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 and host range of the banana Fusarium wilt fungus, Fusarium oxysporum f. sp. cubense, in Asia

Fusarium oxysporum formae specialis cubense (Foc) is a soil-borne fungus that causes Fusarium wilt, which is considered to be the most destructive disease of bananas. The fungus is believed to have evolved with its host in the Indo-Malayan region, and from there it was spread to other banana-growing areas with infected planting material. The diversity and distribution of Foc in Asia was investigated. A total of 594 F. oxysporum isolates collected in ten Asian countries were identified by vegetative compatibility groups (VCGs) analysis. To simplify the identification process, the isolates were first divided into DNA lineages using PCR-RFLP analysis. Six lineages and 14 VCGs, representing three Foc races, were identified in this study. The VCG complex 0124/5 was most common in the Indian subcontinent, Vietnam and Cambodia; whereas the VCG complex 01213/16 dominated in the rest of Asia. Sixty-nine F. oxysporum isolates in this study did not match any of the known VCG tester strains. In this study, Foc VCG diversity in Bangladesh, Cambodia and Sri Lanka was determined for the first time and VCGs 01221 and 01222 were first reported from Cambodia and Vietnam. New associations of Foc VCGs and banana cultivars were recorded in all the countries where the fungus was collected. Information obtained in this study could help Asian countries to develop and implement regulatory measures to prevent the incursion of Foc into areas where it does not yet occur. It could also facilitate the deployment of disease resistant banana varieties in infested areas.S1 Table. Morphological identity, PCR-RFLP clade and lineage identity, vegetative compatibility group, host and host subgroup, location and origin of Fusariumisolates collected in Asia.S2 Table. Vegetative compatibility tester strains used to characterize compatibility to Asian Fusariumoxysporumf. sp. cubense isolates.http://www.plosone.orgam2018Forestry and Agricultural Biotechnology Institute (FABI)Microbiology and Plant Patholog

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

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

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

The distribution of <i>Fusarium oxysporum</i> f. sp. <i>cubense</i> (Foc) vegetative compatibility groups (VCGs) in Bangladesh, India and Sri Lanka.

<p>VCG 0123 is shown in light green, 0124/5 is shown in dark green, VCG 0128 is shown in blue, VCG 01217 is shown in dark grey, VCG 01220 is shown in light grey and VCG 0124/22 is shown in purple.</p

The group, subgroup and common synonyms of <i>Musa</i> cultivars and their relationship with vegetative compatibility groups (VCGs) of <i>Fusarium oxysporum</i> f. sp. <i>cubense</i>.

<p>The group, subgroup and common synonyms of <i>Musa</i> cultivars and their relationship with vegetative compatibility groups (VCGs) of <i>Fusarium oxysporum</i> f. sp. <i>cubense</i>.</p