He will use this knowledge in less than 24 hours.

So back when I was moving in at the start of the school year, I decided to try out my BD player on my hot new Dell monitor. Since I didn’t want to take out my Kara no Kyoukai BDs from their box, I opted to crack open the cheap flimsy Summer Wars case and throw in the disc. I hadn’t actually tried playing it yet anyway. I opted to skip the OZ introduction and go straight to the opening. First of all, I was disappointed that Funimation set the credits in goddamn Arial. Seriously, what? I find it difficult to believe they didn’t have Helvetica already from one of countless projects they could’ve used it on already.

More importantly, Funimation translated the chapter title of the book that Kenji was reading on the train. It turns out he was reading about Shor’s Algorithm. Shor’s Algorithm is the quantum algorithm that can factor integers in polynomial time. I alluded to it when I talked about Summer Wars and cryptography a while back and now’s as good a time as ever to get into it, since, you know, this means Kenji really is a quantum computer and all.

In quantum computing, instead of dealing with bits that are 1s and 0s, we deal with quantum bits, or qubits, that are denoted $|1\rangle$ and $|0\rangle$. These are really just quantum states. Quantum states have interesting properties, the most famous of which is being in superposition. Essentially, what this means is that we can have qubits that are in a state where it represents 1 and 0 simultaneously.

This turns out to be a very neat property to have. If we’ve got $n$ qubits, then we’ve got like $2^n$ inputs simultaneously. That’s great, right? We can compute things exponentially fast now! Well, it turns out it isn’t that simple.

Suppose that we do end up running an algorithm or something on our $2^n$ states. What happens when we want an answer? Well, we measure the state. The problem with that is that on measuring a quantum state, you won’t get all $2^n$ results. You’ll only get one of those results with a certain probability. What’s more is that once you measure a quantum state, you’re stuck with whatever it gave you and you can’t go back and check for another result.

That is obviously no good, because we want not just any answer, but, you know, the right answer. So it looks like approaching our problem from the naive interpretation doesn’t seem to work. And in fact, that immediately tells us that quantum computers aren’t a magic bullet that can help us solve SAT in a reasonable amount of time.

What we can do is manipulate our resultant quantum state so that the actual correct result we want is the one with very, very high probability and reduce the probability of getting any of the other results to near-zero. Of course, the problem is that we don’t know which result is the correct one to begin with. It’s actually this exact hitch that prevents us from efficiently solving general NP-complete problems.

Luckily, if we want to factor some numbers, we can take advantage of this. We can reduce the problem of factoring any old number into prime factors to the problem of order finding, which I’d mentioned before. For an integer $N$, there’s another integer $m$ such that $k^m=1\bmod N$ and we call $m$ the order of $k$ in the integers modulo $N$.

So suppose we have $k\bmod N$ with order $m$ and we decide to keep on taking powers of it, like $k\bmod N,k^2\bmod N, k^3\bmod N$ and so on. Eventually, we’ll get up to $k^m\bmod N$, at which point, that’s just $1$ and we start our cycle over again. The thing to note here is that this sequence is finite and periodic, which narrows down what we have to search through.

So what do we do? We create a quantum state for our period that’s in a superposition over all the powers of $k$ in $\bmod N$. We don’t know what our period is (that’s what we’re trying to find), so we just choose a large enough one that our order will be in the period. Now, in order to find this order, we do some magic with a Quantum Fourier Transform.

What?

So I mentioned at the beginning that when we look at a quantum state in superposition, we can think of it as returning one of the possible states with a certain amount of probability. How the probabilities for these quantum states work is actually more complicated and it’s from this that we can massage an answer out of our magic quantum algorithm.

Quantum states have something called amplitudes, which are kind of like probabilities, except that amplitudes can be basically any complex number as long as all of the amplitudes $a_i$ for a quantum state satisfy $\sum|a_i|^2=1$. If we’re just measuring a quantum state, this really doesn’t change anything, we just get state $|i\rangle$ back with probability $|a_i|^2$. But when we’re manipulating quantum states, there’s more to the amplitudes to consider.

If we think about amplitudes in terms of waves, we can see that we can cancel waves out. For instance, the states $\alpha|1\rangle$ and $-\alpha|1\rangle$ have the same probability if we were to measure them (since $|\alpha|^2=|-\alpha|^2$), but if we put them together and add them, they’d cancel each other out. What’s more is that since amplitudes are complex numbers, so we can think of them as having a direction (like a vector on the complex plane) and we end up with weird stuff like having things partially cancel out.

So just like a regular Fourier transform extracts some information about frequencies from a sequence, we can use the Quantum Fourier Transform to extract some information about all of this amplitude stuff that we can’t see just by measuring the state. And this is exactly how we can manipulate our state so that it increases the probability of the answer we want when we go to measure it and depress the probabilities of all the other answers.

Now after rambling about amplitudes and Fourier transforms and orders, we should probably step back and see what we’ve done in perspective. What we have is a way to find the order $m$ of an element $k \bmod N$. Well, actually, we don’t even have that quite yet. What we get out of all of that quantum stuff is just some information about $m$ that we have to manipulate further, except we just do that with plain old number theory instead of quantum computation. But, what we’ve kind of stumbled through is kind of a vague description of Shor’s algorithm.

Like I mentioned in the first Summer Wars post, this is the only way we know of so far that lets us factor large numbers (especially the ones that are used in cryptography) within a feasible amount of time. Again, Kenji (or at least his brain) is basically a quantum computer. Coincidentally, I played around a bit with what the implications could be if our brains were quantum computers in an earlier post on Kaiba.

But even if Kenji isn’t a quantum computer, he’s goddamn smart for a high school student. The theory behind the computing part of quantum computing is really just a ton of linear algebra, so an understanding of quantum mechanics isn’t really necessary (which is great, because I don’t know any of that crap). The quantum computing course that I took in my undergrad only had second year linear algebra as a formal prerequisite, but that was mainly to allow students from different departments to take it and also implies that you’ve got first and second year math down at the very least.

「魔法少女まどか☆マギカ」/「ゆーげん」

This show was made for me.

I think that should be your sign that you won’t find a paean to the greatness of Madoka and why it is man’s greatest work.

I mean, I was sold on the show already back when it was just “Shinbo makes an original anime”. That’s all it took.

And then it became “Shinbo makes an original magical girl anime” and I’m like, whoa, even better. I mean, I liked Nanoha. More of that? Yes, please.

And then it became “Shinbo makes an original magical girl anime with character designs by Ume Aoki, music by Yuki Kajiura, and script by Urobuchi Gen”. By this point I’d gone nuts. Magical girls! Designed by Ume-tentei! I mean, yeah, Kajiura and Urobuchi were kinda weird choices for a wide face magical girl show, but still, they’re great too. This will be amazing!

And then the characters! The character designs! Aoi Yuuki! Friggin Chiwa Saitou voicing mahou shoujo Senjougahara. Mizuhashi Kaori voicing a magical girl WITH GUNS. KitaEri voicing swords girl. An adorable mascot character!

If the anime that we got was the anime that was presented through only these CMs, I would have been ready to call it anime of the year already.

And then the actual first episode happened. I watched most of the episode going THIS IS AMAZING and then we get to the witch barrier.

FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF

And then Mami comes with her Gate of Babylon and her GUNS.

I mean, you can probably imagine my mental state as I wait for each week of this cartoon. And so this goes on and I flip out each week and once episode 3 rolls around I go, screw this, I’ll just watch the live stream.

Yes, I watched episode 3 raw when it aired.

FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF

And of course, each week while I’m watching this, I just send my friend a series of FFFFFFFFFFFFFFFFFFFFs.

And then we get to episode 10. I don’t know if it was apparent, but my favourite character is Homura, probably largely because Senjougahara is one of my favouritest characters. So yeah, we get to episode 10 and this is the best anime of the year and possibly of all years.

The end.

Yes, I know a lot of people have problems with it and a lot of what people have said about the show is pretty hyperbolic. Maybe there it does strain itself in trying to be deep and all. Maybe the characters weren’t amazing. I will admit that it very likely is not the best anime ever.

But I don’t think there’s been an anime that has been quite so attuned to my brain. And that is why I love it so.

