Empirical Means on Pseudo-Orthogonal Groups

Wang, Jing; Sun, Hongwei; Fiori, Simone

doi:10.3390/math7100940

Cited by 2 publications

(2 citation statements)

References 29 publications

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“…Certainly not, for to do so, it would be necessary to extend the result we have just obtained to any closed paths. This should involve quite an elaborate analysis, stemming from the basic properties of pseudo-orthogonal groups, as in [34].…”

Section: The Additional Group Of Motions and The Twin Paradoxmentioning

confidence: 99%

Noninertial Proper Motions of the Minkowski Metric, the Sagnac Effect, and the Twin Paradox

Popov

Матвеев

2023

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The Sagnac effect and related twin paradox with a rotating disc are analyzed. It may seem that the special theory of relativity gives an easy and exhaustive treatment here. However, such consideration is deceptive since the principles of special relativity are originally established only for the inertial frames of reference, whereas the Sagnac experiment and the twin paradox exist in a noninertial one. We introduce an additional group of motions related to the rotation with uniform angular speed and show that these transformations leave the Minkowski metric invariant. Thus, we can give a firm mathematical ground to a usual easy consideration of the Sagnac effect. It should be noted that the presented result is true for a special case of motions; general coordinate transformations into accelerating frames of reference do not preserve the metric.

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Section: The Additional Group Of Motions and The Twin Paradoxmentioning

confidence: 99%

Noninertial Proper Motions of the Minkowski Metric, the Sagnac Effect, and the Twin Paradox

Popov

Матвеев

2023

Axioms

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“…Euclidean metric: The expression of the geodesic corresponding to the Euclidean metric was derived in [50]. Let γ : [0, 1] → Sp(2n) be a geodesic arc connecting the points X, Y ∈ Sp(2n).…”

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confidence: 99%

Manifold Calculus in System Theory and Control—Fundamentals and First-Order Systems

Fiori

2021

Symmetry

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The aim of the present tutorial paper is to recall notions from manifold calculus and to illustrate how these tools prove useful in describing system-theoretic properties. Special emphasis is put on embedded manifold calculus (which is coordinate-free and relies on the embedding of a manifold into a larger ambient space). In addition, we also consider the control of non-linear systems whose states belong to curved manifolds. As a case study, synchronization of non-linear systems by feedback control on smooth manifolds (including Lie groups) is surveyed. Special emphasis is also put on numerical methods to simulate non-linear control systems on curved manifolds. The present tutorial is meant to cover a portion of the mentioned topics, such as first-order systems, but it does not cover topics such as covariant derivation and second-order dynamical systems, which will be covered in a subsequent tutorial paper.

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Empirical Means on Pseudo-Orthogonal Groups

Cited by 2 publications

References 29 publications

Noninertial Proper Motions of the Minkowski Metric, the Sagnac Effect, and the Twin Paradox

Noninertial Proper Motions of the Minkowski Metric, the Sagnac Effect, and the Twin Paradox

Manifold Calculus in System Theory and Control—Fundamentals and First-Order Systems

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