Hopf normal form with SNS_N symmetry and reduction to systems of nonlinearly coupled phase oscillators

Coupled oscillator models where $N$ oscillators are identical and symmetrically coupled to all others with full permutation symmetry $S_N$ are found in a variety of applications. Much, but not all, work on phase descriptions of such systems consider the special case of pairwise coupling between oscillators. In this paper, we show this is restrictive - and we characterise generic multi-way interactions between oscillators that are typically present, except at the very lowest order near a Hopf bifurcation where the oscillations emerge. We examine a network of identical weakly coupled dynamical systems that are close to a supercritical Hopf bifurcation by considering two parameters, $\epsilon$ (the strength of coupling) and $\lambda$ (an unfolding parameter for the Hopf bifurcation). For small enough $\lambda>0$ there is an attractor that is the product of $N$ stable limit cycles; this persists as a normally hyperbolic invariant torus for sufficiently small $\epsilon>0$. Using equivariant normal form theory, we derive a generic normal form for a system of coupled phase oscillators with $S_N$ symmetry. For fixed $N$ and taking the limit $0<\epsilon\ll\lambda\ll 1$, we show that the attracting dynamics of the system on the torus can be well approximated by a coupled phase oscillator system that, to lowest order, is the well-known Kuramoto-Sakaguchi system of coupled oscillators. The next order of approximation genericlly includes terms with up to four interacting phases, regardless of $N$. Using a normalization that maintains nontrivial interactions in the limit $N\rightarrow \infty$, we show that the additional terms can lead to new phenomena in terms of coexistence of two-cluster states with the same phase difference but different cluster size

Chaos in generically coupled phase oscillator networks with nonpairwise interactions

The Kuramoto-Sakaguchi system of coupled phase oscillators, where interaction between oscillators is determined by a single harmonic of phase differences of pairs of oscillators, has very simple emergent dynamics in the case of identical oscillators that are globally coupled: there is a variational structure that means the only attractors are full synchrony (in-phase) or splay phase (rotating wave/full asynchrony) oscillations and the bifurcation between these states is highly degenerate. Here we show that nonpairwise coupling - including three and four-way interactions of the oscillator phases - that appears generically at the next order in normal-form based calculations, can give rise to complex emergent dynamics in symmetric phase oscillator networks. In particular, we show that chaos can appear in the smallest possible dimension of four coupled phase oscillators for a range of parameter values

Evaluation of the Sigma Quality level for Serum Iron determination by two colorimetric methods, Ferrozine and Ferene S

Iron plays important functions in the body such as the formation and functioning of hemoglobin and it’s disorders are among the most common diseases of human1. It is essential to ensure that its levels determination through laboratory tests are accurate and precise. The participation of laboratories in the External Quality Assessment (EQA) allows the increases of the quality level of the laboratory results and improvement of its performance.2This study was developed in the Portuguese Nacional EQA Program (PNAEQ) concerning the laboratories results from the Clinical Chemistry Scheme. The main objective of this study was to evaluate and improve the sigma quality level regarding the Iron parameter and reduce the variability of the laboratories results participating in the EQA program of Clinical Chemistry of the Nacional External Quality Assessment Program (PNAEQ). The mean Sigma quality level indicated that the Ferrozine method had a better performance compared with Ferene S method. Half of the control samples had a sigma quality level higher than 3.0, which is set as the minimum acceptable quality.3 Despite of the improved of the Sigma quality level in the Pilot Test, the results demonstrated a need to improved the analytical process performance and to identified more potential causes and implement new improvement actions. It becomes necessary to raise awareness with the laboratories, improving the Pilot Test participation frequency, resulting in a recurrent and current assessment of the laboratory activity performance. Developing Six Sigma projects on a periodic basis is important for continuously and progressively increasing the level of Sigma quality in laboratory examinations. The main advantage of quality assessment on the sigma scale is providing evidence of overall laboratory performance, taking into account random and systematic errors.N/

Conditional tests for elliptical symmetry using robust estimators

This paper presents a procedure for testing the hypothesis that the underlying distribution of the data is elliptical when using robust location and scatter estimators instead of the sample mean and covariance matrix. Under mild assumptions that include elliptical distributions without first moments, we derive the test statistic asymptotic behaviour under the null hypothesis and under special alternatives. Numerical experiments allow to compare the behaviour of the tests based on the sample mean and covariance matrix with that based on robust estimators, under various elliptical distributions and different alternatives. This comparison was done looking not only at the observed level and power but we rather use the size-corrected relative exact power which provides a tool to assess the test statistic skill to detect alternatives. We also provide a numerical comparison with other competing tests.Comment: In press in Communications in Statistics: Theory and Methods, 201

Um ambiente virtual interativo com o geogebra e o M3 para um estudo de volume de pirámides

