I somehow managed to miss the notion of totally
bounded when I was learning topology,
and it popped up on stack exchange recently. It is a stronger version
of boundedness for metric spaces: a space !!M!! is totally bounded if, for
any chosen !!\epsilon!!, !!M!! can be covered by a finite family of
balls of radius !!\epsilon!!.
This is a strictly stronger property than ordinary boundedness, so the
question immediately comes up: what is an example of a space that is
bounded but not totally bounded. Many examples are well-known. For
example, the infinite-dimensional unit ball is bounded
but not totally bounded. But I didn't think of this right away.
Instead I thought of the following rather odd example: Let !!S!! be
the closed unit disc and assign each point a polar coordinate
!!\langle r,\theta\rangle!! as usual. Now consider the following metric:
The idea is this: you can travel between points only along the radii
of the disc. To get from !!p_1!! to !!p_2!! that are on different
radii, you must go through the origin:
Now clearly when !!\epsilon < \frac12!!, the !!\epsilon!!-ball that
covers each point point !!\left\langle 1, \theta\right\rangle!! lies
entirely within one of the radii, and so an uncountable number of such
balls are required to cover the disc.
It seems like this example could be useful in other circumstances
too. Does it have a name?
In yesterday's article I
described a simple and useful feature that could have been added to
the standard I/O library, to allow an environment variable to override
the default buffering behavior. This would allow the invoker of a
program to request that the program change its buffering behavior even
if the program itself didn't provide an option specifically for doing
that.
ptyget is a universal pseudo-terminal interface. It is designed to be
used by any program that needs a pty.
ptyget can also serve as a wrapper to improve the behavior of existing
programs. For example, ptybandage telnet is like telnet but
can
be put into a pipeline. nobuf grep is like grep but won't
block-buffer if it's redirected.
Previous pty-allocating programs — rlogind, telnetd, sshd, xterm,
screen, emacs, expect, etc. — have caused dozens of security problems.
There are two fundamental reasons for this. First, these programs are
installed setuid root so that they can allocate ptys; this turns every
little bug in hundreds of thousands of lines of code into a potential
security hole. Second, these programs are not careful enough to
protect
the pty from access by other users.
ptyget solves both of these problems. All the privileged code is in
one
tiny program. This program guarantees that one user can't touch
another
user's pty.
ptyget is a complete rewrite of pty 4.0, my previous pty-allocating
package. pty 4.0's session management features have been split off
into
a separate package, sess.
Leonardo Taccari informed me that NetBSD's stdio actually
has the environment variable feature I was asking for! Christos
Zoulas suggested adding stdbuf similar to the GNU and FreeBSD
implementations, but the NetBSD people observed, as I did, that it
would be simpler to just control stdio directly with an environment
variable, and did it. Here's the relevant part of the NetBSD setbuf(3) man
page:
The default buffer settings can be overwritten per descriptor
(STDBUFn) where n is the numeric value of the file descriptor
represented by the stream, or for all descriptors (STDBUF). The
environment variable value is a letter followed by an optional
numeric value indicating the size of the buffer. Valid sizes range
from 0B to 1MB. Valid letters are:
Finally, Mariusz Ceier pointed out that there is an ancient bug report in
glibc
suggesting essentially the same environment variable mechanism that I
suggested and that was adopted in NetBSD. The suggestion was firmly
and summarily rejected. (“Hell, no … this is a terrible idea.”)
Interesting wrinkle: the bug report was submitted by Pádraig Brady,
who subsequently wrote the stdbuf command I described above.
Some Unix commands, such as grep, will have a command-line flag to
say that you want to turn off the buffering that is normally done in
the standard I/O library. Some just try to guess what you probably
want. Every command is a little different and if the command you want
doesn't have the flag you need, you are basically out of luck.
Maybe I should explain the putative use case here. You have some
command (or pipeline) X that will produce dribbles of data at
uncertain intervals. If you run it at the terminal, you see each
dribble timely, as it appears. But if you put X into a pipeline,
say with
X | tee ...
or
X | grep ...
then the dribbles are buffered and only come out of X when an entire
block is ready to be written, and the dribbles could be very old
before the downstream part of the pipeline, including yourself, sees
them. Because this is happening in user space inside of X, there is
not a damn thing anyone farther downstream can do about it. The only
escape is if X has some mode in which it turns off standard I/O
buffering. Since standard I/O buffering is on by default, there is a
good chance that the author of X did not think to affirmatively add
this feature.
