The Universe of Discourse
https://blog.plover.com
The Universe of Discourse (Mark Dominus Blog)enRecommended reading: Matt Levine’s Money Stuff
https://blog.plover.com/2020/08/06#money-stuff
<p>Lately my favorite read has been Matt Levine’s <em>Money Stuff</em> articles
from Bloomberg News. Bloomberg's web site requires a subscription but
you can also get the <em>Money Stuff</em> articles as an
<a href="http://link.mail.bloombergbusiness.com/join/4wm/moneystuff-signup">occasional email</a>.
It arrives at most once per day.</p>
<p>Almost every issue teaches me something interesting I didn't know, and
almost every issue makes me laugh.</p>
<p>Example of something interesting: a while back it was all over the
news that oil prices were negative. Levine was there to explain what
was really going on and why. Some people manage index funds. They
are not trying to beat the market, they are trying to match the index.
So they buy derivatives that give them the right to buy oil futures
contracts at whatever the day's closing price is. But say they
already own a bunch of oil contracts. If they can get the close-of-day
price to dip, then their buy-at-the-end-of-the-day contracts will all
be worth more because the counterparties have contracted to buy at the
dip price. How can you get the price to dip by the end of the day?
Easy, unload 20% of your contracts at a bizarre low price, to make the
value of the other 80% spike… it makes my head swim.</p>
<p>But there are weird second- and third-order effects too. Normally if
you invest fifty million dollars in oil futures speculation, there is
a worst-case: the price of oil goes to zero and you lose your fifty
million dollars. But for these derivative futures, the price could in
theory become negative, and for short time in April, it did:</p>
<blockquote>
<p>If the ETF’s oil futures go to -$37.63 a barrel, as some futures did
recently, the ETF investors lose $20—their entire investment—and, uh,
oops? The ETF runs out of money when the futures hit zero; someone
else has to come up with the other $37.63 per barrel.</p>
</blockquote>
<p>One article I particularly remember discussed the kerfuffle a while
back concerning whether Kelly Loeffler improperly traded stocks on
classified coronavirus-related intelligence that she received in her
capacity as a U.S. senator. I found
<a href="https://www.bloomberg.com/opinion/articles/2020-05-14/senator-s-stock-trades-make-trouble">Levine's argument</a>
persuasive:</p>
<blockquote>
<p>“I didn’t dump stocks, I am a well-advised rich person, someone
else manages my stocks, and they dumped stocks without any input
from me” … is a good defense! It’s not insider trading if you don’t
trade; if your investment manager sold your stocks without input
from you then you’re fine. Of course they could be lying, but in
context the defense seems pretty plausible. (Kelly Loeffler, for
instance, controversially dumped about 0.6% of her portfolio at
around the same time, which sure seems like the sort of thing an
investment adviser would do without any input from her? You could
call your adviser and say “a disaster is coming, sell everything!,”
but calling them to say “a disaster is coming, sell a tiny bit!”
seems pointless.)</p>
</blockquote>
<p>He contrasted this case with that of Richard Burr, who, unlike
Loeffler, remains under investigation. The discussion was factual and
informative, unlike what you would get from, say, Twitter, or even
Metafilter, where the response was mostly limited to variations on
“string them up” and “eat the rich”.</p>
<p><em>Money Stuff</em> is also very funny. Today’s letter discusses a
disclosure filed recently by
<a href="https://en.wikipedia.org/wiki/Nikola_Corporation">Nikola Corporation</a>:</p>
<blockquote>
<p>More impressive is that Nikola’s revenue for the second quarter was
very small, just
$36,000. Most impressive, though, is how they earned that revenue:</p>
<blockquote>
<p>During the three months ended June 30, 2020 and 2019 the Company
recorded solar revenues of $0.03 million and $0.04 million,
respectively, for the provision of solar installation services to
the Executive Chairman, which are billed on time and materials
basis. … </p>
</blockquote>
<p>“Solar installation projects are not related to our primary operations
and are expected to be discontinued,” says Nikola, but I guess they
are doing one last job, specifically installing solar panels at
founder and executive chairman Trevor Milton’s house? It is a $13
billion company whose only business so far is doing odd jobs around
its founder’s house. </p>
</blockquote>
<p>A couple of recent articles that cracked me up discussed clueless
day-traders pushing up the price of Hertz stock <em>after</em> Hertz had
declared bankruptcy, and how Hertz diffidently attempted to get the
SEC to approve a new stock issue to cater to these idiots. (The SEC
said no.)</p>
<p>One recurring theme in the newsletter is “Everything is Securities
Fraud”. This week, Levine asks:</p>
<blockquote>
<p>Is it securities fraud for a public company to pay bribes to public
officials in exchange for lucrative public benefits?</p>
</blockquote>
<p>Of course you'd expect that the executives would be criminally
charged, as they have been. But is there a cause for the company’s
shareholders to sue? If you follow the newsletter, you know what the
answer will be:</p>
<blockquote>
<p>Oh absolutely…</p>
</blockquote>
<p>because Everything is Securities Fraud.</p>
<blockquote>
<p>Still it is a little weird. Paying bribes to get public benefits is,
you might think, the sort of activity that benefits
shareholders. Sure they were deceived, and sure the stock price was
too high because investors thought the company’s good performance
was more legitimate and sustainable than it was, etc., but the
shareholders are strange victims. In effect, executives broke the
law in order to steal money for the shareholders, and when the
shareholders found out they sued? It seems a little ungrateful?</p>
</blockquote>
<p>I recommend it.</p>
<ul>
<li><a href="https://www.bloomberg.com/authors/ARbTQlRLRjE/matthew-s-levine">Matt Levine on Bloomberg.com</a> </li>
<li><a href="http://link.mail.bloombergbusiness.com/join/4wm/moneystuff-signup">Free email newsletter signup</a></li>
</ul>
<p><a href="https://twitter.com/matt_levine">Levine also has a Twitter account</a> but it
is mostly just links to his newsletter articles.</p>
A maybe-interesting number trick?
