February 2004 Image courtesy Ranier Herges | Aromatic Möbius strip. "Synthesis of a Möbius aromatic hydorcarbon" appeared as a letter to Nature, December 18, 2003. There is a "Hückel rule" that constrains the number of carbon atoms in cyclic hydrocarbon compounds: the number of carbon atoms in an uncharged ring (always even) must be of the form 4n + 2. The most familiar member of this family, benzene, has 6 carbons. The Kiel and Stuttgart-based authors (D. Ajami, O. Oeckler, A. Simon, R. Herges) of this article took up a prediction of E. Heilbronner (1964) that rings of 4n molecules could be stable if they had the topology of a Möbius strip. They found an ingenious method for synthesizing a stable, twisted "annulene" with 16 carbon atoms: surgery between an annulus-like 8-carbon aromatic molecule and a cylinder-like one (in this case, tetradehydrodianthracene). Two 8-carbon aromatic molecules, one annular, one cylindrical, are joined to form a 16-carbon non-orientable molecule. In the second row, the regions in parentheses would each contain n-6 carbon atoms, when the process is generalized to n-atom rings. Image from Nature, used with permission. |
| The Escher-inspired image at the left illustrated the Nature table of contents. | Perelman in the Baltimore Sun. The dateline was Moscow, January 19 2004 for Douglas Birch's piece on Gregory Perelman. "In his office overlooking the faded pastel mansions along a St. Petersburg canal, a young Russian mathematician spent eight solitary years grappling with the Poincare Conjecture, one of the most famous and frustrating conundrums in math." The Poincaré Conjecture "goes to the heart of topology, or the mathematical study of surfaces, which holds that the world consists of two basic shapes, the sphere and the doughnut." Birch did not get this idea from Perelman, who refused to be interviewed. But he did speak with John Milnor ("It's the kind of a subject where it's very easy to make a mistake if you're not careful") and Gennadi A. Leonov, Dean at St. Petersburg State University and one of Perelman's former teachers ("Grigori Perelman is one of the brilliant successors of earlier Petersburg mathematicians") and he quotes Jean-Pierre Serre and Michael Anderson on the importance of the problem and the originality of Perelman's ideas. "Malignant Maths" is the title of a piece in the January 22 2004 Economist. The subtitle is less threatening: "Mathematical models aid the understanding of cancer." The focus is on three works appearing in Discrete and Continuous Dynamical Systems--Series B which is devoting its February issue to the topic. - Zvia Agur and her colleagues (Institute for Medical BioMathematics, Bene Ataroth, Israel) present a model for the workings of angiogenesis (the process by which a tumour creates its own blood vessels). Dr Agur set up a system of differential equations, where the variables are "the number of cells in a tumour, the concentration of the angiogenetic growth factors within it and the volume of the blood vessels." Analysis of this system led to "the discovery that there are circumstances in which a tumour oscillates in size, instead of growing steadily," with clear therapeutic implications.
- Denise Kirschner (University of Michigan) describes her investigations into the use of the immune system to fight tumor growth. A novel treatment, known as small interfering RNA (siRNA) therapy, might suppress the action of a molecule called "transforming growth factor beta" (TGF-beta), which large tumours use to elude the immune system. Dr. Kirchener also uses a differential equation model. Her variables are "the number of immune-system 'effector cells' (those that combat tumours), the number of tumour cells, the amount of interleukin-2 (a protein that enhances the body's ability to fight cancer), and an additional variable to account for the effects of TGF-beta. ... In the model, a daily dose of siRNA over the course of 11 successive days succeeded in counteracting the effects of TGF-beta, and so allowed the immune system to bring the tumour under control"
- Pep Charusanti and his colleagues (UCLA) investigated the action of Gleevec, a drug used against chronic myeloid leukaemia. Gleevec starves cancer cells by inhibiting their metabolism of ATP. The riddle was why Gleevec was ineffective in a "blast crisis," the terminal state of the disease. Charusanti's mathematical model "shows that cells in blast crisis expel the drug too quickly for it to be useful as an ATP-blocker," giving a direction to look for improvements in the therapy.
The article ends by quoting Richard Feynman: "mathematics is a deep way of describing nature, and any attempt to express nature in philosophical principles, or in seat-of-the-pants mechanical feelings, is not an efficient way." Bayesian athletics. The January 20 2004 Science Times section of the New York Times ran an article by John Leonhardt with the title: "Subconsciously, Athletes May Play Like Statisticians." Leonhardt is picking up a letter to the January 15 Nature with the more academic title: "Bayesian integration in sensorimotor learning," by Konrad Körding and Daniel Wolpert (University College, London). It turns out that, like Monsieur Jourdain, we have all been performing Bayesian integration without knowing it. Körding and Wolpert set up the following experiment. - "Subjects reached to a visual target with their right index finger in a virtual-reality set-up that allowed us to displace the visual feedback of their finger laterally relative to its actual location. ...On each movement, the lateral shift was randomly drawn from a prior distribution ... During the movement, visual feedback of the finger position was only provided briefly, midway through the movement. We manipulated the reliability of this visual feedback on each trial."
Pretty fiendish. But in a good cause: to demonstrate "that subjects internally represent both the statistical distribution of the task and their sensory uncertainty, combining them in a manner consistent with a performance-optimizing bayesian process." Körding and Wolpert expect that, more generally, "such a bayesian process might be fundamental to all aspects of sensorimotor control and learning." For example, "taking into account a priori knowledge might be key to winning a tennis match. Tennis professionals spend a great deal of time studying their opponent before playing an important match, ensuring that they start the match with correct a priori knowledge." Bayesian filters for spam. "Bayesian" may be the new geek buzz-word. Here we have Andrew Cantor in his USA Today Cyberspeak column (December 26, 2003) telling us how "The Reverend Thomas Bayes was an 18th century English mathematician who came up with a theorem for determining the probability of an event based on existing knowledge." And how "In August 2002, Paul Graham wrote an article called 'A Plan for Spam'. He suggested using Bayes's techniques to identify the probability of a message being spam. Unlike other spam filters, this would be based on the content of messages you already knew were spam." Cantor mentions some commercial products devised to convert this 18th-century notion into 21st-century cash. Article available online. -Tony Phillips Stony Brook Math in the Media Archive |