Since new posts push old posts down (I believe this would be FILO), comments accompanying old posts get sort of lost in the shuffle and I can't be sure if people read the responses I leave to comments there, so just to clear up some Japanese, I'll make this post.
The title of my blog is Iwakan, in Chinese characters, 違和感, which translates roughly to "feeling out of place." For some pointless but fun (for me) analysis of the word, we break it down into its roots, which for Chinese-root words (漢語) conveniently means by character, since each character roughly represents a single concept, as well as a (possible multiple) reading. I've written the Chinese reading* in parenthesis, just so you can see how the following borrowed readings (音読み) change them a little.
違 - (wéi) i - This character is also read chiga-u, which even people who only know a little Japanese would recognize from Chigaimasu!, meaning wrong or different. My dictionary tells me that you could also use the character 異, (yì) i, for the first character. This has a similar meaning of foreign or different, but is much less common. The only words I can think of that use it are 異人 ijin, meaning "foreigner" or "barbarian" and kotonaru, which isn't very common, but means "to differ."
和 - (hé) wa - This character is important in Japan because it is an old name for Japan, one that dates back to some ancient Chinese tome, but also because it means harmony, which is a central principle here. Social harmony is very important.
感 - (gân) kan - This one is simply feeling. It doesn't have a Japanese reading, at least not one that I've ever heard of being used, but interestingly enough, it is used as a suru-verb, meaning that you can attach the verb suru, to do to the end of it to make it into the verb "to feel," but it is so common that over time this has changed into the easier to say kanjiru, which then conjugates as one would expect a Japanese verb to do.
Now, the word in the URL (I don't remember what this part of a URL is called) is fukakai, written in Chinese characters as 不可解, meaning "incomprehensible." Again, let's break down this word.
不 - (bù) fu - Not. Of the famous "bu yao!"
可 - (kê) ka - Possible.
解 - (jiê)kai - Explain, understand, solve.
So, if you put it all together, you get "not possibly understood," or "incomprehensible."
I thought the names were fitting.
*making the upside-down circonflex is more hassle then it is worth, so you get stuck with thinking that the town I'm implying is up-down, when it's really more like down-up.
Monday, December 27, 2010
Sunday, December 26, 2010
More Math
I said I would get around to another short post, so I'll do that. There should be a Christmas post coming, but I don't have pictures yet, so it wouldn't be as good as it could be.
Anyway, while reading about Bayesian inference on Wikipedia, I came across the Raven paradox, also known as Hempel's paradox. You can just read on there about how it works, or you can read my explanation which will be basically the same thing. So, here goes.
Consider the statement "all ravens are black." As a logical statement, this is the same as "if x is a raven, then x is black." Like all logical statements, it is logically equivalent to its contrapositive, which is "if x is not black, then x is not a raven." If you don't deal with logical statements like that, you should probably think about why statements and their contrapositives are logically equivalent.
Now, if we use the scientific (inductive) method, we can support or disprove this statement (or its contrapositive) with evidence. For example, if we see a raven that is black, we support our statement, and if we see a raven that isn't black, we have disproven the statement. Of course, since the statement and its contrapositive are equivalent, support one is the same as supporting the other and disproving one is the same as disproving the other.
So, what if we see a green apple. It is green, and since an apple is not a raven, it supports the contrapositive "if x is not black, then it is not a raven." So, this observation supports the original statement. But what happens if we start with the (obviously false) statement "if x is a raven, then x is white"? The contrapositive here is "if x is not white, then x is not a raven." Seeing the green apple again supports this contrapositive, so this observation simultaneously supports the (obviously contradictory) statements "all ravens are black" and "all ravens are white."
Weird.
Anyway, while reading about Bayesian inference on Wikipedia, I came across the Raven paradox, also known as Hempel's paradox. You can just read on there about how it works, or you can read my explanation which will be basically the same thing. So, here goes.
Consider the statement "all ravens are black." As a logical statement, this is the same as "if x is a raven, then x is black." Like all logical statements, it is logically equivalent to its contrapositive, which is "if x is not black, then x is not a raven." If you don't deal with logical statements like that, you should probably think about why statements and their contrapositives are logically equivalent.
