Constructing polynomial systems with many positive solutions using tropical geometry

The number of positive solutions of a system of two polynomials in two variables defined in the field of real numbers with a total of five distinct monomials cannot exceed 15. All previously known examples have at most 5 positive solutions. Tropical geometry is a powerful tool to construct polynomial systems with many positive solutions. The classical combinatorial patchworking method arises when the tropical hypersurfaces intersect transversally. In this paper, we prove that a system as above constructed using this method has at most 6 positive solutions. We also show that this bound is sharp. Moreover, using non-transversal intersections of tropical curves, we construct a system as above having 7 positive solutions.Comment: 21 pages, 8 figure

Approximation algorithms for wavelet transform coding of data streams

This paper addresses the problem of finding a B-term wavelet representation of a given discrete function $f \in \real^n$ whose distance from f is minimized. The problem is well understood when we seek to minimize the Euclidean distance between f and its representation. The first known algorithms for finding provably approximate representations minimizing general $\ell_p$ distances (including $\ell_\infty$) under a wide variety of compactly supported wavelet bases are presented in this paper. For the Haar basis, a polynomial time approximation scheme is demonstrated. These algorithms are applicable in the one-pass sublinear-space data stream model of computation. They generalize naturally to multiple dimensions and weighted norms. A universal representation that provides a provable approximation guarantee under all p-norms simultaneously; and the first approximation algorithms for bit-budget versions of the problem, known as adaptive quantization, are also presented. Further, it is shown that the algorithms presented here can be used to select a basis from a tree-structured dictionary of bases and find a B-term representation of the given function that provably approximates its best dictionary-basis representation.Comment: Added a universal representation that provides a provable approximation guarantee under all p-norms simultaneousl

Bistable equilibrium points of mercury body burden

In the last century mercury levels in the global environment have tripled as a result of increased pollution from industrial, occupational, medicinal and domestic uses \cite{BaMe03}. Glutathione is known to be the main agent responsible for the excretion of mercury (we refer to \cite{Thim05}, \cite{ZalBar99} and \cite{Lyn02}). It has also been shown that mercury inhibits glutathione synthetase (an enzyme acting in the synthesization of Glutathione), therefore leading to decreased glutathione levels (we refer to \cite{Thim05}, \cite{GeGe05}, \cite{GeGe06} and \cite{RDeth04}). Mercury also interferes with the production of heme in the porphyrin pathway \cite{WoMaEc93}. Heme is needed for biological energy production and ability to detox organic toxins via the P450 enzymes \cite{Boy06}. The purpose of this paper is to show that body's response to mercury exposure is hysteretic, i.e. when this feedback of mercury on its main detoxifying agents is strong enough then mercury body burden has two points of equilibrium: one with normal abilities to detoxify and low levels of mercury and one with inhibited abilities to detoxify and high levels of mercury. Furthermore, a small increase of body's mercury burden may not be sufficient to trigger observable neurotoxic effects but it may be sufficient to act as a switch leading to an accumulation of mercury in the body through environmental exposure until its toxicity becomes manifest.Comment: To appear in Journal of Biological System

Surviving a Genome Collision: Genomic Signatures of Allopolyploidization in the Recent Crop Species

Polyploidization has played a major role in crop plant evolution, leading to advantageous traits that have been selected by humans. Here, we describe restructuring patterns in the genome of Brassica napus L., a recent allopolyploid species. Widespread segmental deletions, duplications, and homeologous chromosome exchanges were identified in diverse genome sequences from 32 natural and 20 synthetic accessions, indicating that homeologous exchanges are a major driver of postpolyploidization genome diversification. Breakpoints of genomic rearrangements are rich in microsatellite sequences that are known to interact with the meiotic recombination machinery. In both synthetic and natural B. napus, a subgenome bias was observed toward exchanges replacing larger chromosome segments from the C-subgenome by their smaller, homeologous A-subgenome segments, driving postpolyploidization genome size reduction. Selection in natural B. napus favored segmental deletions involving genes associated with immunity, reproduction, and adaptation. Deletions affecting mismatch repair system genes, which are assumed to control homeologous recombination, were also found to be under selection. Structural exchanges between homeologous subgenomes appear to be a major source of novel genetic diversity in de novo allopolyploids. Documenting the consequences of genomic collision by genomic resequencing gives insights into the adaptive processes accompanying allopolyploidization

Counting positive intersection points of a trinomial and a T\mathbf{T}-nomial curves via Groethendieck's dessin d'enfant

