Probability of Coincidence

All process components demonstrate unavailability as a result of a failure. For alarms and emergency systems it is unlikely that these systems will be unavailable when a dangerous process episode occurs. The danger results only when a process upset occurs and the emergency system is unavailable. This requires a coincidence of events. 

Assume that a dangerous process episode occurs pd times in a time interval Tv The frequency of this episode is given by 

The average frequency of dangerous episodes Ad is the number of dangerous coincidences divided by the time period  

For small failure rates U = j/rr, and pA = AT,. Substituting into Equation 11-27 yields 

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An alternative approach is to model the probability of coincidence as a function of either sample concentration or the fraction of time the sensing zone is occupied by passing cells (dead time). The model produces a coincidence-correction factor which is applied to the count. Still another approach is to reduce the size of the sensing zone, which reduces the probability of coincidence. In practice, commercial counters combine the latter two approaches. 

A different approach has been applied for the classification of flame images that provides information on the probability of coincidence with each of the combustion states previously known. This procedure is inspired on the cepstral analysis techniques, commonly used for speech recognition [37]. Although sound records are of a type different from image data, both can be equally transformed into covariance matrices, as required in this kind of method. 

To be considered only if there is either a causal dependence on SL-2 or a high probability of coincidence. 

Folding of a peptide probably occurs coincident with its biosynthesis (see Chapter 38). The physiologically active conformation reflects the amino acid sequence, steric hindrance, and noncovalent interactions (eg, hydrogen bonding, hydrophobic interactions) between residues. Common conformations include a-helices and P pleated sheets (see Chapter 5). 

Usually, experience of the past is used as the basis of failure scenarios, whereas one should look at a process each time again as if all unexpected events could occur. One has to keep in mind that accidents are often due to the highly unlikely coincidence or complex casual chains that seem improbable. It is necessary to examine all failure modes for all possible design alternatives in order to decrease the probability of an incident. 

The instantaneous composition of a copolymer X formed at a monomer mixture composition x coincides, provided the ideal model is applicable, with stationary vector ji of matrix Q with the elements (8). The mathematical apparatus of the theory of Markov chains permits immediately one to wright out of the expression for the probability of any sequence P Uk in macromolecules formed at given x. This provides an exhaustive solution to the problem of sequence distribution for copolymers synthesized at initial conversions p l when the monomer mixture composition x has had no time to deviate noticeably from its initial value x°. As for the high-conversion copolymerization products they evidently represent a mixture of Markovian copolymers prepared at different times, i.e. under different concentrations of monomers in the reaction system. Consequently, in order to calculate the probability of a certain sequence Uk, it is necessary to average its instantaneous value P Uk over all conversions p preceding the conversion p up to which the synthesis was conducted. 

Interestingly enough, quantity Ha (Eq. 84) has a rather transparent probabilistic meaning. In fact, the growth of the terminal a-th type block of a macroradical may be over either by the transition of an active center into another phase, or by its vanishing due to the chain termination reaction. The probabilities of these events, coinciding with the probabilities that a block chosen at random will be either internal or external, are equal to Ha and 1 -Ha, respectively. 

Figure 2.5 Electronic transitions with the greatest probability of absorption from S0(v = 0) (a) where both electronic states have similar geometries, shown by the minima of the curves being coincident (b) where the excited state has a larger intemuclear distance than the ground state...

At present it is universally acknowledged that TTA as triplet-triplet energy transfer is caused by exchange interaction of electrons in bimolecular complexes which takes place during molecular diffusion encounters in solution (in gas phase -molecular collisions are examined in crystals - triplet exciton diffusion is the responsible annihilation process (8-10)). No doubt, interaction of molecular partners in a diffusion complex may lead to the change of probabilities of fluorescent state radiative and nonradiative deactivation. Nevertheless, it is normally considered that as a result of TTA the energy of two triplet partners is accumulated in one molecule which emits the ADF (11). Interaction with the second deactivated partner is not taken into account, i.e. it is assumed that the ADF is of monomer nature and its spectrum coincides with the PF spectrum. Apparently the latter may be true when the ADF takes place from Si state the lifetime of which ( Tst 10-8 - 10-9 s) is much longer than the lifetime of diffusion encounter complex ( 10-10 - lO-H s in liquid solutions). As a matter of fact we have not observed considerable ADF and PF spectral difference when Sj metal lo-... 

So far the quasi-chemical approximation has been shown as a way to deal with 2-site probabilities. For lateral interactions we are usually dealing with probabilities of many more sites (e.g., see the 5-site probability in eqn. (8)). The quasi-chemical approximation becomes then much more cumbersome and is hardly ever used. The approach in Section 3.1.4 presents a way that is straightforward to extend to larger numbers of sites, while it can be made to coincide with the quasi-chemical approximation for 2-site probabilities. 

Now it is probably no coincidence that in addition to occupying the benzodiazepine receptor site on the chloride ion channel protein, these remarkable drugs can also occupy the sedation-convulsant receptor of the same protein. Sedation with benzodiazepines thus goes hand in hand with anticonvulsant power—as if the two processes had some deep mediating mechanism in common with each other. What could that be ... 

In this integral two or more t s might coincide although the value of the Qs in those points has not been defined. Fortunately, the set of such points is of measure zero in s-dimensional space, so that they do not contribute to the integral, provided it is agreed that Qs shall not contain delta functions of the type 8(t — t2). Thus we restrict ourselves to situations in which the dots do not have a positive probability to coincide the serving window must not be the one for marriage licences. 

Exercise. Suppose the dots have a non-zero probability to coincide in pairs. This may be described as a case of two species, namely singles and doubles. Show that the corresponding two-species distribution can be re-arranged as a one-species distribution Qs, which now does involve delta functions. 

We have studied above a model for the surface reaction A + 5B2 -> 0 on a disordered surface. For the case when the density of active sites S is smaller than the kinetically defined percolation threshold So, a system has no reactive state, the production rate is zero and all sites are covered by A or B particles. This is quite understandable because the active sites form finite clusters which can be completely covered by one-kind species. Due to the natural boundaries of the clusters of active sites and the irreversible character of the studied system (no desorption) the system cannot escape from this case. If one allows desorption of the A particles a reactive state arises, it exists also for the case S > Sq. Here an infinite cluster of active sites exists from which a reactive state of the system can be obtained. If S approaches So from above we observe a smooth change of the values of the phase-transition points which approach each other. At S = So the phase transition points coincide (y 1 = t/2) and no reactive state occurs. This condition defines kinetically the percolation threshold for the present reaction (which is found to be 0.63). The difference with the percolation threshold of Sc = 0.59275 is attributed to the reduced adsorption probability of the B2 particles on percolation clusters compared to the square lattice arising from the two site requirement for adsorption, to balance this effect more compact clusters are needed which means So exceeds Sc. The correlation functions reveal the strong correlations in the reactive state as well as segregation effects. 

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