https://blog.plover.com/2020/08/05#GF32
<p>I'm not sure if this is interesting, trivial, or both. You decide.</p>
<p>Let's divide the numbers from 1 to 30 into the following six groups:</p>
<table cellpadding="20em" cellspacing=0 align="center">
<tr bgcolor='white'><td><b>A
<td align='right'>1
<td align='right'>2
<td align='right'>4
<td align='right'>8
<td align='right'>16
<tr bgcolor='pink'><td><b>B
<td align='right'>3
<td align='right'>6
<td align='right'>12
<td align='right'>17
<td align='right'>24
<tr bgcolor='white'><td><b>C
<td align='right'>5
<td align='right'>9
<td align='right'>10
<td align='right'>18
<td align='right'>20
<tr bgcolor='pink'><td><b>D
<td align='right'>7
<td align='right'>14
<td align='right'>19
<td align='right'>25
<td align='right'>28
<tr bgcolor='white'><td><b>E
<td align='right'>11
<td align='right'>13
<td align='right'>21
<td align='right'>22
<td align='right'>26
<tr bgcolor='pink'><td><b>F
<td align='right'>15
<td align='right'>23
<td align='right'>27
<td align='right'>29
<td align='right'>30
</table>
<p>Choose any two rows. Chose a number from each row, and multiply
them mod 31. (That is, multiply them, and if the product is 31 or
larger, divide it by 31 and keep the remainder.)</p>
<p>Regardless of which two numbers you chose, the result will always be
in the same row. For example, any two numbers chosen from rows <em>B</em>
and <em>D</em> will multiply to yield a number in row <em>E</em>. If both numbers
are chosen from row <em>F</em>, their product will always appear in row <em>A</em>.</p>
Flappers
https://blog.plover.com/2020/08/03#flappers
<p><em>Gulliver's Travels</em> (1726), Part III, chapter 2:</p>
<blockquote>
<p>I observed, here and there, many in the habit of servants, with a
blown bladder, fastened like a flail to the end of a stick, which
they carried in their hands. In each bladder was a small quantity
of dried peas, or little pebbles, as I was afterwards informed.
With these bladders, they now and then flapped the mouths and ears
of those who stood near them, of which practice I could not then
conceive the meaning. It seems the minds of these people are so
taken up with intense speculations, that they neither can speak, nor
attend to the discourses of others, without being roused by some
external action upon the organs of speech and hearing… .
This flapper is likewise employed diligently to attend his master in
his walks, and upon occasion to give him a soft flap on his eyes;
because he is always so wrapped up in cogitation, that he is in
manifest danger of falling down every precipice, and bouncing his
head against every post; and in the streets, of justling others, or
being justled himself into the kennel.</p>
</blockquote>
<p>When I first told Katara about this, several years ago, instead of
“the minds of these people are so taken up with intense speculations”
I said they were obsessed with their phones.</p>
<p>Now the phones themselves have become the flappers:</p>
<blockquote>
<p>Y. Tung and K. G. Shin,
<a href="https://rtcl.eecs.umich.edu/rtclweb/assets/publications/2018/yuchih_bumpalert.pdf">"Use of Phone Sensors to Enhance Distracted Pedestrians’ Safety,"</a>
in IEEE Transactions on Mobile Computing, vol. 17, no. 6,
pp. 1469–1482, 1 June 2018, doi: <a href="https://doi.org/10.1109/TMC.2017.2764909">10.1109/TMC.2017.2764909</a>.</p>
</blockquote>
<p>Our minds are not even taken up with intense speculations, but with
Instagram. Dean Swift would no doubt be disgusted.</p>
How are finite fields constructed?
