Generalization of the interaction between the Haar approximation and polynomial operators to higher order methods

International audienceIn applications it is useful to compute the local average of a function f(u) of an input u from empirical statistics on u. A very simple relation exists when the local averages are given by a Haar approximation. The question is to know if it holds for higher order approximation methods. To do so, it is necessary to use approximate product operators defined over linear approximation spaces. These products are characterized by a Strang and Fix like condition. An explicit construction of these product operators is exhibited for piecewise polynomial functions, using Hermite interpolation. The averaging relation which holds for the Haar approximation is then recovered when the product is defined by a two point Hermite interpolation

Reproducibility of a peripheral quantitative computed tomography scan protocol to measure the material properties of the second metatarsal

Background: Peripheral quantitative computed tomography (pQCT) is an established technology that allows for the measurement of the material properties of bone. Alterations to bone architecture are associated with an increased risk of fracture. Further pQCT research is necessary to identify regions of interest that are prone to fracture risk in people with chronic diseases. The second metatarsal is a common site for the development of insufficiency fractures, and as such the aim of this study was to assess the reproducibility of a novel scanning protocol of the second metatarsal using pQCT. Methods. Eleven embalmed cadaveric leg specimens were scanned six times; three times with and without repositioning. Each foot was positioned on a custom-designed acrylic foot plate to permit unimpeded scans of the region of interest. Sixty-six scans were obtained at 15% (distal) and 50% (mid shaft) of the second metatarsal. Voxel size and scan speed were reduced to 0.40 mm and 25 mm.sec-1. The reference line was positioned at the most distal portion of the 2nd metatarsal. Repeated measurements of six key variables related to bone properties were subject to reproducibility testing. Data were log transformed and reproducibility of scans were assessed using intraclass correlation coefficients (ICC) and coefficients of variation (CV%). Results: Reproducibility of the measurements without repositioning were estimated as: trabecular area (ICC 0.95; CV% 2.4), trabecular density (ICC 0.98; CV% 3.0), Strength Strain Index (SSI) - distal (ICC 0.99; CV% 5.6), cortical area (ICC 1.0; CV% 1.5), cortical density (ICC 0.99; CV% 0.1), SSI - mid shaft (ICC 1.0; CV% 2.4). Reproducibility of the measurements after repositioning were estimated as: trabecular area (ICC 0.96; CV% 2.4), trabecular density (ICC 0.98; CV% 2.8), SSI - distal (ICC 1.0; CV% 3.5), cortical area (ICC 0.99; CV%2.4), cortical density (ICC 0.98; CV% 0.8), SSI - mid shaft (ICC 0.99; CV% 3.2). Conclusions: The scanning protocol generated excellent reproducibility for key bone properties measured at the distal and mid-shaft regions of the 2 nd metatarsal. This protocol extends the capabilities of pQCT to evaluate bone quality in people who may be at an increased risk of metatarsal insufficiency fractures

Guest Molecule-Responsive Functional Calcium Phosphonate Frameworks for Tuned Proton Conductivity

We report the synthesis, structural characterization, and functionality of an open-framework hybrid that combines Ca2+ ions and the rigid polyfunctional ligand 5-(dihydroxyphosphoryl) isophthalic acid (PiPhtA). Ca-PiPhtA-I is obtained by slow crystallization at ambient conditions from acidic (pH≈3) aqueous solutions. It possesses a high water content (both Ca coordinated and in the lattice), and importantly, it exhibits water-filled 1D channels. At 75 °C, Ca-PiPhtA-I is partially dehydrated and exhibits a crystalline diffraction pattern that can be indexed in a monoclinic cell with parameters close to the pristine phase. Rietveld refinement was carried out for the sample heated at 75 °C, Ca-PiPhtA-II, using synchrotron powder X-ray diffraction data.All connectivity modes of the “parent” Ca-PiPhtA-I framework are retained in Ca-PiPhtA-II. Upon Ca-PiPhtA-I exposure to ammonia vapors (28% aqueous NH3) a new derivative is obtained (Ca-PiPhtA-NH3) containing 7 NH3 and 16 H2O molecules according to elemental and thermal analyses. Ca-PiPhtA-NH3 exhibits a complex X-ray diffraction pattern with peaks at 15.3 and 13.0 Å that suggest partial breaking and transformation of the parent pillared structure. Although detailed structural identification of Ca-PiPhtA-NH3 was not possible, due in part to nonequilibrium adsorption conditions and the lack of crystallinity, FT-IR spectra and DTA-TG analysis indicate profound structural changes compared to the pristine Ca-PiPhtA-I. At 98% RH and T = 24 °C, proton conductivity, σ, for Ca PiPhtA-I is 5.7 ×10−4 S·cm−1. It increases to 1.3 × 10−3 S·cm−1 upon activation by preheating the sample at 40 °C for 2 h followed by water equilibration at room temperature under controlled conditions. Ca-PiPhtA-NH3 exhibits the highest proton conductivity, 6.6 × 10−3 S·cm−1, measured at 98% RH and T = 24 °C. Ea for proton transfer in the above-mentioned frameworks range between 0.23 and 0.4 eV, typical of a Grothuss mechanism of proton conduction.Proyecto nacional MAT2010-1517