Counting algebraic points in expansions of o-minimal structures by a dense set

The Pila-Wilkie theorem states that if a set $X\subseteq \mathbb R^n$ is definable in an o-minimal structure $\mathcal R$ and contains `many' rational points, then it contains an infinite semialgebraic set. In this paper, we extend this theorem to an expansion $\widetilde{\mathcal R}=\langle \mathcal R, P\rangle$ of $\mathcal R$ by a dense set $P$, which is either an elementary substructure of $\mathcal R$, or it is independent, as follows. If $X$ is definable in $\widetilde{\mathcal R}$ and contains many rational points, then it is dense in an infinite semialgebraic set. Moreover, it contains an infinite set which is $\emptyset$-definable in $\langle \overline{\mathbb R}, P\rangle$, where $\overline {\mathbb R}$ is the real field

Improving Personnel Selection through Value Focused Thinking

Personnel selection has always and will continue to be a challenging endeavor for the military special operations. They want to select the best out of a number of qualified applicants. How an organization determines what makes a successful candidate and how to compare candidates against each other are some of the difficulties that top tier organizations like the special operations face. Value focused thinking (VFT) places criteria in a hierarchal structure and quantifies the values with criteria measurements, known as a decision model. The selection process can be similar to a college selecting their students. This research used college student entry data and strategic goals as a proxy for special operations applicants and standards. It compared two case studies of college admissions selection criteria. A sample pool of 8,000 select and 24,000 non-select candidates was generated from real world datasets. VFT was applied to develop a valid admissions selection process model. The schools admissions documentation was used to build the hierarchies, single attribute value functions (SAVF), multi-attribute value functions (MAVF), and weights. A Monte Carlo simulation was used to sample applicants from the generated pool and examined how accurately the models were able to select the correct applicants