Com a expansão de recursos tecnológicos digitais, como multimídias, o uso desses para os processos de ensino e aprendizagem de Matemática tende a ser cada vez maior. Por outro lado, não sabemos quais são as características dessas multimídias que podem potencializar a aprendizagem. Visando analisar as contribuições de um ambiente virtual interativo, a pergunta norteadora da pesquisa aqui apresentada é: Como um ambiente que contém recursos do M³ e do GeoGeobra pode ser caracterizado construcionista, de forma que esse possa contribuir para a aprendizagem do aluno em um estudo sobre volume de pirâmides? Nesse sentido, essa pesquisa qualitativa foi uma investigação das características desse ambiente composto por atividades relacionadas ao estudo de pirâmides, com ênfase em volume. Para tanto, utilizei como aporte teórico o construcionismo e a teoria cognitiva de aprendizagem multimídia. O ambiente construcionista analisado apresenta características que corroboram com a teoria cognitiva de aprendizagem multimídia

Understanding the Variations in Gibrat's Law with a Markov-Perfect Dynamic Industry Model

Gibrat's Law of proportionate effect, as applied to firms, states that the growth rate of a firm is independent of its size. Empirical work on firm dynamics finds crucial deviations from Gibrat's Law such as smaller firms growing faster than larger firms (Evans, 1987, and Hall, 1987), a negative relationship between the variance of growth rates and size (Dunne and Hughes, 1994), and first-order positive autocorrelation in the growth rates (Kumar, 1995). Moreover, the degree of deviation from Gibrat's Law varies across industries. This paper contributes to our understanding of the forces that make Gibrat's Law a close, but imperfect approximation of firm size distributions and seeks to determine potential sources of cross-industry variation. Here, we employ an extension of the Ericson-Pakes (1995) theoretical framework that allows for firm growth developed by Laincz (2004a). By varying key parameters, the simulations demonstrate potential sources for the various, and sometimes conflicting, results on Gibrat's Law uncovered in the empirical literatureGibrat's Law, Firm Size Distribution, Entry, Exit

Abnormal regulation of Na,K-ATPase in Glucose Intolerant Rats.

Introduction: Glucose is the most important physiological insulin secretagogue. However, the mechanisms underlying glucose-induced insulin release are not fully understood. The role of electrogenic systems such as ionic pumps, to these events remains essentially uninvestigated. Na,K-ATPase, responsible for maintaining Na+ and K+ gradients across the plasma membrane and generates a net outward current, thus changes in its activity may contribute to the early ionic events regulating insulin secretion (Therien and Blostein, 2000). Objective: The aim of this work was to evaluate the regulation of Na,K-ATPase activity by glucose in intact -cells of normal and glucose intolerant (GI) rats and its putative contribution to the regulation of insulin secretion. Material and Methods: Pancreatic -cells, from normal or control or GI rats, were isolated and cultured (48h). Cell batches were pre-incubated (30min) with 2mM glucose to reach basal. Afterwards cells were challenged with glucose in the interval 0-11mM for 60min, for dose-dependence evaluation, or with 8mM glucose for 5-120min, for time-dependence evaluation. ATPase activity was assessed in intact cells by colorimetric quantification of Pi formed in 30min. Na,K-ATPase activity was calculated by the difference between the activities obtained in the absence and in presence the of 1mM ouabain (Costa et al., 2009). Results: In β-cells from normal rats, glucose induced a bimodal regulation of Na,K-ATPase. In the absence of glucose, Na,K-ATPase activity was 0.056±0.015 U/mg. Stimulation with 2mM glucose induced an increase of Na,K-ATPase activity of ~4 fold whereas for [glucose] above 2mM it was observed a significant inhibition of Na,K-ATPase activity (0.061±0.013, 0.080±0.009 and 0.064±0.005 U/mg for 5.6, 8.4 and 11mM glucose, respectively, compared to 0.188±0.035 U/mg observed in 2mM G; n=3-8). β-cells from GI rats does not present this profile; in the absence of glucose, Na,K-ATPase activity was 0.202±0.036 U/mg and no significant differences from this value were observed with the other glucose concentration tested. Addicionally, in β-cells from normal rats, glucose (8mM) induced a time-dependent inhibition, with a biphasic profile, of Na,K-ATPase - it was observed a decrease in the pump activity between 0 and 20min stimulation where it reached a minimum value (77%). For incubation periods over 20min, the pump activity slowly and partially recovered (54%, 55% and 52%, for 30, 60 and 120min, respectively; n=7). In β-cells from GI animals, an less accentuated decrease of Na,K-ATPase activity between 0 ans 20min was also observed (34%), and is not observed further recover in activity. Conclusions: This work demonstrates there Na,K-ATPase is strictly regulated by glucose in pancreatic β-cell. This regulation is unpaired in GI animals. Na,K-ATPase contribution to glucose-induced ionic events and insulin secretion might be relevant and must be explored as a possible therapeutic target in TD2 . 1. Therien AG, Blostein R (2000) Mechanisms of sodium pump regulation. Am J Physiol Cell Physiol 279:C541-C566 2. Costa AR, Real J, Antunes CM, Cruz-Morais J (2009) A new approach for determination of Na,K-ATPase activity: application to intact pancreatic beta-cells. In Vitro Cell Dev Biol Ani