https://blog.plover.com/2020/08/01#finite-fields
<p>Here's another recent Math Stack Exchange answer I'm pleased with.</p>
<p><a href="https://math.stackexchange.com/q/3660825/25554">OP asked</a>:</p>
<blockquote>
<p>I know this question has been asked many times and there is good
information out there which has clarified a lot for me but I still
do not understand how the addition and multiplication tables for
<img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24GF%284%29%24"> is constructed?</p>
<p>I've seen [links] but none explicity explain the construction and
I'm too new to be told "its an extension of <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24GF%282%29%24">"</p>
</blockquote>
<p>The only “reasonable” answer here is “get an undergraduate abstract
algebra text and read the chapter on finite fields”. Because come on,
you can't expect some random stranger to appear and write up a
detailed but short explanation at your exact level of knowledge.</p>
<p>But sometimes Internet Magic Lightning strikes and that's what you do
get! And OP set themselves up to be struck by magic lightning,
because you can't get a detailed but short explanation at your exact
level of knowledge if you don't provide a detailed but short
explanation <em>of</em> your exact level of knowledge — and this person did
just that. They understand finite fields of prime order, but not how
to construct the extension fields. No problem, I can explain that!</p>
<p>I had special fun writing
<a href="https://math.stackexchange.com/a/3660966/25554">this answer</a> because
I just love constructing extensions of finite
fields. (Previously: <a href="https://blog.plover.com/math/z2.html">[1]</a>
<a href="https://blog.plover.com/math/finite-projective-planes.html">[2]</a>)</p>
<hr />
<p>For any given <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24n%24">, there is at most one field with <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24n%24"> elements: only one, if <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24n%24"> is a power of a prime number (<img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%242%2c%203%2c%202%5e2%2c%205%2c%207%2c%202%5e3%2c%203%5e2%2c%2011%2c%2013%2c%20%5cldots%24">) and none otherwise (<img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%246%2c%2010%2c%2012%2c%2014%5cldots%24">). This field with <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24n%24"> elements is written as <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%5cBbb%20F_n%24"> or as <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24GF%28n%29%24">. </p>
<p>Suppose we want to construct <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%5cBbb%20F_n%24"> where <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24n%3dp%5ek%24">. When <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24k%3d1%24">, this is easy-peasy: take the <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24n%24"> elements to be the integers <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%240%2c%201%2c%202%5cldots%20p%2d1%24">, and the addition and multiplication are done modulo <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24n%24">.</p>
<p>When <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24k%3e1%24"> it is more interesting. One possible construction goes like this:</p>
<ol>
<li><p>The elements of <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%5cBbb%20F_%7bp%5ek%7d%24"> are the polynomials $$a_{k-1}x^{k-1} + a_{k-2}x^{k-2} + \ldots + a_1x+a_0$$ where the coefficients <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24a_i%24"> are elements of <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%5cBbb%20F_p%24">. That is, the coefficients are just integers in <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%5c%7b0%2c%201%2c%20%5cldots%20p%2d1%5c%7d%24">, but with the understanding that the addition and multiplication will be done modulo <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24p%24">. Note that there are <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24p%5ek%24"> of these polynomials in total.</p></li>
<li><p>Addition of polynomials is done exactly as usual: combine like terms, but remember that the the coefficients are added modulo <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24p%24"> because they are elements of <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%5cBbb%20F_p%24">.</p></li>
<li><p>Multiplication is more interesting:</p>
<p>a. Pick an irreducible polynomial <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24P%24"> of degree <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24k%24">. “Irreducible” means that it does not factor into a product of smaller polynomials. How to actually locate an irreducible polynomial is an interesting question; here we will mostly ignore it.</p>
<p>b. To multiply two elements, multiply them normally, remembering that the coefficients are in <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%5cBbb%20F_p%24">. Divide the product by <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24P%24"> and keep the remainder. Since <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24P%24"> has degree <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24k%24">, the remainder must have degree at most <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24k%2d1%24">, and this is your answer.</p></li>
</ol>
<hr />
<p>Now we will see an example: we will construct <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%5cBbb%20F_%7b2%5e2%7d%24">. Here <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24k%3d2%24"> and <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24p%3d2%24">. The elements will be polynomials of degree at most 1, with coefficients in <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%5cBbb%20F_2%24">. There are four elements: <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%240x%2b0%2c%200x%2b1%2c%201x%2b0%2c%20%24"> and <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%241x%2b1%24">. As usual we will write these as <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%240%2c%201%2c%20x%2c%20x%2b1%24">. This will not be misleading.</p>
<p>Addition is straightforward: combine like terms, remembering that <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%241%2b1%3d0%24"> because the coefficients are in <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%5cBbb%20F_2%24">:</p>
<p>$$\begin{array}{c|cccc}
+ & 0 & 1 & x & x+1 \\ \hline
0 & 0 & 1 & x & x+1 \\
1 & 1 & 0 & x+1 & x \\
x & x & x+1 & 0 & 1 \\
x+1 & x+1 & x & 1 & 0
\end{array}
$$</p>
<p>The multiplication as always is more interesting. We need to find an irreducible polynomial <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24P%24">. It so happens that <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24P%3dx%5e2%2bx%2b1%24"> is the only one that works. (If you didn't know this, you could find out easily: a <em>reducible</em> polynomial of degree 2 factors into two linear factors. So the <em>reducible</em> polynomials are <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24x%5e2%2c%20x%c2%b7%28x%2b1%29%20%3d%20x%5e2%2bx%24">, and <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%28x%2b1%29%5e2%20%3d%20x%5e2%2b2x%2b1%20%3d%20x%5e2%2b1%24">. That leaves only <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24x%5e2%2bx%2b1%24">.) </p>
