Several papers have highlighted the potential of network science to appeal to a younger audience of high school children and provided lesson material on network science for high school children. However, network science also provides a great topic for outreach activities for primary school children. Therefore, this article gives a short summary of an outreach activity on network science for primary school children aged 8–12 years. The material provided in this article contains presentation material for a lesson of approximately 1 hour, including experiments, exercises, and quizzes, which can be used by other scientists interested in popularizing network science. We then discuss the lessons learned from this material.

In this article, we improve the efficiency of a turbine blade inspection robotic workcell. The workcell consists of a stationary camera and a 6-axis serial robot that is holding a blade and presenting different zones of the blade to the camera for inspection. The problem at hand consists of a 6-DOF (degree of freedom) continuous optimization of the camera placement and a discrete combinatorial optimization of the sequence of inspection poses (images). For each image, all robot configurations (up to eight) are taken into consideration. A novel combined approach is introduced, called blind dynamic particle swarm optimization (BD-PSO), to simultaneously obtain the optimal design for both domains. The objective is to minimize the cycle time of the inspection process, while avoiding any collisions. Even though PSO is vastly used in engineering problems, the novelty of our combinatorial optimization method is in its ability to be used efficiently in traveling salesman problems where the distances between the cities are unknown and subject to change. This highly unpredictable environment is the case of the inspection cell where the cycle time between two images will change for different camera placements.

Planning paths that are length or time optimized or both is an age-long problem for which numerous approaches have been proposed with varied degree of success depending on the imposed constraints. Among classical instances in the literature, the Traveling Salesman Problem and the Vehicle Routing Problem have been widely studied and frequently considered in the realm of mobile robotics. Understandably, the classical formulation for such problems do not take into account many different issues that arise in real-world scenarios, such as motion constraints and dynamic environments, commonly found in actual robotic systems, and consequently the solutions have been generalized in several ways. In this work, we present a broad and comprehensive review of the classical works and recent breakthroughs regarding the routing techniques ordinarily used in robotic systems and provide references to the most fundamental works in the literature.

In this work we consider the mean-field traveling salesman problem, where the intercity distances are taken to be independent and identically distributed with some distribution F. We consider the simplest approximation algorithm, namely, the nearest-neighbor algorithm, where the rule is to move to the nearest nonvisited city. We show that the limiting behavior of the total length of the nearest-neighbor tour depends on the scaling properties of the density of F at 0 and derive the limits for all possible cases of general F.

The traveling salesman problem (TSP) is one of the most fundamental optimizationproblems. We consider the β-metric traveling salesman problem(Δβ-TSP), i.e., the TSPrestricted to graphs satisfying the β-triangle inequalityc({v,w}) ≤ β(c({v,u}) + c({u,w})),for some cost function c and any three vertices u,v,w.The well-known path matching Christofides algorithm (PMCA) guarantees an approximationratio of 3β2/2 and is the best known algorithm for theΔβ-TSP, for 1 ≤ β ≤ 2. Weprovide a complete analysis of the algorithm. First, we correct an error in the originalimplementation that may produce an invalid solution. Using a worst-case example, we thenshow that the algorithm cannot guarantee a better approximation ratio. The example canalso be used for the PMCA variants for the Hamiltonian path problem with zero and oneprespecified endpoints. For two prespecified endpoints, we cannot reuse the example, butwe construct another worst-case example to show the optimality of the analysis also inthis case.

Given a class Γ of curves in [0, 1]2, we ask: in a cloud of n uniform random points, how many points can lie on some curve γ ∈ Γ? Classes studied here include curves of length less than or equal to L, Lipschitz graphs, monotone graphs, twice-differentiable curves, and graphs of smooth functions with m-bounded derivatives. We find, for example, that there are twice-differentiable curves containing as many as OP(n1/3) uniform random points, but not essentially more than this. More generally, we consider point clouds in higher-dimensional cubes [0, 1]d and regular hypersurfaces of specified codimension, finding, for example, that twice-differentiable k-dimensional hypersurfaces in Rd may contain as many as OP(nk/(2d-k)) uniform random points. We also consider other notions of ‘incidence’, such as curves passing through given location/direction pairs, and find, for example, that twice-differentiable curves in R2 may pass through at most OP(n1/4) uniform random location/direction pairs. Idealized applications in image processing and perceptual psychophysics are described and several open mathematical questions are identified for the attention of the probability community.

This paper presents a degree-two probability lower bound for the union of an arbitrary set of events in an arbitrary probability space. The bound is designed in terms of the first-degree Bonferroni summation and pairwise joint probabilities of events, which are represented as weights of edges in a Hamilton-type circuit in a connected graph. The proposed lower bound strengthens the Dawson–Sankoff lower bound in the same way that Hunter and Worsley's degree-two upper bound improves the degree-two Bonferroni-type optimal upper bound. It can be applied to statistical inference in time series and outlier diagnoses as well as the study of dose response curves.