Hyperaromatic stabilization of arenium ions

Benzene-cis- and trans-1,2-dihydrodiols undergo acid-catalyzed dehydration at remarkably different rates: k(cis)/k(trans) = 4500. This is explained by formation of a β-hydroxycarbocation intermediate in different initial conformations, one of which is stabilized by hyperconjugation amplified by an aromatic no-bond resonance structure (HOC(6)H(6)(+) ? HOC(6)H(5) H(+)). MP2 calculations and an unfavorable effect of benzoannelation on benzenium ion stability, implied by pK(R) measurements of -2.3, -8.0, and -11.9 for benzenium, 1-naphthalenium, and 9-phenanthrenium ions, respectively, support the explanation.     

New analysis procedure for fast and reliable size measurement of nanoparticles from atomic force microscopy images

Accurate size measurement during nanoparticle production is essential for the continuing innovation, quality and safety of nano-enabled products. Size measurement by analysing a number of separate particles individually has particular advantages over ensemble methods. In the latter case nanoparticles have to be well dispersed in a fluid and changes that may occur during analysis, such as agglomeration and degradation, will not be detected which could lead to misleading results. Atomic force microscopy (AFM) allows imaging of particles both in air and liquid, however, the strong interactions between the probe and the particle will cause the broadening of the lateral dimension in the final image. In this paper a new procedure to measure the size of spherical nanoparticles from AFM images via vertical height measurement is described. This procedure will quickly analyse hundred of particles simultaneously and reproduce the measurements obtained from electron microscopy (EM). Nanoparticles samples that were difficult, if not impossible, to analyse with EM were successfully measured using this method. The combination of this procedure with the use of a metrological AFM moves closer to true traceable measurements of nanoparticle dispersions.     

Exploring Pi Using the Computer in Middle School Mathematics

The National Council of Teachers of Mathematics (NCTM) in its Curriculum and Evaluation Standards for School Mathematics encourages a middle school curriculum that integrates technology. It recommends that students should be able to identify and use functional relationships and make connections among seemingly diverse concepts and topics. In this activity of exploring the derivation of Pi, students take a constructivist role by collecting data and making conjectures. Using the data they construct graphs and tables and discourse about appropriate algebraic representations. The computer is used as an instructional aid enabling the students to view the data in a variety of forms. They are encouraged to communicate about the connections among the various representations.     

Chaotic scattering in the gravitational three-body problem

We summarize some results of an ongoing study of the chaotic scattering interaction between a bound pair of stars (a binary) and an incoming field star. The stars are modeled as point masses and their equations of motion are numerically integrated for a large number of initial conditions. The global features of the resulting initial-value space maps are presented, and their evolution as a function of system parameters is discussed. We find that the maps contain regular regions separated by rivers of chaotic behavior. The probability of escape within the chaotic regions is discussed, and a straightforward explanation of the scaling present in these regions is reviewed. We investigate a statistical quantity of interest, namely the cross section for temporarily bound interactions, as a function of the third star's incoming velocity and mass. Finally, a new way of considering long-lived trajectories is presented, allowing long data sets to be qualitatively analyzed at a glance.     

A serendipitous route to the benzidene nucleus. Structural elucidation by noeds.

The epimeric indenones (3) and (4) are readily synthesised and rapidly formed by treating bis(p-methoxyphenylcyclopropyl) ketone (2) with stannic chloride in nitromethane. The structure of (3) was determined by X-ray crystallography and that of (4) by NOEDS.     

Translation invariant Julia sets

We show that if the Julia set of a rational function is invariant under translation by one and infinity is a periodic or preperiodic point for , then must either be a line or the Riemann sphere.

     

Holomorphic Mappings on C(I), I uncountable

We investigate the relationship between the various topologies that can be defined on spaces of holomorphic functions on C^(I for I uncountable. Our investigation also leads to some interesting results about the space ${\rm C}^{(I)}{{\hat \otimes} \atop \pi}{\rm C}^{(I)}$ .