Now, if we use the scientific (inductive) method, we can support or disprove this statement (or its contrapositive) with evidence. For example, if we see a raven that is black, we support our statement, and if we see a raven that isn't black, we have disproven the statement. Of course, since the statement and its contrapositive are equivalent, support one is the same as supporting the other and disproving one is the same as disproving the other.
So, what if we see a green apple. It is green, and since an apple is not a raven, it supports the contrapositive "if x is not black, then it is not a raven." So, this observation supports the original statement. But what happens if we start with the (obviously false) statement "if x is a raven, then x is white"? The contrapositive here is "if x is not white, then x is not a raven." Seeing the green apple again supports this contrapositive, so this observation simultaneously supports the (obviously contradictory) statements "all ravens are black" and "all ravens are white."
Weird.
Thursday, December 16, 2010
Bayesian at the Moon
I bet you were expecting more Japanese nonsense, but nah. Today I will talk about something that even mathematicians find boring: statistics.
A friend in the math department was presenting her work to some non-math people a while back, which is of course death for trying to say anything meaningful because an invitation to do so always comes with the caveat of "no math." Anyway, her research is on random compositions. A composition is basically just a set of integers that sum to a given integer, I think ordered from biggest part to smallest or something. I'm not an expert on that stuff, but it's sort of irrelevant because isn't it obvious that this has nothing to do with terrorism? I only ask because someone apparently asked if her work could be applied to something about terrorism. Seriously. It's funny that there's this supposed order of intellectuals looking down on each other that goes something like
mathematicians > physicists > chemists > biologists > psychologists > sociologists > fruity humanities type people
because we really try not to look down on people, but then they go and ask questions like that. Incidentally, if anyone has any additions or revisions to that ordering, let me know in the comments. I'm curious.
So I'm almost don't expositioning (expositing? exposing?). Somebody else asked a more relevant question, as to whether she had considered using Bayesian techniques or something like that. Not knowing what Bayesian really means, she didn't know what to say other than she hadn't used them, and since I am apparently the go to guy for statistics (???) she asked me about it later. All I could tell her was what I knew about Bayesian stats, which is this; in Bayesian statistics, you treat population parameters themselves as random variables. I read some more about it, but that's still my basic understanding of it.
You see, there are two approaches to statistics, frequentist (which is what I teach, sort of) and Bayesian (which is not what I teach). We'll see which makes more sense to you by looking at a problem.
Let's say that there's a population that we're measuring something from, and let's say that there's a distribution to that measurement, which I won't make any assumptions about the shape of, or anything, other than that it has finite mean and variance (this is not really much of an assumption, but if I feel like it, I may talk about a distribution that doesn't have these). Let's call that mean M. Normally, we would use the Greek letter mu, but I can't do that on here. Now, to a frequentist, this mean is just a fixed number that is inherent to the population, and we don't know it. If we want to know it, well, we're out of luck, but we can make a good guess at it.
The way that statisticians guess is by taking a random sample and using an appropriate test statistic or estimator built from that random sample. To estimate the population mean, you can imagine that a good estimator is the sample mean. That is, take n measurements from the population, add them up, and divide by n. Let's call that value m. In fact, m is what's called an unbiased estimator because its expected value is the desired value, M.
You can see now what I meant by M not being random, but m being random. M is fixed, to a frequentist, and so doesn't have probabilities associated with it, but m, being built out of a RANDOM sample, has probabilities. It doesn't have just one value, but a range of them, hopefully with those near the actual value of M having higher probabilities. Using m, then we can estimate M by building a confidence interval around it, which depending on our confidence level will probably contain M, though we can't say where. This is what they mean on the news when they say that some proportion is something plus or minus a margin of error. That's a confidence interval.
Alternately, if we wanted to see whether the M for our population is the same as some given mean, like a national average, or an accepted value of some sort, we can perform a hypothesis test. This is slightly harder to understand, but I think highlights a frequentist way of thinking. What you do here is come up with a null hypothesis that always looks like
H_0: M = (given value)
and assume that null hypothesis is true. Now there's a giant theorem, called the Central Limit Theorem, which states that under certain conditions, such as independent observations and large enough sample size, sample means (which are random, remember!) have a normal distribution centered at M, the population mean, and standard deviation (sigma)/(n^.5), where sigma represents the population standard deviation. The exact value there isn't what's important. What matters is that if we assume that our population mean is a given value, we can find the probability of getting a sample mean like ours (m) [nearly] exactly.
