Square lattice representation

Figure 5. Square lattice representation of two-dimensional space for percolation theory. Black squares represent protein pores white squares represent hydrophobic pol5Tner backbone. Lattices with two different porosities (percentage of sites designated as pores) are shown (a) 20% porosity (b) 60% porosity.

Fig. 2.20. Allowed planar molecular orientations in the representation of double (upper row) and ordinary (lower row) azimuthal angles two rigidly fixed molecular orientations (n = 2) for a square lattice (a) and for a triangular lattice (b), and four discrete orientations (n = 4) on a square lattice (c).

Table I. Configurations, Multiplicities, and Energies Appropriate to the Representation of a Square Lattice...

Note that the classic SLSP model for a square lattice ABCO would provide a percolation trajectory connecting opposite sites OA and CB (see Fig. 32). On the other hand, for visualization of dynamic percolation we shall consider an effective three-dimensional static representation of a percolation trajectory connecting ribs OA and ED spaced distance Lh (see Fig. 33). Such a consideration allows us to return to the lattice OEDA, with the initial dimension <7 = 2, which is non-square and characterized by the two dimensionless sizes Lh 11 and L/l. The new lattice size of the system can be determined from the rectangular triangle OEQ by... 

In lattice models, the location of each element on the lattice can be stored as a vector of coordinates [(X, F,), (X2, Y2), (X3, Y3),..., (Xn, F )], where (X Y,) are the coordinates of element i on a two-dimensional lattice (a three-dimensional lattice will require three coordinates for each element). Since lattices enforce a fixed geometry on the conformations they contain, conformations can be encoded more efficiently by direction vectors leading from one atom (or element) to the next. For example in a two-dimensional square lattice, where every point has four neighbors, a conformation can be encoded simply by a set of numbers (Lu L2, L3,..., L ), where L, g 1, 2,3,4 represents movement to the next point by going up, down, left, or right. Most applications of GAs to protein structure prediction utilize one of these representations. 

Fig. 11 Schematic representation of all the phases considered. Dark a, white b, gray e. (a) Lamellar phase, (b) Coaxed cylinder phase, (c) Lamella-cylinder phase, (d) Lamella-sphere phase, (e) Cylinder-ring phase, (f) Cylindrical domains in a square lattice structure, (g) Spherical domains in the CsCI type structure, (h) Lamella-cylinder-II. (i) Lamella-sphere-II. (j) Cylinder-sphere. (k) Concentric spherical domain in the bcc structure. Reprinted with permission from Zheng et el. [104]. Copyright 1995 American Chemical Society...

Figure 2. Two-dimensional square lattice (a) in the recursion scheme representation (fc). After defining o) = /o> 4o)i I f s)...

The square lattice is only one possible representation of space (Figure 4.20). Every lattice has an associated coordination number, z, which describes the number of bonds emanating from each site for example, the square lattice in Figure 4.19 has a coordination number of 4. In addition there are lattices that have no obvious dimensionality, like the Bethe lattice (Figure 4.20). The Bethe lattice is a homogeneous tree-like structure, in which the number of sites... 

Figure 13.19 Schematic representation of eleven possible tiiphase morphologies. Note black, white, and gray shadings (a) lamellar, (b) coaxial cylinder, (c) lamella-cylinder, (d) lamella-sphere, (e) cylinder-ring, (f) cylindrical domains In a square lattice structure, g) spherical domains in a CsCI-salt lattice structure, (h) lamella-cylinder-ll, (/) lamella-sphere-ll, (y) cylinder-sphere, (k) concentric spherical domains in the bcc stmcture.

This lattice representation provides a powerfull tool for numerical simulations using e.g. Monte Carlo methods. While somewhat out of the scope of this paper, these numerical simulations are interesting to interpret because they evidence the type of liquid flow properties it is possible to account for by means of the percolation concept. Figure 10 illustrates a typical result of these simulations. It represents for different cases, the distribution of irrigated zones in a cross section of the packing. The black squares show the intersections of the liquid flow paths with the cross section. 

