Singularity, definition

The positive adsorption of a solute at the S—MO interfaee results whenever the ehemieal potential ehange (at eonstant temperature and pressure) is negative for a solute moved from a bulk solution and bonded to or adsorbed at a S—MO interfaee. The ehemieal potential imbalanee diminishes as solute adsorption and eoneentration ehanges at the S—MO interfaee. The eomplete disappearanee of a ehemieal potential differenee (at eonstant temperature and pressure) for the solute in the bulk liquid and the S—MO interfaee serves as the definition of adsorption equilibrium for the solute. This singular definition of adsorption has produeed a large number of adsorption models whieh strive to translate the ehemieal potential model of equilibrium adsorption into measurable variables and eonstants. Many of these adsorption equations will be examined in detail later in this ehapter. 

In this figure the next definitions are used A - projection operator, B - pseudo-inverse operator for the image parameters a,( ), C - empirical posterior restoration of the FDD function w(a, ), E - optimal estimator. The projection operator A is non-observable due to the Kalman criteria [10] which is the main singularity for this problem. This leads to use the two step estimation procedure. First, the pseudo-inverse operator B has to be found among the regularization techniques in the class of linear filters. In the second step the optimal estimation d (n) for the pseudo-inverse image parameters d,(n) has to be done in the presence of transformed noise j(n). 

The second integral in Eq. (155) seemed to be singular when n + / + q = 0. However, in this case, (i must be zero, and consequently this term will never contribute to the final result for being suppressed by the prefactor. With the definition in Eq. (132), we can write... 

To continue, we assume the following situation We concentrate on an x-y plane, which is chosen to be perpendicular to the seam. In this way, the pseudomagnetic field is guaranteed to be perpendicular to the plane and will have a nonzero component in the z direction only. In addition, we locate the origin at the point of the singularity, that is, at the crossing point between the plane and the seam. With these definitions the pseudomagnetic field is assumed to be of the form [113]. 

The BFGS method is considered to be superior to DFP in most cases because (a) it is less prone to loss of positive definiteness or to singularity problems... 

The condition number is always greater than one and it represents the maximum amplification of the errors in the right hand side in the solution vector. The condition number is also equal to the square root of the ratio of the largest to the smallest singular value of A. In parameter estimation applications. A is a positive definite symmetric matrix and hence, the cond ) is also equal to the ratio of the largest to the smallest eigenvalue of A, i.e.,... 

The numerical evaluation of definite integrals can be carried out in several ways. However, in all cases it must be assumed that the function, as represented by a table of numerical values, or perhaps a known function, is well behaved. While this criterion is not specific, it suggests that the functions haying pathological problems, e.g. singularities, discontinuities,..may not survive under the treatment in question. 

Newton, the limit h —> 0 is singular. The symmetries underlying quantum and classical dynamics - unitarity and symplecticity, respectively - are fundamentally incompatible with the opposing theory s notion of a physical state quantum-mechanically, a positive semi-definite density matrix classically, a positive phase-space distribution function. 

As mentioned earlier, singular matrices have a determinant of zero value. This outcome occurs when a row or column contains all zeros or when a row (or column) in the matrix is linearly dependent on one or more of the other rows (or columns). It can be shown that for a square matrix, row dependence implies column dependence. By definition the columns of A, a, are linearly independent if... 

Due to the conservation of elements, the rank of J will lie less than or equal to K — E 1 In general, rank(J) = Ny < K - E, which implies that V = K — T eigenvalues of J are null. Moreover, since M is a similarity transformation, (5.51) implies that the eigenvalues of J and those of J are identical. We can thus limit the definition of the chemical time scales to include only the Nr finite ra found from (5.50). The other N components of the transformed composition vector correspond to conserved scalars for which no chemical-source-term closure is required. The same comments would apply if the Nr non-zero singular values of J were used to define the chemical time scales. 

In general, if all (n = l,. .., A7e) are distinct, then A will be full rank, and thus a = A 1 /3 as shown in (B.32). However, if any two (or more) (< />) are the same, then two (or more) columns of Ai, A2, and A3 will be linearly dependent. In this case, the rank of A and the rank of W will usually not be the same and the linear system has no consistent solutions. This case occurs most often due to initial conditions (e.g., binary mixing with initially only two non-zero probability peaks in composition space). The example given above, (B.31), illustrates what can happen for Ne = 2. When ((f)) = ()2, the right-hand sides of the ODEs in (B.33) will be singular nevertheless, the ODEs yield well defined solutions, (B.34). This example also points to a simple method to overcome the problem of the singularity of A due to repeated (< />) it suffices simply to add small perturbations to the non-distinct perturbed values need only be used in the definition of A, and that the perturbations should leave the scalar mean (4>) unchanged. 

The preceding definition of a kinetic SDE reduces to that given by Hiitter and Ottinger [34] in the case of an invertible mobility matrix X P, for which Eq. (2.268) reduces to the requirement that Zap = K. In the case of a singular mobility, the present definition requires that the projection of Z p onto the nonnull subspace of K (corresponding to the soft subspace of a constrained system) equal the inverse of within this subspace, while leaving the components of Z p outside this subspace unspecified. 

In retrospect, it seems unfortunate that in 1971 Morrison and Mosher8 generalized the definition, while keeping the term, an asymmetric synthesis is a reaction in which an achiral unit in an ensemble of substrate molecules is converted by a reactant into a chiral unit in such a manner that the stereoisomeric products arc produced in unequal amounts ( Footnote The substrate molecule must have either enantiotopic or diastereotopic groups or faces) . Obviously the phrase "an achiral unit in an ensemble of substrate molecules is too inexact and requires a great deal of additional explanation, which was partially given by the footnote (note that molecule, i.e., singular, was used ). Currently, the Morrison-Mosher term appears to be equivalent to stereoselective reaction. Unfortunately, this term was only defined in the modem sense by Izumi in 1971, i.e., in the same year the Morrison-Mosher definition was published. 

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