「little world」/「momoc」

Yeah, I made the same joke last year. But who would’ve thought that we’d get two Utena derived shows within a year? Certainly not me.

But yes, Mawaru Penguindrum is a show that has penguins in it.

It’s fascinating how different Penguindrum is from Star Driver but still manage to be so heavily evocative of Utena. Having Penguindrum as a comparison, I realized that there really was no reason to expect that Star Driver would make as many callbacks to Utena as it does. I mean, it’s not like Igarashi and Enokido hadn’t worked on stuff together before.

Like Star Driver, Penguindrum exudes its own sense of style. In Star Driver, a lot of that is constrained to the more fantastical parts of the show, like whenever we’re dealing with Cybodies and the Glittering Crux. There’s no such separation in Penguindrum. Even in the real world, the architecture and shots of the locations are still really surreal. So while in Star Driver, there’s a clear delineation between real and fantasy, we’re often left wondering which is which in Penguindrum.

And the penguins don’t help very much in this department either.

Looking back at Star Driver, there actually isn’t that much weird stuff going on during the school life section of each episode. I mean, yes, weird events happen and all, but what you saw was pretty much what happened. In Utena, weird things where we’re not really sure how to take happen in the school life parts, which is the route that Penguindrum takes.

Structurally, Star Driver resembles Utena much more because of the clearly defined events that take place in each episode. On occasion, we’ll be missing, say, a song or something but for the most part, we know how each episode goes down. Penguindrum is not very repetitive at all, at least not in how each episode goes. Yeah, there’s SEIZON SENRYAKU, but those are really more diversions than a shadow play and then ZETTAI UNMEI MOKUSHIROKU followed by a substantial duel or APPRIVOISE followed by GINGA BISHOUNEN and a substantial duel.

Of course, all three still make use of visual symbols and phrases and keep on repeating them a lot. The obvious messages are pretty obvious because they’re shoved in front of you, but it just sort of piles on to you to the point that there are probably some other messages hidden under this mound of symbols you’ve suddenly got. I mean, every goddamn thing that shows up on the screen or uttered by anyone is probably a symbol. Has anyone dissected the Triple-H/ARB songs yet?

Penguindrum also differs from Utena and Star Driver by not being about a prince who strolls into town and upsets the establishment trying to revolutionize the world. Certainly, we’ve got our shadowy cabal trying to revolutionize the world, but it’s really more about two brothers trying to save their sister, which is much less grand. Of course, the penguins would probably deflate any sense of awe pretty quickly.

Not that I don’t appreciate the penguins. I think they’re an awesome part of the show. It’s an interesting way of adding another layer to all of those symbols we’re buried under, in that they supposedly are reflective of the state of mind of the characters they shadow. More importantly, their antics are just plain enjoyable and the show would be much more depressing without them.

Does the choice of penguins mean anything or does Ikuhara just really, really like them? It is a mystery.

Man, I wish I had a penguin that mirrored my internal thoughts and did errands for me and was invisible to everyone except me.

「うさぎドロップ」/「ゆーた」

Single dad manga? Why not? Yotsuba&!’s great!

I don’t know whether there’s much I can add about Usagi Drop. The whole thing is fairly straightforward. It’s a great story about a guy who ends up having to look after a kid all of a sudden and gets into the ups and downs and details of it. I’m sure we’ve all been told that raising a kid is hard and doubly so if you’re the only doing it. Although I guess Daikichi lucked the hell out because Rin is probably the best kid you could hope for.

Okay, maybe I can say something about the much more contentious second half of the story, in which Rin is no longer an adorable child but is a teenager. The immediate reaction is “NOOOO why can’t Usagi Drop be about Daikichi taking care of little Rin forever?” but you know, it doesn’t work like that. Kids grow up and parents should get a chance to feel proud if their kids turn out to be decent human beings, which Rin is in spades.

But also important is that it has some loose threads that need to be resolved. Does anything end up happening between Rin and her mother? Do Daikichi and Kouki’s mom get together? Or do Rin and Kouki get together? I’d imagine a ton of people were interested in seeing how the last two questions get answered. Which of course leads us to the ending.

I’ve already talked about the ending and I’ve tried to understand where it came from. So a lot of people are pissed at the ending because it’s weird as hell and came out of nowhere. But more importantly, it was basically a punch in the gut for everyone who was waiting on those last two questions because it basically resolved the “problem” (because if you think about it, you can’t really have both happen) in the worst way imaginable.

Really, that’s the only snag of the entire thing and I’ll admit it’s a pretty big one. One of the great things about Usagi Drop is its realism and believability, which this development was not. Of course, this is only a concern if you’re at all interested in the manga. If your only desire is to remain in bliss solely in the realm of a guy tries to raise a kid without knowing how, then stick to the anime and the first half of the manga, which is the most adorable thing.

「Mage Killer」/「三輪」

The first time I read Fate/Zero, I got through about half of the prologue before forgetting about it. That was before I read Fate/stay night.

One of the great things about Fate/Zero is that the particular Holy Grail War it covers is serious business. Instead of having a bunch of high school students kind of flail about, you have some powerful magi all scheming against each other. These dudes know what they’re doing. And actually, that all of these guys basically off each other one way or another kind of explains why we’re left with high school students ten years later.

The main draw of Fate/zero for me has always been that I’d heard it’s particularly brutal and that there was a magus going around ruining everyone with the help of modern technology. After reading Fate/stay night, Kiritsugu’s character became much, much more interesting. As much as people seem to think that Rider is the best (and I’ll admit he is pretty fantastic), Kiritsugu has always been the most compelling character to me.

Those who are familiar with Fate/ will know that it’s all about dealing with ideals, whether it’s defending your ideals, sorting out your ideals, or having your ideals challenged. It’s kind of an easy subject to bring up when fighting for the Holy Grail. Fate/stay night is essentially about Shirou sorting out his ideals. In Fate/zero, we have Kiritsugu, who has the same ideals as Shirou, but ends up choosing a vastly different way of realizing them.

This combined with the high calibre of opponents means that there is a ton of cool stuff that goes down in this story. The fact that these are people prepared for the Grail War and not high school students not only means that their fights are better just because they’re better, but they’re also much better prepared. Calling the Grail War a war in Fate/stay night always seemed a bit silly to me, but in Fate/zero, I think there are enough casualties and heavy weaponry to justify it.

Of course, that’s all after I went through Fate/stay night. So what drove me to try and read the books the first time? Well, I’d just finished playing Saya no Uta at the time. Saya no Uta probably remains one of the most horrifying and disturbing things I’ve read. So obviously, I was looking for more in that vein and found this light novel that the same guy wrote.

On the physical horror front, I think Team Caster’s got it covered. In terms of emotional despair, we’ve got everyone else who’s connected to the events of Fate/stay night. It’s mentioned somewhere in the notes of the first volume that there is basically no happy ending for Fate/zero given what we know plays out in Fate/stay night. Basically, everyone involved bets the house on winning the Grail and ends up ruining their childrens’ lives in the process.

It’s going to be a long three months once the first half of Fate/zero is finished. I guess there’s uncensored Team Caster fun times to look forward to in March, though.

「シュタインズゲート・瓦礫と助手」/「huke」

Even though the track record for anime adaptations of Nitro+ games has been abysmal, I couldn’t help but hope that Steins;Gate would turn out well. After all, it’s the best visual novel ever or something according to Japan. And if it didn’t pan out, I guess I’d just fall back on the visual novel with huke’s pretty art.

And as I was watching it, I thought it was pretty funny and all. I was still trying to figure out why Japan loved this thing so much. It wasn’t until someone mentioned something about 2ch jokes that I think I got it.

So I’d known that Steins;Gate was about time travel, based on the synopsis. What I didn’t know was that the story was basically based on the John Titor story. And it the connection wasn’t superficial either. It wasn’t just, there’s a guy calling himself John Titor from the future on 2ch. A lot of details, like the IBM 5100 and the mechanics of time travel and divergence numbers, are taken straight from the story. You can think of Steins;Gate as one giant John Titor reference.

This is fascinating to me, because even though I’m not intimately familiar with the details of the John Titor story, I’ve been on the Internet long enough to remember this whole thing. It’s kind of weird to see something that was sort of the early-web version of an internet meme be used as the skeleton of a more fleshed-out story. I guess something like Densha Otoko would qualify as well, but I think there’s a lot less you can do with it to make something unique like Steins;Gate.