The square lattice is only one of a myriad possible representations of space. Every lattice has an associated coordination number, z, which describes the number of bonds emanating from each site for example, the square lattice in Figure 4 has a coordination number of 4. In addition there are lattices which have no obvious dimensionality, like the Bethe lattice. The Bethe lattice is a homogeneous tree structure, the number of sites on the surface of the tree increases without bound as the size of the tree grows. The coordination number of the Bethe lattice can be from 2 to oo. There are also lattice representations that are irregular each site does not have the same characteristic shape. Voronoi lattices, both two- and three-dimensional, are constructed by placing points randomly in space and tessellating around these points to construct an internal surface [39, 40]. Some relevant properties of each lattice-dimensionality D, coordination number z, critical probability p for site and bond percolation--are listed in Table 1. 

Fig. 9) [45,46]. Although the schematic representations of motif 2 and 5 in Fig. 9 may suggest the helices of a four-armed junction cross perpendicularly, they actually possess an angle of about 60°. AFM analyses showed that the torsion angles between helices are relatively constant throughout the entire lattice (Fig. 9) [45,46]. By incorporating a protein (RuvA), a square-planar configuration (motif 6 in Fig. 9) has been built [47].

Figure 31 Representation of the one-dimensional cyanide-bridged polymer of stoichiometry [Cu -(tetrenH2)]4[W (CN)8]4 10H2O (pH = 7). The structure is composed of W2CU2 squares joined through CN bridges. The lattice H2O molecules are omitted for clarity.

In this paper, we examine the electron correlation of one-dimensional and quasi-one-dimensional Hubbard models with two sets of approximate iV-representability conditions. While recent RDM calculations have examined linear [20] as well as 4 x 4 and 6x6 Hubbard lattices [2, 57], there has not been an exploration of ROMs on quasi-one-dimensional Hubbard lattices with a comparison to the one-dimensional Hubbard lattices. How does the electron correlation change as we move from a one-dimensional to a quasi-one-dimensional Hubbard model How are these changes in correlation reflected in the required A -repre-sentability conditions on the 2-RDM One- and two-par-ticle correlation functions are used to compare the electronic structure of the half-filled states of the 1 x 10 and 2x10 lattices with periodic boundary conditions. The degree of correlation captured by approximate A -repre-sentability conditions is probed by examining the one-particle occupations around the Fermi surfaces of both lattices and measuring the entanglement with a size-extensive correlation metric, the Frobenius norm squared of the cumulant part of the 2-RDM [23]. 

Fig. 25 A representation of the in-plane X-ray scattering measured at —0.13 V in solution containing 0.1 M HCIO4 - -10 M Cu + -I-10 M KBr. The solid circles correspond to the measured c(2 X 2) reflections and the squares to the location of bulk Pt CTRs and Pt Bragg reflections. The lower figures show scans through the indicated reciprocal lattice points at (a) (1/2, 1/2, 0.1) (b) (0,1,0.1) and (c) (1/2, 3/2, 0.1). In each case, the solid lines are fits of a Lorentzian lineshape to the data.

Quartercore symmetry was used flnoughout these calculations. For the square pitch lattices, the computer representation explicitly modeled each fuel rod, aluminum rod, intermediate band (Core IV), center grid plate (Core V), side sheet, moderator, base plate, and core tank in the X-Y direction. The fuel support structure above the water was not represented in the computer modeL The only difference in the modeling of the triangular pitch lattices (Cores I, II, and III) was that the fuel regions were homogenized except for the external mns in each fuel assembly. The outward half of each of these pins was explicitly modeled (see Table I). 

To construct the lattice in space, at each site, we place either a molecule of solvent or a sequence of a polymer chain, in the knowledge that those sequenees are connected to one another. Figure 3.4 gives a 2D representation of the solution. Each grid square represents a site in the lattiee which contains either a molecule of solvent A (in which case the square is white) or a sequence of the solute B (in which case the square is black). 

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