Differentiability divergence

Two main strategies for the synthesis of dendrimers can be differentiated divergent and convergent methods [123]. The divergent method is particularly suited for the large-scale preparation of dendrimers. However, products often suffer from purification problems and imperfection. In contrast, the convergent... 

The chaotic nature of individual MD trajectories has been well appreciated. A small change in initial conditions (e.g., a fraction of an Angstrom difference in Cartesian coordinates) can lead to exponentially-diverging trajectories in a relatively short time. The larger the initial difference and/or the timestep, the more rapid this Lyapunov instability. Fig. 1 reports observed behavior for the dynamics of a butane molecule. The governing Newtonian model is the following set of two first-order differential equations ... 

Hence, in the limit A->0, the differential cross-section diverges. This is due to the long range character of the Coulomb field, which entails that the expansion (10-259) of KA does not converge. Alternatively, the asymptotic condition is not valid in this case. 

A simple repetition of the iteration procedure (2.20)-(2.22) results in divergence of higher order solutions. However, a perturbation theory series may be summed up so that all unbound diagrams are taken into account, just as is usually done for derivation of the Dyson equation [120]. As a result P satisfies the integral-differential equation... 

Once this divergence happens, further solution of the differential equation is not possible beyond this point, and we have to reformulate the problem. To clarify our idea, let us consider the ID problem. At the turning point, p q) = 0 and A diverges. If we invert A to A = dq/dp, the divergence is removed and the propagation of A proceeds smoothly through the caustics. This inversion is equivalent to the canonical transformation, (p,q) (—q,p). It can be easily... 

As previously indicated, both condensation and addition polymers may be prepared from monomers of functionality exceeding two, with resulting formation of nonlinear polymers. Hence the distinction between linear and nonlinear polymers subdivides both the condensation and the addition polymers, and four types of polymers are at once differentiable linear condensation, nonlinear condensation, linear addition, and nonlinear addition. The distinction between linear and nonlinear polymers is clearly warranted not only by the marked differences in their structural patterns but also by the sharp divergence of their properties. 

This can be inserted into the current expressions (30) and (32), whose divergences enter into the time development equations for the densities. (Note that the denominator of (76) simplifies greatly in all cases except where the silicon is nearly intrinsic. Here again, if the charge states are equilibrated, n+ and n can be eliminated in favor of n0, via (3), (4), and (75). Whether AH and /or nDH must be retained as distinct variables in the system of differential equations, or whether they, too, can be eliminated in favor of 0, will depend on whether or not they are able to come quickly into local equilibrium with the monatomic species. 

The divergence operator is the three-dimensional analogue of the differential du of the scalar function u x) of one variable. The analogue of the derivative is the net outflow integral that describes the flux of a vector field across a surface S... 

The most important quantitative measure for the degree of chaotic-ity is provided by the Lyapunov exponents (LE) (Eckmann and Ru-elle, 1985 Wolf et. al., 1985). The LE defines the rate of exponential divergence of initially nearby trajectories, i.e. the sensitivity of the system to small changes in initial conditions. A practical way for calculating the LE is given by Meyer (Meyer, 1986). This method is based on the Taylor-expansion method for solving differential equations. This method is applicable for systems whose equations of motion are very simple and higher-order derivatives can be determined analytically (Schweizer et.al., 1988). 

The main advantage of NMR spectroscopy is its use with proteins in solution. In consequence, rather than obtaining a single three-dimensional structure of the protein, the final result for an NMR structure is a set of more or less overlying structures which fulfill the criteria and constraints given particularly by the NOEs. Typically, flexibly oriented protein loops appear as largely diverging structures in this part of the protein. Likewise, two distinct local conformations of the protein are represented by two differentiated populations of NMR structures. Conformational dynamics are observable on different time scales. The rates of equilibration of two (or more) substructures can be calculated from analysis of the line shape of the resonances and from spin relaxation times Tj and T2, respectively. 

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