This whole aspect of Steins;Gate got me because I am a nerd and nerds friggin love references. I mean, look at any Shaft anime. But yes, Steins;Gate has tons of references and where there aren’t any, it’ll just create memes. For the most part, these references are all internet memes. If it’s not part of the John Titor mythos, they’ll be working in a 2ch meme. And then there’s stuff like EL PSY CONGROO and tutturu~ which seems like it was designed to be spammed across the internet.

The characters themselves are exactly the same kinds of people. After all, they encounter John Titor on 2ch. What kind of people read and post on 2ch? Daru and Okarin are really obvious otaku. But Mayuri seems relatively normal, except that she works at a maid cafe in Akiba and buys doujin at Toranoana. Okay. Even the most respectable and well-adjusted of the cast, Kurisu-TINA, is a closet VIPPER.

So I guess what’s special about Steins;Gate isn’t that it’s just a good story about a bunch of friends who fall into some time travel conspiracy. It’s that it’s a good story about a bunch of friends who are just like the people watching it who fall into a time travel conspiracy based on an internet urban legend.

「一面の花」/「k野」

I’d been hearing about how awesome Heartcatch Precure was so I’d always intended to eventually watch it. The art was fantastic and I’d seen some great clips of the fights. The problem is that when I found out about it, it was already really far into the 50 or so episodes. So once I got a ton of time after I graduated and was looking for a show to watch, I figured it was time. After all, I rolled through Cardcaptor Sakura pretty quickly.

I’ve been thinking about whether I’d put Heartcatch at the top of my magical girl list, but I think it ends up coming second to Cardcaptor Sakura. Still, this means I think that Heartcatch is pretty damned good. I think it’s a better magical girl show than Madoka or Nanoha and I friggin love those two shows.

What makes Heartcatch so great is that it’s just good. It has a great cast of characters, the story is wonderful, and the production is stellar. Most importantly, it’s just a magical girl show. It isn’t magical girl with lasers. It isn’t magical girl with zombies. It is what it is and it stands toe to toe with all of those magical girl and x shows, where x is something that makes it more palatable to people who need an excuse for why they’re watching shows for little girls.

And Heartcatch is girly as hell. Yeah, the Cures might be throwing some punches around, but the show is tripping flowers and fashion. You can attempt to reframe Nanoha as scifi and talk about intelligent devices and stuff, but the Cures spray perfume to transform. At this point, Nanoha’s pretty much decked out in mecha gear in Force. Meanwhile, this is a show with tambourines and those spinny things that look way too much like their only function is to sell toys.

There’s also the characters’ boundless optimism and energy. There isn’t anything that’s especially dark about is show, even the villains are kind of goofy. Everyone’s got dreams and everyone’s fighting to protect their dreams and it’s terrible that villains use the uncertainty in people’s hearts against them. Here, the objective isn’t to seal away some rogue magics or befriend the opposition with lasers, but to restore people’s hearts.

The other Precure series are going to have a tough time measuring up to the Heartcatch bar, but I hope they do.

「ガンバスター」/「タガナー」

It’s quite illuminating to watch Gunbuster and recognize things that I’ve seen before scattered throughout so many shows just because they decided to pay an homage to this. The obvious case is Gurren Lagann, where certain little things are straight up ripped out from this show. Another case would be Lucky Star’s Bun Guster track. At many of these points, it was like a light flipped on in my head and I went ohhhhhhhhhhhh.

What’s more impressive is that even though this is one of the oldest shows I’ve watched, it still felt new. It was kind of like when I watched Utena for the first time. In this case, the time dilation stuff added a really interesting dimension to the entire plot. While Gurren Lagann relies on physical size alone to give it that grand sense of scale, here, time and distance are used as well. It’s not on the same scale physically, but it definitely matches in the sort of awe you feel when you watch it.

This all makes Diebuster quite interesting as well. While it’s a “sequel”, it feels very different. There are some callbacks and some of the same themes, but it’s largely doing its own thing. I spent a lot of time trying to figure it out and how it connected to Gunbuster but it just ended up giving me all the answers I was looking for. I don’t think there was the same sort of awe as when I watched Gunbuster, but it was more like ‘wow, that’s pretty neat’.

But comparing these two to Gurren Lagann makes their similarities a lot more apparent. Both are about a girl who look up to a girl who’s very successful as their role model. Both of them eventually surpass their role model and achieve their dreams through hard work and guts. And then both of them save the world together with the mentor they’ve surpassed.

Watching these was kind of exciting because everyone loves Gurren Lagann and the question is whether Gainax will be able to top it. I imagine everyone who watched Gunbuster and Diebuster must’ve thought the same thing, that surely they can’t top THIS. I’m glad that they’ve proven us wrong twice now.

「静雄無双」/「巖本英利/旧PNス・タンリー」

I’d planned to write about Durarara!! during the whole twelve days of anime thing, but I dropped it at the last minute to make room for Utena and Star Driver. But now, you’re going, wait a second, there was no Durarararararara!! this year, how can it be a special moment in anime for this year?

Well, you see, it turns out a chunk of the light novels was decently far along. I went through books four through six, which, if we allow ourselves to dream a bit, would correspond to the second season. I liked the anime, but I think everyone is in agreement that the second half is weaker, although whether it was still good or terrible is up for debate. The nice thing about the three books immediately after is that the focus is not on Anri or Kida, so you won’t have to worry about them being a downer in your anime about crazy and exciting things in the big city.

So it should please everyone that a second season of the anime focuses much, much more heavily on everyone’s favourite really angry guy, Heiwajima Shizuo. The fourth book goes back to the Durarara/Baccano standard of telling a story with about a thousand different threads somehow coming together at the end. The fifth and sixth books form one story in which Shizuo is the focus in the same way that Mikado, Anri, and Kida were the main characters of their respective arcs. We’ll also see some familiar faces. An interesting thing about the fourth book is the return of some characters that you may or may not remember and the introduction of several new ones.

Shizuo was a really great side character before. He shows up, gets trolled by Izaya, and proceeds to destroy things, usually with other things. The focus that books 5 and 6 puts on him is great because it develops him from the guy that can be described by the previous sentence into a legitimate badass. I feel like he’s always been thoughtful and that his only real problem is getting pissed off really easily, so I enjoyed watching him put his talents to use in a more focused way.

The thing that I enjoy about Ryohgo Narita’s stuff is his ability to launch a million seemingly unrelated characters and plot threads and be able to tie them together. The one-volume version of that is very neat to read through, but I have a feeling that there’s some longer term stuff that he’s set up in the early books that will come back in a big way and there are already hints of that at the end of the sixth book. This isn’t unlike the Haruhi series, where seemingly inconsequential details come back later in an earthshattering way, although, here the pieces are bigger, I guess.

Really looking forward to that second season.

「ありがとう放浪息子」/「神田川」

The main problem for the characters is some variation of that, isn’t it?

Anime is full of traps and when you sit back and think about it, it’s kind of like, wow, they’re everywhere, aren’t they? Which, if you think about it, is kind of strange that Hourou Musuko feels so different from everything else. I mean, it’s an anime about traps.

An easy description of Hourou Musuko would be that it’s about gender and sexual identity and it does it realistically, unlike the usual fetish or comedy treatment. But what’s more interesting isn’t just that it examines these things in a realistic and mature manner, but that it starts from square one. These are children that are dealing with these things.

So we have these kids and we’re watching them struggle with it on their own. If they’re lucky, they meet someone to struggle along with. Maybe they’re doing okay for a while. Suddenly, things change, and they’re in middle school and have a whole new slew of bullshit to worry about. There’s never really any rest from these things, there are no answers and the kids just have to keep on growing up and dealing with crap that comes their way.

That the anime chose to focus on the middle school parts was something I found kind of unfortunate. Yeah, it’s really exciting that there’s a ton of relationship drama here, but I thought a nice thing about Hourou Musuko was that it took the time to take a look at the slower parts of the characters’ lives as well. Of course, it worked out that the particular slice that they chose to animate has some sort of closure, so that was nice. But just like how the manga starts long before the anime does, now the manga has gone beyond where the anime ends and everyone’s in high school.