<p>To multiply two polynomials, we multiply them normally, then divide by <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24x%5e2%2bx%2b1%24"> and keep the remainder. For example, what is <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%28x%2b1%29%28x%2b1%29%24">? It's <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24x%5e2%2b2x%2b1%20%3d%20x%5e2%20%2b%201%24">. There is a theorem from elementary algebra (the <a href="https://en.wikipedia.org/wiki/Polynomial_greatest_common_divisor#Euclidean_division">“division theorem”</a>) that we can find a unique quotient <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24Q%24"> and remainder <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24R%24">, with the degree of <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24R%24"> less than 2, such that <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24PQ%2bR%20%3d%20x%5e2%2b1%24">. In this case, <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24Q%3d1%2c%20R%3dx%24"> works. (You should check this.) Since <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24R%3dx%24"> this is our answer: <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%28x%2b1%29%28x%2b1%29%20%3d%20x%24">.</p>
<p>Let's try <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24x%c2%b7x%20%3d%20x%5e2%24">. We want <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24PQ%2bR%20%3d%20x%5e2%24">, and it happens that <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24Q%3d1%2c%20R%3dx%2b1%24"> works. So <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24x%c2%b7x%20%3d%20x%2b1%24">. </p>
<p>I strongly recommend that you calculate the multiplication table yourself. But here it is if you want to check:</p>
<p>$$\begin{array}{c|cccc}
· & 0 & 1 & x & x+1 \\ \hline
0 & 0 & 0 & 0 & 0 \\
1 & 0 & 1 & x & x+1 \\
x & 0 & x & x+1 & 1 \\
x+1 & 0 & x+1 & 1 & x
\end{array}
$$</p>
<p>To calculate the unique field <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%5cBbb%20F_%7b2%5e3%7d%24"> of order 8, you let the elements be the 8 second-degree polynomials <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%240%2c%201%2c%20x%2c%20%5cldots%2c%20x%5e2%2bx%2c%20x%5e2%2bx%2b1%24"> and instead of reducing by <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24x%5e2%2bx%2b1%24">, you reduce by <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24x%5e3%2bx%2b1%24">. (Not by <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24x%5e3%2bx%5e2%2bx%2b1%24">, because that factors as <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%28x%5e2%2b1%29%28x%2b1%29%24">.) To calculate the unique field <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%5cBbb%20F_%7b3%5e2%7d%24"> of order 27, you start with the 27 third-degree polynomials with coefficients in <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%5c%7b0%2c1%2c2%5c%7d%24">, and you reduce by <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24x%5e3%2b2x%2b1%24"> (I think).</p>
<hr />
<p>The special notation <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%5cBbb%20F_p%5bx%5d%24"> means the ring of all polynomials with coefficients from <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%5cBbb%20F_p%24">. <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%5clangle%20P%20%5crangle%24"> means the ring of all multiples of polynomial <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24P%24">. (A ring is a set with an addition, subtraction, and multiplication defined.)</p>
<p>When we write <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%5cBbb%20F_p%5bx%5d%20%2f%20%5clangle%20P%5crangle%24"> we are constructing a thing called a “quotient” structure. This is a generalization of the process that turns the ordinary integers <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%5cBbb%20Z%24"> into the modular-arithmetic integers we have been calling <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%5cBbb%20F_p%24">. To construct <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%5cBbb%20F_p%24">, we start with <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%5cBbb%20Z%24"> and then agree that two elements of <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%5cBbb%20Z%24"> will be considered equivalent if they differ by a multiple of <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24p%24">.</p>
<p>To get <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%5cBbb%20F_p%5bx%5d%20%2f%20%20%5clangle%20P%20%5crangle%24"> we start with <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%5cBbb%20F_p%5bx%5d%24">, and then agree that elements of <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%5cBbb%20F_p%5bx%5d%24"> will be considered equivalent if they differ by a multiple of <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24P%24">. The division theorem guarantees that of all the equivalent polynomials in a class, exactly one of them will have degree less than that of <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24P%24">, and that is the one we choose as a representative of its class and write into the multiplication table. This is what we are doing when we “divide by <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24P%24"> and keep the remainder”.</p>
<hr />
<p>A particularly important example of this construction is <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%5cBbb%20R%5bx%5d%20%2f%20%5clangle%20x%5e2%20%2b%201%5crangle%24">. That is, we take the set of polynomials with real coefficients, but we consider two polynomials equivalent if they differ by a multiple of <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24x%5e2%20%2b%201%24">. By the division theorem, each polynomial is then equivalent to some first-degree polynomial <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24ax%2bb%24">. </p>
<p>Let's multiply $$(ax+b)(cx+d).$$ As usual we obtain $$acx^2 + (ad+bc)x + bd.$$ From this we can subtract <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24ac%28x%5e2%20%2b%201%29%24"> to obtain the equivalent first-degree polynomial $$(ad+bc) x + (bd-ac).$$</p>
<p>Now recall that in the complex numbers, <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%28b%2bai%29%28d%20%2b%20ci%29%20%3d%20%28bd%2dac%29%20%2b%20%28ad%2bbc%29i%24">. We have just constructed the complex numbers,with the polynomial <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24x%24"> playing the role of <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24i%24">.</p>
<hr />
<p>[ Note to self: maybe write a separate article about what makes this a good answer, and how it is structured. ]</p>
What does it mean to expand a function “in powers of x-1”?
https://blog.plover.com/2020/07/31#taylor-series
<p><a href="https://math.stackexchange.com/q/3689743/25554">A recent Math Stack Excahnge post</a>
was asked to expand the function <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24e%5e%7b2x%7d%24"> in powers of <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%28x%2d1%29%24"> and
was confused about what that meant, and what the point of it was.