Note that adding the --unbuffered flag to the downstream grep does
not solve the problem; grep will produce its own output timely, but
it's still getting its input from X after a long delay.
One could imagine a program which would interpose a pseudo-tty, and
make X think it is writing to a terminal, and then the standard I/O
library would stay in line-buffered mode by default. Instead of
running
X | tee some-file | ...
or whatever, one would do
pseudo-tty-pipe -c X | tee some-file | ...
which allocates a pseudo-tty device, attaches standard output to it,
and forks. The child runs X, which dribbles timely into the
pseudo-tty while the parent runs a read loop to remove dribbles from
the master end of the TTY and copy them timely into the pipe. This
would work. Although tee itself also has no --unbuffered flag
so you might even have to:
pseudo-tty-pipe -c X | pseudo-tty-pipe -c 'tee some-file' | ...
I don't think such a program exists, and anyway, this is all
ridiculous, a ridiculous abuse of the standard I/O library's buffering
behavior: we want line buffering, the library will only give it to us
if the process is attached to a TTY device, so we fake up a TTY just
to fool stdio into giving us what we want. And why? Simply because
stdio has no way to explicitly say what we want.
But it could easily expose this behavior as a controllable
feature. Currently there is a branch in the library that says how to
set up a buffering mode when a stream is opened for the first time:
if the stream is for writing, and is attached to descriptor 2,
it should be unbuffered; otherwise …
if the stream is for writing, and connects descriptor 1 to a
terminal device, it should be line-buffered; otherwise …
if the moon is waxing …
…
otherwise, the stream should be block-buffered
To this, I propose a simple change, to be inserted right at the beginning:
If the environment variable STDIO_BUF is set to "line", streams
default to ine buffering. If it's set to "none", streams default
to no buffering. If it's set to "block", streams default to block
buffered. If it's anything else, or unset, it is ignored.
Now instead of this:
pseudo-tty-pipe --from X | tee some-file | ...
you write this:
STDIO_BUF=line X | tee some-file | ...
Problem solved.
Or maybe you would like to do this:
export STDIO_BUF=line
which then it affects every program in every pipeline in the rest of
the session:
X | tee some-file | ...
Control is global if you want it, and per-process if you want it.
This feature would cost around 20 lines of C code in the standard I/O
library and would impose only an insigificant run-time cost. It would
effectively add an --unbuffered flag to every program in the
universe, retroactively, and the flag would be the same for every
program. You would not have to remember that in mysql the magic
option is -n and that in GNU grep it is --line-buffered and that
for jq is is --unbuffered and that Python scripts can be
unbuffered by supplying the -u flag and that in tee you are just
SOL, etc. Setting STDIO_BUF=line would Just Work.
Programming languages would all get this for free also. Python
already has PYTHONUNBUFFERED but in other languages
you have to do something or other; in Perl you use some
horrible Perl-4-ism like
{ my $ofh = select OUTPUT; $|++; select $ofh }
This proposal would fix every programming language everywhere. The
Perl code would become:
$ENV{STDIO_BUF} = 'line';
and every other language would be similarly simple:
/* In C */
putenv("STDIO_BUF=line");
[ Addendum 20180521: Mariusz Ceier corrects me,
pointing out that this will not work for the process’ own standard
streams, as they are pre-opened before the process gets a chance to
set the variable. ]
It's easy to think of elaborations on this: STDIO_BUF=1:line might
mean that only standard output gets line-buffering by default,
everything else is up to the library.
This is an easy thing to do. I have wanted this for twenty years.
How is it possible that it hasn't been in the GNU/Linux standard
library for that long?
This weekend I got a very exciting text message from Katara:
I have a math question for you
Oh boy! I hope it's one I can answer.
Okay
there's this game called spot it where you have cards with 8 symbols
on them like so
and the goal is to find the one matching symbol on your card and the
one in the middle
how is it possible that given any pair of cards, there is exactly one
matching symbol
Well, whatever my failings as a dad, this is one problem I can solve.