Put in common sense terms, if M is actually 0, and we take sample means from the population, most of them will be near 0, but not actually zero. Occasionally, we would get a strange sample, but not that often, so if we get a mean that is "far" from 0, we know that our assumption of the null hypothesis must be wrong. (Statisticians supposedly think that rare events do happen, just not to them). What a hypothesis does is quantify how strange test statistics are by putting them in terms of conditional probabilities.
Now, what do Bayesian statisticians do? Sort of the opposite. They say that a population parameter is a random variable and look at the conditional probability that a hypothesis is true given some evidence. They calculate
P(H|E) = P(E|H)P(H)/P(E) = P(E|H)P(H)/[P(E|H)P(H) + P(E|~H)P(~H)]
which should strike you as weird for a couple reasons. Firstly, we are looking for the probability of a hypothesis being true, such as the hypothesis that the earth orbits the sun. It doesn't make sense because it is intuitively not a random thing. It's just something we don't know whether is true or not, at least to me. Secondly, to do that calculation we have to know P(H), which is the probability that the hypothesis is true. That is, we have to assume a distribution for the truth of H going in. Generally, this is something sort of intuitive, like given two options about which we know nothing, there is a 50% chance of either being true.
Maybe this makes more sense to you, if you are scientifically minded. It's like the scientific method in that you go in thinking that a hypothesis is either true or not, with some probabilities assigned, and you do an experiment, and the results of the experiment either makes it seem more or less likely that the hypothesis is indeed true. Anyway, it's sort of interesting, right?
Since I have a bunch of time, I'll mention one distribution that doesn't meet that finite mean and variance condition from before.
Hopefully that picture loaded. I made it for a homework assignment a while ago using the Mac's built in and extremely handing graphing utility. The red curve is a normal (also called Gaussian) distribution, which is extremely useful, but has finite mean and variance and is only there for comparison. The black curve is a Cauchy distribution, which doesn't have finite mean or variance. Notice that the "tails" on the Gaussian distribution go to zero very quickly, and that those on the Cauchy distribution don't go to zero quite as fast. That is, it has "fat tails." This is an actual term in probability. Fat tailed girls make the world go 'round.
The explanation I've heard for a Cauchy distribution is this. Imagine that you set a flashlight up so that it is pointed down a given distance from the ground, and allow it to rotate freely 180 degrees in such a way that the angle from the vertical line from the flashlight to the ground and the ray of light coming from the light is uniformly distributed. What that just means is that the distribution for the angle is just a box; that is the probability is equal all throughout the 180 degrees and zero elsewhere. What is the distribution of the distance (technically displacement) from the inital position of the light to where the light strikes the ground?
You can see that the probability of the light being perfectly parallel to the ground is 0 (since there are an uncountably infinite number of possible angles), but that there is a considerable probability of the angle being near that, so that the displacement is huge. This is what results in the fat tails. The pdf is given by
f(x) = 1/[pi(1+x^2)]
which math nerds can tell you integrates to 1, so it is a legitimate pdf.
What happens when you try to find the mean, though? To do so, you need to integrate
xf(x) = x/[pi(1+x^2)] ~ 1/x
which doesn't integrate. That is, the integral diverges. In other words, even though 1/x -> 0 as x gets big, it doesn't get small enough fast enough. So, a Cauchy distribution doesn't have a finite mean. For variance, you would need to integrate x^2f(x), which you can imagine diverges even faster, so it doesn't have a finite variance, either. Or any well-defined moments for that matter. Booyeah
A friend in the math department was presenting her work to some non-math people a while back, which is of course death for trying to say anything meaningful because an invitation to do so always comes with the caveat of "no math." Anyway, her research is on random compositions. A composition is basically just a set of integers that sum to a given integer, I think ordered from biggest part to smallest or something. I'm not an expert on that stuff, but it's sort of irrelevant because isn't it obvious that this has nothing to do with terrorism? I only ask because someone apparently asked if her work could be applied to something about terrorism. Seriously. It's funny that there's this supposed order of intellectuals looking down on each other that goes something like
mathematicians > physicists > chemists > biologists > psychologists > sociologists > fruity humanities type people
because we really try not to look down on people, but then they go and ask questions like that. Incidentally, if anyone has any additions or revisions to that ordering, let me know in the comments. I'm curious.
