Isc (silently): That... that sounds like Jacob.
Θ: It is Jacob.
Isc: Why is Jacob pretending to be Esau?
Θ: Rebekah's got him all worried up about how our blessing Esau would turn out for him.
Isc: Good Lord, I smell Esau's shirt.
Θ: Yes, and that's not all!
Isc: ... goatskin? Does he really think I can't tell the difference between my son's hands and my goats? I thank You that it isn't raining.
Θ: Indeed, he really thinks to have drawn the wool over your eyes, as it were.
Isc: So, what do I do now?
Θ: You will have to recite the blessing from the other side. It is too bad that neither of them will understand until... well, I still have my plans for them.
Monday, January 14, 2013
Thursday, January 10, 2013
Sounds like a bit much?
Dr. Thursday, that flower of a papist and programmer, writes
I can only imagine that something of the project involved an automation of the code-writing process; that would not only make the typing go swifter, it would also make the proof-checking simpler for, if you are careful, you can be sure that your helper-program only outputs correct code! Now, the Doctor assures us that his programs are doing something... something for him, though I'm quite ignorant as to what, except that one of them I'm pretty sure is writing the last of them.
Now, I don't really know why Dr. Thursday insists on having such a huge program to do I-don't-know-what, because there's a much shorter program that will do everything. Yes, everything. Well, given infinite space to work in, it will eventually do everything that can be done by an ordinary computer in finite time. It's called algorithmic search and takes about $n^2 \log(n)$ steps to run the first $n$ programs about $n$ steps each, unless they stop first. Now, admittedly, this is something of a blunt-force weapon, much like proving Wiles' Theorem That Settled Fermat's Last Puzzle by listing all the proofs possible. Now that we know it has a proof, we know you'll get to a proof eventually, but whether that happens before the next big crunch I don't know.
But, there it is.
Wow.So I did a silly little test: for twenty-seven seconds I typed out as quickly as I could an easy-to-think-of sequence, "one two three four..." and it came to 110 bytes, including some errors. At such a rate, without sleeping eating or other dull interruptions, it would take about two years, I had my machine tell me, to produce a quarter of a gigabyte. I actually don't think I could tolerate more than a couple hours of just typing on just one thing in any given day, so that forty years is a more reasonable estimate.
I have just written the LARGEST piece of source code I have ever done in my entire life. The source code measures about a quarter of a gigabyte, and yes, that does include comments. In fact, one routine probably exceeds all the source code I have ever written, in my BS, my MS, my PhD, and over 30 years of industrial software development.
And even nicer, it is provably correct... the acedemics would drool at that.
I can only imagine that something of the project involved an automation of the code-writing process; that would not only make the typing go swifter, it would also make the proof-checking simpler for, if you are careful, you can be sure that your helper-program only outputs correct code! Now, the Doctor assures us that his programs are doing something... something for him, though I'm quite ignorant as to what, except that one of them I'm pretty sure is writing the last of them.
Now, I don't really know why Dr. Thursday insists on having such a huge program to do I-don't-know-what, because there's a much shorter program that will do everything. Yes, everything. Well, given infinite space to work in, it will eventually do everything that can be done by an ordinary computer in finite time. It's called algorithmic search and takes about $n^2 \log(n)$ steps to run the first $n$ programs about $n$ steps each, unless they stop first. Now, admittedly, this is something of a blunt-force weapon, much like proving Wiles' Theorem That Settled Fermat's Last Puzzle by listing all the proofs possible. Now that we know it has a proof, we know you'll get to a proof eventually, but whether that happens before the next big crunch I don't know.
But, there it is.
Sunday, January 6, 2013
Oh, Hello!
Hello, blog!
Hello, post editor;
Hello, comment form;
Hello, archive;
Hello, saved drafts (not today, though, not today...);
Hello,
Hello, included maths, as
$$ \mathrm{Hom}(\mathrm{Ind}_H^G R, S) \simeq \mathrm{Hom}(R,\mathrm{Res}^G_H S) \tag{Frobenius} $$
Hello, names of regular readers, and the odd passers-by;
Hello, actual readers (because it's personal!);
Hello, New Year and Epiphany!
