Wikipedia:Reference desk/Archives/Mathematics/2007 April 7

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April 7

Poor Little Suzy

One day Little Suzy found a circular spot of bacterial growing in a pond. Deciding to include it in her Science Report, she measured the size of the bacterial everyday. The first day the bacterial patch had a radius of exactly 10cm. The next day it was exactly 11cm. For the next 8 days, she got the following results.

Radius of the circular bacterial patch

r[0] = 10.00 exact

r[1] = 11.00 exact

r[2] = 12.21

r[3] = 13.70

r[4] = 15.58

r[5] = 18.00

r[6] = 21.25

r[7] = 25.76

It soon became clear to her that the radius of the bacterial patch follows the following rules. The first 3 days are a dead give away.

r[0] = 10

r[t] = r[t-1] + ( r[t-1] * r[t-1] ) / 100

For her Science report, she decides to use a differential equation.

She wrote ${\frac {\text{dr}}{\text{dt}}}=k\cdot r^{2}\,$

with the solution $r(t)={\frac {R_{0}}{1-R_{0}\cdot k\cdot t}}$ where $R_{0}$ is r(t=0) which is 10.00

However no matter how hard she tried, she could not find the values of k that will give the results consistent with her measurements.

She tried k=1/100 which gives r(7)=33.33

She tried k=0.00874 which gives r(7)=25.76 but r(3)=13.55

Why can't Little Suzy find a value of k that works? 220.239.107.54 06:31, 7 April 2007 (UTC)

Is this a serious question? The function r, at least for the observed experimental data, satisfies

${\frac {\Delta r}{\Delta t}}=0.01\cdot r^{2}~\mathrm {for} ~\Delta t=1\,.$

In general, the difference quotient will not be the same as the derivative.  --Lambiam^Talk 09:12, 7 April 2007 (UTC)

Indeed. Suzy is approximating a difference equation by a differential equation. In the general case, the solutions to the two equations may have quite different behaviours. In this case, for example, the solution to the differential equation becomes unbounded ("infinitely large") as t approaches 1/kR₀, whereas the solution to the difference equation is clearly bounded for all t. Gandalf61 10:37, 7 April 2007 (UTC)

Tell me. If you are Suzy, what differential equation would you use for your Science Report? It has to be consistent with the actual measurements. 220.239.107.54 12:55, 7 April 2007 (UTC)

If I was Suzy, I would disguise my homework question as a cute little story. − Twas Now ( talk • contribs • e-mail ) 23:06, 7 April 2007 (UTC)

If this is homework, it is an impossibly difficult assignment. Compare the related sequence 1, 2, 6, 42, 1806, 3263442, 10650056950806, ... (sequence A007018 in the OEIS). Although a closed-form expression for the n'th term a_n is given, it cannot in any obvious way be extended to a differentiable function f on the reals such that f(n) = a_n. The relationship becomes obvious if you define a_n = r_n/100, where r_n is Little Suzy's sequence. The recurrence relation is then the same as for A007018: a_n+1 = a_n + a_n². The only difference is the starting value.  --Lambiam^Talk 23:48, 7 April 2007 (UTC)

New exciting developments ?

I was just wondering if any of you helpers here have written any noteworthy papers on mathematics or made any contributions in general to math. I know there are some brilliant minds here like Lambiam and Ksmrq (and many more). Or are you noticing any new exciting developments in the field of mathematics? —Preceding unsigned comment added by 144.132.64.209 (talk • contribs)

Thanks for the compliment. I choose not to reveal my real-world identity, but some publications of which I'm an author or co-author are cited in Wikipedia articles (not added by me). I think some noteworthy papers by KSmrq are also cited, if I'm not mistaken about his/her identity. There are many new and exciting developments (for instance the novel techniques and insights in the proofs of Fermat's Last Theorem and the Poincaré conjecture, or the advances in number theory and other fields of mathematics made possible by fast computers), most of which is regrettably too technical for me to really understand it.  --Lambiam^Talk 13:26, 7 April 2007 (UTC)

Flattery, eh? Mathematics is a field in which humility is recommended; for, the really good problems are so difficult, and the accomplishments of Euler, say, are so overwhelming. (Euler could do great mathematics late in life, blind, with children playing in his lap, across many topics, more prolifically than any other mathematician that ever lived.) Even if you prove Fermat's Last Theorem or the Poincaré conjecture, the world takes only fleeting notice and few will ever understand (or care) what you have done. For Wikipedia, I just try to explain a few things as best I can.

In a field as old as mathematics, most researchers dig deeper and deeper into very specialized topics trying to extract some gem. Broadly speaking, many fascinating recent events have the opposite character, revealing structure connecting specialties. More subtly, I see indications of a renaissance in attitude towards teaching, shaking off some of the bad side effects of the Bourbaki approach, which left students bewildered by seeing the general without the prior motivation of the specific. Also, connections to physics and other applications are no longer quite so unsavory, and we see inspiration and insights flow both directions.

Computers have helped, in ways we tend to overlook. Especially, the web has facilitated world-wide conversation and collaboration, and allowed papers to be seen much more quickly. Computer typesetting, the ubiquitous TeX, has also made publication of results considerably easier than in the days of typewriters (with hand-lettered symbols) and "penalty copy" burdens on publishers. Explorations and experiments can also use computers (sometimes), and a computer algebra system can help us keep our signs straight.

If you would like to keep an eye on advanced developments, try the AMS web site. --KSmrq^T 23:23, 7 April 2007 (UTC)

Perhaps Category:Wikipedians_by_Erdős_number would be helpful to identify Wikipedians in which you may be interested.--Ķĩřβȳ♥♥♥ŤįɱéØ 17:35, 10 April 2007 (UTC)