Hearing molecular drums
Three quantum nano-resonators assembled from carbon monoxide molecules. **a** "Bilby," **b** "Hawk," **c** "Broken Hawk." Each is assembled on a copper crystal by placing 90 CO molecules (black dots) around a polygonal contour. The polygon is geometrically the union of seven identical 30-60-90-degree triangles. In polygons **a** and **b**, any two adjacent triangles are related by reflection across their common border; this does not hold for **c**. The polygonal shapes **a** and **b** are known to be mathematically isospectral (but different in this respect from **c** even though **c** matches, for example, their area and perimeter); the authors exploit this feature to directly access the phase of the quantum-mechanical system formed by the surface electrons trapped inside the CO walls. Image after Manoharan *et al.*
*Science* for February 8, 2008 ran a Report by a 6-member Stanford team entitled "Quantum Phase Extraction in Isospectral Electronic Nanostructures." The team, led by Hari Manoharan, took advantage of the discovery (Carolyn Gordon, David Webb, Scott Wolpert, 1992) of pairs of distinct polygonal shapes *isospectral* in the sense that they had exactly the same vibrational profile: identical responses at every frequency. This discovery was the long-awaited answer to Mark Kac's 1966 question "Can you hear the shape of a drum?" The *Science* authors use carbon monoxide molecules to draw a pair of different but geometrically isospectral shapes on the surface of a copper crystal. Each has area about 57 square nanometers, and encloses about 30 of "the 2D Fermi sea of electrons" that inhabit the surface; this pond of electrons will function as a "vibrating medium." The authors remark that "the time-independent Schrödinger equation is also a wave equation defined by the Laplacian and boundary conditions," i.e. the same equation that governs the sound of a drum, and that therefore the electronic resonances of the set of captured electrons will be the same for the two structures. Their main result is showing that "the complete phase information of wave functions in both structures can be experimentally determined" by "harnessing the topological property of isospectrality as the additional degree of freedom." This is physically significant because the spatial variation of the phase of the wave functions is measured without the usual reliance on interference phenomena. The supplementary information for this report includes a movie with soundtrack where the Schrödinger vibrations of the Bilby, Hawk and Broken Hawk nano-structures can be "heard" (at the rate of 100 THz ~ 1 KHz).
Markov Clusters in the tree of life
"Long-held ideas regarding the evoltionary relationships among animals have recently been upended by sometimes controversial hypotheses based largely on insights from molecular data." So begins the abstract of a paper in the April 10 2008 *Nature*. The authors, an 18-member international team led by Casey Dunn (Brown), present in "Broad phylogenomic sampling improves resolution of the animal tree of life" a new method for selecting the genes to analyze in order to more accurately understand the relative position of species on the evolutionary tree. "We present a new approach to identification of orthologous genes in animal phylogenomic studies that relies on a Markov cluster algorithm to analyse the structure of BLAST hits to a subset of the NCBI HomoloGene Database." BLAST (basic local alignment search tool) is a powerful algorithm, invented in 1990, for locating occurrences of a piece of genetic code in the NCBI (National Center for Biotechnology Information) database. The Markov Cluster Algorithm was devised in 2000 by Stijn van Dongen. It uses a stochastic, dynamic procedure to pinpoint the most significant part of a graph. In Henk Nieland's words: "Simulate many random walks (or flow) within the whole graph, and strengthen flow where it is already strong, and weaken it where it is weak. By repeating the process an underlying cluster structure will gradually become visible." (animated MCA simulation available here).
The Markov Cluster Algorithm at work. Red represents intensity. In Nieland's words: "... flow between different dense regions of nodes which are sparsely connected eventually evaporates, showing cluster structure present in tbe original input graph." Image courtesy of Stijn van Dongen.
The path to algebra: fractions
"News of the Week" in *Science* (March 21, 2008) was a story by Jeffrey Mervis about the National Mathematics Advisory Panel's release the week before of "a 120-page report on the importance of preparing students for algebra ... and its role as a gateway course for later success in high school, college, and the workplace." The report is available online. Mervis spoke with Larry Faulkner, the chair of the panel, and reports that the panel avoided "avoided taking sides in a debilitating 2-decade-long debate on the appropriate balance between drilling students on the material and making sure they understand what they are doing." The recommendations are that "students should memorize basic arithmetic facts and spend more time on fractions and their meaning." But, as Mervis explains, "how teachers achieve those goals is up to them."
- "Why do so many students have trouble with fractions? Faulkner: "Fractions have been downplayed." He mentions the perception that decimals and spreadsheets have eliminated the need for fractions. "But it's important to have an instinctual sense of what a third of a pie is, or what 20% of something is, to understand the ratio of numbers involved and what happens as you manipulate it."
- Q: Was the panel disappointed by the overall quality of existing education research? A: " ... We found a serious lack of studies with adequate scale and design for us to reach conclusions about their applicability for implementation."
- Q: Should the government be spending more money on this research? A:" ... If you want to get the value, you probably need to pay for it."
- Professional development programs? A: "There's tremendous variation in in-service programs. And the evidence is that many are not very effective."
- Calculators? "We feel strongly that they should not get in the way of acquiring automaticity [memorization of basic facts]. But the larger issue is the effectiveness of pedagogical software. At this stage, there's no evidence of substantial benefit or damage, but we wouldn't rule out products that could show a benefit."
Tony Phillips
Stony Brook University
tony at math.sunysb.edu |