Complex Chebyshev polynomials and generalizations with an application to the optimal choice of interpolating knots in complex planar splines

AbstractThis paper contains a brief account on complex planar splines which are complex valued functions defined piecewise on a grid. For noncontinuous (so called nonconforming) splines the problem of the placement of knots at which these splines are required to be continuous is investigated. It is shown that this problem reduces to finding complex Chebyshev polynomials under the additional requirement that the zeros of the polynomials are on the boundary of the corresponding domains. It is proved that the zeros of a generalized Chebyshev polynomial are in the convex hull of the domain on which the Chebyshev polynomials are defined. Some open problems are stated. A numerical and graphical display for the optimal location of three and six points on certain triangles is provided

Cross-Dimensional Mapping of Number, Length and Brightness by Preschool Children

Human adults in diverse cultures, children, infants, and non-human primates relate number to space, but it is not clear whether this ability reflects a specific and privileged number-space mapping. To investigate this possibility, we tested preschool children in matching tasks where the dimensions of number and length were mapped both to one another and to a third dimension, brightness. Children detected variation on all three dimensions, and they reliably performed mappings between number and length, and partially between brightness and length, but not between number and brightness. Moreover, children showed reliably better mapping of number onto the dimension of length than onto the dimension of brightness. These findings suggest that number establishes a privileged mapping with the dimension of length, and that other dimensions, including brightness, can be mapped onto length, although less efficiently. Children's adeptness at number-length mappings suggests that these two dimensions are intuitively related by the end of the preschool years

Implementing a new mathematics curriculum in England: district Research Lesson Study as a driver for student learning, teacher learning and professional dialogue.

Against a backdrop of a transformation in teacher professional development and learning and state school organisation in England this century, this chapter describes a project which harnessed six cycles of Research Lesson Study at school and district level over two years to tailor the implementation of a new statutory curriculum in England to address the professional development needs of teachers and classroom learning needs of London students. It also reports the findings of research carried out during the project into how these teachers learned and developed this new curricular expertise and practice- knowledge through lesson study dialogues that supported student learning. It concludes by proposing future directions for teacher professional learning research and practice

Best interpolation in a strip II: Reduction to unconstrained convex optimization

In this paper, we study the problem of finding a real-valued function f on the interval [0, 1] with minimal L 2 norm of the second derivative that interpolates the points ( t i , y i ) and satisfies e(t) ≤ f(t) ≤ d(t) for t ∈ [0, 1]. The functions e and d are continuous in each interval ( t i , t i +1) and at t 1 and t n but may be discontinuous at t i . Based on an earlier paper by the first author [7] we characterize the solution in the case when e and d are linear in each interval ( t i , t i +1). We present a method for the reduction of the problem to a convex finite-dimensional unconstrained minimization problem. When e and d are arbitrary continuous functions we approximate the problem by a sequence of finite-dimensional minimization problems and prove that the sequence of solutions to the approximating problems converges in the norm of W 2,2 to the solution of the original problem. Numerical examples are reported.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/44775/1/10589_2004_Article_BF00248266.pd

The nature of the association between number line and mathematical performance: An international twin study

Background: The number line task assesses the ability to estimatenumerical magnitudes. People vary greatly in this abilityand this variability has been previously associated with mathematical skills. However, the sources of individual differences in number line estimation and its association with mathematics are not fully understood. Aims: This large scale genetically sensitive studyuses a twin design to estimate the magnitude of the effects of genes and environments on: (1) individualvariation in number line estimation and (2) the co-variation of number line estimation with mathematics. Samples: We used over3,0008-16 years-old twins from US, Canada,UK, and Russia, and a sample of 1,456 8-18 years-old singleton Russian students. Methods: Twins were assessed on: (1)estimation of numerical magnitudes using a numberline task and (2) two mathematics components: fluency and problemsolving. Results: Results suggest that environments largelydrive individual differences in numberline estimation.Both genes and environments contribute to different extents to the number line estimationandmathematics correlation, depending on the sample and mathematics component. Conclusions: Taken together, the results suggest that in more heterogeneous school settings, environments may be more important in driving variation in number line estimation and its associationwith mathematics, whereas in more homogeneous school settings, genetic effects drive the covariation between number line estimationand mathematics. These results are discussed in light of development and educational settings

