Fuzzy Picture

Another way to represent the probability is by a density plot. Suppose we could see exactly where the electron was at a given time and that we marked the spot. If we looked again a little later, the electron would be in a different place—let us mark this spot too. Eventually, if we marked enough spots, we would end up with a fuzzy picture like those shown for the Is and 2s orbitals. Now the density of the dots is an indication of the probability of finding an electron in a given space—the more densely packed the dots (that is, the darker the area), the greater the probability of finding the electron in this area. This is rather like some maps where different altitudes are indicated by different colours. 

One should note, though, that with a well-chosen potential, for instance the field of bare nuclei, the Furry picture is likely a good approximation to the fuzzy picture. 

Charon discovered (James Christy) Charon is discovered as an apparent bulge on a fuzzy picture of Pluto. Its mass is about 12 percent that of Pluto. 

Abstraction is the most useful technique a developer can apply being able to state the important aspects of a problem uncluttered by less-important detail.6 It s equally important to be able to trace how the more-detailed picture relates to the abstraction. We ve already seen some of the main abstraction techniques in Catalysis—the ability to treat a complex system as one object and to treat complex interactions as one action and yet state the outcome precisely. This approach contrasts with more-traditional design techniques in which abstract also tends to mean fuzzy, so you can t see whether a statement is right or wrong because it might have many different interpretations. 

FIGURE 6.18 Cluster validity V(k), see Equation 6.13, for the algorithms fc-means, fuzzy c-means, and model-based clustering with varying number of clusters. The left picture is the result for the example used in Figure 6.8 (three spherical clusters), the right picture results from the analysis of the data from Figure 6.9 (two elliptical clusters and one spherical cluster). 

In structure determination from X-ray diffraction data, it sometimes happens that, on the Fourier maps, parts of the coming out structure are unclear. Fuzzy electron density maps may present problems in determining even the approximate positions of the respective fragments of the structure being analyzed. For example, the layered structure of the inclusion (intercalation) compound formed by Ni(NCS)2 (4-methylpyridine)4 (host) and methylcellosolve (guest) [1], The guest molecules are (Fig. 11.1) located on twofold crystal axes of unit cell symmetry and are orientationally disordered as shown in the picture. 

The quotations give the flavor of the fuzzy word-pictures typical of evolutionary biology. The lack of quantitative details—a calculation or informed estimation based on a proposed intermediate structure of how much any particular change would have improved the active swimming ability of the organism—makes such a story utterly useless for understanding how a cilium truly might have evolved. 

You ll understand the ride of this chapter once you ve finished it. By the way, pi is pronounced pie and represents the Greek letter n. In the meantime, I want you to recall that we picture electrons in atoms as residing in orbitals—those fuzzy things that can be spherical, dumbbell-shaped, or even ring-shaped. 

In discussing and drawing pictures of orbitals, there s potential for confusion in that you might think that if you were small enough to look at an atom directly, you would actually see these fuzzy shapes. Not so. Orbitals represent possible locations of electrons, not a physical thing. This idea is similar to drawing the orbits of the planets around the Sun. We draw those lines to represent the paths of planets, but those lines don t really exist in space. In the same way, atomic orbitals don t exist as physical things but rather are mathematical representations of where we are likely to find electrons. 

Here the term orbit is being used metaphorically, not literally. Though the common picture is to show electrons orbiting, satellite-like, around the nucleus, the space occupied by the electrons cannot be clearly delineated. The best we can do is to describe a sort of fuzzy region of proba-... 

Pictures of high resolution appear crisp, whereas pictures of low resolution appear fuzzy. A decrease of resolution is accompanied by an increase of fuzziness. Consequently, similarity measures based on the minimum level of resolution required to distinguish objects can be formulated in terms of the maximum level of fuzziness at which the objects are distinguishable. Similarity can be regarded as fuzzy equivalence. This principle provides an alternative mathematical basis for using the methods of topological resolution [262] in similarity analysis the theory of fuzzy sets [382-385]. 

This picture of the hemoglobin molecule is certainly, as Perutz says, still a very general and fuzzy one, which omits all the finer details of molecular configuration. However, it is the most specific picture that has yet emerged for any protein. 

If all species consisted of biparental systems with unproblematlcal cohesion, only the boundary cases would spoil the picture, or better stated, would nake the boundary somewhat fuzzy. But there are much more severe cases i... 

In addition to conventional sequence motifs (Prosite, BLOCKS, PRINTS, etc.), the compilation of structural motifs indicative of specific functions from known structures has been proposed [268]. This should improve even the results obtained with multiple (one-dimensional sequence) patterns exploited in the BLOCKS and PRINTS databases. Recently, the use of models to define approximate structural motifs (sometimes called fuzzy functional forms, FFFs [269]) has been put forward to construct a library of such motifs enhancing the range of applicability of motif searches at the price of reduced sensitivity and specificity. Such approaches are supported by the fact that, often, active sites of proteins necessary for specific functions are much more conserved than the overall protein structure (e.g. bacterial and eukaryotic serine proteases), such that an inexact model could have a partly accurately conserved part responsible for function. As the structural genomics projects produce a more and more comprehensive picture of the structure space with representatives for all major protein folds and with the improved homology search methods linking the related sequences and structures to such representatives, comprehensive libraries of highly discriminative structural motifs are envisionable. 

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