8 research outputs found

    Some classifications of biharmonic hypersurfaces with constant scalar curvature

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    We give some classifications of biharmonic hypersurfaces with constant scalar curvature. These include biharmonic Einstein hypersurfaces in space forms, compact biharmonic hypersurfaces with constant scalar curvature in a sphere, and some complete biharmonic hypersurfaces of constant scalar curvature in space forms and in a non-positively curved Einstein space. Our results provide additional cases (Theorem 2.3 and Proposition 2.8) that supports the conjecture that a biharmonic submanifold in a sphere has constant mean curvature, and two more cases that support Chen's conjecture on biharmonic hypersurfaces (Corollaries 2.2,2.7).Comment: 11 page

    Popular progression differences in vector spaces II

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    Green used an arithmetic analogue of Szemer\'edi's celebrated regularity lemma to prove the following strengthening of Roth's theorem in vector spaces. For every α>0\alpha>0, β<α3\beta<\alpha^3, and prime number pp, there is a least positive integer np(α,β)n_p(\alpha,\beta) such that if nnp(α,β)n \geq n_p(\alpha,\beta), then for every subset of Fpn\mathbb{F}_p^n of density at least α\alpha there is a nonzero dd for which the density of three-term arithmetic progressions with common difference dd is at least β\beta. We determine for p19p \geq 19 the tower height of np(α,β)n_p(\alpha,\beta) up to an absolute constant factor and an additive term depending only on pp. In particular, if we want half the random bound (so β=α3/2\beta=\alpha^3/2), then the dimension nn required is a tower of twos of height Θ((logp)loglog(1/α))\Theta \left((\log p) \log \log (1/\alpha)\right). It turns out that the tower height in general takes on a different form in several different regions of α\alpha and β\beta, and different arguments are used both in the upper and lower bounds to handle these cases.Comment: 34 pages including appendi

    Distribution of vegetative compatibility groups of <i>Fusarium oxysporum</i> f. sp. <i>cubense</i> found in Asian countries.

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    <p>The y-axis shows the number of isolates, while the x-axis shows countries represented. The legend corresponds each of the VCGs to a specific colour: VCG 0120/15 (light orange), 0121 (dark orange), 0122 (burgundy), 0123 (light green), 0124/5 (dark green), 0126 (light purple), 0128 (blue), 01213/16 (red), 01217 (dark grey), 01218 (black), 01219 (yellow), 01220 (light grey), 0124/22 (dark purple) and self-incompatible and isolates incompatible to known VCGs (pink).</p
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