Optimal partitioning of an interval and applications to Sturm-Liouville eigenvalues

We study the optimal partitioning of a (possibly unbounded) interval of the real line into $n$ subintervals in order to minimize the maximum of certain set-functions, under rather general assumptions such as continuity, monotonicity, and a Radon-Nikodym property. We prove existence and uniqueness of a solution to this minimax partition problem, showing that the values of the set-functions on the intervals of any optimal partition must coincide. We also investigate the asymptotic distribution of the optimal partitions as $n$ tends to infinity. Several examples of set-functions fit in this framework, including measures, weighted distances and eigenvalues. We recover, in particular, some classical results of Sturm-Liouville theory: the asymptotic distribution of the zeros of the eigenfunctions, the asymptotics of the eigenvalues, and the celebrated Weyl law on the asymptotics of the counting function

Allocation in Practice

How do we allocate scarcere sources? How do we fairly allocate costs? These are two pressing challenges facing society today. I discuss two recent projects at NICTA concerning resource and cost allocation. In the first, we have been working with FoodBank Local, a social startup working in collaboration with food bank charities around the world to optimise the logistics of collecting and distributing donated food. Before we can distribute this food, we must decide how to allocate it to different charities and food kitchens. This gives rise to a fair division problem with several new dimensions, rarely considered in the literature. In the second, we have been looking at cost allocation within the distribution network of a large multinational company. This also has several new dimensions rarely considered in the literature.Comment: To appear in Proc. of 37th edition of the German Conference on Artificial Intelligence (KI 2014), Springer LNC

Towards a Formal Model of Recursive Self-Reflection

Self-awareness holds the promise of better decision making based on a comprehensive assessment of a system\u27s own situation. Therefore it has been studied for more than ten years in a range of settings and applications. However, in the literature the term has been used in a variety of meanings and today there is no consensus on what features and properties it should include. In fact, researchers disagree on the relative benefits of a self-aware system compared to one that is very similar but lacks self-awareness. We sketch a formal model, and thus a formal definition, of self-awareness. The model is based on dynamic dataflow semantics and includes self-assessment, a simulation and an abstraction as facilitating techniques, which are modeled by spawning new dataflow actors in the system. Most importantly, it has a method to focus on any of its parts to make it a subject of analysis by applying abstraction, self-assessment and simulation. In particular, it can apply this process to itself, which we call recursive self-reflection. There is no arbitrary limit to this self-scrutiny except resource constraints

On Partitioning Colored Points

P. Kirchberger proved that, for a finite subset $X$ of $\mathbb{R}^{d}$ such that each point in $X$ is painted with one of two colors, if every $d+2$ or fewer points in $X$ can be separated along the colors, then all the points in $X$ can be separated along the colors. In this paper, we show a more colorful theorem