Higher cumulants

Figure 4.52. Coefficients of variation that reflect both tablet to tablet and analytical variability. For formulation B, particularly strengths 2 and 3, the drop in CV with higher cumulative release (a - b) is marked, cf. Fig, 4.50. When the dissolution rate is high, individual differences dominate, while towards the end analytical uncertainty is all that remains. The very low CVs obtained with strength 3 of formulation A ( 0.7-0.8%, data offset by +10% for clarity) are indicative of the analytical uncertainty. Because content uniformity is harder to achieve the lower the drug-to-excipient ratio, this pattern is not unexpected.

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. 

The required analysis involves higher cumulants and has been carried out in Ref. (T. Bhattacharya et.al., 2000 2003). (Ref. (T. Bhattacharya et.al., 2000 2003) also points to previous literature.) It turns out that the distribution will be localized provided... 

We now turn to the quantum version of these results. In this case, the analogous cumulant expansion gives exactly the same equations for the centroids as above, while the equations for the higher cumulants are different. We can again investigate whether a trajectory limit exists. Localization holds in the weakly nonlinear case if the classical condition above is satisfied. In the case of strong nonlinearity, the inequality becomes... 

The low acute toxicity of LC-AmB would be expected to allow higher cumulative doses of the antibiotic to be administered. This hypothesis was tested in CDl mice, which were given various doses of LC-AmB daily for three weeks. Groups were also treated with Fungizone (0.5mg/kg) and Abelcet (10 mg/kg) according to the same regimen. At the end of the treatment period, mice were sacrificed and various biochemical parameters were measured (28). 

Johnson CK (1969) Addition of higher cumulants to the crystallographic structure-factor equation a generalized treatment for thermal-motion effects. Acta Crystallogr A 25 187-194... 

Other uses of fluoroscopy would have the potential for higher cumulative collar exposures. For example, abdominal interventional and angiography procedures typically use image intensifiers and x-ray field sizes which are larger than those used in a cardiac catheterization laboratory. Depending on the orientation of the primary beam, the scattered radiation from the patient may have greater intensity in these other clinical situations than in a cardiac catheterization laboratory. 

Formally, in the two steps of the DMS-process (dibromocyclopropanation and reaction with alkyllithium) a carbon atom is inserted between the two centers of a double bond. The reaction may be extended to the preparation of still higher cumulated bond systems as well as to numerous other — including functionalized — allenic systems which cannot or only with much effort be prepared by other routes. The examples shown here serve illustrative purposes only, for more extensive coverage of the literature the reader is referred to the various reviews and monographs which have appeared recently [66, 69, 71, 72]. 

Exercise. The cumulants of first and second degree of a multivariate Gaussian distribution are given by (6.9b). Prove that all higher cumulants vanish. 

Alternatively one may postulate that all higher cumulants are zero. This specifies all stochastic properties of L(t) in terms of the single parameter F. The L(t) defined in this way is called Gaussian white noise. From the mathematical point of view it does not really exist as a stochastic function (no more than the delta function exists as a function) and in physics it never really occurs but serves as a model for any rapidly fluctuating force. 

We consider a process A(t) which shares with the Langevin force L(t) the properties (1.2) and (1.3) but not the Gaussian property (3.1). The higher cumulants of A do not vanish but they are supposed to be delta-correlated... 

Summary. The semiclassical Boltzmann-Langevin method is extended to calculations of higher cumulants of current. Rs efficiency is demonstrated for mesoscopic diffusive contacts and chaotic cavities. We show that in addition to a dispersion at the inverse RC time characteristic of charge relaxation, higher cumulants of noise have a low-frequency dispersion at the inverse dwell time of electrons in the system. 

If the voltage is high enough, the noise of isolated contacts can be considered as white at frequencies at which the distribution function / fluctuates. This allows us to consider the contacts as independent generators of white noise, whose intensity is determined by the instantaneous distribution function of electrons in the cavity. Based on this time-scale separation, we perform a recursive expansion of higher cumulants of current in terms of its lower cumulants. In the low-frequency limit, the expressions for the third and fourth cumulants coincide with those obtained by quantum-mechanical methods for arbitrary ratio of conductances Gl/Gr and transparencies Pl,r [9]. Very recently, the same recursive relations were obtained as a saddle-point expansion of a stochastic path integral [10]. 

Of 53 bone marrow transplant recipients in whom ciclosporin was used to suppress graft-versus-host disease, 63% developed acute nephrotoxicity (304). These patients had significantly higher plasma ciclosporin concentrations during the first month after transplantation than those who did not develop acute nephrotoxicity, even though they received the same cumulative dose. Children received a higher cumulative dose, but their plasma concentrations did not differ significantly from the adults, and they suffered less nephrotoxicity. 

In addition to the acute neurotoxic sjmptoms caused by oxaliplatin, about 10-15% of patients develop a moderate neuropathy, particularly after cumulative intravenous doses of 700-800 mg/m. The symptoms of cumulative neuropathy include non-cold-related dysesthesia, paresthesia, superficial and deep sensory loss, and eventually sensory ataxia and functional impairment, which persists between treatment cycles. Most of these sjmptoms usually resolve a few weeks or months after oxaliplatin withdrawal. Lower cumulative doses (for example 510-765 mg/m ) and higher cumulative doses exceeding 1020 mg/m have been associated with incidences of cumulative grade 3 neurotoxicity of 3.2% and 50% respectively (8,9,76,77). In addition, higher cumulative doses, exceeding 1000 mg/m, have been associated with severe, atjrpical neurotoxic sjmptoms, such as micturition... 

Carbamazepine is a potent sodium channel blocker and has therefore been studied in the prevention of oxaliplatin-induced neuropathy (116). The doses of carbamazepine were adjusted to produce serum concentrations in the range 30-60 pg/ml. None of the patients who took carbamazepine reported symptoms of peripheral neurotoxicity however, two patients (one who forgot to take carbamazepine and one who stopped taking it because he felt tired) developed grade-1 peripheral sensory neurotoxicity. These symptoms were abolished when carbamazepine was restarted. One can therefore speculate that the concomitant use of carbamazepine may allow the use of a higher cumulative dose of oxaliplatin without the occurrence of grade-4 neuropathy. However, a multicenter trial is warranted to confirm these encouraging preliminary results (117). 

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