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Math in the Media |
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 1283 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 unresonable computational resources.''
A differential equation for organism
growth. The equation is
dm -- = a m3/4[1-(m/M)1/4] dt |
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=1002 .r=1003 .l=1004 e=1005 ai=1006 o=1007 au=1008 |
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
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