Dynamical mapping

If the dynamic mapping is also area-preserving (i.e., dxdp = dx dp ), (8.44) is satisfied if we choose... 

State diagrams are very useful tools in the characterization of amorphous ingredients and food systems (Roos, 1995 Slade and Levine, 1991). Slade and Levine (1988, 1991), acknowledging the earlier work of Franks et al. (1977) and MacKenzie (1977), formulated a state diagram (called a dynamics map or mobility transformation map ) for food systems that includes four dimensions temperature, concentration, pressure, and time. This state... 

Mobility will be used as a transcendent principle to connote all of these interdependent concepts embodied in the dynamics map in Figure 1. Thus, mobility will be the key to all transformations, as well as the basis for defining appropriate reference states (page 1842). 

Relation (6.40) represents a dynamical mapping which is mediated by Hamilton s equations of motion in the upper state and ultimately by the forces —dV/dR and —dV/dr and the torque —dV/d y. The energy dependence is mainly determined by the slope dV/dR of the potential in the direction of the dissociation path while dV/dr and dV/d y control the vibrational and rotational state distribution of the fragment. 

In order to discriminate this special case from dynamical mapping we introduce the term FC mapping. As an immediate consequence we note that ... 

Cheredilov BF (1967) Seasonal dynamics maps of the Black Sea surface. In Oceanographic investigations of the Black Sea. Naukova Dumka, Kiev, p 119 (in Russian)... 

If we disregard electronic phenomena, ignore the residual coupling terms between atomic quasi-particles, and specialize to the case of a single kind of composite quasi-particle, i.e. a pure chemical substance, at low density, the dynamical map P = W(X) allows us to write the Hamiltonian iC in the form,... 

As an example, consider a two-dimensional dynamical mapping with the simplest periodic orbit—that is, a fixed point at (qo,po)- Suppose that the initial condition (qo,Po) is only infinitesimally shifted from (qo,Po) with = dq p q— po= dp. With one iteration of the map, (qo,Po) evolves to (q, p ), and the initial errors dq and dp are propagated to bq and 8p, which is given by... 

The KAM theorem demonstrates the existence of KAM tori when the perturbations to the motion are small. What happens when a nearly integrable Hamiltonian is strongly perturbed For example, with increasing perturbation strength, what is the last KAM torus to be destroyed and how should we characterize the phase space structures when all KAM tori are destroyed Using simple dynamical mapping systems, which can be regarded as Poincare maps in Hamiltonian systems with two DOFs, MacKay, Meiss, and Percival [8,9] and Bensimon and Kadanoff [10] showed that the most robust KAM curve... 

One more way to conduct a similarity-based virtual screening is to retrieve the structures containing a user-defined set of pharmacophoric features. In the Dynamic Mapping of Consensus positions (DMC) algorithm those features are selected by finding common positions in bit strings for all active compounds. The potency-scaled DMC algorithm (POT-DMC) " is a modification of DMC in which compounds activities are taken into account. The latter two methods may be considered as intermediate between conventional similarity search and probabilistic SAR approaches. 

The functional aspects, in terms of water dynamics and glass dynamics, and the appropriate kinetic description of the nonequilibrium thermomechanical behavior of food systems have been illustrated as a dynamics map, shown in Figure 8.13. 

FIGURE 8.13 A four-dimensional dynamics map with axes of temperature, concentration, time, and pressure, which can be used to describe mobility transformation in nonequilibrium glassy and rubbery systems. Aw water activity RVP, relative vapour pressure. (From Slade, L. and Levin, H., Crit. Rev. Food Sci. Nulr., 30, 115, 1991.)... 

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