On Kakeya maps with regularity assumptions

For a $n-$dimensional Kakeya set $(n\geq 3)$ we may define a Kakeya map associated to it which parametrizes the Kakeya set by $[0,1]\times S^{n-1},$ where $S^{n-1}$ is thought of as the space of unit directions. We show that if the Kakeya map is either $\alpha-$H\"{o}lder continuous with $\alpha>\frac{(n-2)n}{(n-1)^2},$ or continuous and in the Sobolev space $H^{s}$ for some $s>(n-1)/2,$ then the Kakeya set has positive Lebesgue measure

Incidence estimates for α\alpha-dimensional tubes and β\beta-dimensional balls in R2\mathbb{R}^2

We prove essentially sharp incidence estimates for a collection of $\delta$-tubes and $\delta$-balls in the plane, where the $\delta$-tubes satisfy an $\alpha$-dimensional spacing condition and the $\delta$-balls satisfy a $\beta$-dimensional spacing condition. Our approach combines a combinatorial argument for small $\alpha, \beta$ and a Fourier analytic argument for large $\alpha, \beta$.Comment: 18 pages, 8 figure

A note on maximal operators for the Schr\"{o}dinger equation on T1.\mathbb{T}^1.

Motivated by the study of the maximal operator for the Schr\"{o}dinger equation on the one-dimensional torus $\mathbb{T}^1$, it is conjectured that for any complex sequence $\{b_n\}_{n=1}^N$, $$\left\| \sup_{t\in [0,N^2]} \left|\sum_{n=1}^N b_n e \left(x\frac{n}{N} + t\frac{n^2}{N^2} \right) \right| \right\|_{L^4([0,N])} \leq C_\epsilon N^{\epsilon} N^{\frac{1}{2}} \|b_n\|_{\ell^2}$$ In this note, we show that if we replace the sequence $\{\frac{n^2}{N^2}\}_{n=1}^N$ by an arbitrary sequence $\{a_n\}_{n=1}^N$ with only some convex properties, then $$\left\| \sup_{t\in [0,N^2]} \left|\sum_{n=1}^N b_n e \left(x\frac{n}{N} + ta_n \right) \right| \right\|_{L^4([0,N])} \leq C_\epsilon N^\epsilon N^{\frac{7}{12}} \|b_n\|_{\ell^2}.$$ We further show that this bound is sharp up to a $C_\epsilon N^\epsilon$ factor.Comment: 13 page

An 11-bp Insertion in Zea mays fatb Reduces the Palmitic Acid Content of Fatty Acids in Maize Grain

The ratio of saturated to unsaturated fatty acids in maize kernels strongly impacts human and livestock health, but is a complex trait that is difficult to select based on phenotype. Map-based cloning of quantitative trait loci (QTL) is a powerful but time-consuming method for the dissection of complex traits. Here, we combine linkage and association analyses to fine map QTL-Pal9, a QTL influencing levels of palmitic acid, an important class of saturated fatty acid. QTL-Pal9 was mapped to a 90-kb region, in which we identified a candidate gene, Zea mays fatb (Zmfatb), which encodes acyl-ACP thioesterase. An 11-bp insertion in the last exon of Zmfatb decreases palmitic acid content and concentration, leading to an optimization of the ratio of saturated to unsaturated fatty acids while having no effect on total oil content. We used three-dimensional structure analysis to explain the functional mechanism of the ZmFATB protein and confirmed the proposed model in vitro and in vivo. We measured the genetic effect of the functional site in 15 different genetic backgrounds and found a maximum change of 4.57 mg/g palmitic acid content, which accounts for ∼20–60% of the variation in the ratio of saturated to unsaturated fatty acids. A PCR-based marker for QTL-Pal9 was developed for marker-assisted selection of nutritionally healthier maize lines. The method presented here provides a new, efficient way to clone QTL, and the cloned palmitic acid QTL sheds lights on the genetic mechanism of oil biosynthesis and targeted maize molecular breeding

Null structures and degenerate dispersion relations in two space dimensions

Motivated by water-wave problems, in this paper we consider a class of nonlinear dispersive PDEs in 2D with cubic nonlinearities, whose dispersion relations are radial and have vanishing Guassian curvature on a circle. For such a model we identify certain null structures for the cubic nonlinearity, which suffice in order to guarantee global scattering solutions for the small data problem. Our null structures in the power-type nonlinearity are weak, and only eliminate the worst nonlinear interaction. Such null structures arise naturally in some water-wave problems

An Incidence Estimate and a Furstenberg Type Estimate for Tubes in R2\mathbb {R}^2 R 2

Abstract We study the $$\delta$$ δ -discretized Szemerédi–Trotter theorem and Furstenberg set problem. We prove sharp estimates for both two problems assuming tubes satisfy some spacing condition. For both problems, we construct sharp examples that share common features