Phase transitions in least-effort communications

Original article can be found at: http://iopscience.iop.org Copyright IOP PublishingWe critically examine a model that attempts to explain the emergence of power laws (e.g., Zipf's law) in human language. The model is based on the principle of least effort in communications—specifically, the overall effort is balanced between the speaker effort and listener effort, with some trade-off. It has been shown that an information-theoretic interpretation of this principle is sufficiently rich to explain the emergence of Zipf's law in the vicinity of the transition between referentially useless systems (one signal for all referable objects) and indexical reference systems (one signal per object). The phase transition is defined in the space of communication accuracy (information content) expressed in terms of the trade-off parameter. Our study explicitly solves the continuous optimization problem, subsuming a recent, more specific result obtained within a discrete space. The obtained results contrast Zipf's law found by heuristic search (that attained only local minima) in the vicinity of the transition between referentially useless systems and indexical reference systems, with an inverse-factorial (sub-logarithmic) law found at the transition that corresponds to global minima. The inverse-factorial law is observed to be the most representative frequency distribution among optimal solutions.Peer reviewe

Reciprocal Sums as a Knowledge Metric: Theory, Computation, and Perfect Numbers

We first provide a short survey of reciprocal sums. We discuss some of the history of their computation and application, show how they are applied in various modern contexts, and discuss some ways that their values are computed. We give an example of computing a reciprocal sum by providing (we believe) the first computation of the sum of the reciprocals of perfect numbers. Second, we introduce a new use for reciprocal sums; that is, they can be used as a knowledge metric to classify the current state of number theorists’ understanding of a given class of integers