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

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

Orthogonal arrays of the six variables and three levels used to optimize PCR conditions for the real-time detection of tropical strains of <i>Fusarium oxysporum</i> f. sp. <i>cubense</i> race 4.

<p>D/P stands for denaturation/polymerization pattern.</p

Characteristics of primers (F/R) and probes (P) used in this study.

<p>Characteristics of primers (F/R) and probes (P) used in this study.</p

Development of a hydrolysis probe-based real-time assay for the detection of tropical strains of <i>Fusarium oxysporum</i> f. sp. <i>cubense</i> race 4

<div><p><i>Fusarium oxysporum</i> f. sp. <i>cubense</i> (Foc) is one of the most important threats to global banana production. Strategies to control the pathogen are lacking, with plant resistance offering the only long-term solution, if sources of resistance are available. Prevention of introduction of Foc into disease-free areas thus remains a key strategy to continue sustainable banana production. In recent years, strains of Foc affecting Cavendish bananas have destroyed plantations in a number of countries in Asia and in the Middle East, and one African country. One vegetative compatibility group (VCG), 01213/16, is considered the major threat to bananas in tropical and subtropical climatic conditions. However, other genetically related VCGs, such as 0121, may potentially jeopardize banana cultures if they were introduced into disease-free areas. To prevent the introduction of these VCGs into disease-free Cavendish banana-growing countries, a real-time PCR test was developed to accurately detect both VCGs. A previously described putative virulence gene was used to develop a specific combination of hydrolysis probe/primers for the detection of tropical Foc race 4 strains. The real-time PCR parameters were optimized by following a statistical approach relying on orthogonal arrays and the Taguchi method in an attempt to enhance sensitivity and ensure high specificity of the assay. This study also assessed critical performance criteria, such as repeatability, reproducibility, robustness, and specificity, with a large including set of 136 <i>F</i>. <i>oxysporum</i> isolates, including 73 Foc pathogenic strains representing 24 VCGs. The validation data demonstrated that the new assay could be used for regulatory testing applications on banana plant material and can contribute to preventing the introduction and spread of Foc strains affecting Cavendish bananas in the tropics.</p></div

Response of <i>Fusarium</i> strains to some existing <i>Fusarium oxysporum</i> f. sp. <i>cubense</i> detection tools.

<p>Response of <i>Fusarium</i> strains to some existing <i>Fusarium oxysporum</i> f. sp. <i>cubense</i> detection tools.</p

ANOVA table on the effects of dilution (10 times or 100 times), sample type (dry or frozen), and their interaction.

<p>ANOVA table on the effects of dilution (10 times or 100 times), sample type (dry or frozen), and their interaction.</p

Post-hoc Tukey Kramer pairwise comparison test between groups of treatments.

<p>Samples were either dry or frozen and either diluted 10 times or 100 times.</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