Survival probability function

The corresponding credit curves, which are consistent with these survival probabilities, take the form shown in Exhibit 21.20. This shows that the credit curve inversion is consistent with the changes in the survival probability functions. 

In general the survival probability is of interest if a system has to maintain its function during a certain period of time (e.g. a rocket). If, on the other hand, a system has to function on demand, as for example a trip system, the availability is the adequate parameter. Of course, combinations of both parameters can also be appropriate. A standby system like an emergency power supply needs a high availability (probability to start) and a high survival probability (functioning until the grid supply is restored, i.e. until mission time t). 

The density distribution of distance between the chain ends corresponding to SAWs statistics has been presented by Pietronero as multiplication of two functions w R) S (R) G (R). where one is the Gaussian function G (R) - cxp -dR /2a N and determines the probability that the trajectory of end-chain RW via N steps ends in a spherical layer R,R + dR. The survival probability function S(R) selects these trajectories of RW which are not self-avoided. Such function has been determined starting from the following point of view... 

Fig. 18.3. (a) The Weibull distribution function, (b) When the modulus, m, changes, the survival probability changes os shown. 

Figure 2 Survival probability of geminate ion pairs as a function of time. The two solid lines correspond to two different values of the initial electron-cation distance. The broken lines show the asymptotic kinetics calculated from Eq. (25). The value of the escape probability for Tq = O.Sr is indicated by

The probability that the ions have a separation r, given that one ion is at r0, at time t is the density p(r, t r0, t0), which is also related to the Green s function of the diffusion equation (see Sect. 2.3 and the discussion in Appendix A). As before, the survival probability is the integral of the density over all space [eqn. (123)] and this may be related to the flux, J, crossing the encounter surface [eqn. (124)], which is (following Chap. 3, Sect. 1.1)... 

This Green s function displays the same reciprocity to interchange of the co-ordinate r, t with r0, f° as does the single ion-pair Green s function [72, 499] [eqn. (162)]. Using exactly the same techniques as used in Chap. 7, Sect. 2.3, the equation satisfied by the survival probability can be shown to be... 

An isolated ion-pair, of initial separation r0 at time t0 = 0, in a nonpolar solvent may recombine or separate and ultimately escape. At a time t, the probability that the ion-pair will have recombined is < (t r0, t0 = 0) and that it is still extant p(t r0) f0 = 0). A short while later, the probability that the ion-pair has not recombined is p t + df r0, t0 =0). The change in survival probability is the probability that the ion-pair recombined during the time interval t to t + df, that it had a lifetime between t and t -f df. Defining the lifetime distribution function as f(f), then... 

Measurement of the survival probability of Rydberg molecules as a function of delay time for pulsed-field ionization. 

The above-described pair problem is treated by the Smoluchowski equation [3, 19] - see Fig. 1.10. It operates with the probability densities (Fig. 1.11) and contains the recombination rate characterizing particle motion. Knowledge of the probability density to find a particle at a given point at time moment t gives us (by means of a trivial integration over reaction volume) the quantity of our primary interest - survival probability of a particle in the system with... 

If there is no interaction between similar reactants (traps) B, they are distributed according to the Poisson relation, Ab (r, t) = 1. Besides, since the reaction kinetics is linear in donor concentrations, the only quantity of interest is the survival probability of a single particle A migrating through traps B and therefore the correlation function XA(r,t) does not affect the kinetics under study. Hence the description of the fluctuation spectrum of a system through the joint densities A (r, ), which was so important for understanding the A4-B — 0 reaction kinetics, appears now to be incomplete. The fluctuation effects we are interested in are weaker here, thus affecting the critical exponent but not the exponential kinetics itself. It will be shown below that adequate treatment of these weak fluctuation effects requires a careful analysis of many-particle correlations. 

This function has the long-time behavior pa(t) Cat a, where Ca is a constant. The survival probability for the subdiffusive case is plotted in Fig. 3 and compared with the Brownian survival. Clearly, for long times, the survival probability in the subdiffusive system decays in a much slower fashion. 

The only essential components of the mineral deposition mechanism that are fairly certain at this time relate to phosphate. Even for phosphate, alternative mechanisms are proposed, which are not mutually exclusive but probably function in parallel, in the regulation of different aspects of skeletal calcium transport, and to some extent provide redundancy that allows many mineral transport disorders to be survivable. Alkaline phosphatase activity is essential to produce phosphate. Its major substrate is pyrophosphate. In the absence of the alkaline phosphatase, normally highly expressed as an ectoenzyme by osteoblasts, there is little matrix mineralization... 

The function F(t — t ) is related, as with the temporary network model of Green and Tobolsky (48) discussed earlier, to the survival probability of a tube segment for a time interval (f — t ) of the strain history (58,59). Finally, this Doi-Edwards model (Eq. 3.4-5) is for monodispersed polymers, and is capable of moderate predictive success in the non linear viscoelastic range. However, it is not capable of predicting strain hardening in elongational flows (Figs. 3.6 and 3.7). 

Figure 3.27. The separation quantum yield (survival probability at t = oo) as a function of the initial distance between the ions for D = 1CT5 cm2/s and three recombination rates (from top to bottom), Wo = 10,100,1000 ns-1 (a = 5 A, = 10A,L = 5 A). At the top (a) the start from inside the recombination layer related to the left, horizontal branches of the curves (b) the outside start related to the right branches approaching the maximum tp = 1 at ro —> oo. (From Refs. 32 and 158.)...

Figure 3.57. The ion survival probability as a function of time at To = 0.5 ns with a great excess of acceptors. In line with UT and IET (above) and Markovian theory (below) (dashed curve), the contact approximation (dashed-dotted line in the middle) and exponential model with fcjep = A et = 1.0 ns-1 (dotted line) are also shown. The horizontal thick lines indicate the free-ion quantum yield ((). The concentrations and ionization parameters are the same as in Figure 3.56, while wy = 3.4ns-1, D = D = 1.2 X 10-6 cm2/s, k1 — 7S4 A3/ns, and kr — 4S6 A3/ns. (From Ref. 195.)...

To obtain this interesting result we used also Eq (64). In conclusion the function 11(f) coincides with the function T(f) of Eq. (63). It is natural to call this quantity survival probability. 

Figure 42 displays a computational example of P i) (solid line) and the survival probability C(f)p (dashed line) as a function of the total number of states N and the number of the prompt states K. The initial state is taken to have uniform weights for either the K prompt states or the N — K delayed states. It is seen that with fixed K and increasing N the decay of the initial delayed state is shifted to much longer time, whereas the decay of the initial prompt state changes little. 

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