C 1 Quintic Splines on Type-4 Tetrahedral Partitions

Starting with a partition of a rectangular box into subboxes, it is shown how to construct a natural tetrahedral (type-4) partition and associated trivariate C ¹ quintic polynomial spline spaces with a variety of useful properties, including stable local bases and full approximation power. It is also shown how the spaces can be used to solve certain Hermite and Lagrange interpolation problems.     

Super spline spaces of smoothnessr and degreed≥3r+2

The problem of computing the dimension of spaces of splines whose elements are piecewise polynomials of degreed withr continuous derivatives globally has attracted a great deal of attention recently. We contribute to this theory by obtaining dimension formulae for certain spaces of super splines, including the case where varying amounts of additional smoothness is enforced at each vertex. We also explicitly construct minimally supported bases for the spaces. The main tool is the Bernstein-Bézier method.Communicated by Klaus Höllig.     

An extension of the Dirac theory of constraints

Constructions introduced by Dirac for singular Lagrangians are extended and reinterpreted to cover cases when kernel distributions are either nonintegrable or of nonconstant rank, and constraint sets need not be closed.     

Microsoft Excel as a supplement in a high school statistics course

This study was conducted in the spring of 2002 at Arlington High School in a Probability and Statistics course for junior and senior level students. Two sections of the class were compared with respect to the use/non-use of EXCEL in the course. The experimental group (the one using EXCEL) outperformed the control group on five of the six teacher-created unit tests and on both of the criterion referenced assessments used by the district; however, the differences were not statistically significant at the 0.05 level. Furthermore, the experimental group expressed very positive feelings towards the use of the software in a survey administered at the conclusion of the course.     

The Width-Volume Inequality

We prove that a bounded open set U in has k-width less than C(n) Volume(U) ^k/n . Using this estimate, we give lower bounds for the k-dilation of degree 1 maps between certain domains in . In particular, we estimate the smallest (n – 1)-dilation of any degree 1 map between two n-dimensional rectangles. For any pair of rectangles, our estimate is accurate up to a dimensional constant C(n). We give examples in which the (n – 1)-dilation of the linear map is bigger than the optimal value by an arbitrarily large factor. Received: January 2006, Revision: May 2006, Accepted: June 2006     

Trivariate Cr Polynomial Macroelements

Trivariate C^r macroelements defined in terms of polynomials of degree 8r + 1 on tetrahedra are analyzed. For r = 1,2, these spaces reduce to well-known macroelement spaces used in data fitting and in the finite-element method. We determine the dimension of these spaces, and describe stable local minimal determining sets and nodal minimal determining sets. We also show that the spaces approximate smooth functions to optimal order.     

Free-carrier generation in aggregates of single-wall carbon nanotubes by photoexcitation in the ultraviolet regime

We present evidence for the generation of free carriers in aggregated single-wall carbon nanotubes by photoexcitation in the energetic range of the π→π(*) transition associated with the M saddle point of the graphene lattice. The underlying broad absorption culminating at 4.3 eV can be fit well with a Fano line shape that describes strong coupling of a saddle-point exciton to an underlying free electron-hole pair continuum. Moreover, it is demonstrated that transitions in this energetic region autoionize into the continuum by detecting features unique to the presence of free charges in the transient transmission spectra of the continuum-embedded second sub-band exciton, S(2).     

The first total synthesis of lipid II: the final monomeric intermediate in bacterial cell wall biosynthesis

Bacterial peptidoglycan is composed of a network of beta-[1,4]-linked glyan strands that are cross-linked through pendant peptide chains. The final product, the murein sacculus, is a single, covalently closed macromolecule that precisely defines the size and shape of the bacterial cell. The recent increase in bacterial resistance to cell wall active agents has led to a resurgence of activity directed toward improving our understanding of the resistance mechanisms at the molecular level. The biosynthetic enzymes and their natural substrates can be invaluable tools in this endeavor. While modern experimental techniques have led to isolation and purification of the biosynthetic enzymes utilized in peptidoglycan biosynthesis, securing useful quantities of their requisite substrates from natural substrates has remained problematic. In an effort to address this issue, we report the first total synthesis of lipid II (4), the final monomeric intermediate utilized by Gram positive bacteria for peptidoglycan biosynthesis.