So I'm almost don't expositioning (expositing? exposing?). Somebody else asked a more relevant question, as to whether she had considered using Bayesian techniques or something like that. Not knowing what Bayesian really means, she didn't know what to say other than she hadn't used them, and since I am apparently the go to guy for statistics (???) she asked me about it later. All I could tell her was what I knew about Bayesian stats, which is this; in Bayesian statistics, you treat population parameters themselves as random variables. I read some more about it, but that's still my basic understanding of it.
You see, there are two approaches to statistics, frequentist (which is what I teach, sort of) and Bayesian (which is not what I teach). We'll see which makes more sense to you by looking at a problem.
Let's say that there's a population that we're measuring something from, and let's say that there's a distribution to that measurement, which I won't make any assumptions about the shape of, or anything, other than that it has finite mean and variance (this is not really much of an assumption, but if I feel like it, I may talk about a distribution that doesn't have these). Let's call that mean M. Normally, we would use the Greek letter mu, but I can't do that on here. Now, to a frequentist, this mean is just a fixed number that is inherent to the population, and we don't know it. If we want to know it, well, we're out of luck, but we can make a good guess at it.
The way that statisticians guess is by taking a random sample and using an appropriate test statistic or estimator built from that random sample. To estimate the population mean, you can imagine that a good estimator is the sample mean. That is, take n measurements from the population, add them up, and divide by n. Let's call that value m. In fact, m is what's called an unbiased estimator because its expected value is the desired value, M.
You can see now what I meant by M not being random, but m being random. M is fixed, to a frequentist, and so doesn't have probabilities associated with it, but m, being built out of a RANDOM sample, has probabilities. It doesn't have just one value, but a range of them, hopefully with those near the actual value of M having higher probabilities. Using m, then we can estimate M by building a confidence interval around it, which depending on our confidence level will probably contain M, though we can't say where. This is what they mean on the news when they say that some proportion is something plus or minus a margin of error. That's a confidence interval.
Alternately, if we wanted to see whether the M for our population is the same as some given mean, like a national average, or an accepted value of some sort, we can perform a hypothesis test. This is slightly harder to understand, but I think highlights a frequentist way of thinking. What you do here is come up with a null hypothesis that always looks like
H_0: M = (given value)
and assume that null hypothesis is true. Now there's a giant theorem, called the Central Limit Theorem, which states that under certain conditions, such as independent observations and large enough sample size, sample means (which are random, remember!) have a normal distribution centered at M, the population mean, and standard deviation (sigma)/(n^.5), where sigma represents the population standard deviation. The exact value there isn't what's important. What matters is that if we assume that our population mean is a given value, we can find the probability of getting a sample mean like ours (m) [nearly] exactly.
Put in common sense terms, if M is actually 0, and we take sample means from the population, most of them will be near 0, but not actually zero. Occasionally, we would get a strange sample, but not that often, so if we get a mean that is "far" from 0, we know that our assumption of the null hypothesis must be wrong. (Statisticians supposedly think that rare events do happen, just not to them). What a hypothesis does is quantify how strange test statistics are by putting them in terms of conditional probabilities.
Now, what do Bayesian statisticians do? Sort of the opposite. They say that a population parameter is a random variable and look at the conditional probability that a hypothesis is true given some evidence. They calculate
P(H|E) = P(E|H)P(H)/P(E) = P(E|H)P(H)/[P(E|H)P(H) + P(E|~H)P(~H)]
which should strike you as weird for a couple reasons. Firstly, we are looking for the probability of a hypothesis being true, such as the hypothesis that the earth orbits the sun. It doesn't make sense because it is intuitively not a random thing. It's just something we don't know whether is true or not, at least to me. Secondly, to do that calculation we have to know P(H), which is the probability that the hypothesis is true. That is, we have to assume a distribution for the truth of H going in. Generally, this is something sort of intuitive, like given two options about which we know nothing, there is a 50% chance of either being true.
Maybe this makes more sense to you, if you are scientifically minded. It's like the scientific method in that you go in thinking that a hypothesis is either true or not, with some probabilities assigned, and you do an experiment, and the results of the experiment either makes it seem more or less likely that the hypothesis is indeed true. Anyway, it's sort of interesting, right?
Since I have a bunch of time, I'll mention one distribution that doesn't meet that finite mean and variance condition from before.
