Shape space

A variation to the top-feed dmm filter is the dual dmm filter which uses two dmms of the same size in contact with each other and rotating in opposite directions. The feed enters into the V-shaped space formed on top of the two dmms and the cake that starts forming initially contains coarser particles due to the settling which takes place in the feed zone. This is beneficial to the clarity of the filtrate because the coarser particles act as a precoat. Erom the point of view, however, of the final moisture content of the cake the stratification of the soHds in the cake may lead to somewhat wetter cakes. Utilization of the area of the dmms is poor since there are dead spaces under the two dmms. The primary appHcation of the dual dmm filters is in dewatering coarse mineral or coal suspensions at feed concentrations greater than 200 kg/m. ... 

For higher pressures, rotary compressors of the sliding vane type will give delivery pressures up to 1 MN/m2. In a compressor of this type, as illustrated in Figure 8.35, the compression ratio is achieved by eccentric mounting of the rotor which is slotted to take sliding vanes which sub-divide the crescent-shaped space between the rotor and... 

In the p vs. e diagram every structure type is generally characterized by its own individually shaped space-filling curve. The space-filling curves, however, of all binary structures belonging to one homeotect structure set coincide with one curve (see 3.9.2). 

A space similar to shape space is catalytic task space [4], Points in this space represent different reactions that a molecule can catalyze. This is more of an intuitive concept for developing qualitative arguments than a well-developed quantitative space, but it may eventually be made rigorous enough for theoretical molecular diversity applications. 

Fig. 2. Shape space covering by a small spherical region of sequence space. In order to find (at least) one sequence for every common structure, it is only necessary to search a relatively small sphere around an arbitrarily chosen reference point in sequence space. For example, the covering radius for RNA molecules of chain length n= 100 was determined to be rc = 15 the covering sphere thus contains about 4 x 1024 sequences compared to 1.6 10 sequences in the entire sequence space.

Shape space covering by small connected regions in sequence space shows that only a relatively small fraction of all sequences has to be searched in order to find any of the common structure. 

Compared to the diameter of sequence space, the correlation length of structure space for RNA folding is relatively small (Fontana et al., 1993). A small correlation length implies that a small sphere around any sequence can sample all possible secondary structures. The ability to sample many structures from any sequence point is a property of the fitness landscape referred to as shape space covering. Equation (32) predicts shape space covering of structures when the connectivity is greater than Ac (Reidys et al., 1997). The radius of the covering sphere rmv is defined as... 

The equation expresses that the space of all genotypes, the sequence space I, is a discrete space with the Hamming distance as metric. It is mapped onto a discrete space of structures called shape space with the structure distance as metric (We use I rather than 4 in order to indicate different numbering schemes used for sequences and structures). The evolutionarily relevant quantity, the fitness value fk as shown in Fig. 2.3, is derived from the phenotype Sk through evaluation, which can be understood as another mapping, a map from shape space into the positive real numbers including zero, fk = f(Sk). Both maps need not be invertible in the sense that more than one phenotype may have the same fitness value, and more than one sequence may lead to the same structure. We shall study here neutrality induced by the first map, (// in Eq. (6). 

Frank, R., Hoffmann, S., Kiess, M., et al. (1996) Combinatorial synthesis on membrane supports by the SPOT technique Imaging peptide sequence and shape space. In Combinatorial Peptide and Nonpeptide Libraries A Handbook (Jung, G., ed.), VCH, Weinheim, Germany, pp. 363-386. 

Edgington, S.M. (1993). Shape space biopharmaceutical discovery entering anew evolutionary stage Bio/Technology 11, 285 289. 

Consider two different subsets of the same space D, or subsets of two dynamic shape spaces D and D of two different stoichiometric families of molecules. One may compare those domains of the two subsets that belong to the same shape group H 2. Since within these domains the nuclear configuration is not fully specified, that is, there exists some configurational freedom within these domains, the above approach provides a description of the dynamic similarity of molecular shapes. We shall return to the problems of dynamic shape similarity in Chapter 6. 

Each (a,b)-map can be regarded as a subset of the dynamic shape space D. Such a subset contains all points of D where the internal coordinates corresponding to the nuclear arrangement are fixed. 

If the entire range of curvature parameter b is considered, then a list of the finite number of distinct shape matrices and those curvature values bj where a change of the shape matrix occurs, gives a detailed, numerical shape characterization of the MIDCO surface G(a). In the most general case of variations in the two parameters a and b, as well as in the nuclear configuration K, one can study the dynamic shape space invariance domains, the (a,b)-maps, and various projections of the invariance domains of shape matrices, following the principles [158] applied for the shape group invariance domains of the dynamic shape space D. 

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