A Formula for Inserting Point Masses

Let mu be a probability measure on the unit circle and nu be the measure formed by adding a pure point to mu. We give a formula for the Verblunsky coefficients of the perturbed measure, based on a result of Simon

First and second kind paraorthogonal polynomials and their zeros

Given a probability measure $\mu$ with infinite support on the unit circle $\partial\mathbb{D}=\{z:|z|=1\}$, we consider a sequence of paraorthogonal polynomials \h_n(z,\lambda) vanishing at $z=\lambda$ where \lambda \in \T is fixed. We prove that for any fixed z_0 \not \in \supp(d\mu) distinct from $\lambda$, we can find an explicit $\rho>0$ independent of $n$ such that either \h_n or \h_{n+1} (or both) has no zero inside the disk $B(z_0, \rho)$, with the possible exception of $\lambda$. Then we introduce paraorthogonal polynomials of the second kind, denoted \s_n(z,\lambda). We prove three results concerning \s_n and \h_n. First, we prove that zeros of \s_n and \h_n interlace. Second, for $z_0$ an isolated point in \supp(d\mu), we find an explicit radius \rt such that either \s_n or \s_{n+1} (or both) have no zeros inside B(z_0,\rt). Finally we prove that for such $z_0$ we can find an explicit radius such that either \h_n or \h_{n+1} (or both) has at most one zero inside the ball B(z_0,\rt).Comment: To appear in the Journal of Approximation Theor

Generalized Bounded Variation and Inserting point masses

Let $d\mu$ be a probability measure on the unit circle and $d\nu$ be the measure formed by adding a pure point to $d\mu$. We give a simple formula for the Verblunsky coefficients of $d\nu$ based on a result of Simon. Then we consider $d\mu_0$, a probability measure on the unit circle with $\ell^2$ Verblunsky coefficients $(\alpha_n (d\mu_0))_{n=0}^{\infty}$ of bounded variation. We insert $m$ pure points to $d\mu$, rescale, and form the probability measure $d\mu_m$. We use the formula above to prove that the Verblunsky coefficients of $d\mu_m$ are in the form \alpha_n(d\mu_0) + \sum_{j=1}^m \frac{\ol{z_j}^{n} c_j}{n} + E_n, where the $c_j$'s are constants of norm 1 independent of the weights of the pure points and independent of $n$; the error term $E_n$ is in the order of $o(1/n)$. Furthermore, we prove that $d\mu_m$ is of $(m+1)$-generalized bounded variation - a notion that we shall introduce in the paper. Then we use this fact to prove that \lim_{n \to \infty} \vp_n^*(z, d\mu_m) is continuous and is equal to $D(z, d\mu_m)^{-1}$ away from the pure points.Comment: To appear in Constructive Approximatio

World music in the elementary and junior high general music curriculum : an analysis from an ethnomusicological perspective

Time and emphasis are not the only determinants of whether the teaching of world music makes its potential contribution to the goals of music education and the education process in general. It must be used and taught effectively. At present the music teacher's primary source of information concerning world music and how to use it is found in music textbooks. Thus a satisfactory presentation of world music depends primarily upon the quality of the textbooks and secondarily upon the quality of supplemental resources. This is both with regards to their content, and with regards to the assistance they give the teacher in using that content. Besides improvements in the quality of textbooks a broadening of the music teacher's training would obviously be helpful. Exposure to ethnomusicology, and training in the use and teaching of world music could make a significant impact on the quality of music education by making teachers less dependent on the texts as a source of didactic help, and more independent in structuring their teaching to meet various situations. The aims of this paper are to evaluate the present method of teaching world music in the elementary and junior high general music classes, to examine the historical development of world music as a component of general music curriculum, to see how teaching world music fulfills the objectives and philosophy of music education, to ascertain its present status, and to look at the direction it is following or may need to followMusi