I went a little of overboard in my reply:
You need a particular kind of structure called a projective plane.
They only exist for certain numbers of symbols
A simpler example has 7 cards with 3 symbols each.
One thing that's cool about it is that the symbols and the cards are
"dual": say you give each round card a name. Then make up a new deck
of square cards. There's one square card for each symbol. So there's a
square"Ladybug" card. The Ladybug card has on it the names of the
round cards that have the Ladybug. Now you can play Spot with the
square cards instead of the round ones: each two square cards have
exactly one name in common.
In a geometric plane any two points lie on exactly one common line and
any two lines intersect in exactly one common point. This is a sort of
finite model of that, with cards playing the role of lines and symbols
playing the role of points. Or vice versa, it doesn't matter.
More than you wanted to know 😂
ah thank you, I'm pretty sure I understand, sorry for not responding,
my phone was charging
I still couldn't shut up about the finite projective planes:
No problem! No response necessary.
It is known that all finite projective planes have n²+n+1 points for
some n. So I guess the Spot deck has either 31, 57, or 73 cards and
the same number of symbols. Is 57 correct?
Must be 57 because I see from your picture that each card has 8
symbols.
Katara was very patient:
I guess, I would like to talk about this some more when i get home if
that's okay
Anyway this evening I cut up some index cards, and found a bunch of
stickers in a drawer, and made Katara a projective plane of order 3.
This has 13 cards, each with 4 different stickers, and again, every
two cards share exactly one sticker. She was very pleased and wanted
to know how to make them herself.
Each set of cards has an order, which is a non-negative integer.
Then there must be !!n^2 + n + 1!! cards, each with !!n+1!! stickers
or symbols. When !!n!! is a prime power, you can use field theory to
construct a set of cards from the structure of the (unique) field of
order !!n!!.
Fields to projective planes
Order 2
I'll describe the procedure using the plane of order !!n=2!!, which is
unusually simple. There will be !!2^2+2+1 = 7!! cards, each with
!!3!! of the !!7!! symbols.
Here is the finite field of order 2, called !!GF(2)!!:
+
0
1
0
0
1
1
1
0
×
0
1
0
0
0
1
0
1
The stickers correspond to ordered triples of elements of !!GF(2)!!, except that !!\langle 0,0,0\rangle!! is always omitted. So they are:
Of course, you probably do not want to use these symbols exactly.
You might decide that !!\langle 1,0,0\rangle!! is a sticker with a
picture of a fish, and !!\langle 0,1,0\rangle!! is a sticker with a
ladybug.
Each card will have !!n+1 = 3!! stickers. To generate a card, pick
any two stickers that haven't appeared together before and put them
on the card. Say these stickers correspond to the triples !!\langle
a,b,c\rangle!! and !!\langle x,y,z\rangle!!. To find the triple for
the third sticker on the card, just add the first two triples
componentwise, obtaining !!\langle a+x,b+y,c+z\rangle!!. Remember
that the addition must be done according to the !!GF(2)!! addition table
above! So for example if a card has !!\langle 1,0,1\rangle!! and
!!\langle 0,1,1\rangle!!, its
third triple will be
Observe that it doesn't matter which two triples you add; you always get the third one!
Okay, well, that was simple.
Larger order
After Katara did the order 2 case, which has 7 cards, each with 3 of
the 7 kinds of stickers, she was ready to move on to something
bigger. I had already done the order 3 deck so she decided to do
order 4. This has !!4^2+4+1 = 21!! cards each with 5 of the 21 kinds
of stickers. The arithmetic is more complicated too; it's !!GF(2^2)!! instead
of !!GF(2)!!:
+
0
1
2
3
0
0
1
2
3
1
1
0
3
2
2
2
3
0
1
3
3
2
1
0
×
0
1
2
3
0
0
0
0
0
1
0
1
2
3
2
0
2
3
1
3
0
3
1
2
When the order !!n!! is larger than 2, there is another wrinkle.
There are !!4^3 = 64!! possible triples, and we are throwing away
!!\langle 0,0,0\rangle!! as usual, so we have 63. But we need
!!4^2+4+1 = 21!!, not !!63!!.
