Incomplete enumeration

It is unclear whether previously published fire risk analyses have adequately ircaicd dependent failures and systems interaetions. Examples of either experienced or postulated system interactions that have been missed include unrelated systems that share common locations and the attendant spatially related physical interactions arising from fire. Incomplete enumeration of causes of failure and cavalier assumptions of independence can lead to underestimation of accident l rci uencies by many orders of magnitude,... 

Critictil percolation cluster (using Leath algorithm [26]) with incomplete enumeration [10]. Critic il percolation cluster [68]. 

Incomplete enumeration is a quasi-static algorithm that generates a batch of SAWs from the variable-iV, variable-x ensemble (eq. (2.18)) with p = 0. The idea is to take a standard algorithm for systematically enumerating all self-avoiding walks up to some length Nmix, and modify it so that it pursues each branch of the SAW tree only with some probability / < 1 ... 

To perform an incomplete enumeration of the SAW tree beginning at the root , one simply invokes incenum( 0, /J,JVma ), where 0 is the zero-step walk at the origin. Typically one chooses j3 < fie = /IJ-, in this case it is safe to set Nmax = oo. (If one sets N vc = oo when > fic, then with nonzero probability the algorithm will run forever )... 

A non-recursive implementation of incomplete enumeration can be found in Ref. 11, Section 9. 3. 

The CPU time for one invocation of incomplete enumeration is obviously proportional to the total number of walks encountered during the enumeration. On the average this is... 

In any case, the incomplete-enumeration algorithm is potentially quite efficient, and deserves a systematic theoretical and empirical study. 

To obtain the connective constant fjL, use the shthering-tortoise algorithm or the incomplete-enumeration algorithm. 

What is the precise behavior of the incomplete-enumeration algorithm as iV oo (Section 2.5.3)... 

The water analysis is incomplete unless the number of coliform bacteria present is determined as well. A multiple-tube fermentation technique can be used to enumerate positive presumptive, confirmed, and fecal coliform tests. Results of the tests are expressed in terms of the most probable number (MPN). That is, the count is based on a statistical analysis of sets of tubes in a series of serial dilutions. MPN is related to a sample volume of 100 ml. Thus, an MPN of 10 means 10 coliforms per 100 ml of water. 

By the end of the seventeenth century, then, scientists were not really any closer to enumerating the elements than were the Greek philosophers. Yet a hundred years later the British chemist John Dalton (1766-1844) wrote a textbook that outlined a recognizably modern atomic theory and gave a list of elements that, while still very incomplete and sometimes plain wrong, is in content and in spirit a clear precursor to today s tabulation of the hundred and more elements. Why had our understanding of the elements changed so fast ... 

Any attempt to enumerate the areas in which surface and colloid chemical concepts find applicability is bound to be incomplete and quite variable with time because of changing technology. Nonetheless, we conclude this section with a partial listing of such applications. If there is any difficulty in doing this, it is because of the abundance rather than scarcity of such examples. 

Chemical relationships (defined by say having one or more molecules or even molecular substructures in common) may be particularly difficult to detect, even if individual documents are visually inspected, since molecules may not be represented in a consistent, and hence comparable, manner. Thus some molecules may merely be named, either trivially or systematically others may be drawn as two-dimensional line diagrams with perhaps components such as the Markush R convention denoting a variable group which may be enumerated elsewhere. Three dimensional properties such as stereochemistry may be even less precisely or incompletely specified. 

Taking first the important example of zeolitic, tetrahedrally connected nets, many enumerations have been made. These include compilations based on geometric principles, observed structures and the assembly of observed structural building units (such as chains, cages or sheets) via different symmetry operations. Some of the best-known compilations are those derived by Smith. The problem inherent to this approach is that the number of possible nets is infinite so that all compilations of this type will be incomplete. As a result, more systematic computer-based approaches have been taken to enumerate tetrahedrally coordinated crystalline networks. 

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