Which Chessboards have a Closed Knight\u27s Tour within the Cube?

A closed knight\u27s tour of a chessboard uses legal moves of the knight to visit every square exactly once and return to its starting position. When the chessboard is translated into graph theoretic terms the question is transformed into the existence of a Hamiltonian cycle. There are two common tours to consider on the cube. One is to tour the six exterior n x n boards that form the cube. The other is to tour within the n stacked copies of the n x n board that form the cube. This paper is concerned with the latter. In this paper necessary and sufficient conditions for the existence of a closed knight\u27s tour for the cube are proven

Closed Knight\u27s Tours with Minimal Square Removal for All Rectangular Boards

A closed knight\u27s tour of a chessboard uses legal moves of the knight to visit every square exactly once and return to its starting position. In 1991 Schwenk completely classified the rectangular chessboards that admit a closed knight\u27s tour. For a rectangular chessboard that does not contain a closed knight\u27s tour, this paper determines the minimum number of squares that must be removed in order to admit a closed knight\u27s tour. Furthermore, constructions that generate a closed tour once appropriate squares are removed are provided

A Graph Theoretic Summation of the Cubes of the First n Integers

In this Math Bite we provide a combinatorial proof of the sum of the cubes of the first n integers by counting edges in complete bipartite graphs

Which Chessboards have a Closed Knight\u27s Tour within the Rectangular Prism?

A closed knight\u27s tour of a chessboard uses legal moves of the knight to visit every square exactly once and return to its starting position. In 1991 Schwenk completely classified the m x n rectangular chessboards that admit a closed knight\u27s tour. In honor of the upcoming twentieth anniversary of the publication of Schwenk\u27s paper, this article extends his result by classifying the i x j x k rectangular prisms that admit a closed knight\u27s tour

A Product Affinity Segmentation Framework

Product affinity segmentation discovers the linking between customers and products for cross-selling and promotion opportunities to increase sales and profits. However, there are some challenges with conventional approaches. The most straightforward approach is to use the product-level data for customer segmentation, but it results in less meaningful solutions. Moreover, customer segmentation becomes challenging on massive datasets due to computational complexity of traditional clustering methods. As an alternative, market basket analysis may suffer from association rules too general to be relevant for important segments. In this paper, we propose to partition customers and discover associated products simultaneously by detecting communities in the customer-product bipartite graph using the Louvain algorithm that has good interpretability in this context. Through the post-clustering analysis, we show that this framework generates statistically distinct clusters and identifies associated products relevant for each cluster. Our analysis provides greater insights into customer purchase behaviors, potentially helping personalization strategic planning (e.g. customized product recommendation) and profitability increase. And our case study of a large U.S. retailer provides useful management insights. Moreover, the graph application, based on almost 800,000 sales transactions, finished in 7.5 seconds on a standard PC, demonstrating its computational efficiency and better facilitating the requirements of big data

Open access: academic publishing and its implications for knowledge equity in Kenya

Traditional, subscription-based scientific publishing has its limitations: often, articles are inaccessible to the majority of researchers in low- and middle-income countries (LMICs), where journal subscriptions or one-time access fees are cost-prohibitive. Open access (OA) publishing, in which journals provide online access to articles free of charge, breaks this barrier and allows unrestricted access to scientific and scholarly information to researchers all over the globe. At the same time, one major limitation to OA is a high publishing cost that is placed on authors. Following recent developments to OA publishing policies in the UK and even LMICs, this article highlights the current status and future challenges of OA in Africa. We place particular emphasis on Kenya, where multidisciplinary efforts to improve access have been established. We note that these efforts in Kenya can be further strengthened and potentially replicated in other African countries, with the goal of elevating the visibility of African research and improving access for African researchers to global research, and, ultimately, bring social and economic benefits to the region. We (1) offer recommendations for overcoming the challenges of implementing OA in Africa and (2) call for urgent action by African governments to follow the suit of high-income countries like the UK and Australia, mandating OA for publicly-funded research in their region and supporting future research into how OA might bring social and economic benefits to Africa

Young professionals for health development: the Kenyan experience in combating non-communicable diseases

Young individuals (below 35 years) comprise an estimated 60% of the global population. Not only are these individuals currently experiencing chronic, non-communicable diseases (NCDs), either living with or at risk for these conditions, but will also experience the long-term repercussions of the current NCD policy implementations. It is thus imperative that they meaningfully contribute to the global discourse and responses for NCDs at the local level. Here, we profile one example of meaningful engagement: the Young Professionals Chronic Disease Network (YPCDN). The YPCDN is a global online network that provides a platform for young professionals to deliberate new and innovative methods of approaching the NCD challenges facing our societies. We provide a case study of the 2-year experiences of a country chapter (Kenya) of the YPCDN to demonstrate the significance and impact of emerging leaders in addressing the new global health agenda of the 21st century

