Exponential, time-ordered

The important point in the present context is that these cognitive abilities do not come for free. It is clear that high levels of intensionality are extremely difficult to cope with in computational terms. Kinderman et al. (in press), for example, tested normal adults with a series of tests similar to those used in standard ToM tests but which allowed for up to fifth order intensionality (as opposed to the conventional second order of standard ToM false belief tests). At the same time, subjects were also given tests of environmental causal relationships that required only memory of a sequence of events. Memory tests involved causal relationships of up to sixth orders of embeddness ( A caused B which caused C which. caused F ). Error rates on memory tasks varied fairly uniformly between 5-15% across the six levels of embeddness with no significant trends in contrast, error rates on the ToM tasks increased exponentially with order of embeddness (i.e. intensionality). 

Within this approximation, at least half of the photons that contribute to reflectance of a semi-infinite layer have been scattered only once. On (5-irradiation this part of photons is reflected from the illuminated surface with an exponential time profile Jo(t) = 2 xNoSac-exp(-(Sa + K )c-t), where N, is the number of incident photons at t = 0. Considering a weakly absorbing and not too strongly scattering sample (A = 1 cm S 102 cm-1), the decay time of the single-scattered photons is in the order of r = 500 fsec. The second half of photons is multiply scattered and decays... 

Optimal planar dividing smface VTST has been used to study the effects of exponential time dependent friction in Ref 93. The major interesting result was the prediction of a memory suppression of the rate of reaction which occms when the memory time and the inverse damping time (f) are of the same order. When... 

This formula is exact, but less simple than it looks. The time ordering requires that the exponential be expanded in a series and that in each term of that series the operators B are written in chronological order. That means that the multiple integrals have to be broken up in a number of terms for different parts of the integration domain. Before proceeding, however, we collect a number of properties of the time ordering in the form of Exercises. 

It is true that in the time-ordered exponential (4.2) the integral (4.8) does not occur as such, but is broken up in pieces and infiltrated with factors B coming from elsewhere, but that cannot raise its contribution to a higher order than (4.9). Hence the terms omitted in (4.3) are corrections to the coefficient of t in the exponent which are of higher order in / tc. 

Consider again the linear stochastic differential equation (2.1). There is no need now to assume At(t) stationary, nor to eliminate its average as was done in (2.2). Transform (2.1) to the interaction representation (2.3). According to (XV.3.9) a formal solution can be written by means of the time-ordered exponential... 

The difference with the series (2.4) is that now the expansion in a appears in the exponent. Of course, one must bear in mind that this is not an actual exponent because the time ordering can only be carried out after expanding the exponential - which brings us back to (2.4). Yet the following estimates remain true. 

An even better way to determine absolute rate constants is to use pre - steady state kinetics to measure the rate constants for the formation or decay of enzyme-bound intermediates (Chapter 4). The rate constants for first-order exponential time courses are independent of enzyme concentration and so are unaffected by the presence of denatured enzyme. The impurity just lowers the amplitude of the trace. Pre-steady state kinetics are also less prone to artifacts, discussed next, that are caused by the presence of small amounts of contaminants that have a much higher activity than the mutant being analyzed. The steady state kinetics of a weakly active mutant could be dominated by a fraction of a percent of wild type. In pre-steady state kinetics, however, that contaminant would contribute only a fraction of a percent of the amplitude of the trace. This would be either lost in the noise or observed as a minor fast phase. 

The time-evolution generated by the time-dependent hamiltonian is given by a time-ordered exponential form,... 

The time-local approach is based on the Hashitsume-Shibata-Takahashi identity and is also denoted as time-convolutionless formalism [43], partial time ordering prescription (POP) [40-42], or Tokuyama-Mori approach [46]. This can be derived formally from a second-order cumulant expansion of the time-ordered exponential function and yields a resummation of the COP expression [40,42]. Sometimes the approach is also called the time-dependent Redfield theory [47]. As was shown by Gzyl [48] the time-convolutionless formulation of Shibata et al. [10,11] is equivalent to the antecedent version by Fulinski and Kramarczyk [49, 50]. Using the Hashitsume-Shibata-Takahashi identity whose derivation is reviewed in the appendix, one yields in second-order in the system-bath coupling [51]... 

