Universal behavior

Only the prefactors of this law are specific to the polymer-solvent combinations. (Adapted from Fu ita, H., Polymer Solutions, Elsevier, Amsterdam, the Netherlands, 1990.) 

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Generalized Correla.tions. A simple and rehable method for the prediction of vapor—Hquid behavior has been sought for many years to avoid experimentally measuring the thermodynamic and physical properties of every substance involved in a process. Whereas the complexity of fluids makes universal behavior prediction an elusive task, methods based on the theory of corresponding states have proven extremely useful and accurate while still retaining computational simplicity. Methods derived from corresponding states theory are commonly used in process and equipment design. 

It is then assumed that due to this separation in scales, the so-called subgrid scale (SGS) modeling is largely geometry independent because of the universal behavior of turbulence at the small scales. The SGS eddies are therefore more close to the ideal concept of isotropy (according to which the intensity of the fluctuations and their length scale are independent of direction) and, hence, are more susceptible to the application of Boussinesq s concept of turbulent viscosity (see page 163). 

Coarse-grained polymer models neglect the chemical detail of a specific polymer chain and include only excluded volume and topology (chain connectivity) as the properties determining universal behavior of polymers. They can be formulated for the continuum (off-lattice) as well as for a lattice. For all coarse-grained models, the repeat unit or monomer unit represents a section of a chemically realistic chain. MD techniques are employed to study dynamics with off-lattice models, whereas MC techniques are used for the lattice models and for efficient equilibration of the continuum models.36 2 A tutorial on coarse-grained modeling can be found in this book series.43... 

By eomparing the numerieally ealeulated height-height eorrelation function to the expected universal behavior at the roughening transition, we have estimated the roughening temperature of the model to be given approximatey by... 

We may summarize our discussion by noting that, in the parameter region given by Eqs. (1.18) we find universal behavior. Tt takes the form of crossover scaling laws, in limiting situations reducing to. simple povrer laws. Specific microstructure effects are contained in a few nonuniversal constants. 

As evident from Fig. 11, where different symbols are used to distinguish TE data for lamellar (filled symbols) and non-lamellar phases (open symbols), DL/DOPC-DNA complexes in the Lo,c phase closely follow the universal behavior. These are MVLG2/DOPC-DNA complexes at all DL and DNA complexes of the other DLs at low DL. In contrast, TE of the H c phase and the new distorted hexagonal and DNA bundle phases is not only high but independent of lamellar phases therefore coincides with the deviation from the universal TE curve, suggesting a different mechanism of action for the different structures of DNA complexes. 

As we have elaborated in Sects. 2 and 4.3, the TE of lamellar DNA complexes of MVLs and UVLs shows universal behavior when plotted against the membrane charge density, implying that optimized complexes of MVLs and UVLs transfect equally well. Importantly, the fact that the universal curve is bell-shaped (where TE is plotted logarithmically) implies that optimization of the lipid composition is crucial for objectively comparing lipid performance. Interestingly, while complexes in the H c and HnC phase do not follow this universal curve, their TE no more than equals that of optimized lamellar complexes. 

In this section, we will show that the stationary responses obtained at microelectrodes are independent of whether the electrochemical technique employed was under controlled potential conditions or under controlled current conditions, and therefore, they show a universal behavior. In other words, the time independence of the I/E curves yields unique responses independently of whether they were obtained from a voltammetric experiment (by applying any variable on time potential), or from chronopotentiometry (by applying any variable on time current). Hence, the equations presented in this section are applicable to any multipotential step or sweep technique such as Staircase Voltammetry or Cyclic Voltammetry. 

M. J. Feigenbaum Universal behavior in nonlinear systems. Physica 1983, 7D 16-39... 

Los Alamos Science Universal Behavior in Nonlinear Systems (p. 4)... 

Exciton migration experiments on isotopic naphthalene crystals reveal a universal behavior with reduced donor concentra-... 

Discontinuous and continuous collapse of PCEPAC and poly(carboxyethyl 3-cyclohexylaminocrotonate) (PCECHAC) hydrogels was observed [221] in water/acetone and water/ethanol mixtures (Fig. 4). These results confirm the universal behavior of hydrogels with respect to the thermodynamic quality of solvents. 

M. J. Feigenbaum, Universal Behavior in Nonlinear Systems, in Los Alamos Science, Los Alamos, New Mexico, Summer (1980), 4-27. 

Figure 3 (a) Power-law behavior of x- showing the collapse of data for chessboard and random topographies on a single curve for each adsorption regime. Filled [empty] symbols indicate chessboard [randomi topographies, (b) Universal behavior of a vs. w/AE for MC simulation. 

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