Exact enumeration technique

Most or all of these features have also been found (or extrapolated) from exact enumeration and Monte Carlo techniques. 

A good model for this phenomenon is a SAW on some lattice with an absorbing surface (boundary). Every site on surface visited by the polymer contributes an energy —E,. This model has widely been studied on various lattices and via a number of techniques that include exact enumeration [48,49], Monte Carlo [50], transfer matrix [51], renormalization group [52] etc. For a 2-d Euclidean lattice, exact value of found from conformal field theory is 1/2 [53]. 

This Chapter is organized as follows Section 2 revisits general definitions of quantities of interest, for both deterministic and random fractals, and introduces the notation used. Section 3 contains the numerical techniques such as Monte Carlo (MC) methods and the exact enumeration (EE) technique as well as data analysis schemes. Section 4 provides a discussion of SAWs on deterministic fractals, specifically the Sierpinski square and... 

This problem corresponds to an ILP (Integer Linear programming) problem which is a standard optimization problem (Mitchell, 2002). For medium-size systems (about 100 instrumentation points), this problem can be exactly solved by current computation capacity. Branch-and-Bound or Branch-and-Cut algorithms are enumeration techniques that are well-suited to solve this problem and that provide the optimal solution. 

Because of the short chain limitation of the exact enumeration method, Monte Carlo (MC) techniques have been widely applied to the excluded volume problem, notably by Wall and co-workers. One recent paper has investigated the mean square end-to-end distance of chains as a function of concentration and... 

In an exact-enumeration study, one first generates a complete list of all SAWs up to a certain length (usually N w 15-35 steps), keeping track of the properties of interest such as the number of such walks or their squared end-to-end distances. One then performs an extrapolation to the limit N - oo, using techniques such as the ratio method. Fade approximants or differential approximants. ° Inherent in any such extrapolation is an assumption about the behavior of the coefficients beyond those actually computed. Sometimes this assumption is fairly explicit other times it is hidden in the details of the extrapolation method. In either case, the assumptions made have a profound effect on the numerical results obtained. For this reason, the claimed error bars in exact-enumeration/extrapolation studies should be viewed with a healthy skepticism. 

In determining the density of states of protein models by MC methods, a number of other approaches are also useful. For systems for which it is feasible, exhaustive enumeration of all the conformations of the system by computer provides a viable and exact solution to the problem [61-66]. A number of powerful techniques have been developed, some for exhaustive enumeration of all conformations of relatively short lattice chains [61,63] others, for enumeration of protein conformations that are confined to a compact space [62,64-66]. However, in general protein models, the number of conformations is beyond the power of computer enumeration for these systems, MC methods are the main effective techniques. 

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