Gaussian extended cubature formulae for polyharmonic functions

The purpose of this paper is to show certain links between univariate interpolation by algebraic polynomials and the representation of polyharmonic functions. This allows us to construct cubature formulae for multivariate functions having highest order of precision with respect to the class of polyharmonic functions. We obtain a Gauss type cubature formula that uses ℳ values of linear functional (integrals over hyperspheres) and is exact for all 2ℳ-harmonic functions, and consequently, for all algebraic polynomials of n variables of degree 4ℳ - 1.

Public-private contracts: analysis and formulae for sustainability: development and applications in Mexico

Includes bibliography

Fluoride concentrations of milk, infant formulae, and soy-based products commercially available in Brazil

To assess the fluoride (F) content in commercially available milk formulae in Brazil and to estimate the F intake in children from this source in the first year of life. Samples of cow's milk (n = 51), infant formulae (n = 15), powdered milk (n = 13), and soy-based products (n = 4) purchased in Araçatuba (Brazil) had their F content measured using an ion-specific electrode, after hexamethyldisiloxane-facilitated diffusion. Powdered milk and infant formulae were reconstituted with deionized water, while ready-to-drink products were analyzed without any dilution. Using average infant body masses and suggested volumes of formula consumption for infants 1-12 months of age, possible F ingestion per body mass was estimated. Data were analyzed by descriptive analysis. Mean F content ranged from 0.02 to 2.52 mg/L in all samples. None of the cow's milk provided F intake higher than 0.07 mg/kg. However, two infant formulae, one powdered milk, and one soy-milk led to a daily F intake above the suggested threshold for fluorosis when reconstituted with deionized water. Assuming reconstitution of products with tap water at 0.7 ppm F, two infant formulae, five powdered milk, and four soymilks led to daily F intake ranging from 0.108 to 0.851 mg/kg. The results suggest that the consumption of some brands of infant formulae, powdered milk, and soy-based milk in the first year of age could increase the risk of dental fluorosis, reinforcing the need for periodic surveillance of the F content of foods and beverages typically consumed by young children.

Corrected kriging update formulae for batch-sequential data assimilation

Recently, a lot of effort has been spent in the efficient computation of kriging predictors when observations are assimilated sequentially. In particular, kriging update formulae enabling significant computational savings were derived. Taking advantage of the previous kriging mean and variance computations helps avoiding a costly matrix inversion when adding one observation to the TeX already available ones. In addition to traditional update formulae taking into account a single new observation, Emery (2009) proposed formulae for the batch-sequential case, i.e. when TeX new observations are simultaneously assimilated. However, the kriging variance and covariance formulae given in Emery (2009) for the batch-sequential case are not correct. In this paper, we fix this issue and establish correct expressions for updated kriging variances and covariances when assimilating observations in parallel. An application in sequential conditional simulation finally shows that coupling update and residual substitution approaches may enable significant speed-ups.

Justifying induction on modal μ-formulae

We define a rank function for formulae of the propositional modal μ-calculus such that the rank of a fixed point is strictly bigger than the rank of any of its finite approximations. A rank function of this kind is needed, for instance, to establish the collapse of the modal μ-hierarchy over transitive transition systems. We show that the range of the rank function is ωω. Further we establish that the rank is computable by primitive recursion, which gives us a uniform method to generate formulae of arbitrary rank below ωω.