We consider real polynomial systems $f=g=0$ in two variables where $f$ has $t\geq 3$ monomial terms and $g$ has $3$ monomials terms. We prove that the number of positive isolated solutions of such a system does not exceed $3\cdot 2^{t-2} - 1$. This improves the bound $2^t - 2$ obtained in [T.-Y. Li, J.-M. Rojas and X. Wang, 2003]. This also refines for $t=4,\ldots,9$ the bound $2t^3/3 + 5t$ obtained in [P. Koiran, N. Portier and S. Tavenas, 2015]. Our proof is based on a delicate analysis of the Groethendieck's dessin d'enfant associated to some rational function determined by the system. For $t=3$, it was shown in [T.-Y. Li, J.-M. Rojas and X. Wang, 2003] that the sharp bound is five, and if this bound is reached, then the Minkowski sum of the associated Newton triangles is an hexagon. A further analysis of Groethendieck's dessin d'enfant allows us to show that if the bound five is reached, then there exist two consecutive edges of the hexagon which are translate of two consecutive edges of one Newton triangle.Comment: 32 pages, 25 figure

The predictive value of preoperative B-type natriuretic peptide in children undergoing cardiac surgery

Introduction: B-type natriuretic peptide (BNP) is a biomarker of myocardial stress. In children, the value of preoperative BNP on postoperative outcome is unclear. The aim of this study was to determine the predictive value of preoperative NT-proBNP on postoperative outcome in children after congenital heart surgery. Results: Ninety-seven patients were included in the study with a median age of 3.3 years [0.7-5.2]. Preoperative median NT-proBNP was 412 pg/ml [164-1309]. NT-proBNP was above the P95 reference value for age in 56 patients (58%). Preoperative NT-proBNP was significantly higher in patients who had mechanical ventilation duration of more than 2 days (1156 pg/ml [281-1951] vs. 267 pg/ml [136-790], p=0.003) and who stayed more than 6 days in the pediatric intensive care unit (727 pg/ml [203-1951] vs. 256 pg/ml [136-790], p=0.007). However, preoperative NT-proBNP was not significantly higher in patients with an increased inotropic score, a prolonged cardiopulmonary bypass time or an increased surgical risk category. Conclusions: An elevated preoperative NT-proBNP reflects hemodynamic status and cardiac dysfunction, and therefore is a valuable adjunct in predicting a complicated postoperative course. ___________________________________ Introduction: Le peptide natriurétique type B (BNP) est un marqueur reflétant le stress myocardique. Dans la population pédiatrique, la signification des valeurs préopératoire de BNP, en particulier sur l'évolution postopératoire, n'est pas clairement établie. Le but de l'étude est de déterminer la valeur prédictive de la partie NT sérique du BNP (NT-proBNP) sur l'évolution post opératoire d'enfants porteur d'une cardiopathie congénitale et ayant eu une chirurgie cardiaque. Résultats: Nonante-sept enfants ont été inclus dans l'étude, avec un âge médian de 3.3 ans [0.7-5.2]. La valeur médiane du NT-proBNP préopératoire était de 412 pg/ml [164-1309]. Le NT-proBNP préopératoire était supérieur au P95 des valeurs de référence pour l'âge chez 56 patients (58%). Le NT-proBNP préopératoire était significativement plus élevé chez les patients ayant eu plus de deux jours de ventilation mécanique dans la période postopératoire (1156 pg/ml [281-1951] vs. 267 pg/ml [136-790], p=0.003) et ayant été hospitalisés plus de 6 jours dans l'unité de soins intensifs pédiatrique (727 pg/ml [203-1951] vs. 256 pg/ml [136-790], p=0.007). Par contre, le NT-proBNP préopératoire n'était pas significativement plus élevé chez les patients ayant eu un score d'inotrope élevé pendant leur hospitalisation aux soins intensifs, un temps de circulation extracorporelle prolongé ou ayant subi une chirurgie avec un risque chirurgical élevé. Conclusions: Un NT-proBNP sérique élevé en préopératoire reflète l'importance du stress myocardique induit par l'hémodynamique et la dysfonction myocardique, il est un marqueur qui permet d'améliorer l'identification des patients à risque d'avoir une évolution post opératoire compliquée

Dielectric properties of CaCu3Ti4O12 based multiphased ceramics

A “soft chemistry” method, the coprecipitation, has been used to synthesize the perovskite CaCu3Ti4O12 (CCT). Three main types of materials were obtained for both powders and sintered ceramics: a monophased consisting of the pure CCT phase, a biphased (CCT + CaTiO3), and a three-phased (CCT + CaTiO3 + copper oxide (CuO or Cu2O)). These ceramics, sintered at low temperature, 1050 °C, present original dielectric properties. The relative permittivity determined in the temperature range (−150 < T < 250 °C) is significantly higher than the one reported in the literature. Internal barrier layer capacitor is the probable mechanism to explain the particular behaviour. Moreover, the presence of a copper oxide phase beside the perovksite CCT plays an important role for enhancing the dielectric properties