<a href="https://math.stackexchange.com/a/3689929/25554">I wrote an answer I liked</a>,
which I am reproducing here.</p>
<hr />
<p>You asked:</p>
<blockquote>
<p>I don't understand what are we doing in this whole process</p>
</blockquote>
<p>which is a fair question. I didn't understand this either when I first
learned it. But it's important for practical engineering reasons as
well as for theoretical mathematical ones.</p>
<p>Before we go on, let's see that your proposal is the wrong answer to
this question, because it is the correct answer, but to a different
question. You suggested:
$$e^{2x}\approx1+2\left(x-1\right)+2\left(x-1\right)^2+\frac{4}{3}\left(x-1\right)^3$$</p>
<p>Taking <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24x%3d1%24"> we get <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24e%5e2%20%5capprox%201%24">, which is just wrong, since
actually <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24e%5e2%5capprox%207%2e39%24">. As a comment pointed out, the series you
have above is for <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24e%5e%7b2%28x%2d1%29%7d%24">. But we wanted a series that adds up
to <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24e%5e%7b2x%7d%24">.</p>
<p>As you know, the Maclaurin series works here:</p>
<p>$$e^{2x} \approx 1+2x+2x^2+\frac{4}{3}x^3$$</p>
<p>so why don't we just use it? Let's try <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24x%3d1%24">. We get $$e^2\approx
1 + 2 + 2 + \frac43$$</p>
<p>This adds to <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%246%2b%5cfrac13%24">, but the correct answer is actually around
<img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%247%2e39%24"> as we saw before. That is not a very accurate approximation.
Maybe we need more terms? Let's try ten:</p>
<p>$$e^{2x} \approx 1+2x+2x^2+\frac{4}{3}x^3 + \ldots +
\frac{8}{2835}x^9$$</p>
<p>If we do this we get <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%247%2e3887%24">, which isn't too far off. But it was a
lot of work! And we find that as <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24x%24"> gets farther away from zero, the
series above gets less and less accurate. For example, take <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24x%3d3%2e1%24">,
the formula with four terms gives us <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%2466%2e14%24">, which is dead wrong.
Even if we use ten terms, we get <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24444%2e3%24">, which is still way off. The
right answer is actually <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24492%2e7%24">.</p>
<p>What do we do about this? Just add more terms? That could be a lot of
work and it might not get us where we need to go. (Some Maclaurin
series just stop working at all too far from zero, and no amount of
terms will make them work.) Instead we use a different technique. </p>
<p>Expanding the Taylor series “around <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24x%3da%24">” gets us a different series,
one that works best when <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24x%24"> is close to <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24a%24"> instead of when <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24x%24"> is
close to zero. Your homework is to expand it around <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24x%3d1%24">, and I don't
want to give away the answer, so I'll do a different example. We'll
expand <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24e%5e%7b2x%7d%24"> around <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24x%3d3%24">. The general formula is $$e^{2x} \approx
\sum \frac{f^{(i)}(3)}{i!} (x-3)^i\tag{$\star$}\ \qquad \text{(when
$x$ is close to $3$)}$$</p>
<p>The <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24f%5e%7b%28i%29%7d%28x%29%24"> is the <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24i%24">'th derivative of <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%20e%5e%7b2x%7d%24"> , which is
<img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%242%5eie%5e%7b2x%7d%24">, so the first few terms of the series above are:</p>
<p>$$\begin{eqnarray}
e^{2x} & \approx& e^6 + \frac{2e^6}1 (x-3) + \frac{4e^6}{2}(x-3)^2 +
\frac{8e^6}{6}(x-3)^3\\
& = & e^6\left(1+ 2(x-3) + 2(x-3)^2 + \frac34(x-3)^3\right)\\
& & \qquad \text{(when $x$ is close to $3$)}
\end{eqnarray}
$$</p>
<p>The first thing to notice here is that when <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24x%24"> is <em>exactly</em> <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%243%24">,
this series is perfectly correct; we get <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24e%5e6%20%3d%20e%5e6%24"> exactly, even
when we add up only the first term, and ignore the rest. That's a
kind of useless answer because we already knew that <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24e%5e6%20%3d%20e%5e6%24">.