The Radio Observatory on the Lunar Surface for Solar studies

Abstract The Radio Observatory on the Lunar Surface for Solar studies (ROLSS) is a concept for a near-side low radio frequency imaging interferometric array designed to study particle acceleration at the Sun and in the inner heliosphere. The prime science mission is to image the radio emission generated by Type II and III solar radio burst processes with the aim of determining the sites at and mechanisms by which the radiating particles are accelerated. Specific questions to be addressed include the following: (1) Isolating the sites of electron acceleration responsible for Type II and III solar radio bursts during coronal mass ejections (CMEs); and (2) Determining if and the mechanism(s) by which multiple, successive CMEs produce unusually efficient particle acceleration and intense radio emission. Secondary science goals include constraining the density of the lunar ionosphere by searching for a low radio frequency cutoff to solar radio emission and constraining the low energy electron population in astrophysical sources. Key design requirements on ROLSS include the operational frequency and angular resolution. The electron densities in the solar corona and inner heliosphere are such that the relevant emission occurs at frequencies below 10 MHz. Second, resolving the potential sites of particle acceleration requires an instrument with an angular resolution of at least 2°, equivalent to a linear array size of approximately 1000 m. Operations would consist of data acquisition during the lunar day, with regular data downlinks. No operations would occur during lunar night. ROLSS is envisioned as an interferometric array, because a single aperture would be impractically large. The major components of the ROLSS array are 3 antenna arms arranged in a Y shape, with a central electronics package (CEP) located at the center. The Y configuration for the antenna arms both allows for the formation of reasonably high dynamic range images on short time scales as well as relatively easy deployment. Each antenna arm is a linear strip of polyimide film (e.g., Kaptone) on which 16 science antennas are located by depositing a conductor (e.g., silver). The antenna arms can be rolled for transport, with deployment consisting of unrolling the rolls. Each science antenna is a single polarization dipole. The arms also contain transmission lines for carrying the radio signals from the science antennas to the CEP. The CEP itself houses the receivers for the science antennas, the command and data handling hardware, and, mounted externally, the downlink antenna. We have conducted two experiments relevant to the ROLSS concept. First, we deployed a proof-of-concept science antenna. Comparison of the impedance of the antenna feed points with simulations showed a high level of agreement, lending credence to the antenna concept. Second, we exposed a sample of space-qualified polyimide film, with a silver coating on one side, to temperature cycling and UV 2011) 1942-1957 exposure designed to replicate a year on the lunar surface. No degradation of the polyimide film's material or electric properties was found. Both of these tests support the notion of using polyimide-film based antennas. The prime science mission favors an equatorial site, and a site on the limb could simplify certain aspects of the instrument design. A site on the lunar near side is sufficient for meeting the science goals. While the site should be of relatively low relief topography, the entire site does not have to be flat as the fraction of the area occupied by the antenna arms is relatively small ($0.3%). Further, the antenna arms do not have to lay flat as deviations of ±1 m are still small relative to the observational wavelengths. Deployment could be accomplished either with astronauts, completely robotically, or via a combination of crewed and robotic means. Future work for the ROLSS concept includes more exhaustive testing of the radio frequency (RF) and environmental suitability of polyimide film-based science antennas, ultra-low power electronics in order to minimize the amount of power storage needed, batteries with a larger temperature range for both survival and operation, and rovers (robotic, crewed, or both) for deployment. The ROLSS array could also serve as the precursor to a larger array on the far side of the Moon for astrophysical and cosmological studies

Closed Monochromatic Bishops\u27 Tours

In chess, the bishop is unique as it is locked to a single color on the black and white board. This makes a closed tour in which the bishop visits every square on the board exactly once and returns to its starting position impossible. When can two bishops, one black and one white, legally visit every square (of their respective colors) exactly once and return to their starting positions? Such a tour will be called a closed monochromatic bishops\u27 tour. In this article, necessary and sufficient conditions for the existence of a monochromatic bishop\u27s tour for the rectangular m x n board are proven. Furthermore, a monochromatic knight\u27s move is defined for the three dimensional chessboard and a closed monochromatic knight\u27s tour is provided for the cube of side