Integration of polynomials over linear polyhedra in Euclidean three-dimensional space

This paper is concerned with explicit formulas and algorithms for computing integrals of polynomials over a linear polyhedron in Euclidean three-dimensional space. Symbolic formulas for surface and volume integration are given. Two different approaches are discussed: The first algorithm is obtained by transforming a volume integral into a surface integral and then into a parametric line integral while the second algorithm is obtained by transforming a volume integral into a surface integral and then into a parametric double integral. These algorithms and formulas are followed by an application-example for which we have explained the detailed computational scheme. The symbolic results presented in this paper may lead to an easy incorporation of global geometric properties of solid objects, for example, the volume, centre of mass, moments of inertia, required in the engineering design process. © 1995

CAG repeat expansion in Huntington disease determines age at onset in a fully dominant fashion

Objective: Age at onset of diagnostic motor manifestations in Huntington disease (HD) is strongly correlated with an expanded CAG trinucleotide repeat. The length of the normal CAG repeat allele has been reported also to influence age at onset, in interaction with the expanded allele. Due to profound implications for disease mechanism and modification, we tested whether the normal allele, interaction between the expanded and normal alleles, or presence of a second expanded allele affects age at onset of HD motor signs. Methods: We modeled natural log-transformed age at onset as a function of CAG repeat lengths of expanded and normal alleles and their interaction by linear regression. Results: An apparently significant effect of interaction on age at motor onset among 4,068 subjects was dependent on a single outlier data point. A rigorous statistical analysis with a wellbehaved dataset that conformed to the fundamental assumptions of linear regression (e.g., constant variance and normally distributed error) revealed significance only for the expanded CAG repeat, with no effect of the normal CAG repeat. Ten subjects with 2 expanded alleles showed an age at motor onset consistent with the length of the larger expanded allele. Conclusions: Normal allele CAG length, interaction between expanded and normal alleles, and presence of a second expanded allele do not influence age at onset of motor manifestations, indicating that the rate of HD pathogenesis leading to motor diagnosis is determined by a completely dominant action of the longest expanded allele and as yet unidentified genetic or environmental factors. Copyright © 2012 by AAN Enterprises, Inc