Localised model

The chain model corresponding to the closed circuit milling system with localised models of all its elements is presented in Fig. 2. It can be constructed in different ways, but first let us examine the model shown in Fig. 2a. Every column of the set of cells corresponds to an element of the circuit a mill a classifier or an absorber. The cells within columns correspond to fraction numbers with the total number of fractions equal to r. The fraction size decreases with increasing fraction number. The state of the system is characterised by the set of probabilities f, to occupy the cells, every of which can be interpreted as the relative mass content of particles in the cell ij. In particular, the set fj, I = 1,2,.. .r, corresponds to the particle size distribution in the hold-up of the /111 element of the circuit. 

Fowler, P. F. 1992 Localised models and leapfrog structures of fullerenes. J. chem. Soc. Faraday Trans. 88, 145-146. 

Fig. 4. CRTA loss mass curve as a function of temperature for bochmitcs (a) and localisation model of water on crystallites for Bl (b).

It should be mentioned that, in addition to the localised model (when the magnetic entities are well localised in their spatial positions), there exists an itinerant (or band) model which considers that each magnetic carrier is itinerant through the solid. Whereas the localised model is applicable predominantly to insulators and rare-earth metals, the band model is more relevant to 3d metals as an effect of the great diffuseness of the 3d orbitals. 

As mentioned above, the localised model used to describe the quantum states of light atoms in a heavy atom matrix is the Simple Harmonic Oscillator [7]. 

A count of Kekule structures for all 1812 distinct fullerene isomers of Cjo shows that 20 isomers surpass the count of 12500 for icosahedral C q, and demonstrates the lack of correlation between molecular-orbital indices of stability and raw Kekule counts for fullerenes. Analysis of Kekule structures in terms of benzenoid, cyclopentenoid and cyclopentadienoid rings reveals the source of the stability of icosahedral C q in a localised model to be the fact that uniquely amongst the 1812 structural isomers it has a Fries Kekule structure where all hexagons contain three double bonds and all pentagons none. 

M.o. theory and the transition state treatment In 1942 Wheland proposed a simple model for the transition state of electrophilic substitution in which a pair of electrons is localised at the site of substitution, and the carbon atom at that site has changed from the sp to the sp state of hybridisation. Such a structure, originally proposed as a model for the transition state is now known to describe the (T-complexes which are intermediates in electrophilic substitutions... 

Dewar s treatment of transition state structure, using reactivity numbers, has the logical defect that in the intermediate kinds of transition states for which it provides evidence the electron localisation is only partial. However, in obtaining the values of the reactivity numbers (which are approximate localization energies), the process of localization is considered to be complete thus, values of parameters which strictly are relevant only to the Wheland type of transition state are incorporated into a different model. ... 

The model adopted by Ri and Eyring is not now acceptable, but some of the more recent treatments of electrostatic effects are quite close to their method in principle. In dealing with polar substituents some authors have concentrated on the interaction of the substituent with the electrophile whilst others have considered the interaction of the substituent with the charge on the ring in the transition state. An example of the latter method was mentioned above ( 7.2.1), and both will be encountered later ( 9.1.2). They are really attempts to explain the nature of the inductive effect, and an important question which they raise is that of the relative importance of localisation and electrostatic phenomena in determining orientation and state of activation in electrophilic substitutions. 

M.o. theory has had limited success in dealing with electrophilic substitution in the azoles. The performances of 7r-electron densities as indices of reactivity depends very markedly on the assumptions made in calculating them. - Localisation energies have been calculated for pyrazole and pyrazolium, and also an attempt has been made to take into account the electrostatic energy involved in bringing the electrophile up to the point of attack the model predicts correctly the orientation of nitration in pyrazolium. ... 

The activity of antioxidants in food [ 1 ] emulsions and in some biological systems [2] is depends on a multitude of factors including the localisation of the antioxidant in the different phases of the system. The aim of this study is determining antioxidant distributions in model food emulsions. For the purpose, we measured electrochemically the rate constant of hexadecylbenzenediazonium tetrafluorborate (16-ArN,BF ) with the antioxidant, and applied the pseudophase kinetic model to interpret the results. 

So, despite the very small diameter of the MWCNT with respeet to the de Broglie wavelengths of the charge carriers, the cylindrical structure of the honeycomb lattice gives rise to a 2D electron gas for both weak localisation and UCF effects. Indeed, both the amplitude and the temperature dependence of the conductance fluctuations were found to be consistent with the universal conductance fluctuations models for mesoscopic 2D systems applied to the particular cylindrical structure of MWCNTs [10]. 

The beauty of finite-element modelling is that it is very flexible. The system of interest may be continuous, as in a fluid, or it may comprise separate, discrete components, such as the pieces of metal in this example. The basic principle of finite-element modelling, to simulate the operation of a system by deriving equations only on a local scale, mimics the physical reality by which interactions within most systems are the result of a large number of localised interactions between adjacent elements. These interactions are often bi-directional, in that the behaviour of each element is also affected by the system of which it forms a part. The finite-element method is particularly powerful because with the appropriate choice of elements it is easy to accurately model complex interactions in very large systems because the physical behaviour of each element has a simple mathematical description. 

The MaxEnt valence density for L-alanine has been calculated targeting the model structure factor phases as well as the amplitudes (the space group of the structure is acentric, Phlih). The core density has been kept fixed to a superposition of atomic core densities for those runs which used a NUP distribution m(x), the latter was computed from a superposition of atomic valence-shell monopoles. Both core and valence monopole functions are those of Clementi [47], localised by Stewart [48] a discussion of the core/valence partitioning of the density, and details about this kind of calculation, may be found elsewhere [49], The dynamic range of the L-alanine model... 

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