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1. Mixed-Level Covering, Locating, and Detecting Arrays via Cyclotomy

   Akhtar, Yasmeen; Colbourn, Charles J; Syrotiuk, Violet R (June 2024, Springer)

   Hoffman, Frederick; Holliday, Sarah; Rosen, Zvi; Shahrokhi, Farhad; Wierman, John (Ed.)

   For a finite field of order.q, and.v a divisor of.q − 1, additive translates of a cyclotomic vector yield a.q × q cyclotomic array on.v symbols. For every positive integer.t, for certain.q sufficiently large with respect to.v, such a cyclotomic array is always a covering array of strength.t. Asymptotically such arrays have far too many rows to be competitive with certain other covering array constructions. Nevertheless, for small values of .t , this cyclotomic method produces smallest known covering arrays for numerous parameters suitable for practical application. This paper extends these ideas and shows that cyclotomy can produce covering arrays of higher index, and locating and detecting arrays with large separation. Computational results also demonstrate that certain cyclotomic arrays for the same order.q but different values of .v can be juxtaposed to produce mixed-level covering, locating, and detecting arrays. 

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2. Covering action on Conley index theory

   https://doi.org/10.1017/etds.2022.13  

   LIMA, D. V.; R. DA SILVEIRA, M.; VIEIRA, E. R. (January 2022, Ergodic Theory and Dynamical Systems)

   Abstract In this paper we apply Conley index theory in a covering space of an invariant set S , possibly not isolated, in order to describe the dynamics in S . More specifically, we consider the action of the covering translation group in order to define a topological separation of S which distinguishes the connections between the Morse sets within a Morse decomposition of S . The theory developed herein generalizes the classical connection matrix theory, since one obtains enriched information on the connection maps for non-isolated invariant sets, as well as for isolated invariant sets. Moreover, in the case of an infinite cyclic covering induced by a circle-valued Morse function, one proves that the Novikov differential of f is a particular case of the p -connection matrix defined herein. 

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3. Survey of 8 UAV Set-Covering Algorithms for Terrain Photogrammetry

   https://doi.org/10.3390/rs12142285  

   Hammond, Joshua E.; Vernon, Cory A.; Okeson, Trent J.; Barrett, Benjamin J.; Arce, Samuel; Newell, Valerie; Janson, Joseph; Franke, Kevin W.; Hedengren, John D. (July 2020, Remote Sensing)

   Remote sensing with unmanned aerial vehicles (UAVs) facilitates photogrammetry for environmental and infrastructural monitoring. Models are created with less computational cost by reducing the number of photos required. Optimal camera locations for reducing the number of photos needed for structure-from-motion (SfM) are determined through eight mathematical set-covering algorithms as constrained by solve time. The algorithms examined are: traditional greedy, reverse greedy, carousel greedy (CG), linear programming, particle swarm optimization, simulated annealing, genetic, and ant colony optimization. Coverage and solve time are investigated for these algorithms. CG is the best method for choosing optimal camera locations as it balances number of photos required and time required to calculate camera positions as shown through an analysis similar to a Pareto Front. CG obtains a statistically significant 3.2 fewer cameras per modeled area than base greedy algorithm while requiring just one additional order of magnitude of solve time. For comparison, linear programming is capable of fewer cameras than base greedy but takes at least three orders of magnitude longer to solve. A grid independence study serves as a sensitivity analysis of the CG algorithms α (iteration number) and β (percentage to be recalculated) parameters that adjust traditional greedy heuristics, and a case study at the Rock Canyon collection dike in Provo, UT, USA, compares the results of all eight algorithms and the uniqueness (in terms of percentage comparisons based on location/angle metadata and qualitative visual comparison) of each selected set. Though this specific study uses SfM, the principles could apply to other instruments such as multi-spectral cameras or aerial LiDAR. 

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4. Determination of the Starspot Covering Fraction as a function of Stellar Age from Observational Data

   https://doi.org/10.1093/mnras/stz3197  

   Nichols-Fleming, Fiona; Blackman, Eric G (November 2020, Monthly Notices of the Royal Astronomical Society)

   Abstract The association of starspots with magnetic fields leads to an expectation that quantities which correlate with magnetic field strength may also correlate with starspot coverage. Since younger stars spin faster and are more magnetically active, assessing whether starspot coverage correlates with shorter rotation periods and stellar youth tests these principles. Here we analyze the starspot covering fraction versus stellar age for M-, G-, K-, and F-type stars based on previously determined variability and rotation periods of over 30,000 Kepler main-sequence stars. We determine the correlation between age and variability using single and dual power law best fits. We find that starspot coverage does indeed decrease with age. Only when the data are binned in an effort to remove the effects of activity cycles of individual stars, do statistically significant power law fits emerge for each stellar type. Using bin averages, we then find that the starspot covering fraction scales with the X-ray to bolometric ratio to the power λ with 0.22 ± 0.03 < λ < 0.32 ± 0.09 for G-type stars of rotation period below 15 days and for the full range of F- and M-type stars. For K-type stars, we find two branches of λ separated by variability bins, with the lower branch showing nearly constant starspot coverage and the upper branch λ ∼ 0.35 ± 0.04. G-type stars with periods longer than 15 days exhibit a transition to steeper power law of λ ∼ 2.4 ± 1.0. The potential connection to previous rotation-age measurements suggesting a magnetic breaking transition at the solar age, corresponding to period of 24.5 is also of interest. 

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5. Partitioned Search with Column Resampling for Locating Array Construction

   https://doi.org/10.1109/ICSTW.2019.00056  

   Lanus, Erin; Colbourn, Charles J.; Montgomery, Douglas C. (April 2019, IEEE International Conference on Software Testing, Verification and Validation Workshops (ICSTW))

   Locating arrays are designs used in combinatorial testing with the property that every set of d t-way interactions appears in a unique set of tests. Using a locating array to conduct fault testing ensures that faulty interactions can be located when there are d or fewer faults. Locating arrays are fairly new and few techniques have been explored for their construction. Most of the available work is limited to finding only one fault. Known general methods require a covering array of strength t+d and produce many more tests than are needed. We present Partitioned Search with Column Resampling (PSCR), a randomized computational search algorithmic framework to verify if an array is (d t)-locating by partitioning the search space to decrease the number of comparisons. If a candidate array is not locating, random resampling is performed until a locating array is constructed or an iteration limit is reached. Results are compared against known locating array constructions from covering arrays of higher strength and against published results of mixed level locating arrays for parameters of real-world systems. The use of PSCR to build larger locating arrays from a variety of ingredient arrays is explored. 

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