The Bonenblust-Hille inequality for homogeneous polynomials is hypercontractive

The Bohnenblust-Hille inequality says that the $\ell^{\frac{2m}{m+1}}$ -norm of the coefficients of an $m$-homogeneous polynomial $P$ on $\Bbb{C}^n$ is bounded by $\| P \|_\infty$ times a constant independent of $n$, where $\|\cdot \|_\infty$ denotes the supremum norm on the polydisc $\mathbb{D}^n$. The main result of this paper is that this inequality is hypercontractive, i.e., the constant can be taken to be $C^m$ for some $C>1$. Combining this improved version of the Bohnenblust-Hille inequality with other results, we obtain the following: The Bohr radius for the polydisc $\mathbb{D}^n$ behaves asymptotically as $\sqrt{(\log n)/n}$ modulo a factor bounded away from 0 and infinity, and the Sidon constant for the set of frequencies $\bigl\{ \log n: n \text{a positive integer} \le N\bigr\}$ is $\sqrt{N}\exp\{(-1/\sqrt{2}+o(1))\sqrt{\log N\log\log N}\}$.

Gram-negative flagella glycosylation

Protein glycosylation had been considered as an eccentricity of a few bacteria. However, through advances in analytical methods and genome sequencing, it is now established that bacteria possess both N-linked and O-linked glycosylation pathways. Both glycosylation pathways can modify multiple proteins, flagellins from Archaea and Eubacteria being one of these. Flagella O-glycosylation has been demonstrated in many polar flagellins from Gram-negative bacteria and in only the Gram-positive genera Clostridium and Listeria. Furthermore, O-glycosylation has also been demonstrated in a limited number of lateral flagellins. In this work, we revised the current advances in flagellar glycosylation from Gram-negative bacteria, focusing on the structural diversity of glycans, the O-linked pathway and the biological function of flagella glycosylation.

Convergent Approaches for the Synthesis of the Anti-tumoral Peptide, Kahalalide F. Study of Orthogonal Protecting Groups

Kahalalide compounds are peptides that are isolated from a Hawaiian herbivorous marine species of mollusc, Elysia rufescens, and its diet, the green alga Bryopsis sp. Kahalalide F and its synthetic analogues are the most promising compounds of the Kahalalide family because they show anti-tumoral activity. Linear solid-phase syntheses of Kahalalide F have been reported. Here we describe several new improved synthetic routes based on convergent approaches with distinct orthogonal protection schemes for the preparation of Kahaladide analogues. These strategies allow a better control and characterization of the intermediates because more reactions are performed in solution. Five derivatives of Kahalalide F were synthesized using several convergent approaches.

Nonlinear principal and canonical directions from continuous extensions of multidimensional scaling

A continuous random variable is expanded as a sum of a sequence of uncorrelated random variables. These variables are principal dimensions in continuous scaling on a distance function, as an extension of classic scaling on a distance matrix. For a particular distance, these dimensions are principal components. Then some properties are studied and an inequality is obtained. Diagonal expansions are considered from the same continuous scaling point of view, by means of the chi-square distance. The geometric dimension of a bivariate distribution is defined and illustrated with copulas. It is shown that the dimension can have the power of continuum.