Total time ordered cumulant

According to the previous section, we shall start by considering X and P as fast degrees of freedom, relaxing on a much more rapid timescale than the orientational coordinates and momenta of the solute and the solvent cage. Many different projection schemes are available to handle stochastic partial differential operators. Here we choose to adopt a slightly modified total time ordered cumulant (TTOC) expansion procedure, directly related to the well known resolvent approach. In order to make this chapter self-contained, we summarize the method in the Appendices and its application to the cases considered here and in the next section. 

In this appendix we review briefly the TTOC (total time ordered cumul-ant) procedure applied to a general linear time evolution operator. The same technique was used by Stillman and Freed [33] for other details see Yoon et al. [28] and Hwang and Freed [35], and references quoted therein. Also we show how to apply the TTOC procedure for projecting out a subset of fast momenta, from a phase space of coordinates and momenta. 

To process the entire sample volume requires approximately 2 centrifuge run(s) with an estimated total run time of 12 hours, 5 minutes. SpinPro determines how many runs are required to process the entire sample volume. The total run time is estimated. When large sample volumes are involved, and thus many runs are required, the investigator can change the optimization criterion to "minimize number of runs" or "minimize cumulative run time" in order to more efficiently process the sample. Since two runs are required here, the investigator may want to select a larger rotor for use in the Lab Plan. 

Successive orders H(n ) can be shown to correspond to successive orders in a moment (or cumulant) expansion of the propagator, which takes one to increasing times. Truncation of the chain at a given order n (i.e., 3 + 3n modes) leads to an approximate, lower-dimensional representation of the dynamical process, which reproduces the true dynamics up to a certain time. In Ref. [51], we have demonstrated explicitly that the nth-order (3n+3 mode) truncated HEP Hamiltonian exactly reproduces the first (2n + 3)rd order moments (cumulants) of the total Hamiltonian. A related analysis is given in Ref. [73],... 

Water used in the experiments was doubly distilled and passed through an ion exchange unit. The conductivity was approximately 1 x 10"6 S/m. Simulated HLLW consisted of 21 metal nitrates in an aqueous 1.6 M nitric acid solution as shown in Table 1 and was supplied by EBARA Co. (Tokyo, Japan). Concentrations were verified by AA for Na and Cs with 1000 1 dilution and by ICP for the other elements with 100 1 dilution. Total metal ion concentration was 98,393 ppm. The experimental apparatus consisted of nominal 9.2 cm3 batch reactors (O.D. 12.7 mm, I.D. 8.5 mm) constructed of 316 stainless steel with an internal K-type thermocouple for temperature measurement. Heating of each reactor was accomplished with a 50%NaNO2 + 50% KNO 2 salt bath that was stirred to insure uniform temperature. Temperature in the bath did not vary more than 1 K. The reactors were loaded with the simulated HLLW waste at atmospheric conditions according to an approximate calculated pressure. Each reactor was then immersed in the salt bath for 2 min -24 hours. After a predetermined time, the reactor was removed from the bath and quenched in a 293 K water bath. The reactor was opened and the contents were passed through a 0.1 pm nitro-ceflulose filter while diluting with water. Analysis of the liquid was performed with methods in Table 1. Analysis of filtered solids were carried out with X-ray diffraction with a CuK a beam and Ni filter. Reaction time was defined as the time that the sample spent at the desired temperature. Typical cumulative heat-up and cool-down time was on the order of one minute. Results of this work are reported in terms of recoveries as defined by ... 

The normal plot of the effects (Fig. 4A.9) is very easy to interpret, and shows that gel time is totally controlled by three factors (5 = dimethylaniline, 4 = copper octanoate and 1 = hydroquinone, in this order of importance). Factors 4 and 5 contribute to lower the gel time, whereas factor 1 tends to increase it. There is no evidence of significant interactions. This is welcome, because it means the factors can be varied independently of one another, at the customer s convenience. The numerical values on the right side of the plot are the cumulative probabilities corresponding to the z values on the left. 

The current level of lead emissions from mobile sources such as autos, trucks, farm equipment, and aircraft, is a small fraction of the total lead releases nationally and globally that occurred up to the early and mid-1970s. Chapters 3 and 4 detailed and quantified this dramatic reduction. However, lead from mobile sources largely settled in the environmental compartments of soils and dusts, where residence time is on the order of decades. This cumulative tally has been estimated for the United States at over 3 million MT. Furthermore, current lead inputs to existing lead loadings in soils and as dusts even at reduced values occur because of the ready remobilization of small dust particles from leaded soil surfaces. In a number of geographic areas with heavily used traffic arteries, roadside soil lead releases as fine particle dusts can produce significant localized elevations in air lead content. 

In order to control the exhaustion rate, the equations introduced by Nobbs and Ren can be used. These make use of the total elapsed time since the start of the cycle. However, in order to minimise the cumulative errors associated with random deviation of concentration values from the set values, the exhaustion rate can be calculated from the current concentration (or exhaustion). In this method, the consequence of the initial high strikes of some of the profiles at the start of the dyeing is suppressed and the dyeing demonstrates less deviation, since the model restarts at each measured point. Practical determination of the target exhaustion rate takes this into account by replacing e with estart. 

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