Impact of High Mathematics Education on the Number Sense

In adult number processing two mechanisms are commonly used: approximate estimation of quantity and exact calculation. While the former relies on the approximate number sense (ANS) which we share with animals and preverbal infants, the latter has been proposed to rely on an exact number system (ENS) which develops later in life following the acquisition of symbolic number knowledge. The current study investigated the influence of high level math education on the ANS and the ENS. Our results showed that the precision of non-symbolic quantity representation was not significantly altered by high level math education. However, performance in a symbolic number comparison task as well as the ability to map accurately between symbolic and non-symbolic quantities was significantly better the higher mathematics achievement. Our findings suggest that high level math education in adults shows little influence on their ANS, but it seems to be associated with a better anchored ENS and better mapping abilities between ENS and ANS

Ectoparasitos de cães no ambiente doméstico da vertente Caribe, Costa Rica

Reports on ectoparasites on dogs in Central America are scarce. The aim of this study was to identify flea, louse and tick species infesting dogs in home environments on the Caribbean slope of Costa Rica, and determine their frequency and coexistence. Ectoparasites were collected from dogs in 83 rural homes at five study sites. Specimens were identified and separated according to species. Fleas were the most common ectoparasite (G = 22,217, DF = 8, p = 0.004). Ctenocephalides felis and Pulex simulans were found in 83% and 55% of the homes with ectoparasites, respectively. Trichodectes canis (13%), Heterodoxus spiniger (10%), Rhipicephalus sanguineus (18%), Rhipicephalus (Boophilus) microplus (5%) and Amblyomma ovale (8%) were also present. More than one species was collected in most cases (66%), and the most common combination was C. felis and P. simulans (59% of homes with fleas). The high frequency of P. simulans emphasizes the need for adequate identification. This was the first study involving different ectoparasites of dogs in Costa Rica, as well as the first report of T.  canis in this country. The relative frequency and coexistence of these ectoparasites in the home environment may have implications for animal and human health.Relatos de ectoparasitos em cães da América Central são escassos. O objetivo deste estudo foi identificar espécies de pulgas, piolhos e carrapatos que infestam os cães no ambiente doméstico da vertente Caribe da Costa Rica, determinando sua frequência e coexistência. Foram coletados ectoparasitos de cães em 83 casas rurais de cinco sítios do estudo. Os indivíduos foram separados e identificados. Os ectoparasitos mais comuns foram as pulgas (G = 22.217, DF = 8, p = 0.004). Ctenocephalides felis e Pulex simulans foram observados em 83% e 55% das moradias com ectoparasitos, respectivamente. Da mesma forma também foram observados Trichodectes canis (13%), Heterodoxus spiniger (10%), Rhipicephalus sanguineus (18%), Rhipicephalus (Boophilus) microplus (5%) e Amblyomma ovale (8%). Na maioria dos casos, mais de uma espécie foi achada (66%), sendo que a combinação mais comum foi C. felis e P. simulans (59% de moradias com pulgas). A frequência alta de P. simulans ressalta a necessidade de sua adequada identificação. Este é o primeiro estudo que envolve diferentes ectoparasitas de cães na Costa Rica, da mesma forma, é o primeiro relato de T.  canis no país. A frequência relativa e coexistência destes ectoparasitos no ambiente domiciliar poderiam ter implicações na saúde animal e humana.Universidad de Costa Rica/[803-A8-127]/UCR/Costa Rica/[9-N-2008]/NeTropica/Costa RicaUCR::Vicerrectoría de Investigación::Unidades de Investigación::Ciencias de la Salud::Centro de Investigación en Enfermedades Tropicales (CIET)UCR::Vicerrectoría de Investigación::Unidades de Investigación::Ciencias Básicas::Centro de Investigación en Estructuras Microscópicas (CIEMIC)UCR::Vicerrectoría de Docencia::Ciencias Básicas::Facultad de Ciencias::Escuela de Biologí