Asymptotics of orthogonal polynomials and point perturbation on the unit circle

In the first five sections, we deal with the class of probability measures with asymptotically periodic Verblunsky coefficients of p-type bounded variation. The goal is to investigate the perturbation of the Verblunsky coefficients when we add a pure point to a gap of the essential spectrum. For the asymptotically constant case, we give an asymptotic formula for the orthonormal polynomials in the gap, prove that the perturbation term converges and show the limit explicitly. Furthermore, we prove that the perturbation is of bounded variation. Then we generalize the method to the asymptotically periodic case and prove similar results. In the last two sections, we show that the bounded variation condition can be removed if a certain symmetry condition is satisfied. Finally, we consider the special case when the Verblunsky coefficients are real with the rate of convergence being c_n . We prove that the rate of convergence of the perturbation is in fact O(c_n). In particular, the special case c_n = 1/n will serve as a counterexample to the possibility that the convergence of the perturbed Verblunsky coefficients should be exponentially fast when a point is added to a gap.Comment: Published in the Journal of Approximation Theor

One clutch or two clutches? Fitness correlates of coexisting alternative female life-histories in the European earwig

Whether to reproduce once or multiple times (semelparity vs. iteroparity) is a major life-history decision that organisms have to take. Mode of parity is usually considered a species characteristic. However, recent models suggested that population properties or condition-dependent fitness payoffs could help to maintain both life-history tactics within populations. In arthropods, semelparity was also hypothesised to be a critical pre-adaptation for the evolution of maternal care, semelparous females being predicted to provide more care due to the absence of costs on future reproduction. The aim of this study was to characterize potential fitness payoffs and levels of maternal care in semel- and itero-parous females of the European earwig Forficula auricularia. Based on 15 traits measured in 494 females and their nymphs, our results revealed that iteroparous females laid their first clutch earlier, had more eggs in their first clutch, gained more weight during the 2weeks following hatching of the first clutch, but produced eggs that developed more slowly than semelparous females. Among iteroparous females, the sizes of first and second clutches were significantly and positively correlated, indicating no investment trade-off between reproductive events. Iteroparous females also provided more food than semelparous ones, a result contrasting with predictions that iteroparity is incompatible with the evolution of maternal care. Finally, a controlled breeding experiment reported full mating compatibility among offspring from females of the two modes of parity, confirming that both types of females belong to one single species. Overall, these results indicate that alternative modes of parity represent coexisting life-history tactics that are likely to be condition-dependent and associated with offspring development and specific levels of maternal care in earwig

Detection and Monitoring of Marine Pollution Using Remote Sensing Technologies

Recently, the marine habitat has been under pollution threat, which impacts many human activities as well as human life. Increasing concerns about pollution levels in the oceans and coastal regions have led to multiple approaches for measuring and mitigating marine pollution, in order to achieve sustainable marine water quality. Satellite remote sensing, covering large and remote areas, is considered useful for detecting and monitoring marine pollution. Recent developments in sensor technologies have transformed remote sensing into an effective means of monitoring marine areas. Different remote sensing platforms and sensors have their own capabilities for mapping and monitoring water pollution of different types, characteristics, and concentrations. This chapter will discuss and elaborate the merits and limitations of these remote sensing techniques for mapping oil pollutants, suspended solid concentrations, algal blooms, and floating plastic waste in marine waters

Effects of filtration using soil and fibre mediums in improving the quality of stormwater

Water shortage problems occurred when insufficient quantities or only poor quality of water available for consumption . In some places, while fresh water is abundantly available for domestic use, high demand at industrial site and agricultural area requires the need to investigated alternative method of treatment for non-domestic water supply. In this study, effects of treatment unit using Odec and Sulaman soils, fibre and fibre-soil combinations were investigated towards their ability to filtrate and improve the quality of stormwaters collected from SST Lake, Likas River and Odec Sea. For comparison, treatment was also conducted for soy bean water. Stormwater quality tested include the chemical oxygen demand (COD), pH, turbidity and suspended solid. It was found that Sulaman soil medium yielded 100 % removal rate of suspended solid compared to ODEC and fibre medium while fibre medium showed the least effective filter. The removal rate of COD of all medium was low. All the medium failed to filter soya water due to the high concentration of chemical compound. The study concluded that when the parameters measured is chemically-related such as COD and pH, the soil medium filter less effectively. The effectiveness in filtering non-chemical parameter such suspended solid is affected by the grain size and distribution, texture and the structure of the mediums