Just like with Usagi Drop, we’ve had our fun times watching these kids grow up, but that’s not going to last forever. I don’t know whether this mangaka typically focuses on long term things, but I’d really like to see where this crew ends up in five, ten, or more years down the road.

「闇の力を秘めし鍵よ」/「hatsuko」

I don’t think it’s that much of a surprise if I tell you that Madoka made me watch Cardcaptor Sakura. Well, that’s not entirely true. A more accurate phrasing of that statement would be, I watched Cardcaptor Sakura in anticipation of Madoka. Yep, I spent a good chunk of the Christmas break just before Madoka aired watching the entirety of Cardcaptor Sakura.

Cardcaptor Sakura seems to be one of those childhood anime that sits alongside the likes of Dragon Ball or Sailor Moon. I actually had no idea it was dubbed in English and aired here until when I was way too old and cool for that crap. More recently, it went from ‘that show that a few of my friends watched that seemed lame’ to ‘that show that people cite as the epitome of its genre’.

CCS is a pretty lengthy show, at about 70 episodes. After I got into the whole magical girl thing, I’d planned to watch it eventually, but never really had the time or motivation to do so (which is fairly common for shows that I plan to watch because I “should”). But with 2011 and Madoka quickly approaching and having finished all of my grad school applications, I convinced myself to finally start.

Up until then, my benchmark for magical girl shows was Nanoha. As far as Sakura and Nanoha go, they’re pretty similar. For a while, before I watched either of them, I’d get them confused sometimes. Obviously, I knew that CCS wasn’t going to have huge explosions or funnels, but it was sort of surprising because I’d forgotten that magic doesn’t necessarily manifest itself as a giant pink beam of death. It turns out magic has a wide array of uses.

Speaking of magic, I was surprised to learn that Sakura’s magical girl outfit was not and that there wasn’t even a default outfit. I don’t even know if she actually ever wears the one that she’s seen wearing in most promotional art in the show. The only constant in her magical girl wardrobe is her staff. But nope, Sakura’s eclectic fashion is thanks to Tomoyo.

And I’m not entirely sure when it happened but the focus of the show seemed to shift slightly from Sakura learning to be a magical girl to include falling in love. She basically spends the entire show tracking down the Clow Cards in some way, but somewhere along the line we also get scenes of her and Syaoran sorting their feelings out. Something that I didn’t expect was for the entire show to basically consist of Sakura dealing with the Clow Cards. But even then, the last thing that Sakura had to do before everything ended wasn’t to save the world, it was to tell Syaoran that she loved him.

This love thing was new to me, especially coming from Nanoha, where at best, we have some best friends forever stuff crossing the line occasionally with undertones. Girls and boys interested in each other? Why I never! But it was interesting because it actually went there instead of dancing around it. And it was quite nice seeing Sakura and Syaoran growing in that way.

I think this show made me realize it’s okay to watch shows made for little girls even if they’re not wrapped in a layer of beam weaponry and sci-fi military trappings.

P.S. despite this post’s title, the best OP is Platinum

「キャスターと龍之介」/「ラク」

It’s that time of year again, in which I brush off a bunch of drafts that I’d hoped to post over the course of the year but didn’t get around to finishing on time. A funny thing happened on the way here, which is that I was expecting to not have enough to post about. The reason is that I would have been posting consistently about all the things I wanted to post about all this time. But as it turns out, well, “@blkmage ha ha haaaaa”.

Look forward to twelve days of quality handcrafted posts about some japtoons.

An introduction to mathematical writing

In chapter 2 of Suugaku Girl, we’re introduced to the third component of the little love triangle that’s forming. Tetra is the underclassman that the main character is tutoring and she’s one of the many people who think they might like math but school eventually beats that silly thought out of them.

Anyone who’s taken a real math class will know that by the end of it, your notes are essentially a giant list of definitions, theorems, and proofs along with an example thrown in once in a while. By a real math class, I mean a math class that’s actually concerned about reasoning about how we get our theorems instead of focusing on how to use them for practical applications. I did my undergrad at a school where math has its own faculty and where there’s a lot of rivalry between the various faculties. Everyone in math often jokes about how an example suffices as a proof for engineers and their ilk.

I checked your textbook for a proof and it said that we’ve done enough examples for it to be plausible. Must have been written by engineers.
— Vanderburg, PMATH 340 (F09)

Here, Tetra falls into the same trap when the main character asks her to define what prime numbers are. So MC-kun has to explain how definitions work. First of all, definitions have to be precise, just like theorems. It’s pretty easy to lose marks for misstating theorems and definitions by leaving off an edge case for 0 or 1 or something silly like that. Of course, that leads Tetra to question why these things need to be so precise and arbitrary.

One of the things I realized about math is that it’s all about trying to do things with the definitions you start out with. You can kind of see that in how the number system is built up. We can start with plain old numbers that we use to count things. And then we can add things to it. Like, what happens if we have less of a thing, how would we represent that? Tada, we’ve got negative numbers. Okay, now what if we have a part of a thing? Now we’ve got fractions. And so on and so forth. We realize that not every number can be written as a fraction and suddenly we’ve defined real numbers.

I’m sure we’ve all wondered why complex numbers work out the way they do. It’s because we’ve defined everything to work out like that. Some guy tried to take the square root of a negative number and found that it didn’t work out very well, so we defined the square root of -1 to be $i$. Now we have this $i$ thing, what can we do with it? Well, we can just start writing crazy things like $3 + i + 4i^2 + 5i^3 + 9i^4 + 2i^5 + 6i^6$, but then we realize that it just becomes $2-2i$. Well, okay that’s kind of neat.

But now we know that all complex numbers can be written as $a+bi$ with $a,b\in\mathbb{R}$. So someone along the line must’ve been like, what would happen if we tried to graph these things? So we treat $a$ as the $x$ component and $b$ as the $y$ component and it turns out we can think of complex numbers as other structures like a vector or just a 2-tuple or something. And suddenly, this gives us a way to compare complex numbers, by taking the length of the vector that they define. And now that we have vectors, we can do some weird geometry stuff with them. We can think of these things as the length of the vector and the angle that they form. And then you can go crazy and talk about roots of unity or what multiplication of complex numbers might mean.

The point is that all of mathematics is built up like this. You start off with some definitions or premises and you go nuts. Of course, you don’t have to use the same definitions as everyone else. The problem with doing that is you might not end up with anything interesting. Like, what if we did something nuts and defined 1 to be a prime number like so many people do when they leave off that condition? Well, not much, we’d probably just rephrase all of our theorems to exclude 1.

On the other hand, sometimes when we play around with definitions, we do end up with something interesting. For instance, we can define something called the extended complex numbers, which is just the set $\mathbb{C}\cup\{\infty\}$. Yep, we just say okay, infinity is a number now, deal with it. So what can we do now?

Well, we can divide by 0 now.

I imagine there might be a few people who might flip out at this notion, but yes, since $\infty$ is included in our number system, we can define $\frac{1}{0}=\infty$. Of course, we can’t do everything — $0\times\infty$ and $\infty-\infty$ still don’t mean anything. But if you’ve been paying attention, you might be going, okay well, we can divide by 0, but what else can we do? Dividing by 0 is kinda meaningless if there’s nothing new we can do.

As it turns out, the extended complex numbers defines a very different geometric object. If we remember from the example of the complex numbers, we can basically define every complex number as a vector over a two-dimensional plane. Here, with the extended complex numbers, we can define every number as a line passing through something called the Riemann sphere. And like the complex plane, this sphere lets us create weird relationships between numbers and angles and stuff. This turns out to have interesting properties in complex analysis and quantum mechanics.

So yes, definitions are often arbitrary. Why? Because it’s just useful and interesting that way. You could argue that it’s because nature forces us to define things a certain way. Kind of like, of course you can’t take a square root of a negative number, you just can’t do it! What happens, though, is that we always seem to end up finding useful things that line up with our mathematics rather than inventing our mathematics to do useful things with. After all, mathematicians have been playing around with imaginary numbers for at least a few decades before electromagnetism was even discovered.

