2D examples

Let us return to our 2D example to see how those equations are applied. The momenta are defined by... 

A 3D crystal has its atoms arranged such that many different planes can be drawn through them. It is convenient to be able to describe these planes in a systematic way and Fig. 4 shows how this is done. It illustrates a 2D example, but the same principle applies to the third dimension. The crystal lattice can be defined in terms of vectors a and b, which have a defined length and angle between them (it is c in the third dimension). The box defined by a and b (and c for 3D) is known as the unit cell. The dashed lines in Fig. 4A show one set of lines that can be drawn through the 2D lattice (they would be planes in 3D). It can be seen that these lines chop a into 1 piece and b into 1 piece, so these are called the 11 lines. The lines in B, however, chop a into 2 pieces, but still chop b into 1 piece, so these are the 21 lines. If the lines are parallel to an axis as in C, then they do not chop that axis into any pieces so, in C, the lines chopping a into 1 piece and which are parallel to b are the 10 lines. This is a simple rule. The numbers that are generated are known as the Miller indices of the plane. Note that if the structure in Fig. 6.4 was a 3D crystal viewed down the c axis, the lines would be planes. In these cases, the third Miller index would be zero (i.e., the planes would be the 110 planes in A, the 210 planes in B, and the 100... 

The number of Maxwell equations for each of the possible thermodynamic potentials is given by D(D — l)/2, and the number of Maxwell equations for the thermodynamic potentials for a system related by Legendre transforms is [ )(D — 1)/2]2D. Examples are given in the following section. 

Fig. i 2D example of a uniform grid applied to the unit cell. The ions are on grid points 02 41 23... 

The extension of PTC to higher spatial dimensions is straightforward and many examples can be found in the literature (see, for example, Desjardins et al. (2008)). In Figure 8.1 we show a 2D example with two jets crossing. An important question that arises with PTC is whether or not the moment-transport equations can predict it. To answer this question, we return to Eq. (8.3) and integrate over phase space to find the moment-transport equation ... 

Fig. 2 A Port is a compound with two pairs of four Particles. Here, one parr of three points is shown to illustrate this 2D example. Ports are attached to any other Compound, most commonly anchored to a Particle where a chemical bond should exist Compound Cl is a methyl group with a Port anchored to the carbon atom. C2 is a methylene bridge already connected to a hydroxyl group. Cl and C2 are then attached using the equivalence relation described in Eq. (2) to create C3, an ethanol molecule. By default, a Bond is created between the two anchoring carbons. Adapted with permission from Fig. 2 in Sallai, J. et al. (2013) Web- and Cloud-based Software Infrastructure for Materials Design. Procedia Computer Science Elsevier...

Consider the following 2D example of a differential equation system... 

There is a direct analogy between the one dimensional (ID) and 2D examples. The repetition of a square wave leads to discrete sampling in the frequency domain. In the case of the square wave, there is a series of odd harmonics generated. In 2D, these harmonics appear as orders radiating out in the lobes of the sine function from the dimensions of the fundamental aperture or pixel. The more pixels we have in the hologram, the closer we get to the infinite case and spots generated become more like delta functions. 

Nearest neighbor (NN) interpolation and trilinear (bilinear in the present 2D example) interpolation find the reference image intensity value at position p, and update the corresponding joint histogram entry at p, while partial volume (PV) interpolation distributes the contribution of this sample over multiple histogram entries defined by its NN intensities, using the same weights as for bilinear interpolation. 

Equation 2.7 is the equation for an ellipsoid which can be used to compute the maximum and minimum actuators equivalent stiffnesses (see Figure 2.8 for a 2D example). It can be used to compute the minimum apparent stiffness in any direction in a given position or all over the work space as well as the local stiffness isotropy index or its... 

Figure 7 illustrates how, for a 2d example, a single region receives data from 8 surrounding cells, and now contains a set of particle data from other cells as well as from resident atoms. The forces can then be computed. It should be noted that the force evaluation involves some duplication for the boundary atoms. Obviously, duplication can... 

Figure 5. Diagram illustrating tlie decomposition of a 3d simulation box into separate regions, which are to be controlled by separate processors. The 2d example shows the situation for a 4 processor system where each region is subdivided into cells which are slightly larger th tn the cut-off for the nonbonded potential. (Note that for the small number of processors shown here, the linear dimensions of the whole simulation box are spanned by just two processors, so one processor will have the same neighbour to its right and left).

Figure 7. Diagram showing how a central r ion receives data from 8 sucronnding cells prior to the force calculation, when force evaluation strategy 1 is used for a 2d example. On the right of the diagram the boxes correspond to nodes, the drirk circles arc resident particles emd the lighter circles are particles tcmisferred from another node. In 3 dimensions the central box must receive data from 26 other boxes. This is handled in three sets of overlapped +/— communications with neighbouring boxes in the +x/ — x, +y/ — y and +z/ - z directions.

Figure 24.18 2D example of pore definition based on a watershed segmentation (a) binary image of porous structure ... 

It is obvious that also the symmetry of the net and of the empty space within it has a large influence on the interpenetration. Consider the simplifted 2D examples of Figure 11.4 We will make a new net from the starting net and then displace it Ax and Ay. In the (4,4) square grid it is easy to find displacement parameters that avoid collisions between nodes or the alignment of links but much more difficult for the (3 4,5) net. We might ask if there is some specific geometric property that describes this phenomenon ... 

In the last point we have treated 2D examples, which are only of interest for visualization, not yet for real chemistry. The minimum of degrees of freedom of a triatomic molecule is three. We return to Eqs.(14) in Sect. 3.2.1. We have... 

Figure 11.7. The missing row reconstruction in close-packed surfaces, illustrated in a 2D example. Left the unreconstructed, bulk-terminated plane with surface atoms two-fold coordinated. The horizontal dashed line denotes the surface plane (average position of surface atoms) with surface unit cell vector ai. Right the reconstructed surface with every second atom missing and the remaining atoms having either two-fold or three-fold coordination. The horizontal dashed line denotes the surface plane with surface unit cell vector 2ai, while the inclined one indicates a plane of close-packed atoms. The labels of surface normal vectors denote the corresponding surfaces in the 3D FCC structure.

In the oct-tree partition scheme, the set of particles or distributions is enclosed inside a cubic box, then the cubic box is partitioned into eight identical cubic boxes, and so on. This scheme is illustrated in Figure 3 for a 2D example (in this case there are four children instead of eight). 

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