Asymptotic behavior solutions

The solution gives all of the expected asymptotic behaviors for large N—the proportionate pattern spreading of the simple wave if R > 1, the constant pattern if R < 1, and square root spreading for R = 1. 

Knowledge of all terms of the KS potential in Eq. (51) for a large distance from an atomic or molecular center is interesting both for setting the proper asymptotic behavior for the solutions of the KS equations (50), and for checking the accuracy of approximations for v (r) and v (r) in this region. 

The most effective reply to this criticism is a reference to the two-dimensional Ising model for which an analytic solution is available, and coefficients can be computed for all values of n. Taking the triangular lattice, for example, predictions based on values of n up to 15 are quite adequate to represent the asymptotic behavior of the coefficients relating to thermodynamic and correlation properties to a high degree of accuracy. 

It is often the case that after a sufficiently long time, a transient problem approaches a steady-state solution. When this is the case, it can be useful to calculate the steady solution independently. In this way it can be readily observed if the transient solution has the correct asymptotic behavior at long time. 

The scaling argumentation has been outlined in some detail for the following reason the asymptotic behavior is fairly easily, and also correctly, derived by this technique. Often, however, the result is extended into a region of smaller q-values, down to u = 1184), and now the scaling argument comes to a conclusion which deviates strongly from the result of the analytic solution of Eq. (B.45). 

While the early work on molten NH4CI gave only some qualitative hints that the effective critical behavior of ionic fluids may be different from that of nonionic fluids, the possibility of apparent mean-field behavior has been substantiated in precise studies of two- and multicomponent ionic fluids. Crossover to mean-field criticality far away from Tc seems now well-established for several systems. Examples are liquid-liquid demixings in binary systems such as Bu4NPic + alcohols and Na + NH3, liquid-liquid demixings in ternary systems of the type salt + water + organic solvent, and liquid-vapor transitions in aqueous solutions of NaCl. On the other hand, Pitzer s conjecture that the asymptotic behavior itself might be mean-field-like has not been confirmed. 

One exception might be the case where the polydispersity variable a can assume values from an infinite range (as in the length-polydisperse polymer problem treated in Section V.A, where there is no upper limit on o = L). The form of the formally exact solution (59) can then give information about the asymptotic behavior of the extra weight functions. We leave this issue for future work. 

For large values of u our flow equations break down. Qualitatively the flow is towards large u and small K. We can, however, find the asymptotic behavior in this phase by solving the initial model in the strong pinning limit exactly. To find this solution we will assume strong pinning centers and weak thermal fluctuations ... 

We pause now to highlight some features of a two-dimensional instanton. Let us introduce the local normal coordinates Q+ and Q about the potential minimum, which correspond to the higher and lower vibrational frequencies co+ and a>. The asymptotic behavior of the instanton solution at /3—>< >, t—>0, is described by Q+ - Q° cosh(w r) (see Appendix B), where Q°+ goes to zero faster than Q° so as to keep Q finite. Therefore, the asymptotic instanton direction coincides with... 

If k is purely imaginary and positive, then these states correspond to bound states with asymptotic behavior L) bound state solutions are the only solutions with positive imaginary values of the wave vector [31]. 

The second type of solutions are those for which k is purely imaginary and negative. These states are called antibound states and have the asymptotic behavior of 4>( x > L) a e+ww. 

The parameter , which enters into the downstream boundary conditions, can actually be interpreted as = T/,/71,., where T), - L2/D is the hydraulic diffusion time scale. When 1, there exists an intermediate asymptotic behavior corresponding to the zero-flux solution at z = L. When ( -C 1, the solution is essentially the membrane solution. 

P 7] The topic has only been treated theoretically so far [28], A mathematical model was set up slip boundary conditions were used and the Navier-Stokes equation was solved to obtain two-dimensional electroosmotic flows for various distributions of the C, potential. The flow field was determined analytically using a Fourier series to allow one tracking of passive tracer particles for flow visualization. It was chosen to study the asymptotic behavior of the series components to overcome the limits of Fourier series with regard to slow convergence. In this way, with only a few terms highly accurate solutions are yielded. Then, alternation between two flow fields is used to induce chaotic advection. This is achieved by periodic alteration of the electrodes potentials. 

Determination of the full numerical solution, however, was not the emphasis of that paper. Sader et al. [73] focused on the asymptotic behavior of the potential determined essentially by the leading coefficient, A0 at distances much larger than the period of the charge heterogeneities the potential is an exponential function of distance from the surface with decay length equal to the Debye length,... 

Since the analytic solutions have been obtained, it is interesting to examine the asymptotic behavior of particle motion for the case of near contact between the cylinder and the plane (X 1). The approximate expression for the electrophoretic velocity is given by... 

A plain power-series solution is not practical the asymptotic behavior of j/ x) at large x requires instead a power series with a prefactor exp[—(co / ti)x2 / 2). ... 

The function f X2 is not a satisfactory solution because it becomes infinite as —> oo, but the function e /2 is well-behaved. This asymptotic behavior of [Pg.321]

The mathematical structure of the models is their unifying background systems of nonlinear coupled differential equations with eventually nonlocal terms. Approximate analytic solutions have been calculated for linearized or reduced models, and their asymptotic behaviors have been determined, while various numerical simulations have been performed for the complete models. The structure of the fixed points and their values and stability have been analyzed, and some preliminary correspondence between fixed points and morphological classes of galaxies is evident—for example, the parallelism between low and high gas content with elliptical and spiral galaxies, respectively. 

Since all trajectories of the original system are asymptotic to their omega limit set, in analyzing this equation it is sufficient to determine the asymptotic behavior of (2.3). From a more intuitive viewpoint this is merely starting on the manifold 5-f-x= 1, to which all solutions must tend the mathematical support for this is rigorously established later (see the proof of Theorem 5.1 or Appendix F). Define, for aw > 1,... 

In other words, independently of initial conditions, solutions of (1.1) asymptotically approach the plane, S-t Xi-fX2 = 1, at an exponential rate. The asymptotic behavior of (1.1) is therefore determined by the two-dimensional system obtained from (1.1) by deleting the equation for S and replacing S by 1 — Xi —X2, just as in the previous chapters. This yields... 

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