Implementing and Using Mathematics Standards in North Carolina

North Carolina has a history of supporting and using national standards in structuring the state-adopted mathematics curriculum framework. This focus is associated with increased achievement of North Carolina students. The connection of the 1989 revision of the framework to the 1989 National Council of Teachers of Mathematics standards seemed to increase acceptance of the framework by teachers, school boards, and the public; revisions since then have been explicitly responsive to evolving national standards. Plans for the next revision include attending to the Principles and Standards for School Mathematics. To inform leaders in North Carolina about the Principles and Standards, a symposium was held in September 2000. This symposium provided a model for making the Principles and Standards visible to different constituencies. It is also important to involve teacher education institutions in sharing the Principles and Standards with new teachers so that they enter teaching with a deep understanding of and appreciation for standards.     

Relative substituent position on the strength of π-π stacking interactions

It was observed that the relative position of the arene substituents has a profound influence on the strength of π-π stacking in the 9-benzyl-substituted triptycene system. A new series of model compounds (3a-i) capable of revealing quantitatively π-π stacking interactions was studied. This series of compounds (3a-i) has an ortho-substituted methyl group in one of the two interacting arenes and the syn/anti ratios were determined and compared to a series previously studied compounds (4a-i) that have a para methyl group on the corresponding arene. A greater than 50% increase in the strength of π-π stacking interactions was observed with the methyl group in the ortho position comparing to that in the para position. No difference in π-π stacking interactions was observed when the other aromatic ring was a pentafluorobenzoate group.     

Curvature-dependent elastic properties of liquid-ordered domains result in inverted domain sorting on uniaxially compressed vesicles

Using a coarse-grained molecular model we study the spatial distribution of lipid domains on a 20-nm-sized vesicle. The lipid mixture laterally phase separates into a raftlike, liquid-ordered (l(o)) phase and a liquid-disordered phase. As we uniaxially compress the mixed vesicle keeping the enclosed volume constant, we impart tension onto the membrane. The vesicle adopts a barrel shape, which is composed of two flat contact zones and a curved edge. The l(o) domain, which exhibits a higher bending rigidity, segregates to the highly curved edge. This inverted domain sorting switches to normal domain sorting, where the l(o) domain prefers the flat contact zone, when we release the contents of the vesicle. We rationalize this domain sorting by a pronounced reduction of the bending rigidity and area compressibility of the l(o) phase upon bending.     

Determining equilibrium constants for dimerization reactions from molecular dynamics simulations

With today's available computer power, free energy calculations from equilibrium molecular dynamics simulations "via counting" become feasible for an increasing number of reactions. An example is the dimerization reaction of transmembrane alpha-helices. If an extended simulation of the two helices covers sufficiently many dimerization and dissociation events, their binding free energy is readily derived from the fraction of time during which the two helices are observed in dimeric form. Exactly how the correct value for the free energy is to be calculated, however, is unclear, and indeed several different and contradictory approaches have been used. In particular, results obtained via Boltzmann statistics differ from those determined via the law of mass action. Here, we develop a theory that resolves this discrepancy. We show that for simulation systems containing two molecules, the dimerization free energy is given by a formula of the form ΔG ∝ ln(P(1) /P(0) ). Our theory is also applicable to high concentrations that typically have to be used in molecular dynamics simulations to keep the simulation system small, where the textbook dilute approximations fail. It also covers simulations with an arbitrary number of monomers and dimers and provides rigorous error estimates. Comparison with test simulations of a simple Lennard Jones system with various particle numbers as well as with reference free energy values obtained from radial distribution functions show full agreement for both binding free energies and dimerization statistics.     

Inverse solutions to radiative-transfer problems with partially transparent boundaries and diffuse reflection

Analytical techniques are used to solve a class of inverse radiative-transfer problems relevant to finite and semi-infinite plane-parallel media. While the assumption of isotropic scattering is made, diffuse reflection is allowed at the surface, for the semi-infinite case, and at both surfaces for the case of a finite layer. For the general case based on a semi-infinite medium, a cubic algebraic equation is used to define the basic result, but for the specific case of a semi-infinite medium illuminated by a constant incident distribution of radiation, very simple exact expressions are developed for the albedo for single scattering ? and the coefficient for diffuse reflection ρ. Analytical results are also developed (again in terms of a cubic algebraic equation) for the case of a finite layer with equal reflection coefficients relevant to the two surfaces. For the general case of a finite layer with unequal reflection coefficients, two specific formulations are given. The first algorithm is based on a system of three quadratic algebraic equations for the two reflection coefficients ρ₁ and ρ₂ and the single-scattering albedo ?. Secondly, an elimination between these three algebraic equations is carried out to yield two coupled algebraic equations for ρ₁ and ρ₂ plus an explicit expression for ? in terms of ρ₁ and ρ₂. In addition, an exact expression for τ₀, the optical thickness of the finite layer, is developed in terms of ?, ρ₁ and ρ₂. As is typical with the considered class of inverse problems in radiative transfer, all surface quantities are either specified or considered available from experimental measurements. All basic results are tested numerically.     

The viscous-slip, diffusion-slip, and thermal-creep problems for a binary mixture of rigid spheres described by the linearized Boltzmann equation

An analytical version of the discrete-ordinates method (the ADO method) is used with recently established analytical expressions for the rigid-sphere scattering kernels to develop concise and particularly accurate solutions to the viscous-slip, the diffusion-slip, and the half-space thermal-creep problems for a binary gas mixture described by the linearized Boltzmann equation. In addition to a computation of the viscous-slip, the diffusion-slip, and the thermal-slip coefficients, for the case of Maxwell boundary conditions for each of the two species, the velocity, heat-flow, and shear-stress profiles are established for each species of particles. Numerical results are reported for two binary mixtures (Ne–Ar and He–Xe) with various molar concentrations.