Vertex approximation

The mutual correspondence of non-Markovian and Markovian (impact) approximations becomes clear, if the second derivative of K/(t) is considered. It varies differently within three time intervals with the following bounds xc < xj < Tj1 (Fig. 2.5). Orientational relaxation occurs in times Fj1. The gap near zero has a scale of xj. A parabolic vertex of extent xc and curvature I4 > 0 is inscribed into its acute end. The narrower the vertex, the larger is its curvature, thus, in the impact approximation (tc = 0) it is equal to 00. In reality xc =j= 0, and the... 

The curvature k of the interface in two dimensions is calculated in a similar way. Consider a polygon consisting of vertices v,- connected by edges Let us denote by /, the length of the edge between (7 — l)th and th vertex. The curvature k at the /th vertex can be then approximated as follows ... 

Fig. A3.1. Some lowest-order diagrams for the temperature GF (A3.6). The dashed and solid lines correspond to the GF for high-frequency and resonance low-frequency vibrations of a molecular planar lattice in the harmonic approximation (see Eq. (A3.9) and (A3.10)). Each vertex is associated with the factor -y/N, the integration and summation being performed over each vertex coordinates r, from 0 to / , and over all internal wave vectors K. At ptiClK 1, the main contribution is provided by a-type diagrams.184...

The last term is introduced within the self-consistent Hartree approximation (within the functional up to one vertex), //// = 10/9 accounts different coefficients in functional for the self-interaction terms (ri4 - for the given field d and dla)2 d )2 terms), cf [20], We presented da = 2k d t>ke lkfix, ... 

To construct an approximation to the relaxation process in the reaction network iV, we also need to restore cycles, but for this purpose we should start from the whole glued network V on si (not only from fixed points as we did for the steady-state approximation). On a step back, from the set si to si and so on some of glued cycles should be restored and cut. On each step we build an acyclic reaction network, the final network is defined on the initial vertex set and approximates relaxation of if. 

If k >S> fcs4 then almost all concentration in the steady state is accumulated inside A. This vertex is not a glued cycle, and immediately we find the approximate eigenvector for zero eigenvalue, the vector column with coordinates (0,0,0,1,0,0). 

Figure 5. A DNA molecule whose helix axes have the connectivity of a cube. The molecule shown consists of six cyclic strands that have been catenated together in this particular arrangement. They are labeled by the first letters of their positional designations, Up, Down, Front, Back, Left, and Right. Each edge contains 20 nucleotide pairs of DNA, so we expect that their lengths will be approximately 68 A. All of the twisting has been shown in the middle of the edges for clarity, but the DNA is base-paired from vertex to vertex. From model building, the axis-to-axis distance across a square face seems to be approximately 100 A, with a volume (in a cubic configuration) of approximately 1760 nm3 when the cube is folded as shown.

Here q is a wavevector (eqn 1.6), ip(q) is the Fourier transform of />(r), and S(q) is the structure factor (Fourier transform of the two-point correlation function). The cubic term, ft, is zero for a symmetric system and otherwise may be chosen to be positive. The quartic term, y, is then positive to ensure stability. For block copolymers, these coefficients may be expressed in terms of vertex functions calculated in the random phase approximation (RPA) by Leibler (1980). The structure factor is given by... 

From the above expressions it is evident that for the calculations of the vertex function and the propagator the knowledge of four-particle correlation functions are required. For simplification, Gaussian approximation has been assumed and the four-particle correlation function is written as the product of two two-particle correlation functions. 

Next, for simplicity, an approximation for the vertex function is considered. It has been shown by Balucani that y dn q) can approximately be given... 

Note that even without the approximation of the vertex function (given by Eq. (278)], it is possible to demonstrate that Rtt z — 0) oc crjj, but the analysis becomes cumbersome. Thus the approximation for the vertex function is not necessary, but it is made to simplify the analysis. 

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