Removable singularity

In this case the crack is said to have a zeroth opening. The cracks of a zeroth opening prove to possess a remarkable property which is the main result of the present section. Namely, the solution % is infinitely differentiable in a vicinity of T, dT provided that / is infinitely differentiable. This statement is interpreted as a removable singularity property. In what follows this assertion is proved. Let x G T dT and w > (f in O(x ), where O(x ) is a neighbourhood of x. For convenience, the boundary of the domain O(x ) ia assumed to be smooth. 

The crack is said to have a zero opening in this case. As it turned out there is no singularity of the solution provided the crack has a zero opening. What this means is the solution of (3.144), (3.147), (3.148) coincides with the solution of (3.140)-(3.142) found in the domain Q with the initial and boundary conditions (3.144), (3.145) (and without (3.143)). In the last case the equations (3.141), (3.142) hold in Q. This removable singularity property is of local character. Namely, if O(x ) is a neighbourhood of the point and... 

It will be convenient to define a time and solution dependent transformation which proportions grid points on the derivative, will require that the grid will be tmiform in this so called computational space. In order to normalize, allow for optimization, and remove singularities we can write for the transformed coordinate C(x,t) (11) ... 

Note that there are removable singularities at both ends of the instanton path, namely, at z = 0 and z = 1. Thus, in the practical numerical computations some modihcations to remove the singularities are necessary. The details are not given here and the reader should refer to Ref. [30, 31]. Finally, the numerically stable... 

In this example, x( ) as given by Eq. (182) has a pole in the lower complex halfplane (oo = — iy). This pole can be traced back to a removable singularity on the right-hand side of Eq. (181), so that X (m, tw) is analytic everywhere. This is an important drawback, since the information about the nature of the modes of the unperturbed system, which is contained in the poles of the generalized susceptibility, is not contained in the partial Fourier transform X ( , tw) as defined by Eq. (178). 

Example 3.2.8. Diffusion of a Substrate in an Enzyme Catalyzed Reaction - BVPs with Removable Singularity... 

There is a removable singularity at X = 0. This is handled by replacing 0 with e = 10- . 

Obtain the series solutions for this problem using Maple s dsolve command. (Since there is a removable singularity at x = 0, use c(l)=l and D(c)(l) = cl to obtain series solutions and obtain the constant cl using the boundary condition at x = 0). 

The most essential feature of this approach is in exclusion of the singularities of the derivatives from consideration, since there are no singularities in the local domain by definition, and the integration removes singularities between the domains. These equations, however, cannot be apply for materials with very small grain sizes (20-50 nm and less), because these elasticity relationships (3-5) are not valid on atomic distances. 

Concerning the Lam6 functions. Table 1 in Ref. [2] shows their classification, structure, and numbers. In fact, the derivatives of Eq. (8) with respect to k and jx, allow the identification of the removable singularities in the solutions of Eqs. (15 and 16), respectively ... 

With / identically zero, s was varied from 0.98 to 1.04, to investigate expansions of the area and volume fractions in s about s = 1. Strictly speaking, there is a (removable) singularity in the curve of area of a unit cell at s = 1 because of the change in symmetry from tetragonal to cubic (when / is constant), so we choose to report the area of a six-armed cell. It was found that to whithin 0.5%, the value of remained constant at the value... 

We must know Yl u) and Y u) to convert QeC-t Oi o) into Q L,u,X). In the primary chain, excluded-volume interactions can act between infinitesimally close points on the chain, since the chain is allowed to take on conformations which contain infinitesimally small loops. Owing to this property, unless vo is zero, Qb diverges to infinity as e —> 0, as will be illustrated below. However, no such singularity appears in a coarse-grained chain with cut-off length A because intrachain interactions in it do not act between contour points separated by less than A. Therefore, Q(L, u, A) should remain finite in the limit e 0 if A 0. The renormalization constants Yl and can be determined from this physical requirement, as exemplified below. Because of this operation, renormalization is sometimes misunderstood as a mere mathematical maneuver of removing singular behavior of the primary chain in the limit of e —> 0. 

All the terms in this equation are analytical functions of the parameter z except for A(z), which is the numerical solution of Equation (6.181). Equation (7.27) is a system of coupled nonlinear equations with removable singularity at z = -1 that may cause instability of the numerical solution. Note, however, that the nonlinearity of this equation is totally due to the factor 9 z) [see Equation (6.193)], which is related to the normalization condition for the vector U. Thus, for a given mode y the solution can be written in the form... 

Finally, we should pay attention to the removable singularities in Equation (8.41) arising from both ends (z = 0 and 1) of the instanton path, and derive the expression... 

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