Hopefully that picture loaded. I made it for a homework assignment a while ago using the Mac's built in and extremely handing graphing utility. The red curve is a normal (also called Gaussian) distribution, which is extremely useful, but has finite mean and variance and is only there for comparison. The black curve is a Cauchy distribution, which doesn't have finite mean or variance. Notice that the "tails" on the Gaussian distribution go to zero very quickly, and that those on the Cauchy distribution don't go to zero quite as fast. That is, it has "fat tails." This is an actual term in probability. Fat tailed girls make the world go 'round.
The explanation I've heard for a Cauchy distribution is this. Imagine that you set a flashlight up so that it is pointed down a given distance from the ground, and allow it to rotate freely 180 degrees in such a way that the angle from the vertical line from the flashlight to the ground and the ray of light coming from the light is uniformly distributed. What that just means is that the distribution for the angle is just a box; that is the probability is equal all throughout the 180 degrees and zero elsewhere. What is the distribution of the distance (technically displacement) from the inital position of the light to where the light strikes the ground?
You can see that the probability of the light being perfectly parallel to the ground is 0 (since there are an uncountably infinite number of possible angles), but that there is a considerable probability of the angle being near that, so that the displacement is huge. This is what results in the fat tails. The pdf is given by
f(x) = 1/[pi(1+x^2)]
which math nerds can tell you integrates to 1, so it is a legitimate pdf.
What happens when you try to find the mean, though? To do so, you need to integrate
xf(x) = x/[pi(1+x^2)] ~ 1/x
which doesn't integrate. That is, the integral diverges. In other words, even though 1/x -> 0 as x gets big, it doesn't get small enough fast enough. So, a Cauchy distribution doesn't have a finite mean. For variance, you would need to integrate x^2f(x), which you can imagine diverges even faster, so it doesn't have a finite variance, either. Or any well-defined moments for that matter. Booyeah
Wednesday, December 15, 2010
日本語の投稿
日本にいるけん、投稿は日本語での方がいいでしょ？今日、第二故郷帰りをしようと思ってたが、昨晩美絵ちゃんのお母さんが晩ご飯に誘ってくれたから、行かないと思う。好きな奥出雲町は米子市から列車で何時間かかるから、行って、帰ればあそこにいる時間が短いし。美絵ちゃんと飲み会に行くから、明日も無理だそうで、来週にしようかと思ってる。
兎に角、雪が降ってる。近くのコンビニにも行きたくない。顔が凍る感じだ！だけど、ホット•レモンが美味しそう！どうしよう？？？
日本に着いてからの食べたものリスト：
すき家の牛丼
味噌汁
大好きな納豆ごはん
五目飯（なんじゃらご飯だがニュアンス分からない）
カレーライス（ビーフの、もちろん）
たこ焼き （凧じゃなくて、鮹だ）
インスタント焼きそば
鍋
ポテト•サラダ
美味しそうだけん！
兎に角、雪が降ってる。近くのコンビニにも行きたくない。顔が凍る感じだ！だけど、ホット•レモンが美味しそう！どうしよう？？？
日本に着いてからの食べたものリスト：
すき家の牛丼
味噌汁
大好きな納豆ごはん
五目飯（なんじゃらご飯だがニュアンス分からない）
カレーライス（ビーフの、もちろん）
たこ焼き （凧じゃなくて、鮹だ）
インスタント焼きそば
鍋
ポテト•サラダ
美味しそうだけん！
Tuesday, December 14, 2010
Natto
I don't have anything new to post, so I'll post an old picture of an old favorite, something I can't get in America, even in Philly's Chinatown, which otherwise is full of great stuff, not all of it Chinese, including Pho restaurants, Japanese candy and general Korean weirdness. The food I have missed more than any other is probably natto, which most people think is disgusting, but I just ate another bowl of today.
It's just fermented beans, basically, and it smells sort of awful, but it tastes really good, at least to me, and is apparently awesome for you. I think in the picture, it has shouyu (soy sauce) and karashi (mustard), which is my favorite combination of things to eat it with, but I have been eating it with daikon oroshi (ground up daikon [I think we use the word daikon, right?]) {grouping symbols}, which is also good.
