Recursive approach

AUGMENTED SPACE RECURSION APPROACH FOR ALLOY PHASE STABILITY... 

The recursive approach uses an elementary law of conditional expectation. Let A be an event and A its complement. Let Y be a random variable, E Y) its expectation (or average value) and E Y A) is conditional expectation, given that the event A has occurred. P A) is the probability that event A occurs. Then the law of total probability for expectation is [31] ... 

In addition to being a simpler method for obtaining the average properties such as My, and Mn compared to the Flory and similar approaches [31], the recursive approach also more easily allows an evaluation of the effect of unequal reactivity and unequal structural unit molecular weights on the average properties [27,28,33]. 

Problem 5.34 Calculate the gel-point conversions for the systems cited in Problem 5.27 using the recursive approach for comparison with the corresponding values calculated according to Flory-Stockmayer theory. 

Recalculate the gel point conversion of the polymerization system of Exercise 5.22 by the simpler recursive approach and compare the results. Also calculate the M of the polymer formed at 50% conversion of the A groups. Assume that the molecular weights of all the monomers in the polymerization system are equal and have a value of 100. 

Although the relations for probability of a finite chain and cross-link probability derived by the recursive approach remain unchanged on taking into account the formation of condensation by-products, the latter do affect the calculated value of sol fraction. Modify Eq. (5.231) for sol fraction w in homopolymerization of Af (f = 3), taking into account the formation of condensation by-product which is removed from the reaction mixture. 

Problem 5.32 Use the recursive approach to derive expressions for Mw and M,i for a polymerization system composed of equimolar amounts of A—A and B—B. To keep the system as simple as possible take the molecular weights of the two structural units as equal (denoted by Mq). 

In spite of its apparent simplicity, the direct method has several major shortcomings. First, the computational effort involved in solving the matrix eigenproblem in Eq. (53) scales as 0(N3). This feature, along with the storage requirements, which scale as N2, for H, C, and the required workspace, limit N in practice to no more than several thousand. The recursive approach described below is designed to overcome these unfavorable characteristics. 

Augmented Space Recursive Approach for Alloy Phase... 

In Chap. 3 we employed an asymptotic expansion of the conserved energy-like quantity of a numerical method in powers of the stepsize. By positing such an expansion we are able to calculate the term of any order directly, using a recursive approach. Given such an expansion of the Hamiltonian, we have a corresponding... 

Statistical methods which have been used to treat polymerization problems can be grouped into three categories direct, formal Markov chain theory, and recursive approaches. They differ in detail and objectives but all contain the underlying assumption that polymerization follows Markovian statistics. 

Lopez-Serrano, F., Castro, J.M., and Tirrell, M. (1980) Recursive approach to copolymerization statistics. Polymer 21, 263-273. 

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