55) shows that the proposed approach can be integrated into existing applications that use three rotation parameters by simply replacing 13 Eq. (57) with Eqs.…”

Section: Explicit Euler (Rk1)mentioning

confidence: 99%

Time integration of rigid bodies modelled with three rotation parameters

Holzinger

Gerstmayr

2021

Multibody Syst Dyn

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Three rotation parameters are commonly used in multibody dynamics or in spacecraft attitude determination to represent large spatial rotations. It is well known, however, that the direct time integration of kinematic equations with three rotation parameters is not possible in singular points. In standard formulations based on three rotation parameters, singular points are avoided, for example, by applying reparametrization strategies during the time integration of the kinematic equations. As an alternative, Euler parameters are commonly used to avoid singular points. State-of-the-art approaches use Lie group methods, specifically integrators, to model large rigid body rotations. However, the former methods are based on additional information, e.g. the rotation matrix, which must be computed in each time step. Thus, the latter method is difficult to incorporate into existing codes that are based on three rotation parameters. In this contribution, a novel approach for solving rotational kinematics in terms of three rotation parameters is presented. The proposed approach is illustrated by the example of the rotation vector and the Euler angles. In the proposed approach, Lie group time integration methods are used to compute consistent updates for the rotation vector or the Euler angles in each time step and therefore singular points can be surmounted and the accuracy is higher as compared to the direct time integration of rotation parameters. The proposed update formulas can be easily integrated into existing codes that use either the rotation vector or Euler angles. The advantages of the proposed approach are demonstrated with two numerical examples.

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“…We then place each SN on a triangular face, where its position and orientation are defined by the face center coordinates and its normal vector n face , respectively. To direct the node, initially oriented along z, towards the face's normal vector n face = (n x , n y , n z ) T , we apply the Davenport yaw-pitch-roll intrinsic chained rotation [52] to the reference vector z = (0, 0, 1) T , as illustrated in Fig. 5(b).…”

Section: B Sensor Placementmentioning

confidence: 99%

Channel Characterization and Modeling for Optical Wireless Body-Area Networks

Haddad

Khalighi

Zvánovec

et al. 2020

IEEE Open J. Commun. Soc.

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We address channel characterization and modeling for medical wireless body-area networks (WBANs) based on the optical wireless technology. We focus on the intra-WBAN communication links, i.e., between a set of medical sensors and a coordination node, placed on the patient's body. We consider a realistic mobility model, e.g., inside a hospital room, which takes into account the effect of shadowing due to body parts movements and the variations of the underlying channels. To take into account the global and local user mobility, we consider a dynamic model based on a three-dimensional animation of a walk cycle, as well as walk trajectories based on an improved random way-point mobility model. Then, Monte Carlo ray-tracing simulations are performed to obtain the channel impulse responses for different link configurations at different instants of the walk scenarios. We then derive first-and second-order statistics of the channel parameters such as the channel DC gain, delay spread, and coherence time, and furthermore, propose best fit statistical models to describe the distribution of these parameters for a general scenario.

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Rotations about nonorthogonal axes.

Cited by 32 publications

References 4 publications

SO(8) supergravity and the magic of machine learning

SO(8) supergravity and the magic of machine learning

Time integration of rigid bodies modelled with three rotation parameters

Channel Characterization and Modeling for Optical Wireless Body-Area Networks

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