Enumeration of 3-letter patterns in compositions

Let A be any set of positive integers and n a positive integer. A composition of n with parts in A is an ordered collection of one or more elements in A whose sum is n. We derive generating functions for the number of compositions of n with m parts in A that have r occurrences of 3-letter patterns formed by two (adjacent) instances of levels, rises and drops. We also derive asymptotics for the number of compositions of n that avoid a given pattern. Finally, we obtain the generating function for the number of k-ary words of length m which contain a prescribed number of occurrences of a given pattern as a special case of our results.Comment: 20 pages, 1 figure; accepted for the Proceedings of the 2005 Integer Conferenc

Building Nim

The game of nim, with its simple rules, its elegant solution and its historical importance is the quintessence of a combinatorial game, which is why it led to so many generalizations and modifications. We present a modification with a new spin: building nim. With given finite numbers of tokens and stacks, this two-player game is played in two stages (thus belonging to the same family of games as e.g. nine-men's morris): first building, where players alternate to put one token on one of the, initially empty, stacks until all tokens have been used. Then, the players play nim. Of course, because the solution for the game of nim is known, the goal of the player who starts nim play is a placement of the tokens so that the Nim-sum of the stack heights at the end of building is different from 0. This game is trivial if the total number of tokens is odd as the Nim-sum could never be 0, or if both the number of tokens and the number of stacks are even, since a simple mimicking strategy results in a Nim-sum of 0 after each of the second player's moves. We present the solution for this game for some non-trivial cases and state a general conjecture

USE OF ARTIFICIAL FIDUCIAL MARKERS FOR USV SWARM COORDINATION

Typical swarm algorithms (leader-follower, artificial potentials, etc.) rely on knowledge about the pose of each vehicle and inter-vehicle proximity. This information is often obtained via Global Positioning System (GPS) and communicated via radio-frequency means.. This research examines the capabilities and limitations of using a fiducial marker system in conjunction with an artificial potential field algorithm to achieve inter-vehicle localization and coordinate the motion of unmanned surface vessels operating together in an environment where satellite and radio communications are inhibited. Using Gazebo, a physics-based robotic simulation environment, a virtual model is developed for incorporating fiducial markers on a group of autonomous surface vessels. A control framework using MATLAB and the Robot Operating System (ROS) is developed that integrates image processing, AprilTag fiducial marker detection, and artificial potential control algorithms. This architecture receives multiple video streams, detects AprilTags, and extracts pose information to control the forward motion and inter-vehicle spacing in a swarm of autonomous surface vessels. This control architecture is tested for a variety of trajectories and tuned so that the swarm can successfully maintain formation control.Lieutenant, United States NavyApproved for public release. Distribution is unlimited

The socio-economic importance of non-timber forest products for rural livelihoods in West African savanna ecosystems: current status and future trends

For millennia, rural West African communities living in or adjacent of savanna ecosystems have been collecting components of local plant species (e.g. fruits, leaves, bark) in order to fulfil essential household subsistence needs (alimentation, medical care, energy demand etc.), to generate cash income and to overcome times of (financial) crisis. Thus, these non-timber forest products (NTFPs) make a considerable contribution to the well-being of local households. However, climate and land use change severely impact West African savanna ecosystems and, consequently, the safe-guarding of dependent rural livelihoods. The conversion of savanna area into cultivated land for subsistence farming owing to the ongoing population growth, as well as the progressive promotion of cash crops (e.g. cotton) is ever-increasing. As a consequence, present land-use management in West Africa has to cope with serious trade-offs. Within this decision-making NTFPs have been constantly understated due to a lack of appropriate economic figures to use within common cost-benefit analysis, and, thus, have been frequently outcompeted by seemingly more profitable land-use options. Therefore, it is crucial to provide appropriate economic data for NTFPs in order to create positive incentives for both decision-makers and NTFP beneficiaries to conserve NTFP-providing trees. The key finding of this analysis is that income from NTFPs accounts for 39 % on average of an annual total household income in Northern Benin, representing the second largest income share next to crop income and proving the respective households to be economically heavily dependent on NTFPs. Thereby, socio-economic characteristics of NTFP users tremendously shape their preferences for woody species. Particularly ethnicity has a major impact on the species used and the economic return obtained by them. Moreover, the study investigated the impacts of climate and land use change on the economic benefits derived from the three economically most important tree species in the region Vitellaria paradoxa, Parkia biglobosa and Adansonia digitata in 2050: Environmental changes will have primarily negative effects on the economic returns from all the three species. At large, the study underpins the economic relevance of NTFPs for rural communities in West African savannas and, consequently, the necessity to appropriately sustain them in order to safe-guard local livelihoods. Providing key figures on the current and future economic benefits obtained from NTFPs can augment common cost-benefit analysis, and, delivering detailed information about peoples’ use preferences for local species, this study clearly contributes to improve the basis of decision-making with reference to local land-use policies