Popular progression differences in vector spaces II

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 $\alpha>0$, $\beta<\alpha^3$, and prime number $p$, there is a least positive integer $n_p(\alpha,\beta)$ such that if $n \geq n_p(\alpha,\beta)$, then for every subset of $\mathbb{F}_p^n$ of density at least $\alpha$ there is a nonzero $d$ for which the density of three-term arithmetic progressions with common difference $d$ is at least $\beta$. We determine for $p \geq 19$ the tower height of $n_p(\alpha,\beta)$ up to an absolute constant factor and an additive term depending only on $p$. In particular, if we want half the random bound (so $\beta=\alpha^3/2$), then the dimension $n$ required is a tower of twos of height $\Theta \left((\log p) \log \log (1/\alpha)\right)$. It turns out that the tower height in general takes on a different form in several different regions of $\alpha$ and $\beta$, and different arguments are used both in the upper and lower bounds to handle these cases.Comment: 34 pages including appendi

Some classifications of biharmonic hypersurfaces with constant scalar curvature

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

The distribution of <i>Fusarium oxysporum</i> f. sp. <i>cubense</i> (Foc) vegetative compatibility groups (VCGs) in Vietnam.

<p>VCG 0123 is shown in light green, VCG 0124/5 is shown in dark green, VCG 0128 is shown in blue, VCG 01221 is shown in white and VCG 0124/22 is shown in purple.</p

Vegetative compatibility group (VCG) and lineage distribution of <i>Fusarium oxysporum</i> f. sp. <i>cubense</i> isolates in Asia.

<p>Vegetative compatibility group (VCG) and lineage distribution of <i>Fusarium oxysporum</i> f. sp. <i>cubense</i> isolates in Asia.</p

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

<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

The distribution of <i>Fusarium oxysporum</i> f. sp. <i>cubense</i> (Foc) vegetative compatibility groups (VCGs) in Indonesia.

<p>VCG 0120/15 is shown in dark yellow, VCG 0121 is shown in light orange, VCG 0123 is shown in light green, VCG 0124/5 is shown in dark green, VCG 01213/16 is shown in red, VCG 01218 is shown in black and VCG 01219 is shown in light yellow.</p