Each sticker is represented not by one triple, but by three. The
triples !!\langle a,b,c\rangle, \langle 2a,2b,2c\rangle,!! and
!!\langle 3a,3b,3c\rangle!! must be understood to represent the same
sticker, all the multiplications being done according to the table
above. Then each group of three triples corresponds to a sticker, and
we have 21 as we wanted.
Each triple must have a leftmost non-zero entry, and in each group of
three similar triples, there will be one where this leftmost non-zero
entry is a !!1!!; we will take this as the canonical representative of
its class, and it can wear a costume or a disguise that makes it
appear to begin with a !!2!! or a !!3!!.
We can stop there, because everything after
!!\langle 1,3,3 \rangle!!
begins with a !!2!! or a !!3!!, and so is some other triple in
disguise. For example
what sticker goes with
!!\langle 0,2,3 \rangle!!? That's actually
!!\langle 0,1,2 \rangle!! in disguise, it's
!!2·\langle 0,1,2 \rangle!!, which is “dice”. Okay, how about !!\langle 3,3,1
\rangle!!? That's the same as !!3\cdot\langle 1,1,2 \rangle!!,
which is “ladybug”.
There are !!21!!, as we wanted. Note that the !!21!! naturally
breaks down as !!1+4+4^2!!, depending on how many zeroes are at the
beginning;
that's where that comes from.
Now, just like before, to make a card, we pick two triples that have
not yet gone together, say !!\langle 0,0,1 \rangle!! and !!\langle
0,1,0 \rangle!!. We start adding these together as before, obtaining
!!\langle 0,1,1 \rangle!!. But we must also add together the
disguised versions of these triples, !!\langle 0,0,2 \rangle!! and
!!\langle 0,0,3 \rangle!! for the first, and !!\langle 0,2,0 \rangle!!
and !! \langle 0,3,0 \rangle!! for the second. This gets us two
additional sums, !!\langle 0,2,3 \rangle!!, which is !!\langle 0,1,2
\rangle!! in disguise, and !!\langle 0,3,2 \rangle!!, which is
!!\langle 0,1,3 \rangle!! in disguise.
It might seem like it also gets us !!\langle 0,2,2 \rangle!! and
!!\langle 0,3,3 \rangle!!, but these are just !!\langle 0,1,1
\rangle!! again, in disguise. Since there are three disguises for
!!\langle 0,0,1 \rangle!! and three for !!\langle 0,1,0 \rangle!!, we
have nine possible sums, but it turns out the the nine sums are only
three different triples, each in three different disguises. So our
nine sums get us three additional triples, and, including the two we
started with, that makes five, which is exactly how many we need for
the first card. The first card gets the stickers for triples
!!\langle 0,0,1 \rangle,
\langle 0,1,0 \rangle
\langle 0,1,1 \rangle
\langle 0,1,2 \rangle,!! and
!!\langle 0,1,3 \rangle,!! which are apple, bicycle,
carrot, dice, and elephant.
That was anticlimactic. Let's do one more. We don't have a card yet
with ladybug and trombone. These are !!\langle 1,1,2 \rangle!! and
!!\langle 1,3,2 \rangle!!, and we must add them together, and also the
disguised versions:
These nine results do indeed pick out three triples in three disguises
each, and it's easy to select the three of these that are canonical:
they have a 1 in the leftmost nonzero position, so the three sums are
!!\langle 0,1,0 \rangle,!! !!\langle 1,0,2 \rangle,!! and !!\langle
1,2,2 \rangle!!, which are bicycle, hat, and piano. So the one card
that has a ladybug and a trombone also has a bicycle, a hat, and a
piano, which should not seem obvious. Note that this card does have
the required single overlap with the other card we constructed: both
have bicycles.
Well, that was fun. Katara did hers with colored dots instead of
stickers:
The ABCDE card is in the upper left; the
bicycle-hat-ladybug-piano-trombone one is the second row from the
bottom, second column from the left. The colors look bad in this
picture; the light is too yellow and so all the blues and purples look
black.x
After I took this picture, we checked these cards and found a couple
of calculation errors, which we corrected. A correct set of cards is:
This construction always turns a finite field of order !!n!! into a
finite projective plane of order !!n!!.
