Graph pebbling is the study of moving discrete pebbles from certain initial distributions on the vertices of a graph to various target distributions via pebbling moves. A pebbling move removes two pebbles from a vertex and places one pebble on one of its neighbors (losing the other as a toll).For t ≥ 1 the t-pebbling number of a graph is the minimum number of pebbles necessary so that from any initial distribution of them it is possible to move t pebbles to any vertex.We provide the best possible upper bound on the t-pebbling number of a diameter two graph, proving a conjecture of Curtis, et al., in the process. We also give a linear time (in the number of edges) algorithm to t-pebble such graphs, as well as a quartic time (in the number of vertices) algorithm to compute the pebbling number of such graphs, improving the best known result of Bekmetjev and Cusack.Furthermore, we show that, for complete graphs, cycles, trees, and cubes, we can allow the target to be any distribution of t pebbles without increasing the corresponding t-pebbling numbers; we conjecture that this behavior holds for all graphs.Finally, we explore fractional and optimal fractional versions of pebbling, proving the fractional pebbling number conjecture of Hurlbert and using linear optimization to reveal results on the optimal fractional pebbling number of vertex-transitive graphs.
A series of 22 primary reaction-control-system engine attitude-control firings were observed from the Maui Space Surveillance Site during the space shuttle STS-115 mission. The firings occurred during a pass over Maui on 19 September 2006 during which the orbiter was in sunlight and the observatory was in darkness. The observed attitude maneuvers maintained the orbiter in an orientation in which its long axis was aligned with the line of sight from the observatory. This ensured that the thrust vectors of all the observed engine firings were perpendicular to the line of sight, providing an optimal side-on observation of the exhaust. The firings ranged between 80 and 320 ms in duration and involved 2 or 3 engines for pitch, roll, and yaw adjustments. A 0.328 deg field-of-view acquisition scope of the 3.6 m telescope of the Advanced Electro-Optical System provided unfiltered imagery in the near-ultraviolet visible spectral region. The most interesting white-light features were transients, one observed at engine start up and two at shutdown. The analysis of the transient speeds reveals that the startup transient consists of either unburned propellant droplets or higher-pressure gas evaporated from droplets and that the shutdown transients are attributable to a slightly staggered release of unburned oxidizer and fuel, respectively. The first (oxidizer) shutdown transient is the brightest feature, for which an intensity evolution analysis is conducted. The analysis of the groundbased data is fully consistent with spectral features attributable to primary reaction-control-system engine transients observed in previous measurements from the space shuttle bay using an imager spectrograph. Nomenclature A= Brook scaling parameter for density, cm 1 A = Brook scaling parameter for flux, g A 0 = Lorentzian scaling parameter for density, cm 1 deg 2 A 0 = Lorentzian scaling parameter for flux, g deg 2 B= Brook angular distribution parameter, unitless C p = specific heat at constant pressure, J g 1 K 1 D = particle diameter, m D 0 = initial particle diameter, m d = horizontal video-image frame width at range, R, km F = exhaust particle flux, cm 2 s 1 F S = solar spectral flux, W cm 2 g = spectral atmospheric transmission and sensitivity factor, unitless IR e ; x = radiant intensity along image coordinate x perpendicular to the thrust axis at nozzle distance R e , W cm 2 IR e ; y = volume emission rate at nozzle distance R e , where y is the radial distance with respect to the thrust axis, W cm 3 I = angular volume emission-rate distribution, W cm 3 I' = scattered intensity as a function of the solar scattering angle, W sr 1 m = particle mass, g N = number density, particles cm 3 P; T = Planck radiation function, W sr 1 cm 2 Hz 1 P vap = vapor pressure, torr _ Q = heat gain rate, W _ Q Earth = Earthshine heating rate, W _ Q rad = radiative cooling rate, W _ Q sub = heat of sublimation cooling rate, W _ Q sun = solar radiation heating rate, W R = range, km R = range vector, km R e = axial distance to the nozzle exit, m r = particle ...
We investigate generalizations of pebbling numbers and of Graham's pebbling conjecture that π(G × H) ≤ π(G)π(H), where π(G) is the pebbling number of the graph G. We develop new machinery to attack the conjecture, which is now twenty years old. We show that certain conjectures imply others that initially appear stronger. We also find counterexamples that show that Sjöstrand's theorem on cover pebbling does not apply if we allow the cost of transferring a pebble from one vertex to an adjacent vertex to depend on the edge and we describe an alternate pebbling number for which Graham's conjecture is demonstrably false.
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