Exhausting families of representations and spectra of pseudodifferential operators

Families of representations of suitable Banach algebras provide a powerful tool in the study of the spectral theory of (pseudo)differential operators and of their Fredholmness. We introduce the new concept of an exhausting family of representations of a C*-algebra A. An {\em exhausting family} of representations of a C*-algebra A is a set F of representations of A with the property that every irreducible representation of A is weakly contained in some \phi \in F. An exhausting family F of representations of A has the property that `"a \in A is invertible if, and if, \phi(a) is invertible for any \phi \in F." Consequently, the spectrum of a is given by \Spec(a) = \cup_{\phi \in F} \Spec(\phi(a)). In other words, every exhausting family of representations is invertibility sufficient, a concept introduced by Roch in 2003. We prove several properties of exhausting families and we provide necessary and sufficient conditions for a family of representations to be exhausting. Using results of Ionescu and Williams (2009), we show that the regular representations of amenable, second countable, locally compact groupoids with a Haar system form an exhausting family of representations. If $A$ is a separable C*-algebra, we show that a family F of representations of $A$ is exhausting if, and only if, it is invertibility sufficient. However, this result is not true, in general, for non-separable C*-algebras. With an eye towards applications, we extend our results to the case of unbounded operators. A typical application of our results is to parametric families of differential operators arising in the analysis on manifolds with corners, in which case we recover the fact that a parametric operator F is invertible if, and only if, its Mellin transform is invertible. In view of possible applications, we have tried to make this paper accessible to non-specialists in C*-algebras.Comment: Additional reference

Exhaustive families of representations of C∗C^*-algebras associated to NN-body Hamiltonians with asymptotically homogeneous interactions

We continue the analysis of algebras introduced by Georgescu, Nistor and their coauthors, in order to study $N$-body type Hamiltonians with interactions. More precisely, let $Y$ be a linear subspace of a finite dimensional Euclidean space $X$, and $v_Y$ be a continuous function on $X/Y$ that has uniform homogeneous radial limits at infinity. We consider, in this paper, Hamiltonians of the form $H = - \Delta + \sum_{Y \in S} v_Y$, where the subspaces $Y$ belong to some given family S of subspaces. We prove results on the spectral theory of the Hamiltonian when $S$ is any family of subspaces and extend those results to other operators affiliated to a larger algebra of pseudo-differential operators associated to the action of $X$ introduced by Connes. In addition, we exhibit Fredholm conditions for such elliptic operators. We also note that the algebras we consider answer a question of Melrose and Singer.Comment: 5 page

The Universal Language of Music: Why Music is Important in Education

This capstone project focused on the need for increasing music opportunities in schools and the ways in which it increases literacy, particularly in an English Language Learners (ELL) classroom, where increasing English literacy is so vital. After interviewing a district music director, a high school English language development director, and a high school ELL teacher, three action options emerged as ways to provide more music in high school ELL classrooms. Based on the findings, an action was undertaken to help secondary teachers incorporate more music into their daily curriculum

Anthropometric indices and measures to assess change in the nutritional status of a population: a systematic literature review

Background: Undernutrition is a major public health issue highlighted by the 2015 Sustainable Development Goals, with target 2.2 aiming to 'end hunger' by 2030. On-going surveillance is essential detecting nutritional stress in a population and is key to planning consequent interventions. Whilst methodologies of nutritional surveillance systems vary across different settings, organisations and even within the same country, the direct evidence-base underpinning these practices is limited. This paper aims therefore to: 1) compare the performance of different anthropometric indices/measurements for detecting change in the nutritional situation at population level; 2) discuss their properties and appropriateness for use in a surveillance system. Methods: This systematic literature review considered peer-reviewed and grey literature. Evidence was compiled from standard electronic databases, websites and snowballing. The search was performed in November 2015 by a single reviewer using the following terms to capture two concepts: 1) Undernutrition and 2) Nutrition surveillance. The search was limited to children under five and the period considered started in 1980. Languages included English and French. Articles had to assess whether the changes or trend observed at population level were statistically significant. All study designs were included. Results: A total of 4563 articles were retrieved from the electronic database search. Most articles (3137, 89%) were not directly relevant based on title and abstract; 39 articles were reviewed in full. A total of 17 articles met the inclusion criteria and an additional 4 papers were added after snowballing. A number of measures and indices such as weight, weight-for-height/length, triceps skinfold and middle-upper arm circumference performed well in the detection of short term changes in the nutritional situation of a population. Height/Length-for-age responded the most to long term change. Applying a standard set of criteria (simplicity, acceptability, cost, independence of age, reliability and accuracy) to determine which is the most appropriate measure or index identified middle-upper arm circumference as the one with the greatest net benefits. Conclusions: Limited available evidence suggests that mid-upper arm circumference is the best measure to detect short term changes in the nutritional state of a population: this should receive higher priority in surveillance systems

Assessment of the PROBIT approach for estimating the prevalence of global, moderate and severe acute malnutrition from population surveys

Abstract Objective Prevalence of acute malnutrition is classically estimated by the proportion of children meeting a case definition in a representative population sample. In 1995 the WHO proposed the PROBIT method, based on converting parameters of a normally distributed variable to cumulative probability, as an alternative method requiring a smaller sample size. The present study compares classical and PROBIT methods for estimating the prevalence of global, moderate and severe acute malnutrition (GAM, MAM and SAM) defined by weight-for-height Z-score (WHZ) or mid-upper arm circumference (MUAC). Design Bias and precision of classical and PROBIT methods were compared by simulating a total of 1·26 million surveys generated from 560 nutrition surveys. Setting Data used for simulation were derived from nutritional surveys of children aged 6-59 months carried out in thirty-one countries around the world. Subjects Data of 459 036 children aged 6-59 months from representative samples were used to generate simulated populations. Results The PROBIT method provided an estimate of GAM, MAM and SAM using WHZ or MUAC proportional to the true prevalence with a small systematic overestimation. The PROBIT method was more precise than the classical method for estimating the prevalence for GAM, MAM and SAM by WHZ or MUAC for small sample sizes (i.e. n<150 for SAM and GAM; n<300 for MAM), but lost this advantage when sample sizes increased. Conclusions The classical method is preferred for estimating acute malnutrition prevalence from large sample surveys. The PROBIT method may be useful in sentinel-site surveillance systems with small sample size

Comparison of weight and height-based indices for assessing the risk of death in severely malnourished chidren

To compare the effectiveness of treating malnourished children in different centers, the authors believe there is a need to have a simple method of adjusting mortality rates so that differences in the nutritional status of the children are taken into account. The authors compared different anthropometric indices based on weight and height to predict the risk of death among severely malnourished children. Anthropometric data from 1047 children who survived were compared with those of 147 children who died during treatment in therapeutic feeding centers set up in African countries in 1993. The optimal ratio of weight to height determined by logistic regression was weight (kg)/height (m) (95% confidence interval of bêta estimate 1.65-1.84). The receiver operating curves (sensitivity vs. specificity) showed that the body mass index (weight (kg)/height (m) -puissance 2-), optimal ratio of weight to height, and weight/height index expressed as the percentage of the median of the National Center for Health Statistics' standard were equivalent and superior to the weight/height index expressed as the z score of the National Center for Health Statistics' standard to predict death. As the optimal ratio of weight to height is easier to calculate than the weight/height index expressed as the percentage of the median or z score and does not depend upon either standards or tables, the optimal ratio of weight to height could be conveniently used to adjust mortality rates for nutritional status in therapeutic feeding centers. (Résumé d'auteur