Tensor conjugate

The tensor S is symmetric only at a point O unique for each body, this point is called the center of hydrodynamic reaction. This tensor is called the conjugate tensor and characterizes the crossed reaction of the body under translational and rotational motion (the drag moment in the translational motion and the drag force in the rotational motion). For bodies with orthotropic, axial, or spherical symmetry, the conjugate tensor is zero. However, it is necessary to take this tensor into account for bodies with helicoidal symmetry (propeller-like bodies). 

The pseudopotential density-functional technique is used to calculate total energies, forces on atoms and stress tensors as described in Ref. 13 and implemented in the computer code CASTEP. CASTEP uses a plane-wave basis set to expand wave-functions and a preconditioned conjugate gradient scheme to solve the density-functional theory (DFT) equations iteratively. Brillouin zone integration is carried out via the special points scheme by Monkhorst and Pack. The nonlocal pseudopotentials in Kleynman-Bylander form were optimized in order to achieve the best convergence with respect to the basis set size. 5... 

The hyperfine coupling tensors of carotenoids were determined from the HYSCORE analysis of the contour line-shapes of the cross-peaks (Dikanov and Bowman 1995,1998, Dikanov et al. 2000), which provided the principal components of the tensors that appear to be rhombic. Such tensors are characteristic of planar conjugated radicals with the unpaired spin in a pz orbital of the carbon of the C-H group. 

We shall be concerned with the doubled operators describing the energy-momentum tensor of the free Maxwell and Dirac fields according to the tilde conjugation rules, we have, respectively ... 

Since both Z and y are complex vector spaces of functions, they are closed under complex conjugation, and hence so is their tensor product. The tensor product separates points, since any two points of different radius can be separated by Ir and any two points of different spherical angle can be separated by J . Finally, the function... 

The generalization of this conjugation phase condition, to cover the case of tensors with a complex inner structure and half-integer ranks, is presented in [85]. 

The set of (2k + 1) Hermitian conjugated operators does not form an irreducible tensor, but from (2.21) we can easily see that the operators... 

Electron annihilation operators, as Hermitian conjugates of creation operators, are no longer the components of an irreducible tensor. According to (13.40), such a tensor is formed by (21 + l)(2s + 1) components of the operator... 

It follows from (15.49) that the tensors aiqls have the following property with respect to the operation of Hermitian conjugation ... 

The largest component of the Y-tensor is in the conjugation direction. Therefore,.even though no particular bulk symmetry is required for nonzero x > a medium in which all conjugated polymeric chains align in the same direction should have a larger x value along the chain direction relative to that in an amorphous or disordered form of the same polymer. Studies of x in ordered or stretch-oriented polymers as discussed below confirm this prediction. Finally, the polymeric chains should pack as closely as possible in order to maximize the hyperpolarizability density and hence x ... 

For molecules containing several conjugated bonds yn becomes much larger than y°. Of course, y itself is a fourth rank tensor property (analogous to x(3)) and can be specified in the molecular or laboratory reference frames. For an isotropic medium one measures an orientational average of the hyperpolarizability... 

In summary, the in-plane components (a33 and c ) of the shielding tensor depend on the electronic structure changes of the conjugated systems, induced by the inductive effect and the 71-modification effect mentioned above. 

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