62 research outputs found

    Lexicographic Products of Half Linearly Ordered Groups

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    Considerations on the accretion of Uranus and Neptune by mutual collisions of planetary embryos in the vicinity of Jupiter and Saturn

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    Modeling the formation of the ice giants Uranus and Neptune is a long-lasting problem in planetary science. Due to gas-drag, collisional damping, and resonant shepherding, the planetary embryos repel the planetesimals away from their reach and thus they stop growing (Levison et al. 2010). This problem persists independently of whether the accretion took place at the current locations of the ice giants or closer to the Sun. Instead of trying to push the runaway/oligarchic growth of planetary embryos up to 10-15 Earth masses, we envision the possibility that the planetesimal disk could generate a system of planetary embryos of only 1-3 Earth masses. Then we investigate whether these embryos could have collided with each other and grown enough to reach the masses of current Uranus and Neptune. Our results point to two major problems. First, there is typically a large difference in mass between the first and the second most massive core formed and retained beyond Saturn. Second, in many simulations the final planetary system has more than two objects beyond Saturn. The growth of a major planet from a system of embryos requires strong damping of eccentricities and inclinations from the disk of gas. But strong damping also favors embryos and cores to find a stable resonant configuration, so that systems with more than two surviving objects are found. In addition to these problems, in order to have substantial mutual accretion among embryos, it is necessary to assume that the surface density of the gas was several times higher than that of the minimum-mass solar nebula. However this contrasts with the common idea that Uranus and Neptune formed in a gas-starving disk, which is suggested by the relatively small amount of hydrogen and helium contained in the atmospheres of these planets. Only one of our simulations "by chance" successfully reproduced the structure of the outer Solar System.Comment: 16 pages; 15 color figures; accepted in Astronomy & Astrophysics; shortened abstrac

    A Study of the Critical Factors that Affect the Achievement of Probation Students

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    It is the purpose of this study to present the results of an investigation of the high school and college records of the 600 students who have been on probation at the Western Kentucky State Teachers College during the school years 1932-32, 1932-33 and 1933-34; and, if possible, to set up some standards by which these students may be given the advice and guidance that will best serve them in their preparation for life

    Radical classes of distributive lattices having the least element

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    summary:Let D\mathcal D be the system of all distributive lattices and let D0\mathcal D_0 be the system of all L∈DL\in \mathcal D such that LL possesses the least element. Further, let D1\mathcal D_1 be the system of all infinitely distributive lattices belonging to D0\mathcal D_0. In the present paper we investigate the radical classes of the systems D\mathcal D, D0\mathcal D_0 and D1\mathcal D_1
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