A finite field of order !!n!! exists exactly when !!n!! is a prime
power and then there is exactly one finite
field. So this construction gives us finite projective planes of
orders !!1,2,3,4,5,7,8,9,11,13,16!!, but not of orders
!!6,10,12,14,15!!. Do finite projective planes of those latter
orders exist?
Is this method the only way to construct a finite projective plane? Yes,
when !!n<9!!. But there are four non-isomorphic projective planes of
order !!9!!, and this only constructs one of them.
Last month I regretted making only 22 posts but I promised:
April will be better; I'm on pace to break the previous volume record,
and I've also been doing a good job of promoting better posts to the
major leagues.
I blew it! I tied the previous volume record. But I also think I
did do a decent job promoting the better posts. Usually I look over
the previous month's posts and pick out two or three that seem to be
of more interest than the others. Not this month! They are all
shit, except the one
ghostwritten by Anette Gordon-Reed. If this keeps up, I will stop
doing these monthly roundup posts.
Andrew Rodland and Adam Vartanian explained ramp
metering. Here's M. Rodland's
explanation:
ramp metering is the practice of installing signals on freeway
onramps that only allow one car every few seconds, so that cars
enter the freeway at evenly-spaced intervals instead of in bunches
and don't cause as many problems merging.
He added that it was widely used in California. McCain is
headquartered in California, and mentions frequently on their web
site that their equipment conforms to Caltrans standards.
M. Vartanian and Richard Soderberg also suggested an explanation for
why the traffic control system might also control sprinklers and
pumps. M. Soderberg says:
DOTs in California and presumably elsewhere often have a need for
erosion control on the steep inclines of earth surrounding their
highway ramps. So any time you see a 45-degree incline covered in
greenery, chances are it has a
sprinkler system attached and carefully maintained by the DOT. Those
same sprinklers are often within a few feet of the ramp's metering
lights…
That makes perfect sense! I had been imagining fire sprinklers, and
then I was puzzled: why would you need fire sprinklers at an
intersection?
Hogs are kept in poor and inhumane conditions (often true, but
their accommodations must still be much more expensive than the
flies’)
Hog farmers are exempted from paying for the negative
externalities of their occupation such as environmental
degradation and antibiotic resistance (often true, but the fly
farmers cannot be paying that much to offset externalities)
Slaughterhouse waste and rotten fruit are more expensive than the
corn and soy on which hogs are fed (I think slaughterhouse waste
and waste fruit are available essentially free to anyone who wants
to haul them away)
The drying process is difficult and expensive (but the listed
price for undried maggots is twice as high)
But I find Marcel Fourné's suggestion much more plausible: the pork
price is artificially depressed by enormous government subsidies.
I started looking into the numbers on this, and got completely
sidetracked on something only peripherally related:
According to the USDA Census of Agriculture for
2012,
in 2012 U.S. farms reported an inventory of 66 million pigs and hogs,
and sales of 193 million pigs and hogs. (Table 20, page 22.)
When I first saw this, I thought I must be misunderstanding the
numbers. I said to myself:
!!\frac{193}{66}\approx 3!!, so the inventory must be turning over
three times a year. But that means that the average hog goes to
market when it is four months old. That can't be right.
Of course it isn't right, it isn't even close, it's complete
nonsense. I wrote up my mistake but did not publish it, and while I
was doing that I forgot to finish working on the subsidy numbers.
James Kushner directed my attention to the MUTCD news
feed and in particular
this amusing item:
the FHWA issued Official Interpretation
4(09)-64
to clarify that the flash rate for traffic control signals and
beacons is a single repetitive flash rate of approximately once
per second, and that a combination of faster and slower flash
rates that result in 50 to 60 flashes per minute is not compliant…
James writes:
I imagined a beacon that flashed once every ten seconds; after
five such iterations, there was one iteration where the beacon
would flash fifty times in a second. "But it flashes 55 times
every minute, so, you know, it, uh, conforms to the standard..."
But the Official Interpretation also says
You asked whether the FHWA would be willing to consider
experimentation with alternative flash rates for warning
beacons. Any requests for experimentation would be evaluated on
their merits and would be addressed separately from this official
ruling.
so there is still hope for James’ scheme.
Two readers suggested musical jokes.
Jordan Fultz asks:
Q: How does Lady Gaga like her steak?
A: Raw, raw, raw-raw-raw!