Who knows what'll happen? God alone. Bless you, all.
Hello, post editor;
Hello, comment form;
Hello, archive;
Hello, saved drafts (not today, though, not today...);
Hello,
King John was not a good man... blockquote;
He had his little ways...
Hello, included maths, as
$$ \mathrm{Hom}(\mathrm{Ind}_H^G R, S) \simeq \mathrm{Hom}(R,\mathrm{Res}^G_H S) \tag{Frobenius} $$
for instance;
Hello, names of regular readers, and the odd passers-by;
Hello, actual readers (because it's personal!);
Hello, New Year and Epiphany!
Who knows what'll happen? God alone. Bless you, all.
Sunday, December 16, 2012
Some Rejoiceables
The U-Bend is re-organizing their whole public interface, with the (unintended?) effect that it's hard to tell what will and won't work, so I'll just link to this, one of the best performance captures of one of my favourites. Oh, and Gaudete!
Thursday, December 13, 2012
It did rain pennies from heaven
... and might again.
Reported here; in my own abstract: any metals denser than basalts and that can be mined fell from solar orbit onto the Earth long after its crust formed and mostly settled-down. And remember that there's still plenty of stuff out there!
I thought this might interest some of my theologically-minded fellow web-loggers.
the triviafer
Reported here; in my own abstract: any metals denser than basalts and that can be mined fell from solar orbit onto the Earth long after its crust formed and mostly settled-down. And remember that there's still plenty of stuff out there!
I thought this might interest some of my theologically-minded fellow web-loggers.
the triviafer
Saturday, December 8, 2012
Poco a poco diminuendo...
(Incidentally, if things seem to be slowing down, I suppose I might blame my recent foray into tumblng, but as that's been less than a week it doesn't really answer. No, I shan't post a link here. Send a message to /dev/null ... not your own, of course, but to the address on that pseudonym's profile page. But it's teribly mathy, so... so I don't know what.)
Update Whee! it works nicely now! Thanks, screencountry!/Update
Since everyone now agrees that the particular warranty wasn't worth the electrons it jiggled and juggled, I don't mind telling everyone (anonymously, at least) that my shiny laptop screen became cracked after a thoroughly-preventable fall of about four feet onto concrete, and that I really should have thought to zip up the carrying bag, but on the other hand I don't see the practical use of a clamshell design like that. Perhaps I shall add gores later. Anyways, the endpoint-purveyor offered to charge me more than 33% more than the original price for it to replace the screen, which offer I declined with some irritation. This was after the manufacturer-designated service centre took four days to notice it had been sent to them (though they were expecting it) and a further week to decide that no, they wouldn't do anything helpful, not even offer to charge me more than &c. So, it turns out there's a friendly business with warehouses in Seattle and in Vancouver that sell ... surplus? excess? anyways, replacement lcd screens. If the thing I'm expecting next wednesday works, I might just tell you all about it. But really I mustn't drop the thing again, that was the worst, and this sort of rigor mallearum is expensive, and would be again even knowing the short-cut at the end. It's a nifty thing, but the screen was the only broken thing I found, after the drop, so things might have gone much worse. I didn't even stub a toe!
Well, your patience during this little ramble has been most appreciated, believe you me.
God bless you all!
Update Whee! it works nicely now! Thanks, screencountry!/Update
Since everyone now agrees that the particular warranty wasn't worth the electrons it jiggled and juggled, I don't mind telling everyone (anonymously, at least) that my shiny laptop screen became cracked after a thoroughly-preventable fall of about four feet onto concrete, and that I really should have thought to zip up the carrying bag, but on the other hand I don't see the practical use of a clamshell design like that. Perhaps I shall add gores later. Anyways, the endpoint-purveyor offered to charge me more than 33% more than the original price for it to replace the screen, which offer I declined with some irritation. This was after the manufacturer-designated service centre took four days to notice it had been sent to them (though they were expecting it) and a further week to decide that no, they wouldn't do anything helpful, not even offer to charge me more than &c. So, it turns out there's a friendly business with warehouses in Seattle and in Vancouver that sell ... surplus? excess? anyways, replacement lcd screens. If the thing I'm expecting next wednesday works, I might just tell you all about it. But really I mustn't drop the thing again, that was the worst, and this sort of rigor mallearum is expensive, and would be again even knowing the short-cut at the end. It's a nifty thing, but the screen was the only broken thing I found, after the drop, so things might have gone much worse. I didn't even stub a toe!
