Consequence Space

One major class of consequences is the adverse health effects including acute, latent fatalities and injuries. Another class of consequences addresses the so called socioeconomic costs of the ERP including psychological effects on the population subject to the ERP, social and economic effects of disruption of normal, everyday activities, etc. A set of distinct attributes, each measuring the degree to which each area of concern is affected as a result of the established ERP, is thus determined. As a result, each decision leads to a multidimensional consequence. 

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From the dose calculation, the risk of specific adverse effect can be calculated and hence the consequences in the multidimensional space discussed above. Thus, given a specific ERP, the consequences in the form of specific values of the established set of attributes are determined and this creates the desired mapping between the decision and the consequence space. 

In this geometric approach the consequent space is overlaid upon the premise coordinate system and is in effect partitioned into seven small nonoverlapping regions, where each region represents a consequent fuzzy set (see Figure 1). To design... 

Slope ofthe Consequent Line Angle (CA) It has been used to create different output spaee partitions. The angle is encoded to cover angles between 0 -180°. As the consequent space is S5munetric and the output u(t) ranges between [0, I], 0-180° is equivalent to 0 - 360°. 

Consider that at low temperatures, a lubricant is a poor solvent for polymer chains. When the temperature increases, interactions between polymer chains decrease the space occupied by the polymer ball takes on greater volume and consequently, the viscosity decrease due to the lubricant temperature increase is compensated by the unfolding of the polymer chain and the result is a reduction of the difference between the viscosities at low and high temperature, and therefore an Increase in viscosity index. 

Let us consider the consequence of mechanics for the ensemble density. As in subsection A2.2.2.1. let D/Dt represent differentiation along the trajectory in F space. By definition,... 

In other words, if we look at any phase-space volume element, the rate of incoming state points should equal the rate of outflow. This requires that be a fiinction of the constants of the motion, and especially Q=Q i). Equilibrium also implies d(/)/dt = 0 for any /. The extension of the above equations to nonequilibriiim ensembles requires a consideration of entropy production, the method of controlling energy dissipation (diennostatting) and the consequent non-Liouville nature of the time evolution [35]. 

The velocity Verlet algorithm may be derived by considering a standard approximate decomposition of the Liouville operator which preserves reversibility and is symplectic (which implies that volume in phase space is conserved). This approach [47] has had several beneficial consequences. 

In tlie previous section we showed tliat because tlie stmcture space is very sparse tliere have to be many sequences tliat map onto tlie countable number of basins in tlie stmcture space. The kinetics here shows tliat not all tlie sequences, even for highly designable stmctures, are kinetically competent. Consequently, the biological requirements of stability and speed of folding severely restrict tlie number of evolved sequences for a given fold. This very important result is schematically shown in figure C2.5.4. 

Consequently, Eqs. (43) and (59) are identical, for applications in a 3D parameter space, except that the vector product in the former is expressed as a commutator in the latter. Both are computed as diagonal elements of combinations of strictly off-diagonal operators and both give gauge independent results. Equally, however, both are subject to the limitations with respect to the choice of surface for the final integration that are discussed in the sentence following Eq. (43). 

Calculating points on a set of PES, and fitting analytic functions to them is a time-consuming process, and must be done for each new system of interest. It is also an impossible task if more than a few (typically 4) degrees of freedom are involved, simply as a consequence of the exponential growth in number of ab initio data points needed to cover the coordinate space. 

The literature on ergodic theory contains an interesting theorem concerning the spectrum of the Frobenius-Perron operator P. In order to state this result, we have to reformulate P as an operator on the Hilbert space L P) of all square integrable functions on the phase space P. Since and, therefore, / are volume preserving, this operator P L P) —+ L r) is unitary (cf. [20], Thm. 1.25). As a consequence, its spectrum lies on the unit circle. 

The exact propagator for a Hamiltonian system for any given time increment At is symplectic. As a consequence it possesses the Liouville property of preserving volume in phase space. 

Caustics The above formulae can only be valid as long as Eq. (9) describes a unique map in position space. Indeed, the underlying Hamilton-Jacobi theory is only valid for the time interval [0,T] if at all instances t [0, T] the map (QOi4o) —> Q t, qo,qo) is one-to-one, [6, 19, 1], i.e., as long as trajectories with different initial data do not cross each other in position space (cf. Fig. 1). Consequently, the detection of any caustics in a numerical simulation is only possible if we propagate a trajectory bundle with different initial values. Thus, in pure QCMD, Eq. (11), caustics cannot be detected. 

When, in a column headed M.p., a value is given in parenthesis, it indicates that the compound is liquid at room temperature and that the value given is consequently the boiling-point. Conversely in a column headed B.p., values given in parenthesis are those of the melting-point. A blank space indicates that the compound has not apparently been recorded. 

Stereochemistry refers to chemistry in three dimensions Its foundations were laid by Jacobus van t Hoff and Joseph Achille Le Bel m 1874 Van t Hoff and Le Bel mde pendently proposed that the four bonds to carbon were directed toward the corners of a tetrahedron One consequence of a tetrahedral arrangement of bonds to carbon is that two compounds may be different because the arrangement of their atoms m space IS different Isomers that have the same constitution but differ m the spatial arrangement of their atoms are called stereoisomers We have already had considerable experience with certain types of stereoisomers—those involving cis and trans substitution patterns m alkenes and m cycloalkanes... 

Following the pioneer work of Beebe in 1945, the adsorption of krypton at 77 K has come into widespread use for the determination of relatively small surface areas because its saturation vapour pressure is rather low (p° 2Torr). Consequently the dead space correction for unadsorbed gas is small enough to permit the measurement of quite small adsorption with reasonable precision. Estimates of specific surface as low as 10 cm g" have been reported. Unfortunately, however, there are some complications in the interpretation of the adsorption isotherm. 

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