Square Network

The ability of the SOM to learn from a random starting point is illustrated by Figure 3.8, in which a square network of one hundred nodes is trained using data points selected randomly from within a unit square. 

Fujita M et al (1994) Preparation, clathration ability, and catalysis of a 2-dimensional square network material composed of cadmium(II) and 4,4 -bipyridine. J Am Chem Soc 116 1151-1152... 

Figure 1.6. Carbon nanotube structures obtained by chemical vapor deposition synthesis, (a) SEM image of self-oriented MWNT arrays. Each tower-like structure is formed by many closely packed multiwalled nanotubes, (b) SEM top view of a hexagonal network of SWNTs (line-like structures) suspended on top of silicon posts (bright dots), (c) SEM top view of a square network of suspended SWNTs, (d) Side view of a suspended SWNT power line on silicon posts (bright) and (e) SWNTs suspended by silicon structures (bright regions). Reproduced from reference 3 with permission from American Chemical Society.

If a large number of open-ended cylindrical pore segments (like the one in Fig. 2) are interconnected such that the diameter of any pore is independent of the size of neighbor pores, a so-called randomized, or stochastic, pore network is formed. Such a set can be assembled from a cohort obeying any stipulated pore diameter distribution function. If all the pore segments are of equal length with a connectivity of 4, a square network... 

The regular square network of Fig. 5 can be modified in a number of simple ways in order to depart from the perfectly random assumption. A simple and elegant way of allowing pore lengths to vary randomly, with or without any correlation with pore diameter, can be achieved by permitting the nodes in the square network to undergo random... 

In this treatment, we will consider a square network of volume L. The area of one floe face scales as and the number of floes per chain (A/ ) is approximated by U%. Also, the Young s modulus of the floes is related to the corresponding force constant of the floes by = k A, where corresponds to the area of a floe face upon which the force is applied (A ). Considering the above in light of Equations 4 and 5, the G of the system in the strong-link regime can then be expressed as ... 

Figure 6.9 pictures the example of the triangular-square network taken from [6.38]. Some 4-coordinated sites are seen comprising inner boundaries of this LRC. It is easy to notice that there are two 5-coordinated atoms in the first coordination sphere of each 5-coordinated atom in LRC. This circumstance follows from the fact that, in P-polyhedra, we have an even number of squares leading to the formation of the MRO, which manifests itself in the formation of chains of 5-coordinated atoms. Collins and Kawamura studied the thermodynamic properties of triangular-square lattices. Kawamura established the existence of the first-order phase transition connected with the transformation of the crystalline structure into a topologically disordered one. 

Reconsider the infinitely long cylinder of Section 3.3 (Fig. 3.14). Subdivide the cross section of the cylinder by straight lines separated both vertically and horizontally Al distance apart, forming the so-called square network [Fig. 4.9(a)]. Significant differences in temperature gradients dT/dx and BT/dy) or geometry may dictate the use of a rectangular network instead (i.e., the use of Ax Ay). 

Fignre 26 (a) STM image of the square network formed by iron and 4,l, 4, l"-terphenyl-l,4"-dicarboxylic acid (b) STM image... 

Fig. 4 Reduced storage modulus [G ( )] plotted on double logarithmic scales versus the reduced frequency (uto. Shown are results for a linear Rouse chain of 250 beads dashed line), for a topologically-square network (250 x 250), solid line with stars, and for a topologically cubic network (150 x 150 x 150), solid line. For simplicity, the characteristic time To = f/fC was chosen to be the same for aU three systems...

Fig. 5 Sketch of a square network built from beads each with friction constant which are connected by elastic springs each with elasticity constant K...

The dynamics of a topologically square network can be treated along the lines used for the cubic network above. The Langevin equations of motion for a square network are similar to those of the cubic network ... 

Analysis of the dynamical viscoelastic quantities shows that the relaxation spectrum H r) of the two-dimensional network goes as H(t) 1/r [65,68-70]. Hence 2-D networks do indeed show dynamical behavior intermediate between that of linear chains and that of 3-D networks. Moreover, in a fractal picture, square networks may be viewed as being fractals and as having a spectral dimension of 2. Now H(r) 1/t leads to an -behavior for the storage modulus G (a>), see Eig. 4, and to G(f) 1/t. 

Fig. 9 Schematic representation of a regular network built from complex cells. For clarity s sake a two-dimensional square network is shown. Each cell has some arbitrary internal GGS structm-e (see magnifying glass), which is identical in all cells...

In real polymer networks the mobility of the cross-Hnk points is intermediate between immobile and free, leading to cooperative modes which involve simultaneous, correlated motions of several CBDW. To model this situation one can use the approach based on regular networks built from complex cells, which was discussed in Sect. 5.3. More specifically, a topologically-square network formed from CBDW (see Fig. 32) can model a CBDW network [244]. In the framework of Sect. 5.3 the elementary cell consists of a cross-link point and two CBDW. 

The profile of potential changes inside the contour is not known. To determine the potential changes, a square network of side h is determined inside contour C. The problem of determination of the profile of potential changes inside contour C is reduced to determination of potential values in each node of the network. The potential distribution determined in this way is discrete in... 

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