Gradient operation

An alternative notation involves the use of the so-called gradient operator V (pronounced del ),... 

Gradient operator Laplace operator Dot product Cross product Divergence operator Curl operator Vector transposition Complex conjugate... 

Our final form for the Boltzman equation is then obtained by substituting these last two expressions for the gain and loss terms into equation 9.29 and adding the term Vijf x, v, t) to the LHS for the case where there is an external force F Vjj is the gradient operator with respect to v and m is the hard-sphere mass) ... 

Adsorption chromatography using small particle silica or alumina has also been employed in the separation of biologically meaningful substances. Phospholipids, for example, have been separated on silica (38). One of the big problems for such substances is detection, since many of the compounds are not U.V. active. Generally, the refractive index detector is employed for isocratic operation, and the moving wire detector for gradient operation. Formation of U.V.-active derivatives is also possible (39). 

F-ATPases (including the H+- or Na+-translocating subfamilies F-type, V-type and A-type ATPase) are found in eukaryotic mitochondria and chloroplasts, in bacteria and in Archaea. As multi-subunit complexes with three to 13 dissimilar subunits, they are embedded in the membrane and involved in primary energy conversion. Although extensively studied at the molecular level, the F-ATPases will not be discussed here in detail, since their main function is not the uptake of nutrients but the synthesis of ATP ( ATP synthase ) [127-130]. For example, synthesis of ATP is mediated by bacterial F-type ATPases when protons flow through the complex down the proton electrochemical gradient. Operating in the opposite direction, the ATPases pump 3 4 H+ and/or 3Na+ out of the cell per ATP hydrolysed. 

S. Abel, M. Mazzotti and M. Morbidelli, Solvent gradient operation of simulated moving beds. I. Linear isotherms. J. Chromatogr.A 944 (2002) 23-39. 

The peak capacity of a gradient depends on the gradient operating conditions. This is discussed in detail in Reference 9. Here we will just give the relevant equation for gradients over a broad range of solvent composition without going into further detail ... 

The absolute value of the pressure gradient is used for all dissipation calculations. Although the pressure-induced flow is in different directions for a positive- and negative-valued pressure gradient operation, the dissipation level, however, is identical. Viscous energy dissipation is always positive definite. 

Furthermore, it should be noted that interpreting polymer spectra in solvent mixtures typical of gradient operation in liquid chrcmatography may be an additional problem. Although all solvents... 

The notation 8/8R designates the gradient operator, also called nabla (V j). We thus finally obtain from Eqs. 2.35 and 2.36 [296]... 

For more complex vector-field potentials depending on the relative orientation as well as the separation of the two particles, the corresponding vector expression is F = — V V, where V = (d/dx, d/dy, d/dz) is the gradient operator. Such vectorial aspects of intermolecular forces are obviously important for real molecules of nonspherical shape.]... 

In (22.37) ip is the wave function of an atom with motionless nucleus. The one-electronic submatrix element of the gradient operator (n/ V ni/i) is non-zero only for h = / 1. Therefore, the matrix element of operator (22.38) inside a shell of equivalent electrons vanishes and one has to account for this interaction only between shells. For the configuration, consisting of j closed and two open shells, it is defined by the following formula [156] ... 

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