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

- Moving with VIGRE
- A new kind of science?
- Math teachers with no math
- New / old math probes the big bang
- A differential equation for organism growth
- Trig tables in verse

**Moving with VIGRE.** Vertical Integration of Research and Education is a 4-year old NSF program aimed improving the way higher mathematics education works in the United States. It's the focus of ``NSF Moves With VIGRE to Force Changes in Academia'' by Dana Mackenzie, in the May 24 2002 *Science*. The idea is to structure mathematics students' access into the profession, beginning with undergraduates, following the pattern of the more obviously experimental sciences like biology. The ``tetrahedral research groups'' devised at the University of Colorado, Boulder, for their successful VIGRE proposal exemplify vertical integation: each includes faculty, postdocs, graduate students and undergrads. The program is being administered with rigor. Of the twelve universities starting awards in 1999, three were not renewed: Rutgers, Carnegie-Mellon and Berkeley. How could this happen? Mackenzie quotes Calvin Moore, the Chair at Berkeley: ``One of our goals is to cultivate self-reliance. ... Some [students] thrive and others don't.'' A nice take on the program philosophy comes from Skip Garibaldi, a VIGRE-supported postdoc at UCLA whose research mentor has arranged for him to work with Jean-Piere Serre. As quoted by Mackenzie, he says: ``You can't get much above Serre, and you can't get much below me. So that's an example of vertical integration.''

**A new kind of science?** ``By relying on mathematical equations to describe the world, scientists for centuries have grossly limited their powers of explanation, asserts Stephen Wolfram'' is the start of Richard Monastersky's piece (*Chronicle of Higher Edcation*, May 17 2002) on the publication of Wolfram's long-awaited opus, "A New Kind of Science." The book is described by Jim Giles (*Nature* May 16 2002) as ``a call for researchers to turn away from calculus and other conventional mathematical tools ... .'' What is to replace calculus? Since John Conway's ``Game of Life'' (with roots in von Neumann's work in the 1940s, but first brought to wide attention by Martin Gardner in the October 1970 *Scientific American*) we have all known that a cellular automaton can start from a couple of simple rules and generate patterns of amazing complexity. Wolfram's fundamental innovation, as best reported by Edward Rothstein (*New York Times* ``Arts & Ideas'' section, May 11, 2002) is to posit that such automata are actually at work behind the complex systems (turbulence, consciousness, the local structure of space-time) that currently baffle scientific inquiry. ``Not only can complex designs and processes arise from the simplest of rules, but ... simple rules actually lie beind the most sophisticated processes in the universe.'' And the corollary: some complex processes cannot be handled by scientific laws in the way we know them. ``All we can do in such cases is discover the simple rules that give birth to the complexity. ... Everything else can be found only by `experiment': the process must run its course.''

**Math teachers with no math.** This may only be a problem in New York, but that's unlikely. The problem is the ``alternate route'' to high school teacher certification: a teacher can become certified with a Bachelor's degree in any subject and 36 credits of college-level mathematics courses. As reported by Alfred S. Posamentier, Dean of the School of Education at City College (Op-Ed, *New York Times*, May 11 2002), those math courses can actually be in accounting, finance, economics or engineering. ``Mathematics is one of the most important subjects in the curriculum, a necessary foundation for many other areas of study, and we are allowing people who may know precious little about it to teach it.'' The remedy? Reinforce the State's new ``math-immersion'' route to secondary school mathematics certification: ``offer college graduates with good academic records in quantitative fields of study a special sequence of courses in the foundations of arithmetic, geometry, algebra, trigonometry, combinatorics, probability and statistics.'' And boost teacher's salaries to competitive levels.

