Point process = random dots

In an RHM the distribution of the possible structures could be uniform, where uniformity is a systematic parameter defined in the computational domain only. This observation allows both stochastic and deterministic conditions, as well as isotropic or anisotropic features to be considered. By reaching this point, the randomness of the final structure wRl be affected by initial conditions that introduce some deterministic touches to the final reconstructed heterogeneous material. Some approaches have proposed the use of electrodes with rather deterministic structures for electrochemical processes, for example, parallel and aligned CNTs, dots of controlled diameter catalysts supported on flat substrates, etc. 

It is understood that P1(1 = 0 for n2 < n. Thus each sample function y(t) is a succession of steps of unit height and at random moments. It is uniquely determined by the time points at which the steps take place. These time points constitute a random set of dots on the time axis. Their number between any two times tl912 is distributed according to the Poisson distribution (2.6). Hence Y(t) is called Poisson process and describes the same situation as (II.2.6). 

Fig. 49a. A representative configuration of block copolymers on the lattice (For clarity a square lattice is shown, while all work refers to a simple cubic lattice). Three symmetric diblock copolymers are shown, each of chain length N = 10. The two monomeric species are labeled A-type (full dots) and B-type (open dots). The vacancies are not shown explicitly, but are assumed to reside on each lattice site left unoccupied by either of the two species of monomer. A volume fraction of < >v = 0.2 is used, since experience with blends [107] has shown that such a system behaves like a very dense melt. The energy contributions eAA, eBB and eAB are shown, b Examples of typical slithering-snake [298,299] motion monomer situated at point labelled by 5 is removed, and one of sites 1,2,3 is randomly chosen for occupation. Note that unlike Refs. [298,299] also the junction point needs to be displaced accordingly, as shown in the figure. For the reverse process, monomer at 3 is removed and the sites 4,5,6 are considered for attachment (of course, a move to site 6 is rejected due to excluded volume constraints), c Interchange of A-Block and B-Block of a diblock copolymer chain. From Fried and Binder [325],...

Figure 5.12 General pnx edure of Jf-means. The colored dots represent the centroids of clusters. To cluster the points into predefined three clusters 1. The process is initialized hy randomly selecting three objects as cluster centroids. 2. After the first iteration, points are assigned to the closest centroid, and cluster centroids are recalculated. 3. The iterative procedures of point reassignment and centroid update are repeated. 4. Centroids are stable the termination condition is met. (See color insert)...

Fig. 3.1 Gaussian Process regression in one dimension. The original function (dotted line) was sampled at ten random points (open squares). The predicted function values (solid line) and the errors (dashed line) are shown...

For the gel point prediction which is ap>phcable to the experimental data, it is necessary to introduce solvent molecules to the lattice model shown above. The introduction of solvent is the process to lay both monomer and solvent molecules randomly to lattice dots (that is, sites), which is known as the site percolation. In the lattice model, the site percolation assumes that, whenever a monomer is laid in the neighbour, the mutual bond is necessarily produced between neighbours. The ptrobabiUty of the site occupetion by the monomer is the dominant factor of the gelation In the experimental system, the probability of occupation correspends with the monomer concentration The cluster of infinite size appears above the threshold of the probability of occupjation... 

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