Contrast structures critical case

Construction of an Asymptotic Expansion for the Parabolic Problem Other Problems with Corner Boundary Layers Nonisothermal Fast Chemical Reactions Contrast Structures in Partial Differential Equations A. Step-Type Solutions in the Noncritical Case Step-Type Solutions in the Critical Case Spike-Type Solutions Applications... 

Progress in nanotechnology also depends critically on new developments in microscopy [42-45]. Compared to other investigation methods that help to explore the relation between the molecular structure and macroscopic properties, microscopy gives the most direct information. Particularly, in the case of disordered or aperiodic structures, visualisation of the structure is often more useful than indirect measurement and interpretation of its scattering properties. In practice, the utilisation and value of microscopes depends on their spatial resolution, the contrast and the imaging conditions. 

In contrast, for the off-critical quench ((iff) 0 when v < v ) regular stripe patterns with domains parallel to the quench interface were always found (Fig. 32d-f). This situation is covered by a ID analysis presented before where the period of the structure is uniquely determined by the velocity of the quench interface. In the limit v —> 0 the periods of the patterns found in our numerical simulations agree with (73). For v > v irregular coarsening patterns, similar to the case of a spatially homogeneous quench, are observed. 

Tc. The two power-law exponents are not independent but depend on a single parameter, the so-called critical exponent X, which is specific for a given interaction potential (e.g., hard spheres). Actually, the interaction potential enters the MCT equations only indirectly via the structure factor S(q), which fixes the nonlinear coupling in the generalized oscillator equation. It is important to note that the MCT exponents are not universal in contrast to those of second-order phase transitions. In the case of hard spheres, for example, S(q) can be calculated via the Percus-Yevick approximation [26], and the full time and -dependence of < >(q. f) were obtained. As an example, Fig. 10 shows the susceptibility spectra of the hard-sphere system at a particular q. Note that temperature cannot be defined in the hard-sphere system instead, the packing fraction cp is used as a parameter. Above the critical packing fraction 0), which corresponds to T < Tc in systems where T exists, the a-process is absent (frozen) and only the fast dynamics is present. At cp < tpc the a-peak and the concomitant susceptibility minimum shift to lower frequencies with increasing cp, so that the closer cp is to the critical value fast dynamics can be identified (curve c in Fig. 10). 

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