title: Existence, mixing and approximation of invariant densities for expanding maps on Rr

authors: Murray, Rua

subjects: mathematics

abstract: This paper generalises Gora and Boyarsky’s bounded variation(BV) approach to the ergodic properties of expanding transformations, and analysies the convergence of Ulam’s method for the numerical approximation of absolutely continuous invariant measures. We first prove an existence theorem for BV invariant densities for piecewise expanding maps on subsets of Rr; the maps must be C2, but may have infinitely many branches and need not be Markov. Under and additional “onto” assumption, explicit bounds on the spectral gap in the associated Perron-Frobenius operator are proved. The corresponding contraction rates are in the BV norm, rather than a projective metric. With this quantitative information, we are then able to prove convergence and explicit upper bounds on the approximation error in Ulam’s method for approximating invariant measures. Because the BV approach is rather concete, the methods of this paper can be applied in practice; this is illustrated by an application of the main results to the Jacobi-Perron transformation on R2.

text: Existence, mixing and approximation of invariant densities for expanding maps on Rr This paper generalises Gora and Boyarsky’s bounded variation(BV) approach to the ergodic properties of expanding transformations, and analysies the convergence of Ulam’s method for the numerical approximation of absolutely continuous invariant measures. We first prove an existence theorem for BV invariant densities for piecewise expanding maps on subsets of Rr; the maps must be C2, but may have infinitely many branches and need not be Markov. Under and additional “onto” assumption, explicit bounds on the spectral gap in the associated Perron-Frobenius operator are proved. The corresponding contraction rates are in the BV norm, rather than a projective metric. With this quantitative information, we are then able to prove convergence and explicit upper bounds on the approximation error in Ulam’s method for approximating invariant measures. Because the BV approach is rather concete, the methods of this paper can be applied in practice; this is illustrated by an application of the main results to the Jacobi-Perron transformation on R2.

year: 2001

title: The effects of promotional content on the nutritional quality of children’s breakfast cereals from Australia, New Zealand, the United Kingdom , Canada and the United States.

authors: Chepulis, Lynne Merran

abstract: Breakfast is considered to be an essential meal for children, offering valuable nutrition for growing bodies. However, the nutritional value of pre-packaged children’s cereals has been questioned recently, particularly as reduced nutritional quality can contribute to childhood obesity.

text: The effects of promotional content on the nutritional quality of children’s breakfast cereals from Australia, New Zealand, the United Kingdom , Canada and the United States. Breakfast is considered to be an essential meal for children, offering valuable nutrition for growing bodies. However, the nutritional value of pre-packaged children’s cereals has been questioned recently, particularly as reduced nutritional quality can contribute to childhood obesity.

year: 2019

title: A Māori and Pacific lens on infant and toddler provision in early childhood education

authors: Rameka, Lesley Kay

abstract: While New Zealand has made major advances in educational provision, Mäori and Pacific chil- dren continue to lag behind other groups, achieving disproportionately lower results on national averages (Marriot & Sim, 2014). Key to educational success for Mäori and Pasifika children is the acknowledgement that they are culturally located and the recognition that effective education must embrace culture. Early childhood education has an important role in building strong learn- ing foundations for young children; however, achieving this is a complex process, especially with regard to Mäori and Pacific children in early childhood education. This article outlines findings from a 2014 nationwide online survey conducted with Mäori and Pacific teachers working in Mäori and Pacific early childhood services and language nests. Respondents were asked about traditional and contemporary caregiving practices for infants and toddlers and what barriers and enablers were faced in implementing these caregiving practices within their services.

text: A Māori and Pacific lens on infant and toddler provision in early childhood education While New Zealand has made major advances in educational provision, Mäori and Pacific chil- dren continue to lag behind other groups, achieving disproportionately lower results on national averages (Marriot & Sim, 2014). Key to educational success for Mäori and Pasifika children is the acknowledgement that they are culturally located and the recognition that effective education must embrace culture. Early childhood education has an important role in building strong learn- ing foundations for young children; however, achieving this is a complex process, especially with regard to Mäori and Pacific children in early childhood education. This article outlines findings from a 2014 nationwide online survey conducted with Mäori and Pacific teachers working in Mäori and Pacific early childhood services and language nests. Respondents were asked about traditional and contemporary caregiving practices for infants and toddlers and what barriers and enablers were faced in implementing these caregiving practices within their services.

year: 2015