Linear patch

The simple intuition behind LLE is that each input point, x , and its fe-nearest neighbours are locally linear, that is, they lie on or near a linear patch of the manifold. By making this assumption the local geometry can be characterised by linear reconstruction weights, W, that reconstruct each point x from its A -nearest neighbours. The weights are measured by the squared distance cost function... 

Qualitative examples abound. Perfect crystals of sodium carbonate, sulfate, or phosphate may be kept for years without efflorescing, although if scratched, they begin to do so immediately. Too strongly heated or burned lime or plaster of Paris takes up the first traces of water only with difficulty. Reactions of this type tend to be autocat-alytic. The initial rate is slow, due to the absence of the necessary linear interface, but the rate accelerates as more and more product is formed. See Refs. 147-153 for other examples. Ruckenstein [154] has discussed a kinetic model based on nucleation theory. There is certainly evidence that patches of product may be present, as in the oxidation of Mo(lOO) surfaces [155], and that surface defects are important [156]. There may be catalysis thus reaction VII-27 is catalyzed by water vapor [157]. A topotactic reaction is one where the product or products retain the external crystalline shape of the reactant crystal [158]. More often, however, there is a complicated morphology with pitting, cracking, and pore formation, as with calcium carbonate [159]. 

Fig. 21.5. Levamisole-activated single-channel currents activated by 30 pM levamisole in a cell-attached patch and current voltage relationship. The rectangular current pulses were recorded at different patch potentials to determine the relationship between channel current and potential. The slope was linear with a conductance of 34 pS.

The angular diameter of the observed patches is about 1°. Their (real) linear diameter can be estimated as ct, where c is the speed of light and t = 300 000 years is the time when the Universe became transparent. An angular diameter of 1° measured 14 billion years later means that the light rays remained parallel over the whole path and hence that the Universe is globally Euclidean. 

Other sets of hair cells are formed in specialized parts of the inner ear.964 The three semicircular canals detect angular acceleration in three directions, while the sac-like utricle and saccule detect linear acceleration including gravitational attraction. These two organs each contain a patch of hair cells whose tips project into a gelatinous layer, which is overlain by a field of small crystals of calcium carbonate. These little stones (otoliths) provide an inertial mass, which resists movement causing the hair cell tips to bend and activate mechanoreceptors to send information about balance and orientation to the brain. 

For reactions of the type A + B = AB (or a+P = y), the situation is different. If one has a linear reaction geometry and the y product forms at different times and locations on the A/B interface, the patches of y eventually merge by fast lateral (interface) transport. Eventually, a full y layer is formed between a and / . At first, this layer has a non-uniform thickness (Fig. 6-4). In Chapter 11 we will show, however, that the uneven a/y and y/p interfaces are morphologically stable and become smooth during further growth. This leads to constant boundary conditions for the y formation after some time of reaction and eventually results in a parabolic rate law, as will be discussed later. 

Local space average color, combined white patch retinex and gray world assumption, linear output. 

Figure 4. Open circles, angular dependence of the second-order scattered light intensity (hyper-Rayleigh scattering from 1064 to 532 nm) for a suspension of purple membrane patches of bacteriorhodopsin before any solubilization has taken place. Solid curve best fit to the data by combining the nonlinear hyperpolarizability of retinal with the linear refractive index of the upo-protein.

After eqn.(3.14) turned out to be obeyed by many systems in practice, a model was developed that could provide a physical picture. This so-called diachoric model [306] explains the fact that the two components of the mixed phase behave independently by demixing on a microscopic scale. Hence, the stationary phase is assumed to consist of little patches or droplets of either pure A or pure B. Obviously, such a model does explain obeyance of eqn.(3.14), while it also gives a handle to explain deviations from linearity in terms of complete mixing of the two phases. 

Maxted4 and others have found that, judged on any one of a number of catalytic reactions, the activity of platinum black decreases linearly with the amount of a poison added, up to a critical concentration of the poison, when the activity is usually diminished to about one-quarter of that of the clean catalyst if further quantities of poison are now added, the activity decreases much more slowly. These observations are quantitative, and are considered to throw some doubt on whether Vavon and Husson s qualitative work really proves the presence of as many as three different kinds of catalysing patches on platinum but they do not deal with precisely the same reactions. A more exact reexamination of these reactions seems desirable but it is difficult to see how there can be less than two or three different kinds of surface, while there may be more. 

Linear, one-dimensional interfaces. Perhaps the most interesting theory of promoter action is that the linear boundary on the surface between two surfaces of different composition is the seat of the catalysis, i.e. the active patch. There is abundant evidence that such linear interfaces often possess unusual reactive powers. They are undoubtedly... 

Schwab and Ketsch4 have developed a theory of the kinetics of contact catalysis based on the assumption that the active patches are located at such interfaces, using the term adlineation for adsorption there but though their assumption seems a very probable one, at any rate for many cases of promoter action, it is scarcely possible to test it by comparison of rates of reaction with a theoretically derived equation there are too many adjustable constants, and other factors. One of these is the rate of surface diffusion to the active linear interfaces, or other active patches, which as many writers have pointed out, may be very considerable.5... 

Another modification easily assessed by immersion microcalorimetry is the change in hydrophobicity of a surface, e.g. by oxidizing a graphitized carbon surface. The energy of immersion in water was shown to increase almost linearly with decrease in hydrophobicity (Young et al., 1954) and the energies of immersion of hydrophobic and hydrophilic patches were estimated to be 31 and 730 J m-2, respectively (Healy et ah, 1955). 

Fine wall mesh schemes have been used to avoid this patching process. It is critical to use a good implicit-difference scheme in this case. Mellor (Ml) developed a good linearized iteration technique which has since been adopted by others. 

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