Affine and Projective Tree Metric Theorems

The tree metric theorem provides a combinatorial four-point condition that characterizes dissimilarity maps derived from pairwise compatible split systems. A related weaker four point condition characterizes dissimilarity maps derived from circular split systems known as Kalmanson metrics. The tree metric theorem was first discovered in the context of phylogenetics and forms the basis of many tree reconstruction algorithms, whereas Kalmanson metrics were first considered by computer scientists, and are notable in that they are a non-trivial class of metrics for which the traveling salesman problem is tractable. We present a unifying framework for these theorems based on combinatorial structures that are used for graph planarity testing. These are (projective) PC-trees, and their affine analogs, PQ-trees. In the projective case, we generalize a number of concepts from clustering theory, including hierarchies, pyramids, ultrametrics, and Robinsonian matrices, and the theorems that relate them. As with tree metrics and ultrametrics, the link between PC-trees and PQ-trees is established via the Gromov product.     

Unprecedented "In Water" imidazole carbonylation: paradigm shift for preparation of urea and carbamate

The first "In Water" imidazolecarbonylation of amine is described. A one pot reaction of carbonylimidazolide in water with a nucleophile provides an efficient and general method for the preparation of urea, carbamates and thiocarbamates. Use of an anhydrous solvent and an inert atmosphere could be avoided. Product precipitate out from the reaction mixture and can be obtained in high purity by filtration, resulting in a simple and scalable method.     

Spectrally resolved flow imaging of fluids inside a microfluidic chip with ultrahigh time resolution

Microfluidics has advanced to become a complete lab-on-a-chip platform with applications across many disciplines of scientific research. While optical techniques are primarily used as modes of detection, magnetic resonance (MR) is emerging as a potentially powerful and complementary tool because of its non-invasive operation and analytical fidelity. Two prevailing limitations currently inhibit MR techniques on microfluidic devices: poor sensitivity and the relatively slow time scale of dynamics that can be probed. It is commonly assumed that the time scale of observation of one variable limits the certainty with which one can measure the complementary variable. For example, short observation times imply poor spectral resolution. In this article, we demonstrate a new methodology that overcomes this fundamental limit, allowing in principle for arbitrarily high temporal resolution with a sensitivity across the entire microfluidic device several orders of magnitude greater than is possible by direct MR measurement. The enhancement is evidenced by recording chemically resolved fluid mixing through a complex 3D microfluidic device at 500 frames per second, the highest recorded in a magnetic resonance imaging experiment. The key to this development is combining remote detection with a time ‘slicing’ of its spatially encoded counterpart. Remote detection circumvents the problem of insensitive direct MR detection on a microfluidic device where the direct sensitivity is less than 10^-5 relative to traditional NMR, while the time slicing eliminates the constraints of the limited observation time by converting the time variable into a spatial variable through the use of magnetic field gradients. This method has implications for observing fast processes, such as fluid mixing, rapid binding, and certain classes of chemical reactions with sub millisecond time resolution and as a new modality for on-chip chromatography.     

The effect of substitution on protonation sites: evidence for protonation at N(3) in N(7)-substituted adenine

Several experimental and theoretical studies have shown that N(1) is the first site for protonation in adenine and N(9)-substituted adenine derivatives. N(7) is considered the site for the second protonation to yield dipositive cations. Results are reported here which indicate that this protonation pattern is altered in N(7)-substituted adenine derivatives. In particular, an X-ray diffraction analysis of the structure of 7-methyladenine dihydrochloride, [C₆N₅H₇]Cl₂, definitively shows that the sites for protonation are N(3), as opposed to N(1), and N(9). Theoretical calculations of the molecular electrostatic potential in various systems suggest that such changes in preferred protonation sites should be expected, in general, when a modification of the molecular structure creates adjacent sites with similar reactivity, such as the N(3) and N(9) sites in 7-methyladenine.     

Near ab-initio methods for the calculations of large molecules: Comparison of pseudopotential and ab-initio results

Results from pseudopotential calculations on 5-hydroxyindole, tryptamine, 5-hydroxytryptamine, 6-hydroxytryptamine and the imidazolium cation are compared to full ab-initio calculations. The localization of all molecular orbitals is found to be identical with the two methods. Orbital energies from the two methods are found to be linearly correlated for all molecules: the slope is close to unity and the orbital energies from the pseudopotential calculation show a very slight and virtually constant shift to lower values. Electron density maps and molecular electrostatic potential maps calculated from the wavefunctions show that the charge distributions and molecular reactivity characteristics predicted by the two methods are nearly identical.     

A misconception concerning the electronic density distribution of an atom

It is shown that the electronic charge density of a ground-state atom decreases monotonically as a function of radial distance from the nucleus, contrary to the widespread belief that the shell structure is reflected by relative maxima in the density. Any proposed relationship between chemical bonding and the maxima in the radial density functions of atoms should therefore be regarded with caution. It is proven that the electrostatic potential of an atom must be monotonically decreasing. The changes in charge distribution upon molecule formation are also discussed.     

A DNR perspective on mathematics curriculum and instruction. Part II: with reference to teacher’s knowledge base

Two questions are on the mind of many mathematics educators; namely: What is the mathematics that we should teach in school? and how should we teach it? This is the second in a series of two papers addressing these fundamental questions. The first paper (Harel, 2008a) focuses on the first question and this paper on the second. Collectively, the two papers articulate a pedagogical stance oriented within a theoretical framework called DNR-based instruction in mathematics. The relation of this paper to the topic of this Special Issue is that it defines the concept of teacher’s knowledge base and illustrates with authentic teaching episodes an approach to its development with mathematics teachers. This approach is entailed from DNR’s premises, concepts, and instructional principles, which are also discussed in this paper.