Last Monday some folks were working on this thing on Walnut Street. I
didn't remember having seen the inside of one before, so I took some
pictures of it to look at later.
Thanks to the Wonders of the Internet, it didn't take long to figure
out what it is for. It is a controller for the traffic lights at the
intersection.
The descriptions of the controllers are written in a dense traffic
control jargon that I find fascinating but opaque. For example, the
170 controller's product
description
reads:
The McCain 170E, 170E HC11, and 170 ATC HC11 controllers’
primary design function is to operate eight-phase dual ring
intersections. Based on the software control package utilized,
the 170’s control applications can expand to include: ramp
metering, variable message signs, sprinklers, pumps, and
changeable lane control.
I think I understand what variable message signs are, and I can guess
at changeable lane control, but what are the sprinklers and pumps for?
What is ramp metering?
The eight-phase dual ring intersection, which I had never heard of
before, is an important topic in the traffic control world. I gather
that it is a four-way intersection with a four-way traffic light that
also has a left-turn-only green arrow portion. The eight “phases”
refer to different traffic paths through the intersections that must
be separately controlled: even numbers for the four paths through the
intersection, and odd numbers 1,3,5,7 for the left-turn-only paths
that do not pass through. Some phases conflict; for example phase 5
(left-turning in some direction, say from south to east), conflicts
with phase 6 (through-passing heading in the opposite direction) but
not with phase 1 (left-turning from north to west).
There's plenty of detailed information about this available.
For example, the U.S. Federal Highway Administration publishes their
Traffic Signal Timing
Manual.
(Published in 2008, it has since been superseded.) Unfortunately,
this seems to be too advanced for me! Section 4.2.1 (“Definitions and
Terminology”) is the first place in the document that mentions the
dual-ring layout, and it does so without explanation — apparently this
is so elementary that anyone reading the Traffic Signal Timing Manual
will already be familiar with it:
Over the years, the description of the “individual movements” of the
dual-ring 8-movement controller as “phases” has blurred into common
communicated terminology of “movement number” being synonymous as
“phase number”.
But these helpful
notes explain
in more detail: a “ring” is “a sequence of phases that are not
compatible and that must time sequentially”.
Then we measure the demand for each phase, and there is an interesting
and complex design problem: how long should each phase last to
optimize traffic flow through the intersection for safety and
efficiency? See chapter
3a
for more details of how this is done.
I love when I discover there is an entire technical domain that I
never even suspected existed. If you like this kind of thing, you may
enjoy geeking out over the Manual of Uniform Traffic Control
Devices,
which explains what traffic signs should look like and what each one
means. Have you ever noticed that the green guide signs on the
highway have up-pointing and down-pointing arrows that are totally
different shapes?
That's because they have different meanings: the up-pointing arrows
mean “go this way” and the down-pointing arrows mean “use this lane”.
The MUTCD says what the arrows should look like, how big they should
be, and when each one should be used.
The MUTCD is the source of one of my favorite quotations:
Regulatory and warning signs should be used conservatively because
these signs, if used to excess, tend to lose their effectiveness.
Words to live by! Programmers in particular should keep this in mind
when designing error messages. You could spend your life studying
this 864-page manual, and I think some people do.
Related geekery: Geometric highway
design: how
sharply can the Interstate curve and still be safe, and how much do
the curves need to be banked? How do you design an interchange
between two major highways? How about a highway exit?
Here's a
highway off-ramp, exit 346A on Pennsylvania I-76 West:
Did you know that the long pointy triangle thing is called a “gore”?
What happens if you can't make up your mind whether to stay on the
highway or take the exit, you drive over the gore, and then smack into
the thing beyond it where the roads divides? Well, you might survive,
because there is a thing there that is designed to crush when you hit
it. It might be a QuadGuard Elite Crash Cushion
System,
manufactured by Energy Absorption Systems,
Inc..
In August 2011, on a particular famous discussion
forum (brought up on this blog again
and again) an individual A, notorious for such acts, posts a
quasi-philosophical
inquiry, incurring
unpopularity, antagonism, and many bad marks, although also a
surprising quantity of rational discussion, including a thoughtful
solution or two.
