and Applied Analysis 3 Proof. We can write terms of a period-three solution of 1.1 as x−1 a, x0 b, x1 ab − 1, x2 b ab − 1 − 1 a ⇐⇒ ab2 − b − 1 a ⇐⇒ ab2 a b 1, x3 a ab − 1 − 1 b ⇐⇒ a2b − a − 1 b ⇐⇒ a2b a b 1. 2.4 Therefore, this is indeed a solution of period three if the system below is satisfied. ab2 a b 1, a2b a b 1. 2.5 Hence we see that ab2 a2b. Thus, this is true if a 0, or if b 0, or if a...

Traditionally, transportation road networks are designed for minimal congestion. Unfortunately, such approaches do not guarantee minimal vehicle emissions. To fill this apparent gap in network design research, an emissions network design problem and solution method is proposed in this article for the purposes of comparing to the traditional network design results. Three air pollutants are consi...

The goal of minimal attribute reduction is to find the minimal subset R of the condition attribute set C such that R has the same classification quality as C. This problem is well known to be NP-hard. When only one minimal attribute reduction is required, it was transformed into a nonlinearly constrained combinatorial optimization problem over a Boolean space and some heuristic search approache...

Constraint satisfaction has become ubiquitous in the modern world, able to model and offer solution techniques for many of the most pressing problems. However, no matter how good the techniques become, some problems just do not have a satisfiable solution, and are referred to as inconsistent. Yet even when such problems are considered, one is still interested in finding the minimal subset of co...

A large number of absolute pose algorithms have been presented in the literature. Common performance criteria are computational complexity, geometric optimality, global optimality, structural degeneracies, and the number of solutions. The ability to handle minimal sets of correspondences, resulting solution multiplicity, and generalized cameras are further desirable properties. This paper prese...

Temporal query checking is an extension of temporal model checking where one asks what propositional formulae can be inserted in a temporal query (a temporal formula with a placeholder) so that the resulting formula is satisfied in the model at hand. We study the problem of computing all minimal solutions to a temporal query without restricting to so-called “valid” queries (queries guaranteed t...