Singular limit

Tanford, C., Kirkwood, J. G. Theory of protein titration curves. I. General equations for impenetrable spheres. J. Am. Chem. Soc. 79 (1957) 5333-5339. 6. Garrett, A. J. M., Poladian, L. Refined derivation, exact solutions, and singular limits of the Poisson-Boltzmann equation. Ann. Phys. 188 (1988) 386-435. Sharp, K. A., Honig, B. Electrostatic interactions in macromolecules. Theory and applications. Ann. Rev. Biophys. Chem. 19 (1990) 301-332. 

Bornemann, F. A., Schiitte, Ch. On the Singular Limit of the Quantum-Classical Molecular Dynamics Model. Preprint SC 97-07 (1997) Konrad-Zuse-Zentrum Berlin. SIAM J. Appl. Math, (submitted)... 

F.A. Bornemann and Ch. Schiitte. On the singular limit of the quantum-classical molecular dynamics model. Preprint SC 97-07, ZIB Berlin, 1997. Submitted to SIAM J. Appl. Math. 

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. 

The singular limit (S 10.2-6), in which one of the / independent R-) is a zero vector (implying that the original basis vectors R,) were linearly dependent), occurs at a critical state Sc, where the number of phases p and dimension/ are changing. Critical state limits will be examined in Chapter 11. 

The shear stress o and the normal stress differences Ny and N2 are all predicted to be linear in the shear rate y. This scaling has indeed been observed for LCPs in Region II (Kiss and Porter 1980b Moldenaers and Mewis 1986 Back et al. 1993b). In contrast, for isotropic polymeric materials, is proportional to y at low shear rates. As noted earlier, if s is identically zero, then there are no domain interactions, and hence the stresses never reach a steady state. However, in the singular limit that approaches zero without reaching it, from Eq. (11-35) we find that a /r -f iiikll and N N2 0 hence only a is nonzero as this limit is approached. 

The branch of mathematics that deals with singular limits is called singular perturbation theory. See Jordan and Smith (1987) or Lin and Segel (1988) for an introduction, Another problem with a singular limit will be discussed briefly in Section 7.5. 

Z.i.b At. model problem about singular limits) Consider the linear differential equation... 

Assuming that all hyperplanes H near H 0) are of the form H(z), this proves that the 2n — 2 branches 7, with coordinate functions Xi,—yi satisfy the condition of the theorem Analytically prolonging, the ji,Si therefore are part of a curve C C P and one goes on to prove that C is a canonically embedded curve of genus n (or a singular limit of such) and H is its theta divisor. 

The first term is positive and is more than the second one it takes into account the all trajectories of walk with imposed on them singular limitation of the connectedness of the links into a chain, and doesn t accept the reverse step. The second term is negative (co 2N) < 1) it takes into account the additional limitations on the trajectories of walk by requirement of their self-intersection absence. At this, the first term at given data s, N, d is... 

This type of equation is also encountered in other areas, such as nonlinear waves, nucleation theory, and phase field models of phase transitions, where it is known as the damped nonlinear Klein-Gordon equation, see for example [165, 355, 366]. In the (singular) limit r 0, (2.15) goes to the reaction-diffusion equation (2.3). Front propagation in HRDEs has been studied analytically and numerically in [149, 150, 152, 151, 374]. The use of HRDEs in applications is problematic. Such equations are obtained indeed very much in an ad hoc manner for reacting and dispersing particle systems, and they can be derived neither from phenomenological thermodynamic equations nor from more microscopic equations, see below. 

A. J. M. Garrett and L. Poladian, Ann. Phys., 188, 386 (1988). Refined Derivation, Exact Solutions, and Singular Limits of the Poisson-Boltzmann Equation. 

The presence of this singularity limits the usability of the short pitch method to substances with tilt angle 6 up to about 35-37°. In practice, however, this is not too serious, as materials with a tilt approaching 40° are rare. 

Singular Limit of the Eigenvalue Problem for Single-Front Solutions a. The e —> 0 Limit Eigenvalue Problem... 

In order to make the calculations more transparent, we will consider the three quantities t),t and D as independent [62, 104] we will restore the relation r = / > only at the end of the calculations. We will be mainly interested in the limit e —> 0, which is equivalent to rj 0. Note that, due to the relation T = fejD, if = 0, then r = 0. In fact, we will see that (e = 0, r = 0) is a singular limit, in the sense that one of the eigenvalues goes to infinity. In order to avoid this singularity, we will keep r away from zero. 

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