「希望」/「niuya」

I’d been rewatching Kaiba with a bunch of people a while ago. Kaiba’s a great show to watch if you want to talk about things, especially if you like dealing with the sorts of moral and ethical issues that come about in a world that’s so starkly different from ours. The obvious things to talk about revolve around identity and existence. Being the computer scientist and the guy who likes to make laboured analogies to things I like, I began to think about memories and cognition from a CS point of view.

Kaiba’s already done a lot of the “what if” in exploring the idea of memories that are as easy to work with as data on today’s computers. There was an observation in one of the episodes that it seemed like one of the characters’ had too many memories and seemed to be unable to create anymore since there was no more room to store them. The obvious answer to me seemed to be just to throw in another disk or something. Of course, another thing we can do, since we can manipulate memories like data, is to selectively choose less important things to throw onto that extra “disk”. In a way, that’s kind of like what we do already, except instead of having a machine suck out our memories, we do things like write them down or take pictures of them. And in the case of photos, it kind of is like having a machine suck a memory out. In either case, that relieves us of the burden of having to keep it in our brain.

In light of this, it’s kind of a good thing that our brain cells do decay eventually and we do forget things. Otherwise, without a way to manipulate our memories like they do in Kaiba, we’d actually run out of room. Usually, we like to think of data or information as this ethereal sort of thing, but realistically, it’s constrained by the physical world. Those bits have to be written down somewhere, whether it’s on a magnetic platter on a hard disk or in a cell in our brains or in a photon in a quantum computer or in a weird capsule like it is in Kaiba.

Recently, someone linked to something about solving NP-complete problems in polynomial time by taking advantage of the many-worlds interpretation of quantum mechanics. This lead me to some of Scott Aaronson’s papers. Scott Aaronson is a complexity theorist who does quantum stuff at MIT and writes a neat blog. I read through the lecture notes for this course he did while he was a postdoc at Waterloo called Quantum Computing Since Democritus and a paper called Why Philosophers Should Care About Computational Complexity. These two things attempt to tie computational complexity and quantum computing with philosophy (duh). The Democritus stuff is pretty accessible if you have a bit of mathematical background and it introduces concepts like computability, complexity, and quantum computation if you’re not familiar with them. The complexity for philosophers paper assumes you’ve done some computational complexity.

So in Kaiba, we can send our minds around and stick them in other bodies or even in machines. But if we can stick them in machines, what’s running it? We’re not sure, but if we can build machines to interface with our minds or run those processes, then we’ve got a computer that can simulate our brains. This is where computational complexity comes into play.

Aaronson notes that a lot of people who think they consider AI a metaphysical impossibility really haven’t thought it through that far and are only considering the practical impossibility. For instance, what can we extract from the fact that Kaiba has computers that can simulate brains? One thing is that simulating a brain can be done efficiently, presumably in polynomial time.

See, in computer science, a lot of things (like simulating a brain) are possible if you just hardcode everything into a lookup table. Luckily, like I mentioned before, there’s a finite amount of stuff we can hold in our heads. So yeah, you could simulate a brain like that and it’ll work because the table’s finite, but you’d probably run out of atoms in the universe to hold all of that data and the possible choices that would branch out that the computer could make as it went along. Of course, that’s only a practical concern.

Which is why Kaiba’s brain simulating must be efficient, otherwise it wouldn’t be physically feasible. This would seem to imply that simulating a brain didn’t turn out to be that hard. Of course, this means that we’ve got a way to formally describe and model the way brains work too. Anyhow, this is all to ask, if we did have a way of doing this, would a computer simulating a person be that person? And then, we can go a bit further and ask whether a person that we’ve fabricated the data for and is simulated by the computer also constitutes a person.

It’s kind of the same question as whether clones would be people, isn’t it? Except instead of creating perfect copies of the entire person, we can just create perfect copies of the person’s memories and throw it into a computer. Of course, this leads us to the implication that brains are deterministic. Otherwise, this entire memory transfer thing wouldn’t work. You wouldn’t be able to designate a person as their memories if how they behaved changed with the brain that they occupied.

We do have some indication that the brains seem to function differently based on the body that they’re controlling. When Kaiba is in Chroniko’s body and meets the girl who’s in the huge guy’s body, they mention something about getting used to the body. It does make sense that biological processes are governed by the brain and isn’t tied to the person’s identity, but then we see how the body interacts with the person when they start to have ~feelings~ for each other. Of course we can consider the body as another part of the input into the brain and then that discrepancy goes away.

So the thing is that in Kaiba, we’ve got a way of creating something that’s literally indistinguishable from a person. Well, okay, we did start with the premise that it’s possible to simulate a person’s brain, which is kind of a stretch in the real world. But now we ask, what properties would human consciousness need to have in order for a computer to be unable to efficiently simulate it?

It turns out there are some possible ways for this to happen. For instance, if we could solve NP-complete problems or do quantum computations, then a classical computer wouldn’t be able to simulate our brains. Of course, that’s assuming P≠NP, so if it turns out that P=NP, then we might have to rethink this condition, and a quantum computer will be able to do quantum computations, so if we ever built one of those, then that also reduces our options. And of course, the bigger problem is that it assumes we can solve NP-complete problems or do quantum computations in our head.

On a slight tangent, this idea of quantum computations in our head leads to the question of what happens if our brains stored quantum information. That would make it very difficult to make perfect copies of our memories because of the no-cloning theorem in quantum information. The theorem is exactly what it sounds like: it’s impossible to make a perfect copy of an unknown quantum state. What this would imply is that if you tried to make a copy of yourself, it would not be 100% identical to you. Admittedly, it’s pretty unlikely that our brains can store any quantum information.

To close, something that’s slightly related, since it happened around the same time as I was watching, was a department seminar I went to where the guy was talking about neuroscience and brain imaging and tying it into graph theory. Essentially, what he did was he took a bunch of MRI scans of someone’s brain and created some graphs on them somehow. He then ran a bunch of algorithms on the graphs for a bunch of metrics to see if graph theory were any use for neuroscience purposes.

What he found was that the brain activity that could be seen from the scans translated to changes on the graph. Those graph transformations acted on the graphs in a predictable way from the metrics. He raised the possibility of linking thought or computations in the brain with graph transformations. The nice thing about graphs is that, even though there are plenty of problems that are hard, we still know a lot about them and we’ve got a ton of graph algorithms already.

Who knows, maybe we’re further along in figuring this brain thing out than we might think.

That really isn't enough terms to identify as Fibonacci

Suugaku Girl is a manga about people who really like math. The great thing about it is that it contains a substantial amount of math. This is also great because it will give me plenty to blog about.

The first chapter of Suugaku Girl is about the main character’s initial meeting with Milka, our socially retarded genius meganekko. She just starts spouting out numbers and, for whatever reason, he feels compelled to guess what comes next. And that is apparently the beginning of this love story.

In the chapter, we’re introduced to a few sequences, some of which are famous and some of which you might not have considered a sequence. It’ll probably help to understand what a sequence is beyond just being a bunch of numbers in a prescribed order.

Formally, we define a sequence as a function $a:S\to R$. The set $S$ is basically our index and is $\{1,2,…,n\}$ if $a$ is a finite sequence or the set of natural numbers $\mathbb{N}$ if we’re dealing with an infinite sequence. Normally, functions are written as $a(n)$, but, as was alluded to before, we refer to terms by index as in $a_n$.

However, $R$ can be any set. In the case of Suugaku Girl, it seems to be sticking with integers, but we can have sequences of bitstrings, vectors, complex numbers, functions, or whatever. What this also means is that a sequence doesn’t necessarily need to have a “pattern”, but can really be any ordered list of numbers (or functions or vectors or…).

Milka also brings up the idea of infinite sequences. A lot of the time, people will try to “solve” a sequence by completing it when they’re given only the first few terms. But, like Milka suggests, what they’re doing in that case is assuming that the rest of the sequence goes on as initially implied. Really, we can define any sequence we like with any behaviour we like. Again, remember that a sequence can be anything we want it to be. In fact, the sequence that Milka defines using the digits of π is kind of like that in that it’s completely arbitrary and doesn’t really have the kind of sequence definitions we’re used to seeing.