But that's not what this series is for. The whole point of <em>this</em>
series is to tell us how different <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24e%5e%7b2x%7d%24"> is from <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24e%5e6%24"> when <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24x%24">
is close to, but not equal to <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%243%24">. </p>
<p>Let's see what it does at <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24x%3d3%2e1%24">. With only four terms we get
$$\begin{eqnarray}
e^{6.2} & \approx& e^6(1 + 2(0.1) + 2(0.1)^2 + \frac34(0.1)^3)\\
& = & e^6 \cdot 1.22075 \\
& \approx & 492.486
\end{eqnarray}$$</p>
<p>which <em>is</em> very close to the correct answer, which is <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24492%2e7%24">. And
that's with only four terms. Even if we didn't know an exact value
for <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24e%5e6%24">, we could find out that <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24e%5e%7b6%2e2%7d%24"> is about <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%2422%2e075%5c%25%24">
larger, with hardly any calculation.</p>
<p>Why did this work so well? If you look at the expression <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%28%5cstar%29%24">
you can see: The terms of the series all have factors of the form
<img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%28x%2d3%29%5ei%24">. When <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24x%3d3%2e1%24">, these are <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%280%2e1%29%5ei%24">, which becomes very
small very quickly as <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24i%24"> increases. Because the later terms of the
series are very small, they don't affect the final sum, and if we
leave them out, we won't mess up the answer too much. So the
series works well, producing accurate results from only a few
terms, when <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24x%24"> is close to <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%243%24">.</p>
<p>But in the <em>Maclaurin</em> series, which is around <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24x%3d0%24">, those
<img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%28x%2d3%29%5ei%24"> terms are <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24x%5ei%24"> terms intead, and when <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24x%3d3%2e1%24">, they are
<em>not</em> small, they're very large! They get <em>bigger</em> as <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24i%24">
increases, and very quickly. (The <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%20i%21%20%24"> in the denominator wins,
eventually, but that doesn't happen for many terms.) If we leave
out these many large terms, we get the wrong results.</p>
<p>The short answer to your question is:</p>
<blockquote>
<p>Maclaurin series are only good for calculating functions when <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24x%24">
is close to <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%240%24">, and become inaccurate as <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24x%24"> moves away from
zero. But a Taylor series around <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24a%24"> has its “center” near <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24a%24">
and is most accurate when <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24x%24"> is close to <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24a%24">. </p>
</blockquote>
I screw up buying a marker
https://blog.plover.com/2020/07/29#copic
<p>Toph left the cap off one of her fancy art markers and it dried out,
so I went to get her a replacement. The marker costs $5.85, plus tax,
and the web site wanted a $5.95 shipping fee. Disgusted, I resolved
to take my business elsewhere.</p>
<p>On Wednesday I drove over to a local art-supply store to get the
marker. After taxes the marker was somehow around $8.50, but I also
had to pay $1.90 for parking. So if there was a win there, it was a
very small one.</p>
<p>But also, I messed up the parking payment app, which has maybe the
worst UI of any phone app I've ever used. The result was a $36
parking ticket.</p>
<p>Lesson learned. I hope.</p>
Zuul crurivastator
https://blog.plover.com/2020/07/29#zuul
<p>Today I learned:</p>
<ul>
<li><p>There is a genus of ankylosaurs named <em>Zuul</em> after “demon and
demi-god Zuul, the Gatekeeper of Gozer, featured in the 1984 film
Ghostbusters”.</p></li>
<li><p>The type species of <em>Zuul</em> is <em>Zuul crurivastator</em>, which means
“Zuul, destroyer of shins”. <a href="https://en.wikipedia.org/wiki/Zuul">Wikipedia says</a>:</p>
<blockquote>
<p>The epithet … refers to a presumed defensive tactic of
ankylosaurids, smashing the lower legs of attacking predatory
theropods with their tail clubs.</p>
</blockquote>
<p>My eight-year-old self is gratified that the ankylosaurids are believed to attack their enemies’ ankles.</p></li>
<li><p>The original specimen of <em>Z. crurivastator</em>, unusually-well
preserved, was nicknamed “Sherman”.</p></li>
</ul>
<p><a href="https://www.youtube.com/watch?v=nDjnHbM-oT4">Here is a video of Dan Aykroyd discussing the name, with Sherman</a>.</p>
More trivia about megafauna and poisonous plants
https://blog.plover.com/2020/07/15#urushiol-2
<p>A couple of people expressed disappointment with yesterday's article,
which asked
<a href="https://blog.plover.com/bio/urushiol.html">were giant ground sloths immune to poison ivy?</a>,
but then failed to deliver on the implied promise. I hope today's
article will make up for that.</p>
<h3>Elephants</h3>
<p>I said:</p>
<blockquote>
<p>Mangoes are tropical fruit and I haven't been able to find any
examples of Pleistocene megafauna that lived in the tropics…</p>
</blockquote>
<p>David Formosa points out what should have been obvious: elephants are
megafauna, elephants live where mangoes grow (both in Africa and in
India), elephants love eating mangoes
<a href="https://twitter.com/susantananda3/status/1260191081037103110">[1]</a>
<a href="https://www.upi.com/Odd_News/2020/01/13/Elephant-climbs-5-foot-wall-to-steal-mangoes-from-lodge/3411578953055/">[2]</a>
<a href="https://www.littlethings.com/mango-loving-elephants/">[3]</a>, and, not
obvious at all…</p>
<p><a href="https://globalelephants.org/elephants-and-toxic-plants/"><em>Elephants are immune to poison ivy!</em></a></p>
<blockquote>
<p>Captive elephants have been known to eat poison ivy, not just a little
bite, but devouring entire vines, leaves and even digging up the
roots. To most people this would have cause a horrific rash …
To the elephants, there was no rash and no ill effect at all…</p>
</blockquote>
<p>It's sad that we no longer have <em>megatherium</em>. But we do have
elephants, which is pretty awesome.</p>
<h3>Idiot fruit</h3>
<p>The idiot fruit is just another one of those legendarily awful
creatures that seem to infest every corner of Australia (see also: box
jellyfish, stonefish, gympie gympie, etc.); Wikipedia says:</p>
<blockquote>
<p>The seeds are so toxic that most animals cannot eat them without
being severely poisoned.</p>
</blockquote>
<p>At present the seeds are mostly dispersed by gravity. The plant is
believed to be an evolutionary anachronism. What Pleistocene
megafauna formerly dispersed the poisonous seeds of the idiot fruit?</p>
<p>A wombat. <a href="https://en.wikipedia.org/wiki/Diprotodon">A <em>six-foot-tall</em> wombat</a>.</p>
<p>I am speechless with delight.</p>
Were giant ground sloths immune to poison ivy?
