Generalised

The non-parametric model is conceptually simpler as it loosely follows the same steps as LLE. Given a new datapoint, x and a set of previously seen data points X, the low dimensional representation y is found using the following steps. Firstly, the -nearest neighbours of x in X are found. The weights, w, that best reconstruct x  

Finally, the low-dimensional embedding y is found by y = 22 w y where the neighbours indexed in Y are the same as those computed from X. As can be seen, this method reconstracts x in the low-dimensional space as a linear combination of the neighbourhood weights measured in the high-dimensional space. 

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Using the theorem that the sufficiency condition for mathematical correctness in 3D-reconstruction is fulfilled if all planes intersecting the object have to intersect the source-trajectory at least in one point [8], it is possible to generalise Feldkamp s method. Using projection data measured after changing the sotuce-trajectory from circular to spiral focus orbit it is possible to reconstruct the sample volume in a better way with the Wang algorithm [9]. 

Many ionic halides dissolve in water to give hydrated ions. The solubility of a given halide depends on several factors, and generalisations are difficult. Ionic fluorides, however, often differ from other halides in solubility. For example, calcium fluoride is insoluble but the other halides of calcium are highly soluble silver fluoride. AgF, is very soluble but the other silver halides are insoluble. 

The breaking of a strategic bond and the generation of synthesis precursors defines a synthesis reaction. In the simplest case, the reaction is already known from literature. In most cases, however, the rcaaion step obtained has to be generalised in order to find any similar and successfully performed reactions with a similar substituent pattern or with a similar rearrangement of bonds. One way of generalizing a reaction is to identify the reaction center and the reaction substructure of the reaction. This defines a reaction type. 

The derivation of th is result introduces a generalisation of the Mulliken appro,xirn ation ... 

Extended Hiickel theory Generalised valence bond model Hartree-Fock... 

One widely used valence bond theory is the generalised valence bond (GVB) method of Goddard and co-workers [Bobrowicz and Goddard 1977]. In the simple Heitler-London treatment of the hydrogen molecule the two orbitals are the non-orthogonal atomic orbitals on the two hydrogen atoms. In the GVB theory the analogous wavefunction is written ... 

Tie generalised Bom equation has been widely used to represent the electrostatic contribution... 

In the generalised Born approach the total electrostatic energy is written as a sum of tin terms, the first of which is the Coulomb interaction between the charges in vacuo ... 

The generalised Born equation has been incorporated into both molecular mechan calculations (by Still and co-workers [Still et al. 1990 Qiu et al. 1997]) and semi-empiri quantum mechanics calculations (by Cramer and Truhlar, in an ongoing series of mod called SMI, SM2, SM3, etc. [Cramer and Truhlar 1992 Chambers et al. 1996]). In th( Ireafirients, the two terms in Equation (11.61) are combined into a single expression of 1 following form ... 

A generalisation of the similarity formulae for binary data can be derived, based on the work of Tversky [Tversky 1977 Bradshaw 1997]. This takes the form ... 

The following rough generalisations may assist the student in the selection of a solvent for recrystallisation, but it must be clearly understood that numerous exceptions are known (for a more detailed discussion, see Section XI,4) ... 

Synthon A generalised fragment, usually an ion, produced by a disconnection. (Some people also use synthon for a synthetic equivalent). 

The following generalisations provide a rough guide to the selection of a suitable solvent ... 

In order to answer these questions as directly as possible we begin by looking at diffusive and displacive transformations in pure iron (once we understand how pure iron transforms we will have no problem in generalising to iron-carbon alloys). Now, as we saw in Chapter 2, iron has different crystal structures at different temperatures. Below 914°C the stable structure is b.c.c., but above 914°C it is f.c.c. If f.c.c. iron is cooled below 914°C the structure becomes thermodynamically unstable, and it tries to change back to b.c.c. This f.c.c. b.c.c. transformation usually takes place by a diffusive mechanism. But in exceptional conditions it can occur by a displacive mechanism instead. To understand how iron can transform displacively we must first look at the details of how it transforms by diffusion. 

It is sometimes stated that a given material has a good chemical resistance , or alternatively the material may be stated to be poor or excellent in this respect. Such an all-embracing statement can be little more than a rough generalisation, particularly since there are many facets to the behaviour of polymers in chemical environments. 

Many studies have been made to try to assess the propensity to hydrogen bonding of chemical structures. As a result the following broad generalisations may be made ... 

In spite of these problems, polymer melts have been sufficiently studied for a number of useful generalisations to be made. However, before discussing these it is necessary to define some terms. This is best accomplished by reference to Figure 8.2, which schematically illustrates two parallel plates of very large area A separated by a distance r with the space in between filled with a liquid. The lower plate is fixed and a shear force F applied to the top plate so that there is a shear stress (t = F/A) which causes the plate to move at a uniform velocity u in a direction parallel to the plane of the plate. 

The second generalisation relates to the effect of temperature on flow. An increase in temperature increases the rate of flow. It also increases the rate of cross-linking. It is commonly observed that at low temperatures the effect of temperature on viscosity predominates and the total flow occurring before cross-linking increases with temperature. 

Because of the wide range of possible formulations it is difficult to make generalisations about the properties of PVC compounds. This problem is illustrated in Table 12.4, which shows some differenees between three distinct types of eompound. 

Since the characteristic grouping of the resins discussed in this chapter largely disappears on cross-linking it is difficult to make simple generalisations relating structure to properties. 

But 1 am informed that theoretical chemists tend to prefer JCP. The path of the generaliser is a difficult one. 

The factors that favour successful industrial innovation have been memorably analysed by a team at the Science Policy Research Unit at Sussex University, in England (Rothwell et al. 1974). In this project (named SAPPHO) 43 pairs of attempted similar innovations one successful in each pair, one a commercial failure - were critically compared, in order to derive valid generalisations. One conclusion was The responsible individuals (i.e., technical innovator, business innovator, chief executive, and - especially - product champion) in the successful attempts are usually more senior and have greater authority than their counterparts who fail . 

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