For finite sequences, we can just list all of the terms of our sequence like $(1,1,1,1,1,1000000,1)$. Obviously, we can’t do that for an infinite sequence. Luckily, a lot of the time we define sequences that have some sort of useful pattern that we can represent in a succinct way. Sometimes, like the digits of π sequence, this is harder.

We can try to formally define all of the sequences that were given in the manga. For instance, the Fibonacci sequence $(a_n)_{n=1}^\infty$ is commonly defined as $a_n = a_{n-2} + a_{n-1}$, but we have to give the first two terms $a_1 = 1, a_2 = 1$. The second sequence (which we’ll call $(b_n)_{n=1}^\infty$) takes a bit more work to define. We’ll need to define $p_n$ to be the $n$th prime number and then we can define $b_n = p_n \times p_{n+1}$. The third sequence $(c_n)_{n=1}^\infty$ is easy, it’s just $c_n = n^n$. And we can formalize the last one, which I’ll call $(d_n)_{n=1}^\infty$, just like the first two with a few more words. We let $\pi = q_1q_2q_3\cdots$ be the decimal expansion of π. Then $d_n = 2\cdot q_n$.

So that’s all fine, but what exactly are sequences used for? I’m pretty sure everyone’s learned about arithmetic and geometric sequences in grade school. Obviously, we can study sequences and their behaviour on their own. We can talk about whether they increase or decrease or how fast they grow or whether they converge. Apart from that, I don’t remember seeing sequences used for something besides sequences until analysis.

Analysis is basically the field of pure math that formalizes the concepts that we’re introduced to in calculus and generalizes them to spaces. Limits are a fundamental idea in calculus and analysis and these are defined by how a sequence converges. And this is where those weird sequences of vectors or functions comes into play, since we can talk about the convergence of a sequence of vectors or a sequence of functions in these other spaces that we want to do analysis in.

That’s probably the easiest example of an “application” of sequences. For myself, over the last few months I’ve read about automatic sequences, which are sequences that can be generated by a deterministic finite automaton. This gives us a way to relate automata theory to number theory and algebra. For instance, once we have k-automatic sequences, we can talk about k-regular sequences and come up with power series with respect to certain rings and fields and bla bla bla.

If you ever want to find out what crazy sequence you might have a part of, check out the Online Encyclopedia of Integer Sequences.

「うさぎドロップ」/「 LC斐爾 」

One of the few famous quotes in Canadian politics is one by Brian Mulroney, leader of the Progressive Conservative Party of Canada and the 18th Prime Minister of Canada, during the 1984 campaign. As the leader of the Liberal Party of Canada that took over after Pierre Trudeau resigned, John Turner inherited Trudeau’s premiership (yes, we can swap prime ministers without elections) and government in the latter part of his term. Just before resigning, Trudeau had appointed a ton of loyalists to various positions.

Technically, it is the Governor General of Canada (who acts as the representative the Crown) who makes these appointments, but he or she only does so on the advice of the prime minister. In this case, the appointments had only been recommended by Trudeau but weren’t finalized by the governor general when Turner became prime minister, so he could have stopped them. During the 1984 leaders’ debate, Mulroney attacked Turner for allowing these appointments and Turner replied that he had no option. This was the lead-in to Mulroney’s famous line:

You had an option, sir. You could have said, ‘I am not going to do it. This is wrong for Canada, and I am not going to ask Canadians to pay the price.’ You had an option, sir — to say ‘no’ — and you chose to say ‘yes’ to the old attitudes and the old stories of the Liberal Party. That sir, if I may say respectfully, that is not good enough for Canadians.

This left Turner shattered in the debate and pretty much won the election for Mulroney. Unfortunately, ever since then, media coverage of electoral debates has always focused on finding that snappy equivalent knockout line, but no one has produced one since.

I like Usagi Drop. Yes, I have read the entire thing. No, the ending was not what I would have liked, but I’m not particularly angry about it. I mean, 95% of it was great, which is a far better ratio than most stuff out there. What’s more is that I think the initial reaction absent the surrounding context made it seem much worse than it actually was, especially with the important reveal that Rin isn’t actually Grandpa’s biological kid.

In fact, I’d say that I don’t really have a problem with most of the ending “arc” and it’s only the very last chapter that makes me raise an eyebrow. I can buy that Rin could have developed romantic feelings for Daikichi. After all, I think it’s an important note that she sees him as a guardian and not her father. He’s always simply been Daikichi to her.

The problem, then, and the part I find hard to believe is Daikichi’s response, which pretty much amounts to a ‘welp, guess I have no choice’. It’s really strange because, even though he was clearly uncomfortable the idea, he didn’t put up that much resistance. It’s strange enough that he offers to wait two years and hopes the problem might go away. After the two year wait period, Rin kind of goes ‘hey, it’s been two years’ and he’s sort of resigned to the whole thing.

What I’m struck by with Usagi Drop’s ending is that Daikichi had an option: he could have said no, but he chose to say yes. It would have been perfectly fine for him to say no. Rin couldn’t have seriously been expecting Daikichi to reciprocate. She would’ve been fine if things continued the way they were. She was prepared to have things stay the same. Things might’ve been awkward for a while, but a ‘no’ would have closed the matter. Instead, we’re left with, well, whatever it is that we have now.

I’d like to stress again that I still think the manga is excellent, even the second half of it. However, Daikichi’s response is baffling and his rationale is unconvincing. Quite frankly, it’s not good enough, especially for such an otherwise great manga. I’ll defend it, but I’ll concede that the end is not at all what I wanted.

「Steins;Gate」/「いさりび」

In Steins;Gate 12, the Future Gadget Lab is faced with a problem. They need to send a copy of a person’s memory, or about 3.24 TB, to the past, but they can only send 36 bytes. What is a MADDO SCIENTISTO to do? What they end up doing is sending that data to the Large Hadron Collider to compress with a black hole. Problem solved! Or is it?

Obviously, using a black hole to compress data is silly, since a black hole would probably just destroy the data. In fact, it’s probably unnecessary, since if it were physically possible to achieve that kind of compression, you’d be able to do it on any computer because of the Church-Turing thesis. The only thing that would change is how quickly the algorithm would run. Luckily, in theoretical computer science, when dealing with the question of whether something is possible or not, we don’t care how long it’ll take. So, is there a compression algorithm that can give us the desired result?

Basically what we’re doing when were compressing stuff is we’re trying to rewrite a bunch if zeroes and ones, or strings, into a smaller bunch of zeroes and ones that mean the same thing. In general, this is impossible to do, since there are less strings of zeroes and ones when you have less digits, so you can’t arbitrarily stuff, say eight bits of information into seven bits. What you can do is create a way to describe the zeroes and ones in a way that is smaller than what you start out with.

For instance, we can describe strings of zeroes and ones in plain English. It’s shorter to write “one hundred zeroes followed by one hundred ones” than writing

00000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000
11111111111111111111111111111111111111111111111111
11111111111111111111111111111111111111111111111111

The obvious problem with our compression scheme so far is that there are some strings that are not very easy to describe and it might be longer to describe in English than it would be to just write the bits themselves.

Again, it appears to be difficult to create a compression scheme that works for any old string. It’s much easier to create such an algorithm when you know something about the kind of string you want to compress, like if it’s a certain size or if it has any patterns or something. In the case of the FG lab, we know they won’t need to worry about compressing strings of length 288 or less. Unfortunately, there isn’t much more we know about the strings they want to compress other than the fact that they’re huge.

Actually, there’s another problem, which is that they need a way to decompress the string they’ve sent. If I give you a description of the string in English, it’s not going to do you any good unless you already know English. So in addition to sending a compressed string, they need to send information about the compression scheme they used, all in 36 bytes.

In information theory, the length of the shortest description of a string $x$ is its Kolmogorov complexity, which we’ll denote $C(x)$. Formally, $C(x)$ is defined in terms of encodings of Turing machines that generate $x$, but it’s pretty easy to argue that the choice of how we describe $x$ doesn’t matter up to a constant term.

We know that $C(x) \leq |x| + O(1)$ (where $|x|$ is the length of $x$), since we can always just write out $x$. We also know that for every length $n$, we can always find a string $x$ such that $C(x) \geq |x|$ since there are $2^n$ strings of length $n$ but only $2^n-1$ shorter descriptions of those strings. These strings that can’t be compressed are defined to be random.

