Gradient difference vector, direct

Robb, Bemaidi, and Olivucci (RBO) [37] developed a method based on the idea that a conical intersection can be found if one moves in a plane defined by two vectors xi and X2, defined in the adiabatic basis of the molecular Hamiltonian H. The direction of Xi corresponds to the gradient difference... 

At the present time it must be stated that the continuous acceleration of the flame front cannot be explained on the basis of the unburned gas turbulence as mentioned above, and that there must be turbulence generating effects inside the reaction zone. One possible mechanism might be the Taylor-instability " , which can lead to an exponential growth of an initial disturbance at a continuously accelerated interface separating two fluids of different densities (positive density gradient in the direction of the acceleration vector). 

Here (j is the CG update parameter. In the above equations, e = e (tj) o vector notation for the discretized electric field strength, = g (fj) o objective functional J with respect to the field strength (evaluated at a field strength of e t) and dk = d (t ) o search direction at the feth iteration. The time has been discretized into N time steps, such as that tj=jx )t, where j = 0,1,2, , N. Different CG methods correspond to different choices for the scalar (j. ... 

Recall that the operation of V on a scalar quantity is the gradient, which is a vector. For example, if V is operated on a scalar pressure field P, then V P is the pressure gradient vector field, which can have different values in the three spatial directions. The operation of V on a vector field can either be the divergence or the curl of the vector field. The former is obtained by the dot product (also called the scalar product) as V v or div 1 , where the result is a scalar whereas, the latter is obtained by the cross product (also called the vector product) V / v. or curl v, and the result is a vector field. 

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