On the Bohnenblust-Hille inequality and a variant of Littlewood's 4/3 inequality

The search for sharp constants for inequalities of the type Littlewood's 4/3 and Bohnenblust-Hille, besides its pure mathematical interest, has shown unexpected applications in many different fields, such as Analytic Number Theory, Quantum Information Theory, or (for instance) in deep results on the $n$-dimensional Bohr radius. The recent estimates obtained for the multilinear Bohnenblust-Hille inequality (in the case of real scalars) have been recently used, as a crucial step, by A. Montanaro in order to solve problems in the theory of quantum XOR games. Here, among other results, we obtain new upper bounds for the Bohnenblust-Hille constants in the case of complex scalars. For bilinear forms, we obtain the optimal constants of variants of Littlewood's 4/3 inequality (in the case of real scalars) when the exponent 4/3 is replaced by any $r\geq4/3.$ As a consequence of our estimates we show that the optimal constants for the real case are always strictly greater than the constants for the complex case

There exist multilinear Bohnenblust-Hille constants (Cn)n=1(infinity) with limn ->infinity(Cn+1-Cn)=0

The n-linear Bohnenblust-Hille inequality asserts that there is a constant C-n is an element of [1, infinity) such that the l(2n/n+1)-norm of (U(e(i1), ..., e(in)))(i1, ...,in=1)(N) is bounded above by C-n times the supremum norm of U, for any n-linear form U :C-N x ... x C-N -> C and N is an element of N (the same holds for real scalars). We prove what we call Fundamental Lemma, which brings new information on the optimal constants, (K-n)(n=1)(infinity) for both real and complex scalars. For instance, Kn+1 - K-n = 2. We study the interplay between the Kahane-Salem-Zygmund and the Bohnenblust-Hille (polynomial and multilinear) inequalities and provide estimates for Bohnenblust-Hille-type inequality constants for any exponent q is an element of [2n/n+1, infinity). (C) 2012 Elsevier Inc. All rights reserved

A multi-integrated approach on toxicity effects of engineered TiO2 nanoparticles

The new properties of engineered nanoparticles drive the need for new knowledge on the safety, fate, behavior and biologic effects of these particles on organisms and ecosystems. Titanium dioxide nanoparticles have been used extensively for a wide range of applications, e.g, self-cleaning surface coatings, solar cells, water treatment agents, topical sunscreens. Within this scenario increased environmental exposure can be expected but data on the ecotoxicological evaluation of nanoparticles are still scarce. The main purpose of this work was the evaluation of effects of TiO2 nanoparticles in several organisms, covering different trophic levels, using a battery of aquatic assays. Using fish as a vertebrate model organism tissue histological and ultrastructural observations and the stress enzyme activity were also studied. TiO2 nanoparticles (Aeroxide® P25), two phase composition of anatase (65%) and rutile (35%) with an average particle size value of 27.6±11 nm were used. Results on the EC50 for the tested aquatic organisms showed toxicity for the bacteria, the algae and the crustacean, being the algae the most sensitive tested organism. The aquatic plant Lemna minor showed no effect on growth. The fish Carassius auratus showed no effect on a 21 day survival test, though at a biochemical level the cytosolic Glutathione-S-Transferase total activity, in intestines, showed a general significant decrease (p<0.05) after 14 days of exposure for all tested concentrations. The presence of TiO2 nanoparticles aggregates were observed in the intestine lumen but their internalization by intestine cells could not be confirmed

Macroecological links between the Linnean, Wallacean, and Darwinian shortfalls

Species are the currency of most biodiversity studies. However, many shortfalls and biases remain in our biodiversity estimates, preventing a comprehensive understanding of the eco-evolutionary processes that have shaped the biodiversity currently available on Earth. Biased biodiversity estimates also jeopardize the effective implementation of data-driven conservation strategies, ultimately leading to biodiversity loss. Here, we delve into the concept of the Latitudinal Taxonomy Gradient (LTG) and show how this new idea provides an interesting conceptual link between the Linnean (i.e., our ignorance of how many species there are on Earth), Darwinian (i.e., our ignorance of species evolutionary relationships), and Wallacean (i.e., our ignorance on species distribution) shortfalls. More specifically, we contribute to an improved understanding of LTGs and establish the basis for the development of new methods that allow us to: (i) better account for the integration between different shortfalls and, (ii) estimate how these interactions may affect our understanding about the evolutionary components of richness gradients at macroecological scales.This manuscript is partially derived from a working group on “Biodiversity Shortfalls” held in November 2019 and sponsored by the National Institutes for Science and Technology (INCT) in Ecology, Evolution, and Biodiversity Conservation (CNPq proc. 465610/2014-5 and FAPEG proc. 201810267000023). JJMG and LEF are supported by Ph.D. and M.Sc. scholarships from CAPES, while LM and RBP are supported by postdoctoral fellowships from CAPES (PNPD). JS was funded by the funded by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Action (grant agreement #843234; project: TAXON-TIME) and by the Spanish Council for Scientific Research (IF_ERC). GT and LJ are supported by a DTI fellowships from CNPq, while JAFD-F, LGL, and CJBC are supported by Productivity Grants from CNPq.Peer reviewe