We can quantify this idea of compressibility by saying that a string $x$ is $c$-compressible if $C(x) \leq |x| – c$. And so, we can argue about how likely it is that we have a string that can be compressed to the degree that we want. For strings of length $n$, we have that at least $2^n – 2^{n-c+1} + 1$ of them are incompressible by $c$.

We have some numbers, so let’s play around with them. Remember from earlier that the approximate size of Okarin’s memories is 3.24 TB, or $3.24 \times 2^{40} \times 8$ bits, while the limit we can send is 36 $\times$ 8 = 288 bits. If we want to squeeze Okarin’s memories into a text message, we’ll have $c = 3.24 \times 2^{40} \times 8 – 288 = 2.849934139166592 \times 10^{13}$. WolframAlpha helpfully tells me that this is slightly more than 1.4 times as many red blood cells in the human body.

We can calculate the probability that we have a compressible string by figuring out the probability that incompressible, which we can do with our lower bound on the number of incompressible strings we have. Taking $n = 3.24 \times 2^{40} \times 8$, the probability is given by $1 – \frac{2^n – 2^{n-c+1} + 1}{2^n} = \frac{1}{2^{c-1}} – \frac{1}{2^n}$. Plugging our numbers in, we get $\frac{1}{2^{3.24 \times 2^{40} \times 8 – 288 – 1}} – \frac{1}{2^{3.24 \times 8 \times 2^{40}}}$.

And unfortunately, after this step, WolframAlpha kind of refused to try to do the computation. I gave it a shot on my laptop in Python but all that happened was CPU usage went up to 100% for a few minutes and it started eating all of my memory. I kind of wanted to keep using my computer instead of waiting it out for a MemoryError exception, so I interrupted it. Doing some further estimation, I threw $2^{2^{40}}$ at WolframAlpha, since it’s kind of about the same neighbourhood in magnitude as the number we’re trying to get. Apparently, it’s a number that’s about 300 billion digits long, so one in that are the kind of odds you’re looking at.

I guess this is another way of saying there’s no chance in hell that what the FG Lab tried to do would ever work. And remember that we’re not talking about any particular compression algorithms or technologies here. We can’t come up with a better algorithm or scheme. We can’t use faster computers. We can’t use quantum computers (Holevo’s theorem says that we can’t, in general, use less qubits than classical bits for the same string). From an information theoretic standpoint, it is impossible.

But I’m pretty sure everyone already could’ve guessed that anyway.

A typical 19th century French galerie... or is it?

Ikoku Meiro no Croisée, or La croisée dans un labyrinthe étranger, was a pleasant surprise and one of my favourite shows this season. Even though Junichi Sato is only doing series composition, it still feels like Aria. We have girls leaving their homes to go to a strange foreign land and learn about it, set to classy music. All we have to do is trade MANHOME for Japan, Space-Venice for Paris, 23rd century for 19th century, and strings for clarinets.

But there was one thing that caught my eye and has bugged me ever since.

Why does this text on my window look so round?

Now, I’m not a graphic design student or anything. I just think typography is cool, so please bear with me as I play amateur type historian. I’m also not that confident in my ability to identify typefaces on the spot. I’m relying heavily on Linotype’s excellent sans serif history articles and Robert Bringhurst’s excellent The Elements of Typographic Style for more general stuff. I’ve been using a combination of Identifont and FontBook to attempt to identify stuff. Anyhow, I expect I will say some wrong things.

A few months ago, I got the chance to read through some type history for a paper. While most of it was on neoclassical type, which was late 1700s/early 1800s, I did skim through 19th and 20th century stuff. Armed with that knowledge, I thought it was a bit weird seeing stores with sans serif signage in general. But other than the problem of sans serifs possibly existing before they were created, they just felt wrong because they looked too “new”. Croisée is set some time in the second half of the 19th century and a quick search revealed that the earliest sans serif faces showed up early in the 19th century but didn’t get big until almost 100 years later.

An attempted reproduction of the above

So I’m pretty sure this is Futura. If it’s not, it’s definitely a geometric sans serif. And as it turns out, geometric sans serifs are products of the early 20th century, with Futura being created in the 1920s. Geometric faces are born from attempts to form idealized letterforms out of circles and lines, coming out of the whole Constructivist movement. It’s a pretty huge contrast to anything from the Romantic era, which is why it look so strange hiding among all of the old shops in the Galerie.

Definitely not Arial

Here, we have what appears to be Helvetica, that venerable Swiss typeface that is used everywhere nowadays, from subway signage to being slapped on photos on tumblr. Or, at least, I’m pretty sure it’s Helvetica, based on the G. The problem is that Helvetica was created in the 1950s. Of course, that’s not a lot of letters to go by and it could be one of Helvetica’s ancestors, but the earliest of those seems to have appeared in the 1880s or 1890s at the earliest. These typefaces, called Grotesques, were pretty popular in Germany, but didn’t catch on elsewhere until the 20th century. This might make its appearance in Paris a little strange, although I’m not sure of the exact timeline.

The lowercase letters were especially helpful

On this cart, we have a gothic sans serif. Gothics are a British creation from the late 19th century and early 20th century. These are recognizable as the ones that show up in old-timey newspaper headlines. Again, I’m not sure about the exact timeline so I’m not sure how it propagated in Europe, but they were pretty popular in the States. However, it’s my understanding that very few gothics were drawn before 1900.

It was annoying trying to take screens with this lady in the way

Finally, here’s what appears to be a humanist sans serif, based on the R and the C. Humanists are supposed to resemble writing more than the mechanical construction of geometrics or grotesques, but it’s hard to tell from just the capital letters. And again, since there aren’t too many letters to go by, it could turn out to be something else, but I don’t recall seeing anything that resembles that C when going through the other early 20th century typefaces. Anyhow, I feel like that could’ve come straight off of a Windows 7 screen or something.

What other typographical shenanigans are hiding on those storefronts?

I focused on sans serif typefaces because they’re relatively easy to pick out and differentiate between. I’m not pro enough to be able to distinguish between serifed fonts yet. But, this little exercise makes me wonder if there isn’t any typographical weirdness lurking among the signage set in serif faces.

I got started on this entire thing because I remembered Mark Simonson writing about the fonts on the maps in the Indiana Jones movies. I actually didn’t think about it until that Le Papillon d’Or sign caught my eye and made me go ‘waaaaaaait a second’. It’s the first time I remember doing that for anime, probably because most of the time, the signs are in Japanese. I guess the fact that it’s trying to present an accurate historical depiction of Paris also creates a sort of uncanny valley effect and things that would normally get passed over start to pop out.

Anyway, everyone should watch this show, because clarinets and cheese and baguettes and 19th century weeaboos and Yuneeeeeeeeee.

Yeah, the first time I went, I basically went on a Saturday and wandered around the dealers’ room with no money, so that probably had something to do with my lacklustre experience. This time, I’d be staying at a hotel with my software engineering pals and would try to actually go to some stuff. Also, I had monies and had an idea of what I wanted to acquire.

Friday started out with discovering that none of the big events actually starts on time. The first thing we went to see (other than the last half of the opening ceremonies because they couldn’t finish on time) was The 404s. They were okay, I guess, but I don’t know if watching Whose Line coloured my impressions of their show. The audience suggestions kinda sucked too so they didn’t have the greatest stuff to work with. Basically, it was not hilarious enough to compel me to watch their show again on Saturday.

After dinner, we managed to spend some time wandering around the dealers’ room, where I scoped out some stuff I wanted to come back for on Saturday after checking prices on Amazon once I got back to the hotel. I ended up with a few CDs (Houkago Tea Time II, cosMo/初音ミクの消失, and buzzG/Symphony) by the time we moved on to line up for Anime Hell. Anime Hell was actually pretty terrible, in large part because about 10% of the content was anime. The rest was just videos on the internet that the MC thought were hilarious. And actual pacing and comedic timing issues aside, it felt like I could’ve spent two hours on /b/ for the same experience.

