Math in the Media

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Tony PhillipsTony Phillips' Take on Math in the Media
A monthly survey of math news

December 2002

*Locust neuron math is reported in the November 21 2002 Nature. "Multiplicative computation in a visual neuron sensitive to looming," by a team at Cal Tech (F. Gabbiani, H. G. Krapp, C. Koch and G. Laurent), investigates the workings of the locust lobula giant movement detector (LGMD). This large neuron "responds vigorously to solid objects approaching on a collision course with the animal." The LGMD receives exitatory inputs "that convey, to a first approximation, the angular velocity of the approaching object" as well as an inhibitory input "related to object size." The authors performed a series of experiments and conclude that the results "are consistent with an implementation of multiplication based on dendritic subtraction of two converging inputs encoded logarithmically, followed by exponentiation through active membrane conductances." They add "These results imply that multiplication occurs within LGMD itself."

*The topology of NOT. In a (classical) logical circuit, a NOT gate is a device that outputs 0 when the input bit is 1, and 1 when the input bit is 0. In a quantum computer, a quantum bit (qubit) can exist in a superposition of two logical states |0> and |1>: its state can be represented as a complex linear combination z0 |0> + z1 |1>. In "Experimental realization of the quantum universal NOT gate" (Nature, October 24, 2002) an international team (F. de Martini, V. Buzek, F. Sciarrino and C.Slas) begin by discussing the problem of implementing NOT in this context. This would involve flipping a state (z0,z1) to its orthogonal state (z*1,-z*0), where the * is complex conjugation. The state space of a qubit can be identified with a 2-dimensional sphere, where the south pole (0,1) is |0>, the north pole (1,0) is |1>, and the other points correspond to z0/z1 by stereographic projection from the sphere to the complex plane.

The "Poincaré sphere" represents the possible states of a qubit. |0> is at the south pole, |1> at the north, and along the equator lie states like |0> + i |1>, and its orthogonal state i |0> + |1>,   with |z0/z1| = 1.

In this representation the NOT of (z0/z1) is (-z*1/z*0): on the sphere, each point is flipped to its antipode. Here is where the topology of the 2-sphere comes into play, because its antipodal map cannot be realized by a rotation in 3-space. In physical language this means that the antipodal map cannot be realized by a unitary operation, and that therefore the universal NOT gate, which would transform an arbitrary qubit to its negation, cannot be realized by a physical device. The article reports that the theoretically best possible approximation to the universal NOT gate has been devised (using photons and lasers) and experimentally tested.

*Elevens and nines on NPR. Episode 1748 of "Engines of our Ingenuity," which aired on November 22, 2002, featured Andrew Boyd as guest speaker. Boyd recounted a conversation with an autistic boy who told him that their ages (14 and 41) were "reversed" and that they would be reversed again at 25 and 52, and again at 36 and 63. The boy hinted that the phenomenon had to do with "mutiples of nine," which led our host to the general statement: "When two people share an age difference that is a multiple of nine, they will find the digits in their ages periodically reversed. When the age difference is not a multiple of 9, they will never experience reversed ages." Boyd goes on to comment about the sometimes unusual abilities of autists, and that "by 'thinking differently,' we are all capable of remarkable feats of ingenuity." The online text version of the episode has a mathematical appendix working out Boyd's assertion and some related questions.

*Computational Biology in Nature. The November 14 2002 Nature features an "Insight" section devoted to "a collection of reviews showing how sophisticated mathematical concepts have illuminated and continue to illuminate the principles underlying biology at a genetic, molecular, cellular and even organismal level." The reviews start with an overview, "Computational systems biology" by H. Kitano and include "The language of genes" (D. B. Searls: "can the methods developed for analysing languages be applied to molecular biology?"), and "Computational approaches to cellular rhythms" (A. Goldbeter: "I shall consider, in turn, oscillations of intracellular calcium, pulsatile signaling in intercellular communication, and circadian rhythms.").

-Tony Phillips
Stony Brook

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