Well, your patience during this little ramble has been most appreciated, believe you me.
God bless you all!
Tuesday, November 27, 2012
More topological reflections of arithmetic
For this part of the story I want a really-pretty diagram that even the mathjax still doesn't understand yet how to draw, so I've prepared a dead picture of it.
The doubled-arrows say what kinds of squares they cross --- what sort of equations. The whole thing is basically a fancy elaboration of the idea $ \Phi + X = Z + Y $ and $ \Psi Z = C \Phi $ and $ A Z = X C $ and $ A Y = X B $ and $ B \Phi = Y \Psi $; and it says that if what it means by those are true, then (that's the dotted double arrow) also what it means by $ \Psi + A = C + B $ is true. Note, however, that not all arithmetically true things are topologically true; in arithmetic we have: if $ A B = 1 $ and $ C B = 1 $, then $A = C$; but this isn't true in figures: one would need the right diagram to make it work.
The best known (if not best-known) generality described by the above diagram is known as “Mather's Second Cube Theorem”, and it's at the heart of a lovely little construction I mentioned earlier, which you get with $ \Psi = * $ . There's essentially only one way for a space to “be $ * $”, so these situations are rather special; on the other hand, as long as $ \Phi $ all comes in one piece, there's exactly one way to fit any $A,B,C$ as in the diagram over any compatible $X, Y, Z$. What it meant for the earlier story is that one doesn't need to worry about $ \Phi$ , because it might as well be $ * $. Making that change doesn't even change the ... well, I can't quite say what it didn't change, because I haven't spelled it, here. But when you can simplify the problem without changing the solution, you're making progress!
There are lots more nifty diagrams involved, probably the prettiest being
--- the $\gg$ here is doing what the $\Downarrow$ was above, never mind. Anyways, I was going to write a fun little paper about it, but then I found this fine gem, which I think describes about the same argument, but better! This happens to me a lot, as you might imagine. One day...
Anyway, if anyone wants more, just ask me!
The doubled-arrows say what kinds of squares they cross --- what sort of equations. The whole thing is basically a fancy elaboration of the idea $ \Phi + X = Z + Y $ and $ \Psi Z = C \Phi $ and $ A Z = X C $ and $ A Y = X B $ and $ B \Phi = Y \Psi $; and it says that if what it means by those are true, then (that's the dotted double arrow) also what it means by $ \Psi + A = C + B $ is true. Note, however, that not all arithmetically true things are topologically true; in arithmetic we have: if $ A B = 1 $ and $ C B = 1 $, then $A = C$; but this isn't true in figures: one would need the right diagram to make it work.
The best known (if not best-known) generality described by the above diagram is known as “Mather's Second Cube Theorem”, and it's at the heart of a lovely little construction I mentioned earlier, which you get with $ \Psi = * $ . There's essentially only one way for a space to “be $ * $”, so these situations are rather special; on the other hand, as long as $ \Phi $ all comes in one piece, there's exactly one way to fit any $A,B,C$ as in the diagram over any compatible $X, Y, Z$. What it meant for the earlier story is that one doesn't need to worry about $ \Phi$ , because it might as well be $ * $. Making that change doesn't even change the ... well, I can't quite say what it didn't change, because I haven't spelled it, here. But when you can simplify the problem without changing the solution, you're making progress!
There are lots more nifty diagrams involved, probably the prettiest being
--- the $\gg$ here is doing what the $\Downarrow$ was above, never mind. Anyways, I was going to write a fun little paper about it, but then I found this fine gem, which I think describes about the same argument, but better! This happens to me a lot, as you might imagine. One day...
Anyway, if anyone wants more, just ask me!
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