Saturday was better. I’d went and got those things I’d left behind on Friday in the dealers’ room (Hidamari Sketch for $30/volume and Mameshiba). AN Idol was equal parts decent and eyebrow-raising. Four of the six performances stuck out: the girl with the shiny pants who sang the least best Card Captor Sakura opening, the guy who was off beat and off key for the first half of some One Piece song (who ended up winning), the kid who sang that Full Moon o Sagashite song that inadvertently revealed all the shotacons in the audience, and the break-dancing Gackt guy who forgot the lyrics.

The AMV contest results were okay, other than there being a really low proportion of AMVs that I actually liked. It mostly reinforced my dislike of the entire concept of AMVs. Like, cutting Final Fantasy FMV sequences to a sweeping orchestral track is not something that is hard to do. There was even one of those stereotypical Linkin Park AMVs. How the hell did that happen?

After waiting a few minutes for onigiri and finding out that they forgot to turn on the rice cooker, it was off to wait for the Masquerade thing to start. It was kind of irritating because the MC was really slow and people in the audience don’t stfu and shout generally unfunny things. Most of the costumes were nice and there were a few super impressive ones. Unfortunately, no one really did any characters I really wanted to see. I really liked the FF9 cosplayers though.

Sunday was panel day. AI in anime was kinda dry and I couldn’t hear Refighting WWII because people were being loud assholes right outside the door. Pokemon Biology was the most fantastic thing ever. It involved extrapolating hilarious things from game mechanics and lore, which is exactly the sort of thing I love to do. Topics included the evolution of humans from Mews, Arceus deniers, and Tangela’s ballin’ shoes. The visual novel panel was okay, I guess. There wasn’t much new, but it was kind of neat to see people talking about visual novels, I guess. And then it was back to the dealers’ room to look for artbooks and finding some super expensive pixiv catalogues and not buying them.

I think if I go again, I’ll stay away from the big highlight events and check out more panels or video rooms. I think staying in a hotel that was a 30 minute walk was a huge factor. Of course, it beats going back home across the city each night, but I think I would’ve been much more open to exploring stuff if I’d stayed at the Doubletree.

Loots are: an edamame Mameshiba, Houkago Tea Time II, cosMo/初音ミクの消失, buzzG feat. 初音ミク×VOCALISTS/Symphony, Hidamari Sketch, and Hidamari Sketch ×365.

One of the reasons I enjoyed Summer Wars so much is because the main character’s superpower is math. Well, okay, you say, he’s really good at math, but so what? A lot of people complain about the implausibility of OZ, but those of us with a basic understanding of cryptography and number theory will have been drawn to Kenji’s quick problem solving work with an eyebrow raised. So let’s talk about why Kenji is a wizard.

Kenji doing some friggin mathematics

We’ll start with modular arithmetic, which Kenji mentions to Natsuki on the train ride to Ueda. When we divide numbers, we often end up with remainders. Suppose we divide some integer $k$ by $N$ and we get a remainder of $r$. Then we say that $k$ and $r$ are equivalent $\bmod{N}$ and we denote that by $k = r \bmod{N}$. Because it’s how division works, for any integer $k, r$ will be some number from $0$ to $N-1$. It turns out a lot of arithmetic operations work the same way in modular arithmetic: adding, subtracting, and multiplying numbers and then taking the modulus of the result will give you the same number as adding, subtracting, and multiplying the moduli of the numbers you started out with.

However, division doesn’t work as we would expect it to. So we have to think about division (or the equivalent operation) differently. Instead of thinking of division as splitting a group of stuff into smaller groups, we’ll think of it as multiplying by an inverse. What’s an inverse? Well, we can try thinking of it in terms of addition. It’s pretty intuitive that subtraction is the opposite of addition. If we have some integer $k$, then the additive inverse of $k$ is $-k$. When we add $k$ and $-k$, we get $0$, the additive identity. The identity is just the special number that we can add to anything and get that same thing back unchanged ($n+0 = n$). In the same way, if we multiply $k$ by its inverse, $k^{-1}$, then we’ll get $1$, since $k \times 1$ is just $k$ again. What this means is that the inverse of $k \bmod{N}$ is just some other number $j$ from $0$ to $N-1$ such that $j\cdot k = 1 \bmod{N}$ and it’s just multiplication again.

Now, the problem with this is that it’s not guaranteed that there’s always an inverse hanging around in $\bmod{N}$. In particular, if $k$ and $N$ share any divisors, then $k$ won’t have an inverse $\bmod{N}$. This is interesting because it also tells us that if we consider integers mod a prime number $P$, then every integer $\bmod{P}$ has an inverse, since $P$ doesn’t share any divisors with any integers from $0$ to $P-1$. We call these things that have inverses units. So if we have a unit $k$, then $k^m$ is also a unit, for any integer $m$. We even have a funny function $\phi$ defined such that $\phi(n)$ is the number of units in $\bmod{n}$.

Love Machine solicits help

So taking everything we’ve learned, we can set up a cryptosystem! The one we’ll be looking at is called RSA, after the guys who invented it. We have Bob who wants to securely send a message $M$ to Alice. Alice chooses two prime numbers $p$ and $q$ and figures out $m = pq$. She also goes and figures out $\phi(m)$, which happens to be $(p-1)(q-1)$. Finally, she picks some integer $k$, a unit in $\bmod{\phi(m)}$. She lets everyone know $m$ and $k$, but she keeps $p$, $q$, and $\phi(m)$ secret.

So Bob wants to send $M$, which is just his message conveniently in number form. He makes $M$ into a number between $0$ and $m$, and if $M$ is too big, he can just break it up into chunks. Bob figures out the smallest $b$ such that $b = a^k \bmod{m}$ and sends $b$ over to Alice. Now, since Alice has $k$ and $\phi(m)$, she can also find $k^{-1}$ pretty easily. Once she has that, she can get the original message by figuring out $b^{k^{-1}} = (M^k)^{k{-1}} = M \bmod{m}$, since $kk^{-1} = 1 \bmod \phi(m)$.

The interesting thing here is that all of the information is out there for someone to encrypt a message to send to Alice, but no one is able to decrypt it. Well, they’re able to decrypt it if they know what $p$ and $q$ are, since once they’ve got that, they can get $\phi(m)$. But it turns out getting $p$ and $q$ from $m$ (which Alice just throws up on the interwebs) is really hard. And it really works for reals, because RSA is pretty widely deployed for things like keeping your credit card information safe while you send it through the tubes to Amazon.

A conveniently displayed ciphertext

Let’s go back and think about units some more. Of course, there are only $N$ numbers in the integers $\bmod{N}$, so there’s a point at which $k^m$ is just $1$ again and starts over. If $k^m = 1 \bmod{N}$, we say that $m$ is the order of $k$. But why do we care about finding the order of $k$?

It turns out finding the order of elements is very, very similar to factoring an integer into primes and other related problems, like discrete logarithms. If we can find orders of elements, it won’t be too hard to figure out how to factor a number. In this case, the eavesdropper wants to figure out what $p$ and $q$ are, so they’ll want to factor $m$. And it turns out a lot of other public-key cryptosystems (like elliptic curves) are based on the difficulty of factoring.

How hard could it be? Well, we could just check every possibility, which doesn’t seem that bad for a number like 48, but once we get into numbers that are hundreds of digits long, that might start to suck. It turns out the fastest known algorithms for order finding take approximately $e^{O(\log N \log \log N)^{\frac{1}{2}}}$ steps. Current key lengths for RSA are at least 1024 bits, which would give us about 4.4 x 10²⁹ operations. Assuming three trillion operations per second (3 GHz), it’d take a PC about 4.7 billion years. Sure, you could just throw more powerful computers at it, but they’d just double the key size and suddenly, you’d need to do 10⁴⁴ operations.

It's a lot easier to write math than typeset it

Well, that’s not entirely true. One of the breakthroughs in quantum computing was coming up with a fast algorithm for factoring. It turns out quantum order finding takes $O((\log N)^2 \log \log N \log \log \log N)$ steps, which, for a 1024-bit key is just over 60 operations. Doubling the key-size to 2048 bits only increases the number of operations by just over 20. Unfortunately (or fortunately, because we’d be pretty screwed if someone could easily break RSA right now), we haven’t built any quantum computers that large yet, nor are we capable of doing so anytime soon.

tl;dr – Kenji is a quantum computer.