Combinatorial problem

Linking this molecular model to observed bulk fluid PVT-composition behavior requires a calculation of the number of possible configurations (microstmctures) of a mixture. There is no exact method available to solve this combinatorial problem (28). ASOG assumes the athermal (no heat of mixing) FIory-Huggins equation for this purpose (118,170,171). UNIQUAC claims to have a formula that avoids this assumption, although some aspects of athermal mixing are still present in the model. 

This paper presents a continuation of work done by Cayley. Cayley has repeatedly investigated combinatorial problems regarding the determination of the number of certain treesK Some of his problems lend themselves to chemical interpretation the number of trees in question is equal to the number of certain (theoretically possible) chemical compounds. 

The combinatorial problem on permutation groups stands out for its generality and the simplicity of the solution. The following... 

The solution of the combinatorial problem is determined by the following rule introduce... 

Polya s paper, translated here for the first time, was a landmark in the history of combinatorial analysis. It presented to mathematicians a unified technique for solving a wide class of combinatorial problems " a technique which is summarized in Polya s main theorem, the "Hauptsatz" of Section 16 of his paper, which will here be referred to as "Polya s Theorem". This theorem can be explained and expounded in many different ways, and at many different levels, ranging from the down-to-earth to highly abstract. It will be convenient for future reference to review the essentials of Polya s Theorem, and to this end I offer the following, rather mundane, way of looking at the type of problem to which the theorem applies and the way that it provides a solution. 

Among other things, Redfield s paper led to a heightened awareness of something that was already beginning to be realized, namely the interrelationship between Polya s Theorem (and other enumeration theorems) on the one hand, and the theory of symmetric functions, -functions, and group characters on the other it helped to show the way to the use of cycle index sums in the solution of hitherto intractable problems and in a more nebulous way it provided a refreshing new outlook on combinatorial problems. 

HarF64 Harary, F. Combinatorial problems in graphical enumeration. Applied Combinatorial Mathematics (E. F. Beckenbach, ed.) Wiley, New York (1964) 185-217. 

The protein structure prediction problem refers to the combinatorial problem to calculate the 3D structure of a protein from its sequence alone. It is one of the biggest challenges in structural bioinformatics. 

The combinatorial problem is represented by a discrete decision process (DDP) (Ibaraki, 1978) where the underlying information in the problem is captured by an explicit state-space model (Nilsson, 1980). 

Conformational analysis is a combinatorial problem. The number of conformations for a molecule with n rotatable bonds is given by ... 

As alternatives that avoid the combinatorial problem of selecting the active set, interior point (or barrier) methods modify the NLP problem (3-85) to form... 

L.M. Adlernan, Molecular computation of solutions to combinatorial problems. Science 266, 1021-1024 (1994). 

Heuristic search procedures can be applied to certain types of combinatorial problems when BB and OA are difficult to apply or converge too slowly. In these problems, it is difficult or impossible to model the problem in terms of a vector of decision variables, which must satisfy bounds on a set of constraint functions, as required by OA. One example is the traveling salesman problem, in which the feasible region is the set of all tours in a graph, that is, closed cycles or paths that visit every node only once. The problem is to find a tour of minimal distance or cost,... 

These combinatorial problems, and many others as well, have a finite number of feasible solutions, a number that increases rapidly with problem size. In a job-shop scheduling problem, the size is measured by the number of jobs. In a traveling salesman problem, it is measured by the number of arcs or nodes in the graph. For a particular problem type and size, each distinct set of problem data defines an instance of the problem. In a traveling salesman problem, the data are the travel times between cities. In a job sequencing problem the data are the processing and set-up times, the due dates, and the penalty costs. 

Genetic Algorithms (GA) are used in case of large combinatorial problems and can be applied e g. for example in complex value chain network design decisions (Chan/Chung 2004)... 

Branching and crosslinking processes can be treated as a combinatorial problem which is not too complicated when the functionalities are equally reactive and unreacted functionalities are considered as the only kind of defects. If loop formation (intra-molecular cyclization) is involved, the complexity increases considerably. The same holds if the monomers contain groups of unequal reactivity or if the reactivity is influenced by substitution effects. 

By assigning, for instance, binary variables to represent the existence or not of process units in a total process system, the potential matches of hot and cold streams so as to reduce the utility consumption or the existence of trays in a distillation- based separation system, the resulting total number of binary variables can grow to 1000 or more which implies that we have to deal with a large combinatorial problem (/. ., 21000 process flowsheets). 

Ql How can we deal with the large combinatorial problem effectively ... 

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