Ordering map

Sub-regions of biosphere are the second-order mapping units. There are two main groups of sub-regions ... 

Figure 3 shows an ordering map for this Lennard-Jones system, with the translational order t (of Eq. [2]) plotted against the bond-orientational order Qs (of Eq. [5]). It can be observed that the data, collected over a wide range of temperatures and densities, collapse onto two distinct equilibrium branches... 

Figure 3 Comparison of Lennard-Jones (solid line) and hard-sphere (circles) model ordering maps for the equilibrium fluid and the fee crystal. (Adapted from Ref. 32.)... 

The conditions on the phase diagram for which this anomalous behavior occurs has been termed water s structurally anomalous region. Inspection of the order map (Figure 4) reveals a dome of structural anomalies within the temperature-density plane, bounded by loci of maximum tetrahedral order (at low densities) and minimum translational order (at high densities) as shown in Figure 5. Also marked on Figure 5 are regions of diffusive anomalies,... 

Like the retina, the visual cortex is a layered structure, in v hich different cell groups perform different tasks. Along its tv o-dimensional surface, one finds an orderly mapped representation of the outside world. Unlike the retina, the cortex consists of multiple areas, hierarchically organized, each of which performs a partial processing task in the analysis of the scene around us. At the present time, over 30 visual cortical areas per hemisphere are recognized in monkey, and a smaller number of more expanded areas are thought to exist in humans. 

Higher order map matrices are —> power matrices derived by a map matrix M by either using the standard matrix multiplication of linear algebra or by using the —> Hadamard matrix product, which leads to matrices whose elements are defined as where k is an integer exponent. 

The above discussion is consistent with the possible existence of two well-defined classes of liquids simple and water like. The formers interact via spherically symmetric nonsoftened potentials and do not exhibit thermodynamic or dynamic anomalies. One can calculate translational and orientational order parameters t and ( ), and project equilibrium state points onto the (f, q) plane thereby generating what is termed the Errington Debenedetti (ED) order map [24]. In water like liquids, interactions are orientation dependent these liquids exhibit dynamic and thermodynamic anomalies, and their ED order map is in general two dimensional but becomes linear (or quasi linear) when the liquid exhibits structural, dynamic, or thermodynamic anomalies. 

The SOM is a model that emphasizes particular aspects of biological neural nets, especially those ordered maps found experimentally in the cortex. The SOM captures some of the fundamental processing principles of the brain. The SOM is a model of unsupervised machine learning and also an adaptive knowledge representation scheme. The SOM is a tool for statistical analysis and visualization. The SOM is both a projection method that maps high-dimensional data space into low-dimensional space, and a clustering method in which similar data samples tend to be mapped to nearby neurons. 

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