https://blog.plover.com/2020/07/14#urushiol
<p>The skin of the mango fruit contains <a href="https://en.wikipedia.org/wiki/urushiol">urushiol</a>, the same
irritating chemical that is found in poison ivy. But why? From the mango's
point of view, the whole point of the mango fruit is to get someone to come
along and eat it, so that they will leave the seed somewhere else. Posioning
the skin seems counterproductive.</p>
<p>An analogous case is the chili pepper, which contains an irritating
chemical, capsaicin. I think the answer here is believed to be that
while capsaicin irritates <em>mammals</em>, birds are unaffected. The
chili's intended target is birds; you can tell from the small seeds,
which are the right size to be pooped out by birds. So chilis have a
chemical that encourages mammals to leave the fruit in place for
birds.</p>
<p>What's the intended target for the mango fruit? Who's going to poop
out a seed the size of a mango pit? You'd need a very large animal,
large enough to swallow a whole mango. There aren't many of these
now, but that's because they became extinct at the end of the
Pleistocene epoch: woolly mammoths and rhinoceroses, huge crocodiles,
giant ground sloths, and so on. We may have eaten the animals
themselves, but we seem to have quite a lot of fruits around that
evolved to have their seeds dispersed by Pleistocene megafauna that
are now extinct. So my first thought was, maybe the mango is
expecting to be gobbled up by a giant gound sloth, and have its giant
seed pooped out elsewhere. And perhaps its urushiol-laden skin makes
it unpalatable to smaller animals that might not disperse the seeds as
widely, but the giant ground sloth is immune. (Similarly, I'm told
that goats are immune to urushiol, and devour poison ivy as they do
everything else.)</p>
<p>Well, maybe this theory is partly correct, but even if so, the animal
definitely wasn't a giant ground sloth, because those lived only in
South America, whereas the mango is native to South Asia. Ground
slots and avocados, yes; mangos no.</p>
<p>Still the theory seems reasonable, except that mangoes are tropical
fruit and I haven't been able to find any examples of Pleistocene
megafauna that lived in the tropics. Still I didn't look very hard.</p>
<p>Wikipedia has an article on
<a href="https://en.wikipedia.org/wiki/Evolutionary_anachronisms">evolutionary anachronisms</a> that
lists a great many plants, but not the mango.</p>
<p>[ Addendum: I've eaten many mangoes but never noticed any irritation from the peel. I speculate that cultivated mangoes are varieties that have been bred to contain little or no urushiol, or that there is a post-harvest process that removes or inactivates the urushiol, or both. ]</p>
<p>[ Addendum 20200715: I know this article was a little disappointing and that it does not resolve the question in the title. Sorry.
But I wrote <a href="https://blog.plover.com/bio/urushiol-2.html">a followup that you might enjoy anyway</a>. ]</p>
Ron Graham has died
https://blog.plover.com/2020/07/08#graham
<p>Ron Graham has died. He had a good run. When I check out I will
probably not be as accomplished or as missed as Graham, even if I make
it to 84.</p>
<p>I met Graham once and he was very nice to me, as he apparently was to
everyone. I was planning to write up a reminiscence of the time, but
I find <a href="https://blog.plover.com/math/erdos.html">I've already done it</a> so you can read
that if you care.</p>
<p>Graham's little book <em>Rudiments of Ramsey Theory</em> made a big impression
on me when I was an undergraduate. Chapter 1, if I remember
correctly, is a large collection of examples, which suited me fine.
Chapter 2 begins by introducing a certain notation of Erdős and Rado:
<img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%5cleft%5c%5b%7b%5cBbb%20N%5catop%20k%7d%5cright%5d%24"> is the family of subsets of <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%5cBbb%0aN%24"> of size <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24k%24">, and</p>
<p>$$\left[{\Bbb N\atop k}\right] \to \left[{\Bbb N\atop k}\right]_r$$</p>
<p>is an abbreviation of the statement that for any <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24r%24">-coloring of
members of <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%5cleft%5c%5b%7b%5cBbb%20N%5catop%20k%7d%5cright%5d%24"> there is always an
infinite subset <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24S%5csubset%20%5cBbb%20N%24"> for which every member of
<img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24%5cleft%5c%5b%7bS%5catop%20k%7d%5cright%5d%24"> is the same color. I still do not find
this notation perspicuous, and at the time, with much less experience,
I was boggled. In the midst of my bogglement I was hit with the next
sentence, which completely derailed me:</p>
<p><img src="https://pic.blog.plover.com/math/graham/no-confusion-hl.png" class="center"
alt="Scan of two lines from _Rudiments of Ramsey Theory_
including the sentence “We will occasionally use this arrow notation unless there is danger
of no confusion.”" /></p>
<p>After this I could no longer think about the mathematics, but only
about the sentence.</p>
<!--
I think Michael Hutchings once told me that Graham had said
that one a scale of one to ten, juggling three balls is a 1, juggling
4 balls is a 3, and juggling five balls is a 57. Graham could juggle
seven balls.)
