Rational invariants of even ternary forms under the orthogonal group

In this article we determine a generating set of rational invariants of minimal cardinality for the action of the orthogonal group $\mathrm{O}_3$ on the space $\mathbb{R}[x,y,z]_{2d}$ of ternary forms of even degree $2d$. The construction relies on two key ingredients: On one hand, the Slice Lemma allows us to reduce the problem to dermining the invariants for the action on a subspace of the finite subgroup $\mathrm{B}_3$ of signed permutations. On the other hand, our construction relies in a fundamental way on specific bases of harmonic polynomials. These bases provide maps with prescribed $\mathrm{B}_3$-equivariance properties. Our explicit construction of these bases should be relevant well beyond the scope of this paper. The expression of the $\mathrm{B}_3$-invariants can then be given in a compact form as the composition of two equivariant maps. Instead of providing (cumbersome) explicit expressions for the $\mathrm{O}_3$-invariants, we provide efficient algorithms for their evaluation and rewriting. We also use the constructed $\mathrm{B}_3$-invariants to determine the $\mathrm{O}_3$-orbit locus and provide an algorithm for the inverse problem of finding an element in $\mathbb{R}[x,y,z]_{2d}$ with prescribed values for its invariants. These are the computational issues relevant in brain imaging.Comment: v3 Changes: Reworked presentation of Neuroimaging application, refinement of Definition 3.1. To appear in "Foundations of Computational Mathematics

Health behaviour modelling for prenatal diagnosis in Australia: a geodemographic framework for health service utilisation and policy development

BACKGROUND: Despite the wide availability of prenatal screening and diagnosis, a number of studies have reported no decrease in the rate of babies born with Down syndrome. The objective of this study was to investigate the geodemographic characteristics of women who have prenatal diagnosis in Victoria, Australia, by applying a novel consumer behaviour modelling technique in the analysis of health data. METHODS: A descriptive analysis of data on all prenatal diagnostic tests, births (1998 and 2002) and births of babies with Down syndrome (1998 to 2002) was undertaken using a Geographic Information System and socioeconomic lifestyle segmentation classifications. RESULTS: Most metropolitan women in Victoria have average or above State average levels of uptake of prenatal diagnosis. Inner city women residing in high socioeconomic lifestyle segments who have high rates of prenatal diagnosis spend 20% more on specialist physician's fees when compared to those whose rates are average. Rates of prenatal diagnosis are generally low amongst women in rural Victoria, with the lowest rates observed in farming districts. Reasons for this are likely to be a combination of lack of access to services (remoteness) and individual opportunity (lack of transportation, low levels of support and income). However, there are additional reasons for low uptake rates in farming areas that could not be explained by the behaviour modelling. These may relate to women's attitudes and choices. CONCLUSION: A lack of statewide geodemographic consistency in uptake of prenatal diagnosis implies that there is a need to target health professionals and pregnant women in specific areas to ensure there is increased equity of access to services and that all pregnant women can make informed choices that are best for them. Equally as important is appropriate health service provision for families of children with Down syndrome. Our findings show that these potential interventions are particularly relevant in rural areas. Classifying data to lifestyle segments allowed for practical comparisons of the geodemographic characteristics of women having prenatal diagnosis in Australia at a population level. This methodology may in future be a feasible and cost-effective tool for service planners and policy developers

Arthrodesis of the proximal interphalangeal joints of a hindlimb in a heifer

A two-year-old Braunvieh heifer was presented with a traumatic luxation of the second phalanx of the medial digit and concurrent subluxation of the second phalanx of the lateral digit of the right hindlimb. Closed reduction of both luxations was possible. Surgical arthrodesis was achieved using one narrow 4.5 mm three-hole equine locking compression plate for each joint. Placement of the bone plates resulted in stable arthrodesis of both proximal interphalangeal joints of the right hindlimb but there was persistent residual lameness. The heifer delivered a healthy calf but was slaughtered eight months after surgery because of varus deformity of the contralateral limb. Radiographs taken post-mortem revealed pronounced periosteal reactions involving both proximal interphalangeal joints and only partial bony bridging of the joint spaces

Radiographic measurement of hindlimb digit length in standing heifers

The digital skeleton of the hindlimbs was evaluated radiographically in 27 standing Brown Swiss heifers. The lateral digital skeleton was significantly longer than its medial counterpart on both a hard (wooden block) and a softer (wooden block covered with a thin rubber mat) surface. There were no significant differences between lengths measured on the hard and the soft surface. The difference between the length of the lateral and medial digits originated at the level of the condyles and increased with P1 and P2. It was partially compensated by a higher third phalanx (P3) in the medial digit, but this did not offset the overall length difference. The findings of this study confirmed that the length asymmetry of the paired digits of cattle documented previously in post-mortem specimens is also present in living cattle. Further investigation is required to determine the clinical relevance of the length asymmetry of the digital skeleton in cattle