Domain expansion

Results of section 7 proved that it was reasonable to not try to estimate indicator on an unlearned space. However it was found that the estimation error is function of the distance of the learning space. Thus it may be possible to authorize the indicator estimation if the data are not far from the learning space according to a predefined threshold. The One class SVM theory enables to define the vahdity domain in function of the learning data set. The increasing error study should help to determine the threshold corresponding to the domain expansion authorized in order to keep the mean squared error, MSE, imder a value which does not disturb the detection process. 

The domain expansion is defined mainly by the competition between the surface (r p2) and the field (r p ) energies, because the elastic term retains its behavior Ka = p and... 

Let H and L be two characteristic lengths associated with the channel height and the lateral dimensions of the flow domain, respectively. To obtain a uniformly valid approximation for the flow equations, in the limit of small channel thickness, the ratio of characteristic height to lateral dimensions is defined as e = (H/L) 0. Coordinate scale factors h, as well as dynamic variables are represented by a power series in e. It is expected that the scale factor h-, in the direction normal to the layer, is 0(e) while hi and /12, are 0(L). It is also anticipated that the leading terms in the expansion of h, are independent of the coordinate x. Similai ly, the physical velocity components, vi and V2, ai e 0(11), whei e U is a characteristic layer wise velocity, while V3, the component perpendicular to the layer, is 0(eU). Therefore we have... 

At the beginning of this section we enumerated four ways in which actual polymer molecules deviate from the model for perfectly flexible chains. The three sources of deviation which we have discussed so far all lead to the prediction of larger coil dimensions than would be the case for perfect flexibility. The fourth source of discrepancy, solvent interaction, can have either an expansion or a contraction effect on the coil dimensions. To see how this comes about, we consider enclosing the spherical domain occupied by the polymer molecule by a hypothetical boundary as indicated by the broken line in Fig. 1.9. Only a portion of this domain is actually occupied by chain segments, and the remaining sites are occupied by solvent molecules which we have assumed to be totally indifferent as far as coil dimensions are concerned. The region enclosed by this hypothetical boundary may be viewed as a solution, an we next consider the tendency of solvent molecules to cross in or out of the domain of the polymer molecule. 

In a good solvent, the end-to-end distance is greater than the 1q value owing to the coil expansion resulting from solvent imbibed into the domain of the polymer. The effect is quantitatively expressed in terms of an expansion factor a defined by the relationship... 

Although the emphasis in these last chapters is certainly on the polymeric solute, the experimental methods described herein also measure the interactions of these solutes with various solvents. Such interactions include the hydration of proteins at one extreme and the exclusion of poor solvents from random coils at the other. In between, good solvents are imbibed into the polymer domain to various degrees to expand coil dimensions. Such quantities as the Flory-Huggins interaction parameter, the 0 temperature, and the coil expansion factor are among the ways such interactions are quantified in the following chapters. 

Next we use the Flory-Huggins theory to evalute AG by Eq. (8.44). As noted above, the volume fraction occupied by polymer segments within the coi domain is small, so the logarithms in Eq. (8.44) can be approximated by the leading terms of a series expansion. Within the coil N2 = 1 and Nj = (1 - 0 VuNa/Vi, where is the volume of the coil domain. When all of these considertions are taken into account, Eq. (8.108) becomes... 

The parameter a which we introduced in Sec. 1.11 to measure the expansion which arises from solvent being imbibed into the coil domain can also be used to describe the second virial coefficient and excluded volume. We shall see in Sec. 9.7 that the difference 1/2 - x is proportional to. When the fully... 

The expansion of the coil domain produces an elastic restoring force which opposes the expansion by tending to restore the molecule to its most probable conformation. 

Finite Difference Method To apply the finite difference method, we first spread grid points through the domain. Figure 3-49 shows a uniform mesh of n points (nonuniform meshes are possible, too). The unknown, here c(x), at a grid point x, is assigned the symbol Cj = c(Xi). The finite difference method can be derived easily by using a Taylor expansion of the solution about this point. Expressions for the derivatives are ... 

Galerldn Finite Element Method In the finite element method, the domain is divided into elements and an expansion is made for the solution on each finite element. In the Galerldn finite element method an additional idea is introduced the Galerldn method is used to solve the equation. The Galerldn method is explained before the finite element basis set is introduced, using the equations for reaction and diffusion in a porous catalyst pellet. 

Generating functions are used in calculating moments of distributions for power series expansions. In general, the nth moment of a distribution,/fxj is E x ") = lx" f x) dx, where the integration is over the domain of x. (If the distribution is discrete, integration is replaced by summation.)... 

Generally speaking, the outcome of any digital computation is a set of numbers in machine representation. Often the problem as originally formulated mathematically is to obtain a function defined over some domain, but the computation itself can give only (approximations to) a finite number of its functional values, or a finite number of coefficients in an expansion, or some other form of finite representation. At any rate, each number y in the finite set of numbers explicitly sought can be thought of, or perhaps even explicitly represented as, some function of the input data x ... 

As described above, protein domains that provide discrete biological cues (e.g., cell binding) or mechanical properties (e.g., expansion or contraction with temperature changes) can be borrowed from nature and designed into synthetic polypeptides or joined with other polymers to provide bio-inspired function in new... 

The requirement I > 2 can be understood from the symmetry considerations. The case of no restoring force, 1=1, corresponds to a domain translation. Within our picture, this mode corresponds to the tunneling transition itself. The translation of the defects center of mass violates momentum conservation and thus must be accompanied by absorbing a phonon. Such resonant processes couple linearly to the lattice strain and contribute the most to the phonon absorption at the low temperatures, dominated by one-phonon processes. On the other hand, I = 0 corresponds to a uniform dilation of the shell. This mode is formally related to the domain growth at T>Tg and is described by the theory in Xia and Wolynes [ 1 ]. It is thus possible, in principle, to interpret our formalism as a multipole expansion of the interaction of the domain with the rest of the sample. Harmonics with I > 2 correspond to pure shape modulations of the membrane. 

Here e is the new value of the energy splitting, the co, are the ripplon frequencies, and the A,- are tunneling amplitudes of transitions that excite the corresponding vibrational mode of the domain wall. Those amplitudes will be discussed in due time for now, we repeat, the expression above will be correct in the limit Ai/Ha>i —> 0. Finally, the renormalized value e was used in the denominator. While, according to Feenberg s expansion [118], including e in the resolvent is actually more accurate, we do it here mostly for convenience. 

A van der Waals attraction between the domain walls undergoing tunneling motions was argued to contribute to the puzzling negative expansivity, observed in a number of low T glasses. 

Since it is the first derivative with respect to r that we are interested in, we only need the 1=1 term from this expansion. The angular part contributes only to the overall constant, but it is the spherical function j (kr) that sets the cutoff value of the wavevector, above which the phonons do not produce significant linear uniform stress on the domain. In Fig. 24, we plot the derivative dji x)/dx (or, rather, we plot the square of it, which enters into all the final expressions). 

While we are at it, we estimate the interaction of the domain with the higher order strain, at least due to the term (B.l), in the frequency region of interest. The next order term in the k expansion in the surface integral from Eq. (B.2) has the same structure but is scaled down from the linear term by a factor of kR. At the plateau frequencies 0C) )/3O, kR < 0.5 as immediately follows from the previous paragraph. While this is not a large number, it is not very small either. Therefore this interaction term is of potential importance. 

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