Many months forward, a distinct party B puts up a substantial
bounty on this inquiry, saying:
I would like a complete
answer to this question which does not use the letter "e" at any
point.
(My apology for any anguish you may go through at this point in my
story on account of this quotation and its obvious and blatant faults.
My wrongdoing was involuntary, but I had no way to avoid it and still
maintain full accuracy.)
Here is a list of March's
shitposts. I don't recall what
my excuse was for there being only 22, but in my defense, I will add
that they were almost all terrible. There was one decent math post I
maybe should have promoted.
(And also Nancy and Squid, which was
awesome, and also 100% Grade A shitpost. I thought when I posted it a
crowd of people would burst into the room and carry me off on their
shoulders. Instead, nobody seems to have noticed.)
April will be better; I'm on pace to break the previous volume record,
and I've also been doing a good job of promoting better posts to the
major leagues.
There have been recurring news stories about the use of dried maggots
as protein supplement in animal feed. For example Insects could feed
the animals of tomorrow’s meat
industry
(maggots fed on slaughterhouse waste, particularly blood) or
Insect farms gear up to feed soaring global protein
demand
(maggots fed on rotten fruit). Then they dry the larvae and
either use them whole or grind them into meal. In particular the fly
meal can be used as a replacement for fish meal, which is ground dried
fish that is used as feed for fish in fish farms. (Yep, we grind up
fish we don't like, to feed to other, better fish.)
I was referred to that second article by
Metafilter
and The Google, in its infinite wisdom, decided to show me an
advertisement for dried fly larvae. The ad was for
NaturesPeck, which sells bagged fly larvae
and fly meal for use as poultry feed or wild bird feed. They have a
special “value pack” that contains 16 pounds of dried larvae for $88.
That is $5.50 a pound! Holy cow, WTF? How can that even be possible
when my local grocery store is selling boneless center cut pork chops
for $2.50 per pound?
Okay, I thought maybe NaturesPeck was some sort of boutique operation,
charging a high markup for small quantities, maybe they claim to have
sustainably-harvested fly meal from free-range organically-fed flies
or something. So I went looking for an industrial wholesaler of bulk
fly meal and quickly found Haocheng Mealworms
Inc. in Xiangtan,
China. This is definitely
what I was looking for; they will be glad to sell you a standard
40-foot shipping container full of dried maggots or other larvae. The
quoted price for dried mealworm larvae is $8400 per metric tonne, plus
shipping ($170–200 per tonne).
Prices, converted to U.S. dollars per pound, are as follows:
So it wasn't just that NaturesPeck was marking them up. Even the
least expensive product costs as much as retail pork chops.
I don't get it. There must be some important aspect of this that I am
missing, because a market failure of this magnitude is impossible.
BTW, Haocheng recommends that:
As a source of high protein additive, put [mealworms] into the
bread, flour, instant noodle, pastry, biscuit, candy, condiment, and
direct into the dishes on the dining table like the foodstuff, and
process into the health care nourishment to improve the human body
immunity ability.
[ Warning: this article is mathematically uninteresting. ]
I woke up in the middle of the night last night and while I was
waiting to go back to sleep, I browsed math Stack Exchange. At four in
the morning I am not at my best, but sometimes I can learn something
and sometimes I can even contribute.
The question that grabbed my attention this time was Arithmetic
sequence where every term is
prime?. OP wants to
know if the arithmetic sequence $$d\mapsto a+d b$$
contains composite elements for every fixed positive integers !!a,b!!.
Now of course the answer is yes, or the counterexample would give us a
quick and simple method for constructing prime numbers, and finding
such has been an open problem for thousands of years. OP was
certainly aware of this, but had not been able to find a simple proof.
Their searching was confounded by more advanced matters relating to
the Green-Tao theorem and such like, which, being more interesting,
are much more widely discussed.
There are a couple of remarkable things about the answers that were
given. First, even though the problem is easy, the first two answers
posted were actually wrong, and another (quickly deleted) was so
complicated that I couldn't tell if it was right or not.
One user immediately commented:
What happens if !!d=a!!?
which is very much to the point; when !!d=a!! then the element of the
sequence is !!a+ab!! which is necessarily composite…
…unless !!a=1!!. So the comment does not quite take care of the
whole question. A second user posted an answer with this same
omission, and had to correct it later.
