Covariant derivative

Summary

In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given by a principal connection on the frame bundle – see affine connection. In the special case of a manifold isometrically embedded into a higher-dimensional Euclidean space, the covariant derivative can be viewed as the orthogonal projection of the Euclidean directional derivative onto the manifold's tangent space. In this case the Euclidean derivative is broken into two parts, the extrinsic normal component (dependent on the embedding) and the intrinsic covariant derivative component. The name is motivated by the importance of changes of coordinate in physics: the covariant derivative transforms covariantly under a general coordinate transformation, that is, linearly via the Jacobian matrix

Official source

https://en.wikipedia.org/wiki/Covariant_derivative

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Complex magnetoelectric effect in multiferroic composites: the case of PFN- PT/(Co,Ni)Fe2O4

Hadi Papi

We report a systematic study of the magnetoelectric (ME) voltage coefficient as a complex quantity in the particulate composite of ferroelectric solid solution 0.94Pb(Fe1/2N1/2)O-3-0.06PbTiO(3)(PFN-PT) with CoFe2O4(CFO) and NiFe2O4 (NFO) ferrites. The results show that the real part of the ME voltage coefficient (alpha') is highly influenced by the magnetostrictive phase through lambda(H). NFO produces larger alpha' at a lower magnetic field, which originates from the softer magnetic properties. In addition, alpha' was found to be positive for NFO composite, while CFO composite shows a negative ME voltage coefficient at high magnetic fields. We argue that the field dependence of alpha' can be interpreted using the dynamic piezomagnetic coefficient, q(ac) = partial derivative lambda(ac)/partial derivative H. The imaginary part of the ME voltage coefficient (alpha '') was also determined for all composites. Both the alpha' and alpha '' show a peak at the same magnetic field, which is attributed to the maximum dynamic piezomagnetism when the magnetic domains are collectively rotated by the dc bias magnetic field. Our results show that the ME voltage coefficient in the composites of PFN-PT/(Co,Ni)Fe2O4 is highly influenced by the content, magnetic softness and field dependence behaviour of magnetostriction of the piezomagnetic phase.

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Insertion of germanium atoms in high- nuclearity rhodium carbonyl compounds: synthesis, characterization and preliminary biological activity of the heterometallic [ Rh13Ge( CO) 25] 3-, [ Rh14Ge2( CO) 30] 2-and [ Rh12Ge( CO) 27] 4-clusters+

Paul Joseph Dyson, Silvia Ruggieri

The reaction of the Rh-7(CO)(16) cluster anion with Ge2+ salts, in different stoichiometric ratios and under different atmospheres, leads to the formation of new heterometallic Rh-Ge clusters, representing the first known examples of Rh carbonyl compounds containing interstitial germanium atoms. More specifically, under N-2 the reaction progressively affords the new Rh13Ge(CO)(25) and Rh14Ge2(CO)(30) clusters in good yields, with the Ge atoms located in cubic and square-anti-prismatic cavities, respectively. However, under a CO atmosphere, the Rh13Ge(CO)(25) derivative undergoes a significant structural and chemical rearrangement giving the new compound, Rh12Ge(CO)(27), where Ge is hosted in an icosahedral metal cage. All species have been characterized by IR spectroscopy and ESI mass spectrometry, and their structures have been elucidated by single-crystal X-ray diffraction. Additionally, preliminary biological tests on the Rh13Ge(CO)(25) and Rh14Ge2(CO)(30) clusters show that they are cytotoxic to various cell lines.

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A functional approach to the numerical conformal bootstrap

Bernardo Zan

We apply recently constructed functional bases to the numerical conformal bootstrap for 1D CFTs. We argue and show that numerical results in this basis converge much faster than the traditional derivative basis. In particular, truncations of the crossing equation with even a handful of components can lead to extremely accurate results, in opposition to hundreds of components in the usual approach. We explain how this is a consequence of the functional basis correctly capturing the asymptotics of bound-saturating extremal solutions to crossing. We discuss how these methods can and should be implemented in higher dimensional applications.

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