Mass singularity

By assumption, the mass ratio = m/M is a small parameter. Thus, rescaling the Schrbdinger equation properly in time and potential transforms it into the singularly perturbed equation... 

In the following we devise, following [14], an efficiently implementable scheme which leads to favorable error bounds independently of the highest frequencies under the mere assumption that the system has bounded energy. The scheme will be time-reversible, and robust in the singular limit of the mass ratio m/M tending to 0. 

Level of enforcement of the incompressibility condition depends on the magnitude of the penalty parameter. If this parameter is chosen to be excessively large then the working equations of the scheme will be dominated by the incompressibility constraint and may become singular. On the other hand, if the selected penalty parameter is too small then the mass conservation will not be assured. In non-Newtonian flow problems, where shear-dependent viscosity varies locally, to enforce the continuity at the right level it is necessary to maintain a balance between the viscosity and the penalty parameter. To achieve this the penalty parameter should be related to the viscosity as A = Xorj (Nakazawa et al, 1982) where Ao is a large dimensionless parameter and tj is the local viscosity. The recommended value for Ao in typical polymer flow problems is about 10. ... 

The singularity of MAbs and the ease of mass production appeared to be the answer to rapid development of highly specific immunoassays. Companies were formed to produce MAbs and incorporate them into assays. In fact, such assays have been developed and have proved very successful for infectious diseases, hormones, and other clinical analytes. 

C. F. Moore, "AppHcation of Singular Value Decomposition to the Design, Analysis, and Control of Industrial Processes," Proceeding of American Control Conference, Boston, Mass., 1986, p. 643. 

Whitaker, S, The Species Mass Jump Condition at a Singular Surface, Chemical Engineering Science 47, 1677, 1992. 

Here Lqp is the distance between points q and p. Note that G q, p) is called a Green s function. There are an infinite number of such functions and all of them have a singularity at the observation point p. Inasmuch as the second Green s formula has been derived assuming that singularities of the functions U and G are absent within volume V, we cannot directly use this function G in Equation (1.99). To avoid this obstacle, let us surround the point by a small spherical surface S and apply Equation (1.99) to the volume enclosed by surfaces S and S, as is shown in Fig. 1.9. Further we will be mainly interested by only cases, when masses are absent inside the volume V, that is. 

Indeed, it satisfies Laplace s equation everywhere except at the point p, since it describes up to a constant the potential of a point mass located at the point p. Also, it has a singularity at this point and provides a zero value of the surface integral over the hemisphere when its radius r tends to infinity. Correspondingly, we can write... 

Unlike prisms, in this class of bodies uniqueness requires knowledge of the density. This theorem was proved by P. Novikov. The simplest example of starshaped bodies is a spherical mass. Of course, prisms are also star-shaped bodies but due to their special form, that causes field singularities at corners, the inverse problem is unique even without knowledge of the density. It is obvious that these two classes of bodies include a wide range of density distributions besides it is very possible that there are other classes of bodies for which the solution of the inverse problem is unique. It seems that this information is already sufficient to think that non-uniqueness is not obvious but rather a paradox. 

As noted earlier, the sum of the mass fractions is unity and thus Eq. (86) will be consistent with Eq. (85) only if the sum of the correction term b m over all chemical species a = 1,..., K is null. In general, this will not be the case if Eq. (89) is used. Another difficulty that can arise is that the mass fractions in two environments may be equal, e.g., i = a2, and thus the coefficient matrix in Eq. (89) will be singular. This can occur, for example, in the equilibrium-chemistry limit where the compositions depend only on the mixture fraction, i.e., (j) — co(0- F°r chemical species that are not present in the feed streams, the equilibrium values for = 0 and = 1 are zero, but for intermediate values of the mixture fraction, the equilibrium values are positive. This implies that the equilibrium values will be the same for at least two values of the mixture fraction in the range 0< < 1. Thus, in the equilibrium limit it is inevitable that two environments will have equal mass fractions for certain species at some point in the flow field. Since singularity implies an underlying correlation between... 

The correction terms can be computed using either Yx or Xa. However, it is clear that the correction terms will depend on which choice is used since the moments controlled by DQMOM are different (i.e., (Y ) vs. (X(")). Thus, for example, with fixed N one set of moments may lead to singular correction terms, but not the other. In general, we can continue to solve for the mass fractions in the CFD model, but with the correction terms computed using the cumulative mass fractions as follows. 

Thus, if mentioned above is to be taken into consideration that absorption band in the region of 330-340 nm reflects to a certain extent the degree of fullerene molecules association in solution, we can come to the conclusion that the less the PVP molecular mass and the less the fullerene contents is the more fullerene molecules are in low associated state. It is quite probable that the increase of the fullerene contents in the complex brings to the formation of adducts, where not single fullerene C60 molecules are bonded with PVP but their associates. It can be one of the reasons of the observed fact of the difference of UV-VIS spectra of C60/PVP complex with PVP of different molecular mass (up to spectra crossover, which is a singular evidence that these compounds are nonidentical). 

In contrast, photolysis of methoprene in true aqueous solution gave a simpler distribution of different products (40). Five major products (25, 11, 13, 13, and 8% yield) were separated, but could not be positively identified due to lack of sufficient quantity (methoprene water solubility = 1.4 mg/1) and the singularly uninformative mass spectral fragmentations of the products. 

We thus arrive at the following composition for the ancestral cloud that spawned the Solar System in 1 gram of matter, we find 0.72 g of hydrogen, 0.26 g of helium and 0.02 g of heavier elements. Despite the superb efforts of past generations of stars, the Sun, like its nebulous father, is singularly poor in metals, since these make up a mere 2% mass fraction of its matter. This, however, is a small fortune compared with the ancient stars in the galactic halo. 

Since in this problem not only the limit but also the character of convergence matters we conclude that consistent homogenization of the micromodel should lead to a description in a broader functional space than is currently accepted. One interesting observation is that the concave part of the energy is relevant only in the region with zero measure where the singular, measure valued contribution to the solution is nontrivial (different from point mass). We remark that the situation is similar in fracture mechanics where a problem of closure at the continuum level can be addressed through the analysis of a discrete lattice (e.g. Truskinovsky, 1996). 

Here 6 is the total initial mass ( amount of heat etc.) and o is the concentration at oo. (The concentration at —oo has been assumed to be zero to take into consideration the singular value of interest in (3.2.2a).)... 

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