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

On the road to personalised and precision geomedicine: medical geology and a renewed call for interdisciplinarity

Our health depends on where we currently live, as well as on where we have lived in the past and for how long in each place. An individual’s place history is particularly relevant in conditions with long latency between exposures and clinical manifestations, as is the case in many types of cancer and chronic conditions. A patient’s geographic history should routinely be considered by physicians when diagnosing and treating individual patients. It can provide useful contextual environmental information (and the corresponding health risks) about the patient, and should thus form an essential part of every electronic patient/health record. Medical geology investigations, in their attempt to document the complex relationships between the environment and human health, typically involve a multitude of disciplines and expertise. Arguably, the spatial component is the one factor that ties in all these disciplines together in medical geology studies. In a general sense, epidemiology, statistical genetics, geoscience, geomedical engineering and public and environmental health informatics tend to study data in terms of populations, whereas medicine (including personalised and precision geomedicine, and lifestyle medicine), genetics, genomics, toxicology and biomedical/health informatics more likely work on individuals or some individual mechanism describing disease. This article introduces with examples the core concepts of medical geology and geomedicine. The ultimate goals of prediction, prevention and personalised treatment in the case of geology-dependent disease can only be realised through an intensive multiple-disciplinary approach, where the various relevant disciplines collaborate together and complement each other in additive (multidisciplinary), interactive (interdisciplinary) and holistic (transdisciplinary and cross-disciplinary) manners

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

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