-->
<p>Outside the mathematical community Graham is probably best-known for
juggling, or for
<a href="https://en.wikipedia.org/wiki/Graham%27s_number">Graham's number</a>, which Wikipedia describes:</p>
<blockquote>
<p>At the time of its introduction, it was the largest specific
positive integer ever to have been used in a published mathematical
proof.</p>
</blockquote>
<p>One of my better Math Stack Exchange posts was in answer to the
question
<a href="https://math.stackexchange.com/q/163423/25554">Graham's Number : Why so big?</a>.
I love the phrasing of this question! And that, even with the strange
phrasing, there is an answer! This type of huge number is quite
typical in proofs of Ramsey theory, and
<a href="https://math.stackexchange.com/a/1046098/25554">I answered in detail</a>.</p>
<p>The sense of humor that led Graham to write “danger of no confusion”
is very much on display in the paper that gave us Graham's number.
If you are wondering about Graham's number, check out my post.</p>
Addendum to “Weirdos during the Depression”
https://blog.plover.com/2020/07/08#weirdos
<p>[ <a href="https://blog.plover.com/book/weirdos.html">Previously</a> ]</p>
<p><a href="http://www.ranprieur.com/">Ran Prieur</a> had
<a href="https://old.reddit.com/r/ranprieur/comments/gwb4sg/weirdos_during_the_depression/fsvmmri/">a take on this</a>
that I thought was insightful:</p>
<blockquote>
<p>I would frame it like this: If you break rules that other people are
following, you have to pretend to be unhappy, or they'll get really
mad, because they don't want to face the grief that they could have
been breaking the rules themselves all this time.</p>
</blockquote>
Weird constants in math problems
https://blog.plover.com/2020/07/07#odd-constants
<p><a href="https://gottwurfelt.com/2020/07/06/how-full-is-the-pool/">Michael Lugo recently considered a problem</a>
involving the allocation of swimmers to swim lanes at random, ending
with:</p>
<blockquote>
<p>If we compute this for large <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24n%24"> we get <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24f%28n%29%20%5csim%200%2e4323n%24">,
which agrees with the Monte Carlo simulations… The
constant <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%240%2e4323%24"> is $$\frac{(1-e^{-2})}2.$$</p>
</blockquote>
<p>I love when stuff like this happens. The computer is great at doing a
quick random simulation and getting you some weird number, and you
have no idea what it really means. But mathematical technique can
unmask the weird number and learn its true identity. (“It was Old Man
Haskins all along!”)</p>
<p>A couple of years back Math Stack Exchange had
<a href="https://math.stackexchange.com/q/2991304/25554">Expected Number and Size of Contiguously Filled Bins</a>,
and although it wasn't exactly what was asked, I ended up looking into
this question: We take <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24n%24"> balls and throw them at random into <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24n%24">
bins that are lined up in a row. A maximal contiguous sequence of
all-empty or all-nonempty bins is called a “cluster”. For example,
here we have 13 balls that I placed randomly into 13 bins:</p>
<p><img style="width: 100%;" class="center" alt="13 boxes, some with blue balls. The boxes
contain, respectively, 1, 0, 3, 0, 1, 2, 1, 1, 0, 1, 2, 1, 0 balls."
src="https://pic.blog.plover.com/math/odd-constants/random-bins.svg" /></p>
<p>In this example, there are 8 clusters, of sizes 1, 1, 1, 1, 4, 1,
3, 1. Is this typical? What's the expected cluster size?</p>
<p>It's easy to use Monte Carlo methods and find that when <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%24n%24"> is
large, the average cluster size is approximately <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%242%2e15013%24">. Do you
recognize this number? I didn't.</p>
<p>But it's not hard to do the calculation analytically and discover that
that the <em>reason</em> it's approximately <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%242%2e15013%24"> is that the actual
answer is $$\frac1{2(e^{-1} - e^{-2})}$$ which is approximately <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%242%2e15013%24">.</p>
<p>Math is awesome and wonderful.</p>
<p>(Incidentally, I tried the
<a href="http://wayback.cecm.sfu.ca/cgi-bin/isc/lookup?number=2.150129268&lookup_type=simple">Inverse Symbolic Calculator</a>
just now, but it was no help. It's also not in Plouffe's
<a href="http://www.gutenberg.org/ebooks/634"><em>Miscellaneous Mathematical Constants</em></a>)</p>
<p>[ Addendum 20200707: <a href="https://www.wolframalpha.com/input/?i=2.15013">WolframAlpha</a>
does correctly identify the <img src="https://chart.apis.google.com/chart?chf=bg,s,00000000&cht=tx&chl=%242%2e15013%24"> constant. ]</p>