Tuplets

The linear space of all n-tuplets of complex numbers becomes an inner-product space if the scalar product of the two elements u and v is defined as the complex number given by... 

A complete unitary space is called a Hilbert space. The unitary spaces of finite dimension are necessarily complete. For reasons of completeness the vector space of all n-tuplets of rational numbers is not a Hilbert space, since it is not complete. For instance, it is possible to define a sequence of rational numbers that approaches the irrational number y/2 as a limit. The set of all rational numbers therefore does not define a Hilbert space. Similar arguments apply to the set of all n-tuplets of rational numbers. 

Consider a chain of N sites between which bonds are formed. A single unbonded site is referred to as a singlet, a run of two unbonded sites as a doublet, etc. In general, a run of n unbonded sites is termed an n-tuplet. It should be noted that an n-tuplet may contain two distinct (n — l)-tuplets, three distinct (n — 2)-tuplets, etc. This is illustrated in Figure 2. 

Consider an ensemble of M identical chains of N sites. Denote the number of n-tuplets in the /th chain at time t by CJ (t). Let k(t) dt be the probability that a bond forms between two unreacted neighbors in the time interval (f, t + dt), the same for all pairs of neighbors. The rate of change of CJ (t) with time is... 

The minus sign appears because the reaction is irreversible and n-tuplets can only be destroyed, never created. This, in fact, is exactly the useful... 

If we denote the mean number of n-tuplets averaged over the M identical chains by... 

Initially there arc N — n + 1 n-tuplets so that the initial conditions are... 

The fraction of M-tuplets which survive, or the probability of survival of an n-tuplet, is... 

We are interested in determining the time dependence of the number of unreacted groups. We now introduce the idea of a /-tuplet again. We denote the number of /-clusters by Bs and the number ofy -tuplets by Ps. Since there are j 1-tuplets in a /-cluster, j-1 2-tuplets (doublets) in a y -cluster, etc., one gets for the relation between the Pj and Bj,... 

It is now most convenient to proceed by deriving a set of equations for the rate of change of the number of /-tuplets, dPjjdt, in terms of the number of /-clusters, Bj, and then transforming this into a set of equations for the dPjdt in terms of the P. Figure 2 serves as a guide for the derivation of the first few equations. Clearly,... 

ChemX/ChemDiverse, PharmPrint, OSPPREYS, 3D Keys, Tuplets... 

Here ftj is the coagulation coefficient between an i-tuplet and a single particle, n0 is the initial single particle concentration, and (7)) is the average dissociation time for an i-tuplet to dissociate to an (i — I )-tuplet. 

The dynamic equations for the dimensionless concentrations of single particles and stable ic-tuplets can be written as... 

Let us now consider the change of j-tuplet probalMlities with time ... 

The complete solution to the problem of describing the sequence lengths distribution includes the calculation of the probability of finding any sequence of n units A and B. Equation (12) ifflows one to calculate the piobabiMes of mch sequences of unreacted units - nx-tuplets. Eridently this is not sufficient for tiie... 

The use of the idea of n x-tuplets is convenient because their probabilities can only decrease with time, and the probability of any sequence consisting of units A and B can be expressed with the aid of the probabilities of n x n -tuplets. For... 

P(AXA) can decrease with time due to the fact that two fixed A units can be converted into B with rate constants ko, kj and k2 dependent on the nature of the neighbors. As the site X can be occupied either by an A unit or by a B unit 4iidi can be on either side of the run AXA, P(AXA) is composed of 2 = 8 p-obabilities of different S-tuplets ... 

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