I might not have picked up on this case either, during the daytime.
But at 4 AM I was not immediately certain that !!a+ab!! was composite
and I think about it. I factored it to get !!a(1+b)!! and then I saw
that if !!a=1!! or !!1+b=1!! then we lose. (!!1+b=1!! is impossible.
!!a+ab!! might of course be composite even if !!a=1!!, but further
argumentation is needed.)
So I did pick up on this, and gave a complete answer, of which the
important part is:
Just take !!d=kb+k+a!! for any !!k!! whatever. Then the element is
$$kb^2+(k+a)b+a = (kb+a)(b+a)$$ which is composite.
Okay, fine. But OP asked how I came up with that and if it was pure
“insight”, so I thought I'd try to reconstruct how I got there at 4
AM. The problem is simple enough that I think I can remember most of
how I got to the answer.
As I've mentioned before, I am not a pure insight kind of person.
While better mathematicians are flitting swiftly from peak to peak, I
plod along in the dark and gloomy valleys. I did not get !!d=kb+k+a!!
in a brilliant flash of inspiration.
Instead, my thought process, as well as I can remember it, went like this:
“That doesn't work when !!a=1!!. But there must be composite numbers in
sequences of the form !!d b+1!!. I wonder what they look like?”
“I guess I'll try an example. How about !!4d+1!!?”
“Hmm, it starts at 5, 5 is composite… No it isn't.”
“But it's increasing by 4 each time so it must hit each mod-5
residue class in some order.”
I should cut in at this point to add that my thinking was nowhere near
this articulate or even verbal. The thing about the sequence hitting
all the residue classes was more like a feeling in my body, like when
I am recognizing a familiar place. When !!a!! and !!b!! are
relatively prime, that means that when you are taking steps of size
!!b!!, you hit all the !!a!!'s and don't skip any; that's what
relatively prime is all about.
So maybe that counts as “insight”? Or “intuition about relative
primality”? I think that description makes it sound much more
impressive than it really is. I do not want a lot of credit for this.
Maybe a better way to describe it is that I had been in this familiar
place many times before, and I recognized it again.
Anyway I continued something like this:
“If it hits every residue class over and over in sequence it
must hit residue class !!0!! infinitely often. How does that go? It
hits !!5!!, so it hits !!5+4·5 = 25!! also.”
That was good enough for me; I did not even consider the next hit,
!!45!!, perhaps because that number was too big for me to calculate at
that moment.
I didn't use the phrase “residue class” either. That's just my verbal
translation of my 4AM nonverbal thinking. At the time it was more
like: there are some good things to hit and some bad ones, and the
good ones are evenly spaced out, so if we hit each position in the
even spacing-ness we must periodically hit some of the good things.
“Okay, then the first element of !!1+d b!! is !!1+b!!, and then
every !!1+b!! steps after that it hits another multiple of !!1+b!!, so we
want every !!1+b!! elements, so take !!d = 1 + k(1+b)!!.”
Then I posted the answer, saying that when !!a=1!! you take
!!d=1+k(1+b)!! and the sequence element is !!1+b+kb+kb^2 =
(1+b)(1+kb)!!.
Then I realized that I had the same feeling in my body even when
!!a≠1!!, because it only depended on the way the residue classes
repeated, and changing !!a!! doesn't affect that, it just slides
everything left or right by a constant amount. So I went back to edit
the !!1+k(1+b)!! to be !!a+k(1+b)!! instead.
I have no particular conclusion to draw about this.
where we allow an arbitrarily large number of M's on the front, so
that every number has a unique representation. For example the number
10,067 is represented by !!\text{MMMMMMMMMMLXVII}!!.
Now sort the list into alphabetical order.
It is easy to show that it begins with !!\text{C}, \text{CC}, \text{CCC}, \text{CCCD},
\ldots!! and ends !!\text{XXXVII},
\text{XXXVIII}!!. But it's still an infinite list! Instead of
being infinite at one end or the other, or even both, like most infinite
lists, it's infinite in the middle.
Of course once you have the idea it's easy to think of more examples
(!!\left\{ \frac1n\mid n\in\Bbb Z, n\ne 0\right\}!! for instance) but
I hadn't seen anything like this before and I was quite pleased.
