INDEX motion space

If, in a vector space of an infinite number of dimensions the components Ai and Bi become continuously distributed and everywhere dense, i is no longer a denumerable index but a continuous variable (x) and the scalar product turns into an overlap integral f A(x)B(x)dx. If it is zero the functions A and B are said to be orthogonal. This type of function is more suitable for describing wave motion. 

In the conventional theory, a saddle with index 1 corresponds to a transition state. Near a saddle, the NHIM Mq exists above it in the phase space. The NHIM Mo consists of those orbits with q = 0 and p = 0—that is, the vibrational motions involving qn,Pn) for n = 2,...,N above the saddle. Thus, its... 

Remember further that each reciprocal lattice point represents a vector, which is normal to the particular family of planes hkl (and of length 1 /d u) drawn from the origin of reciprocal space. If we can identify the position in diffraction space of a reciprocal lattice point with respect to our laboratory coordinate system, then we have a defined relationship to its family of planes, and the reciprocal lattice point tells us the orientation of that family. In practice, we usually ignore families of planes during data collection and use the reciprocal lattice to orient, impart motion to, and record the three-dimensional diffraction pattern from a crystal. Note also that if we identify the positions of only three reciprocal lattice points, that is, we can assign hkl indexes to three reflections in diffraction space, then we have defined exactly the orientation of both the reciprocal lattice, and the real space crystal lattice. 

Particle swarm optimization consists of a swarm of particles, each of which represents a candidate solution. Each particle is represented as a D-dimensional vector w, with a corresponding D-dimensional instantaneous trajectory vector Aw t), describing its direction of motion in the search space at iteration t The index i refers to the ith particle. The core of the PSO algorithm is the position update rule (Eq. (4)) which governs the movement of each of the n particles through the search space. 

We pointed out earlier that for equations of motion of constrained mechanical systems written in index-1 form the position and velocity constraints form integral invariants, see (5.1.16). Thus the coordinate projection and the implicit state space method introduced in the previous section can be viewed as numerical methods for ensuring that the numerical solution satisfies these invariants. 

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