**New / old math probes the big bang.** ``A reconstruction of the initial conditions of the universe by optimal mass transportation'' is the title of an article in the May 16 2002 *Nature* by an international team mostly based at the CNRS Observatoire de la Côte d'Azur in Nice. ``We show that, with a suitable hypothesis, the knowledge of both the present non-uniform distribution of mass and of its primordial quasi-uniform distribution uniquely determines'' a map from present positions to the respective initial ones. The mathematics they use, which they call the Monge-Ampère-Kantorovich (MAK) method, goes back in part to Monge's solution of how best to move a pile of dirt from one location to another: you construct a ``cost'' function and minimize it. They have tested the MAK reconstruction on ``data obtained by a cosmological N-body simulation with 128^{3} particles,'' and exhibit the results. Caution: they note that ``when working with the catalogues of several hundred thousand galaxies that are expected within a few years, a direct application of the assignment algorithm in its present state would require unreasonablecomputational resources.''

**A differential equation for organism growth. ** The equation is

dm -- = a m |

makhi bhakhi phakhi dhakhi .nakhi ~nakhi "nakhi hasjha skaki ki.sga "sghaki kighva ghlaki kigra kakya dhaki kica sga "sjha "nva kla pta pha cha kala-ard.ha-jyaa.h

ka=1 kha=2 ga=3 gha=4 "na=5 ca=6 cha=7 ja=8 jha=9 ~na=10 .ta=11 .tha=12 .da=13 .dha=14 .na=15 ta=16 tha=17 da=18 dha=19 na=20 pa=21 pha=22 ba=23 bha=24 ma=25 | ya=30 ra=40 la=50 va=60 "sa=70 .sa=80 sa=90 ha=100 | a=1 i=100 u=100 |

The following information is necessary to translate the table into modern notation. Angles are measured in minutes of arc, and the table gives sines for multiples of 225 minutes between 0 and 5400 (a right angle). The sines are also measured in minutes (this is the reverse of radian measure!); since a length of 1 along a unit circle corresponds to 180x60/pi=3438 minutes of arc, the sines calculated from the couplet must be divided by 3438 to match modern usage. Finally, the couplet gives the *differences* between consecutive sines. In the following table, the running total of the AArya-bha.tiiya entries is divided by 3438 and compared with the sine given by a calculator, both rounded off to 4 decimal places.

angle sine sine/ sine in from 3438 from minutes verse calculator 0 0 0 0 makhi 225 khi=200 ma=25 225 225 0.0654 0.0654 bhakhi 224 khi=200 bha=24 450 449 0.1306 0.1305 phakhi 222 khi=200 pha=22 675 671 0.1952 0.1951 dhakhi 219 khi=200 dha=19 900 890 0.2589 0.2588 .nakhi 215 khi=200 .na=15 1125 1105 0.3214 0.3214 ~nakhi 210 khi=200 ~na=10 1350 1315 0.3825 0.3827 "nakhi 205 khi=200 "na=5 1575 1520 0.4421 0.4423 hasjha 199 ha=100 sa=90 jha=9 1800 1719 0.5 0.5 skaki 191 ki=100 sa=90 ka=1 2025 1910 0.5556 0.5556 ki.sga 183 ki=100 .sa=80 ga=3 2250 2093 0.6088 0.6088 "sghaki 174 ki=100 "sa=70 gha=4 2425 2267 0.6594 0.6593 kighva 164 ki=100 va=60 gha=4 2700 2431 0.7071 0.7071 ghlaki 154 ki=100 gha=4 la=50 2925 2585 0.7519 0.7518 kigra 143 ki=100 ra=40 ga=3 3150 2728 0.7935 0.7934 hakya 131 ha=100 ya=30 ka=1 3375 2859 0.8316 0.8315 dhaki 119 dha=19 ki=ka+i=100 3600 2978 0.8662 0.8660 kica 106 ka=1 i=100 ca=6 3825 3084 0.8970 0.8969 sga 93 sa=90 ga=3 4050 3177 0.9241 0.9239 "sjha 79 "sa=70 jha=9 4275 3256 0.9471 0.9469 "nva 65 "na=5 va=60 4500 3321 0.9660 0.9659 kla 51 ka=1 la=50 4725 3372 0.9808 0.9808 pta 37 pa=21 ta=16 4950 3409 0.9916 0.9914 pha 22 5175 3431 0.9980 0.9979 cha 7 5400 3438 1. 1. |

-*Tony Phillips Stony Brook *