Construction of C 2 Pythagorean-hodograph interpolating splines by the homotopy method

The complex representation of polynomial Pythagorean-hodograph (PH) curves allows the problem of constructing a C 2 PH quintic “spline” that interpolates a given sequence of points p 0 , p 1 ,..., p N and end-derivatives d 0 and d N to be reduced to solving a “tridiagonal” system of N quadratic equations in N complex unknowns. The system can also be easily modified to incorporate PH-spline end conditions that bypass the need to specify end-derivatives. Homotopy methods have been employed to compute all solutions of this system, and hence to construct a total of 2 N +1 distinct interpolants for each of several different data sets. We observe empirically that all but one of these interpolants exhibits undesirable “looping” behavior (which may be quantified in terms of the elastic bending energy , i.e., the integral of the square of the curvature with respect to arc length). The remaining “good” interpolant, however, is invariably a fairer curve-having a smaller energy and a more even curvature distribution over its extent-than the corresponding “ordinary” C 2 cubic spline. Moreover, the PH spline has the advantage that its offsets are rational curves and its arc length is a polynomial function of the curve parameter.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/41719/1/10444_2005_Article_BF02124754.pd

Rational conchoid and offset constructions: algorithms and implementation

This paper is framed within the problem of analyzing the rationality of the components of two classical geometric constructions, namely the offset and the conchoid to an algebraic plane curve and, in the affirmative case, the actual computation of parametrizations. We recall some of the basic definitions and main properties on offsets (see [13]), and conchoids (see [15]) as well as the algorithms for parametrizing their rational components (see [1] and [16], respectively). Moreover, we implement the basic ideas creating two packages in the computer algebra system Maple to analyze the rationality of conchoids and offset curves, as well as the corresponding help pages. In addition, we present a brief atlas where the offset and conchoids of several algebraic plane curves are obtained, their rationality analyzed, and parametrizations are provided using the created packages

Dislocation-Mediated Melting: The One-Component Plasma Limit

The melting parameter $\Gamma_m$ of a classical one-component plasma is estimated using a relation between melting temperature, density, shear modulus, and crystal coordination number that follows from our model of dislocation-mediated melting. We obtain $\Gamma_m=172\pm 35,$ in good agreement with the results of numerous Monte-Carlo calculations.Comment: 8 pages, LaTe

The long-term survival chances of young massive star clusters

We review the long-term survival chances of young massive star clusters (YMCs), hallmarks of intense starburst episodes often associated with violent galaxy interactions. We address the key question as to whether at least some of these YMCs can be considered proto-globular clusters (GCs), in which case these would be expected to evolve into counterparts of the ubiquitous old GCs believed to be among the oldest galactic building blocks. In the absence of significant external perturbations, the key factor determining a cluster's long-term survival chances is the shape of its stellar initial mass function (IMF). It is, however, not straightforward to assess the IMF shape in unresolved extragalactic YMCs. We discuss in detail the promise of using high-resolution spectroscopy to make progress towards this goal, as well as the numerous pitfalls associated with this approach. We also discuss the latest progress in worldwide efforts to better understand the evolution of entire cluster systems, the disruption processes they are affected by, and whether we can use recently gained insights to determine the nature of at least some of the YMCs observed in extragalactic starbursts as proto-GCs. We conclude that there is an increasing body of evidence that GC formation appears to be continuing until today; their long-term evolution crucially depends on their environmental conditions, however.Comment: invited refereed review article; ChJA&A, in press; 33 pages LaTeX (2 postscript figures); requires chjaa.cls style fil

Taxonomic classification of bacterial 16S rRNA genes using short sequencing reads: evaluation of effective study designs.

Massively parallel high throughput sequencing technologies allow us to interrogate the microbial composition of biological samples at unprecedented resolution. The typical approach is to perform high-throughout sequencing of 16S rRNA genes, which are then taxonomically classified based on similarity to known sequences in existing databases. Current technologies cause a predicament though, because although they enable deep coverage of samples, they are limited in the length of sequence they can produce. As a result, high-throughout studies of microbial communities often do not sequence the entire 16S rRNA gene. The challenge is to obtain reliable representation of bacterial communities through taxonomic classification of short 16S rRNA gene sequences. In this study we explored properties of different study designs and developed specific recommendations for effective use of short-read sequencing technologies for the purpose of interrogating bacterial communities, with a focus on classification using naïve Bayesian classifiers. To assess precision and coverage of each design, we used a collection of ∼8,500 manually curated 16S rRNA gene sequences from cultured bacteria and a set of over one million bacterial 16S rRNA gene sequences retrieved from environmental samples, respectively. We also tested different configurations of taxonomic classification approaches using short read sequencing data, and provide recommendations for optimal choice of the relevant parameters. We conclude that with a judicious selection of the sequenced region and the corresponding choice of a suitable training set for taxonomic classification, it is possible to explore bacterial communities at great depth using current technologies, with only a minimal loss of taxonomic resolution

Construction of Rational Curves with Rational Rotation-Minimizing Frames via Möbius Transformations

Abstract. We show that Möbius transformations preserve the rotationminimizing frames which are associated with space curves. In addition, these transformations are known to preserve the class of rational Pythagorean-hodograph curves and also rational frames. Based on these observations we derive an algorithm for G 1 Hermite interpolation by rational Pythagorean-hodograph curves with rational rotation-minimizing frames. Key words: rational rotation-minimizing frame, Möbius transformations, Pythagorean-hodograph curve