Cumulative evolution

To gain mechanistic insight into the reduction of allylic alcohol 46, the reaction was monitored by measuring the hydrogen evolution and ReactIR3). The addition of 0.5 equiv of Red-Al resulted in the evolution of 2 mol of H2 and the formation of the dimeric intermediate 83 (Scheme 7.26) [19b, 30], Figure 7.2 shows the cumulative evolution of hydrogen (mmol) and the cumulative amount of Red-Al ... 

Figure 19.2 Cumulative evolution of hydration heat in different types of Portland cement.

Figure 8.2. Cumulative evolution of HjS by ECU 18 ( ) or UCD 522 ( during fermentation of a synthetic grape juice containing variable concentrations of YAN and pantothenic acid. Within a given yeast, means with different letters are significantly different at p < 0.05 for ECU 18 (letters a to e) and UCD 522 (letters u to z). Adapted from Wang et al. (2003) with the kind permission of Blackwell Publishing.

Fig. 1. The time evolution (top) and average cumulative difference (bottom) associated with the central dihedral angle of butane r (defined by the four carbon atoms), for trajectories differing initially in 10 , 10 , and 10 Angstoms of the Cartesian coordinates from a reference trajectory. The leap-frog/Verlet scheme at the timestep At = 1 fs is used in all cases, with an all-atom model comprised of bond-stretch, bond-angle, dihedral-angle, van der Waals, and electrostatic components, a.s specified by the AMBER force field within the INSIGHT/Discover program.

Figure 3.42 Evolution of a pulse at the entrance of a micro channel for different diffusion coefficients. Calculated concentration profile (left) and cumulative residence time distribution curve (channel 300 pm x 300 pm x 20 mm flow velocity 1 m s f = 10 s) [27],...

Table 2 Cumulative CO, Evolution from Root Residues (RM), Soil Organic Matter (SOM), and Glucose During 24 Days of Incubation...

The relative skin (5 - S0) evolution with respect to the cumulative acid volume is given in Figure 3 (solid line). Peaks at about 0.5 and 3.5 m3 are artifacts due to hammering effects following rapid variations of the injection rate. According to Equations 6 and 13, the skin evolution, if the formation were a primary porosity one, should be ... 

The second considered example is described by the monostable potential of the fourth order (x) = ax4/4. In this nonlinear case the applicability of exponential approximation significantly depends on the location of initial distribution and the noise intensity. Nevertheless, the exponential approximation of time evolution of the mean gives qualitatively correct results and may be used as first estimation in wide range of noise intensity (see Fig. 14, a = 1). Moreover, if we will increase noise intensity further, we will see that the error of our approximation decreases and for kT = 50 we obtain that the exponential approximation and the results of computer simulation coincide (see Fig. 15, plotted in the logarithmic scale, a = 1, xo = 3). From this plot we can conclude that the nonlinear system is linearized by a strong noise, an effect which is qualitatively obvious but which should be investigated further by the analysis of variance and higher cumulants. 

It is clear that the strong form of the QCT is impossible to obtain from either the isolated or open evolution equations for the density matrix or Wigner function. For a generic dynamical system, a localized initial distribution tends to distribute itself over phase space and either continue to evolve in complicated ways (isolated system) or asymptote to an equilibrium state (open system) - whether classically or quantum mechanically. In the case of conditioned evolution, however, the distribution can be localized due to the information gained from the measurement. In order to quantify how this happ ens, let us first apply a cumulant expansion to the (fine-grained) conditioned classical evolution (5), resulting in the equations for the centroids (x = (t), P= (P ,... 

Fig. 2. Graphical representations of the evolution of cumulative concentrations of sulfur, calcium and magnesium collected throughout the kinetic test.

Figure 12. Time evolution of the internalised cumulated uptake 4>u (see equation (10)), total cumulated uptake

Figure 13. Detail of Figure 12, showing the asymptotic behaviour of the line A ni lf + jf l (dash-dotted line) with respect to the total cumulated uptake dashed line corresponds with the steady-state surface concentration A iii m. The evolution of the surface concentration F(t) = AficMfi o, t) is also shown... 

Figure 2. Carbon dioxide evolution from films containing cornstarch, LDPE, and EAA during soil burial. Circles 13% (v/v) starch squares 20% starch trianges 28% starch. values were obtained by linear regression of time versus cumulative CO2 evolved.

The reaction was followed continuously by measurement of H2 vented, through an electronic pressure controller (Rosemount Inst, model 5866) set at 9 bar, using a wet gas meter. Liquid samples, of 0.5 mL each, preceded by 0.5 mL to flush the line, were removed for NMR analysis at times corresponding to a spread of conversions from 10 to 90% as estimated from the cumulative hydrogen evolution. Spectra were recorded on 0.1 mL samples accurately diluted with 0.5 mL of D2O containing... 

The mathematics of diffusion at flat wall boundaries has been derived in Section 18.2 (see Fig. 18.5a-c). Here, the well-mixed system with large diffusivity corresponds to system B of Fig. 18.5 in which the concentration is kept at the constant value Cg. The initial concentration in system A, CA, is assumed to be smaller than Cg. Then the temporal evolution of the concentration profile in system A is given by Eq. 18-22. According to Eq. 18-23 the half-concentration penetration depth , x1/2, is approximatively equal to (DAt)m. The cumulative mass flux from system B into A at time t is equal to (Eq. 18-25) ... 

There is no need to worry about the existence of the reciprocal in (4.4) as we are merely concerned with a bookkeeping method for obtaining the successive terms of the cumulant expansion. Nor do we have to worry about the dependence of K on t0, because we have seen that this dependence disappears as soon as t — t0 > tc. After this transient the fluctuations around are determined by A(t) alone and hence also their influence on the evolution of itself. In this regime, u moves surrounded by a cloud of fluctuations, not unlike an electron surrounded by its cloud of virtual photons. 

The abundance of lithium in stellar atmospheres presents an important observational constraint to the hydrodynamical models of the outer layers of stars. It can be considered as a cumulative measure of the extent of matter exchange between surface and deeper layers during the stellar evolution. 

Upon completion of a measurement, the raw data were plotted as volts vs. time. The rate of water evolution or the cumulative water evolved was plotted as a function of temperature. The data were normalized by subtracting the corresponding blank measurement, and dividing by the weight of the sample. The quantities of water obtained by integrating under each desorption peak were tabulated as micrograms of water per gram of sample, and as molecules of water per square nanometer of surface. 

Figure 8. Evolution of carbon and heat from Douglas-fir foliage (A) in temperature intervals and (B) cumulative, based on dry weight of the unextracted...

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