Binary Site Variables

In the case of binary variables a G 0,1, we can let density p t) at time t represent the probability of finding value a = 1 at any site at time t. A simple counting of all possible configurations in an arbitrary neighborhood Af gives us the general mean-field expression for the time-dependence of p  

---

Consider an isotropic version of the elementary peripheral PCA introduced above, this time also with binary site variables time steps according to a set of rules determined by three conditional probabilities, po, p and p2 -... 

Suppose we wanted to model an Ising model in the most straightforward possible manner using a CA approach. For argument s sake, let s consider modeling a basic Ising model without any external fields. Using binary site variables <7 = (1 — 5) to make any possible connection to conventional CA transparent, our task is to come up with a CA rule ,... 

Boolean Network with connectivity k- or N, )-net - generalizes the basic binary k = 2) CA model by evolving each site variable Xi 0,1 of according to a randomly selected Boolean function of k inputs ... 

In Franchini et al. (2004) we introduced four Lick/IDS index-index diagrams, i.e. NaD vs Ca4227, NaD vs Mg2, NaD vs Mgb, and NaD vs CaMg, to identify SSA and a-enhanced stars irrespectively of their Teg, log g and [Fe/H]. By applying this method to the 84 normal (i.e. excluding binaries and variable) stars from the S4N web site with [Fe/H] determined by AP04, it results that 8 stars are... 

Working out the historical quasi-chemical approximation in the present language for the two-dimensional Ising model of a binary solution will give perspective on the developments of this chapter. The model system is depicted in Fig. 7.15. Each site of the lattice possesses a binary occupancy variable, 5-, = —1,1 for the /th site. This will be interpreted so that Sj = 1 indicates occupancy of the /th site by one species, e.g. W (water), and = — 1 then indicates occupancy of that site by the other species, say O (oil). We write... 

Each warehouse site has a fixed capital cost independent of the quantity stored, and a variable cost proportional to the quantity shipped. Thus the total cost of opening and operating a warehouse is a nonlinear function of the quantity stored. Through the use of binary integer variables, the warehouse location-distribution problem can be formulated as an integer program. 

Let the binary integer variable 5 denote the decision to select site i. In other words,... 

We define a fee lattice and affect at each site n, a spin or an occupation variable <7 which takes the value +1 or —1 depending on whether site n is occupied by a A or B atom. Within the generalized perturbation method , it has been shown that substitutional binary alloys AcBi-c may be described within a Ising model with effective pair interactions with concentration dependence. Thus, the energy of a configuration c = (<7i,<72,- ) among the 2 accessible configurations for one system can be written... 

Multithreshold rules are natural extensions of the (binary-valued) generalized threshold rule defined in equation 5.121. Using fc-state variables a G 0,1,..., A - 1 and using the j entry of the A threshold vector to define the j threshold at that site (= bij), the generalized multithreshold rule is defined as follows ... 

Finally, it may be noted that, although variable valency in binary oxides is important, it is not a sufficient requirement, as can be concluded from the fact that even in the systems Bi—Mo—O and U—Sb—O not every crystal phase is active and selective [294], What matters is the configuration of the ions at the active site. Apparently, the character of the typical Me—O bands is a function of the situation of oxygen in the lattice. 

The hyperstructure can be mathematically represented by the DARC/PELCO matrix, there being n rows, the number of experimental compounds, and Ns columns, the topochromatic sites of the hyperstructure. Each ith row of the DARC/PELCO matrix is called the topochromatic vector I, and directly accounts for the overall topology of the ith molecule. Therefore, the DARC/PELCO descriptors of the /th molecule are the elements of its topochromatic vector, being binary variables I,-, where I,j. = 1 if the sth topochromatic site of the hyperstructure corresponds to an atom of the i th molecule and I,., = 0 otherwise. 

Fujita-Ban analysis is a modified Free-Wilson analysis where the activity contribution of each substituent is relative to the activity of a - reference compound [Fujita and Ban, 1971]. Any compound can be chosen as the reference, but usually the H-sub-stituted compoimd (all R = H) is adopted. The Free-Wilson matrix in the Fujita-Ban analysis does not contain the descriptors corresponding to the substituents of the reference compound, i.e. the number of indicator variables is diminished by the number of sites S with respect to the corresponding original Free-Wilson approach. Moreover the row vector corresponding to the reference compound is characterized by all the descriptor binary values equal to zero. 

The MTD-MC method is a modified version of the MTD method, accounting for the existence of several low-energy conformations of molecules used to derive the hypermolecule by overlap. Each molecule is described by a vector of binary variables Ii( )s equal to one if the 5th vertex of the hypermolecule is occupied by the ith molecule in the A th conformation. If more than one low-energy conformation is allowed for a molecule, the conformations considered will be the one which best fits the binding site cavity, i.e. the one with the lowest MTD value ... 

Moreover, molecular descriptors different from Free-Wilson descriptors were calculated by transformation of the Free-Wilson matrix through Fourier analysis [Flolik and Halamek, 2002], In this case, Fourier analysis is used to change site- and substituent-oriented binary variables into a few real numbers [Holik and Halamek, 2002]. 

Transformations of a set of molecular descriptors are often performed when there is the need of a —> variable reduction or the need to modify binary vectors, such as site and substituent-oriented variables, into real-valued variable vectors. The milestone of these techniques is the —> Principal Component Analysis (PCA), but also —> Fourier analysis and —> Wavelet analysis are often used, especially for spectra descriptors compression. 

Fourier analysis was, for instance, applied to change site and substituent-oriented binary variables in the —> Free-Wilson analysis, into a few latently dependent real coefficients [Holik and Halamek, 2002]. 

Consider a binary alloy of fixed composition in which there are two types of crystallographic sites (geologists may prefer to consider the K-feldspars, which have variable ordering of the A1 atoms in tetrahedral and octahedral sites). The two types of atoms may arrange themselves so that each occupies only one type of site (complete order) they may be distributed at random (complete disorder) or there may be a degree of disordering specified by a parameter 4>. 

This model suggests selection of F new facilities using X, binary variables. X, will be equal to one if the facility on site i is selected. Yjj variables are introduced to... 

The lattice model for ternary amphiphilic systems can now be constructed [33,36]. A binary variable 5, = 1 indicates whether a lattice site i is occupied by a polar or hydrophobic species, and a second variable /, = 1,0 indicates whether that species is part of a surfactant molecule or not. The Hamiltonian for these variables is... 

---