Besides P, which showed not be confused with the momentum operator P of the H-bond bridge, is the Dyson time-ordering operator [57] acting on the Taylor expansion terms of the exponential operator in such a way so that the time arguments involved in the different integrals will be t > t > t". 

Were it not for the coupling terms, kbAcD and k fAcA, Eq. 4.34 would have the same form as Eq. 1.55 (neglecting its constant term on the right side), with an exponential-decay solution typical of first-order reactions (Eqs. 1.56 and 4.19). Because the coupling terms are linear in the Ac, however, it is always possible to find a solution to Eq. 4.34 by postulating that a pair of time constants, r, and r2, exists such that the Ac still show an exponential time dependence ( rel axation ) 19... 

The approach that led us to the generalized master equation (2.13) can readily be repeated provided that the usual exponentials are replaced by time-ordered exponentials. We thus obtain... 

The roughest approximation consists of replacing the time-ordered exponential appearing in the second term on the right-hand side of Eq. (2.18) by 1. Since we intend to explore cases where the inhomogeneous term should not play any significant role (see Chapter II), Eq. (2.18) becomes... 

In the framework of many-body perturbation theory, one first defines the scattering matrix. S as a time-ordered exponential in terms of the perturbing Hamiltonian and field operators [471. Then, one considers the matrix elements corresponding to the proccs.s in which the recoiless probe particle carries the system either from an initial state a to a final state af >(, (single excitation) or from an initial state to a final state a/a (I>(f... 

When a drug is administered as an i.v. bolus, the entire dose of the drug is injected straight into the blood. Therefore, the absorption process is considered to be completed immediately, and the concentration-time profile of fhe drug in plasma will be determined by the rate of distribution and elimination. When the distribution of the drug is very fast, the plasma concentration-time curve is determined only by the elimination rate and shows a mono-exponential (first-order) decline (a theoretical example is shown in Figure 31.7a ... 

In this case, no back reaction occurs since Ru(bipy)3+ is reduced to the 2 + state by EDTA. While in the absence of catalyst the MV+ absorption is stable, addition of collodal Pt induces a decay of the signal, the rate of which increases sharply with Pt concentration. From a fitting of the absorbance decay curves to an exponential time law, one obtains the rate constants which are plotted as a function of Pt concentration in the lower part of Fig. 7.3. The ascent of the curve is steeper than linear, indicating that the reaction order is greater than one with respect to the Pt concentration. At a concentra-... 

Contrast the above situation with that of wide line FT NMR. First, we need the instrumental capabilities mentioned earlier. Secondly, often we are trying to obtain the shape of a single line or those of a small number of lines which may be complex. Therefore, we cannot presume to know the line shape in order to phase it properly. Similarly, an exponential time window function may alter the lineshape information. Third, the delay time used to avoid the pulse breakthrough in the FID is almost certain to be a significant fraction of the total acquisition time and must be taken into account. Let us deal with each of these difficulties in order. 

The optimal planar dividing surface VTST has been used to study a variety of problems. A study of space-independent but exponential time-dependent friction was presented in Ref. 43. The major interesting result was the prediction of a memory suppression of the rate of reaction which occurs when the memory time and the inverse damping time (l/-y) are of the same order. When this happens, the time it takes the particle to diffuse over the barrier is similar to the memory time and the particle feels the nonlinearity in the potential of mean force. This leads to substantial reduction of the rate relative to the parabolic barrier estimate. 

They compared the stretched exponential time dependence of i(f) they obtained by MD simulations with the formal Eq. (5.42), and asserted that the integral in the equation must have the time dependence of in order that they be the same. However, this argument is tantamount to a conjecture that remains to be verified. 

This procedure is nothing more than application of a digital filter with an exponential time constant equal to twice the muon lifetime. Figure 7 shows the effect of this transformation on the data of Figure 6. Shorter time constants for the exponential filter may be appropriate for short-lived signals, in order to discriminate against noise at later times however, broadening of the peaks in the Fourier spectrum is an unavoidable and undesirable consequence. 

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