Grid path

Definition 20.15. Consider the action of the group hxT, on the plane, where the standard generators ofhxh act hy vector translations, with vectors ( — 1,1) and m,n). An (m, n)-torus front (or sometimes simply torus front is an orbit of this (Z x h)-action on the set of all (m, n)-grid paths. 

The reason for choosing the word torus here is that the quotient space of the plane by this (Z x Z)-action can be viewed as a torus, where each (m, n)-grid path yields a loop following the grid in the northeasterly direction see Figure 20.4. We notice that every torus front has a unique representative starting from the point (0,0). It will soon become clear why we choose to consider the orbits of the action rather than merely considering these representatives of the orbits. 

If the flips are done on torus fronts instead of grid paths, then we do not need to consider different special cases, which is the main reason why we chose to replace the paths by the orbits of the (Z x Z)-action. ... 

The reaction coordinate is one specific path along the complete potential energy surface associated with the nuclear positions. It is possible to do a series of calculations representing a grid of points on the potential energy surface. The saddle point can then be found by inspection or more accurately by using mathematical techniques to interpolate between the grid points. 

This is an open area packing with multiple layers of lattice-type panels. This grid, as described by the manufacturer s bulletin, consists of vertical, slanted, and horizontal planes of metal. The vertical strips have horizontal flanges oriented alternately right and left. Due to the random overlap, the vapor path must zig-zag through the bed. 

Thus even if the mean free path is small compared to the cell length, particle (or equivalently grid) shifting will cause particles to collide with molecules in nearby cells, thereby reducing the effects of locally correlated collision events in the same cell. 

The colhsional contribution arises from grid shifting and accounts for effects on scales where the dimensionless mean free path is small, 1. The discrete Green-Kubo derivation leading to Eq. (52) involves a number of subtle issues that have been discussed by Ihle, Tiizel, and Kroll [26]. 

These expressions for the shear viscosity are compared with simulation results in Fig. 5 for various values of the angle a and the dimensionless mean free path X. The figure plots the dimensionless quantity (v/X)(x/a2) and for fixed y and a we see that (vkin/A,)(x/a2) const A, and (vcol/A)(r/u2) const/A. Thus we see in Fig. 5b that the kinetic contribution dominates for large A since particles free stream distances greater than a cell length in the time x however, for small A the collisional contribution dominates since grid shifting is important and is responsible for this contribution to the viscosity. 

For a grid, achieving equal distribution of gas flow through many parallel paths requires equal resistances and sufficient resistance to equal or exceed the maximum value of any unsteady-state pressure fluctuation. It has been determined experimentally that the head of solids in some fluidized beds above an upwardly-directed grid port can vary momentarily by as much as 30%. This is due to large fluctuations in the jet penetration for an upwardly-directed jet as discussed in the previous section. The equivalent variation downstream of a downwardly-directed port is less than 10%. Thus, as a rule of thumb, the criteria for good gas distribution based on the direction of gas entry are ... 

Fig. 9.12 Top view of folded path waveguide layouts using (a) a double spiral and (b) a grid configuration. Both images are taken using an InGaAs infra red camera while X 1550 nm light is coupled into the silicon waveguide chip...

Normal water paths are always in the direction of least resistance, in other words, generally from the larger mains to the smaller mains. By choosing two hydrants on a large section of pipe (within a loop or grid) and estimating the water flow direction, a test can be conducted. 

Figure 1. A sample field model constructed to find the breach path for length is 5 m., width 2 m., boundary 1 m., and grid size 1 m. (N = 8, M = 3).

The weakest breach path problem can now be defined as finding the permutation of a subset of grid points V = [vo, v, ..., Vk with which a target traverses from the starting point to the destination point with the least probability of being detected, where Vq = 0 is the starting point and Vk = NM+1 is the destination point. Here we can define the breach probability P of the weakest breach path V as... 

In order to solve the weakest breach path problem, where we construct a graph to model the field, Dijkstra s shortest path algorithm can be employed [40]. The detection probabilities associated with the grid points cannot be directly used as weights of the grid points, and consequently they must be transformed to a new measure dv. Specifically, let... 

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