Speaking of grouping symbols, did you know that there is such a thing as a Lie bracket. It looks just like a bracket, but it is named after Lie. Here is an example:
[Y,L]
You would think that this is a closed interval containing the endpoints Y and L, but maybe it is an binary operator, defined in some crazy way with a sum of partial derivatives, at least if we are talking about a finite-dimensional vector space.
Math is crazy like that; you always think you know a bunch, and then it turns out you don't know anything.
It's just fermented beans, basically, and it smells sort of awful, but it tastes really good, at least to me, and is apparently awesome for you. I think in the picture, it has shouyu (soy sauce) and karashi (mustard), which is my favorite combination of things to eat it with, but I have been eating it with daikon oroshi (ground up daikon [I think we use the word daikon, right?]) {grouping symbols}, which is also good.Speaking of grouping symbols, did you know that there is such a thing as a Lie bracket. It looks just like a bracket, but it is named after Lie. Here is an example:
[Y,L]
You would think that this is a closed interval containing the endpoints Y and L, but maybe it is an binary operator, defined in some crazy way with a sum of partial derivatives, at least if we are talking about a finite-dimensional vector space.
Math is crazy like that; you always think you know a bunch, and then it turns out you don't know anything.
Monday, December 13, 2010
Winter Trip
Well, I've made it to Japan and am currently messing around in Yonago. There's not much going on, but the trip should involve an "illumination cruise" in Osaka, a trip to the Tottori sand dunes, and playing the role of Santa Claus himself. There's not much to add, but it's going great.
Saturday, November 6, 2010
Well, it's Another Saturday Night
...and I ain't got nobody
I got no money because I ain't got paid
Oh, how I wish I had something to do
But sit here and grade
I got no money because I ain't got paid
Oh, how I wish I had something to do
But sit here and grade
Saturday, October 30, 2010
Halloween
It's a little late, but happy birthday, John (not Lennon). Also, to everyone else, Happy Halloween. It's that magical time of year where college students get so drunk that they pass out and don't wake up and then their friends pull the fire alarm so everyone has to leave the building at 2am to stand in the cold wind. Then when the poor girl gets to walk embarrassed to the ambulance, people take out their anger at her for having the audacity to not wake up when she's had to much to drink by yelling at her. Oh, college.
Saturday, October 9, 2010
Happy Birthday, John
I guess I am a little late, but Google told me that today (yesterday now) is (was) John Lennon's birthday, so I might as well put up a little post. A very little post. He wasn't really even my favorite Beatle, but he was pretty darn good, you have to admit. So the question on everyone's mind must be what is my favorite song of his. Maybe it's weird, but it's "Oh, Yoko." He's usually so angry or depressed, but when he's talking about Yoko he's just so happy, and it's so upbeat and simple that I can't help liking it. Maybe it's because I've got my own little Yoko. On a sort of related note, her name is actually 洋子, which should be transliterated Youko or Yooko, because it has a long vowel, so when he sings, "Oh, Yo-o-ko," it is actually the right pronunciation.
Do you have a favorite Lennon song?
Do you have a favorite Lennon song?
Tuesday, September 28, 2010
Classes have begun, and as such, I have less time to put up stupid posts that nobody reads. Anyway, I am taking a topology course, which so far has been nothing new, but this professor doesn't seem to get that this is a grad course and he shouldn't be giving us two assignments/week, especially when it mostly amounts to busy work, since it's stuff we've seen before, but I think maybe he just underestimated how many people have heard of closure operators, or really topology in general (it uses an undergrad book). I am also taking a prob/stat class, which is also pretty boring and elementary thus far, and it is notable basically only because I don't want to buy the book because it has nothing new for me (or really most people), but I need the problems out of it, so I need to MacGyver a solution somehow. I am also taking an ODE class. For those of you who don't know what ODE stands for, it means Awkward Russian guy talking for three straight hours and there not being a projector, which he did not anticipate, leading him to have to hold up papers with figures on them. So-so.
I know I am complaining about things, but this is probably the last time I will do it, and that's what blogs are for, anyway. The worst part is that there is one guy who is in all my classes and constantly complains to me about how the classes are too easy and that "graduate students shouldn't be allowed to take these classes because they are too easy." Maybe a direct quote. I realize I am complaining about the same thing, sort of, but I don't really mind them being too easy. I am more annoyed by having to do homework, but it's not like it takes a huge amount of time. I'm more more annoyed by students (and professors!) who think classes are beneath them. Students, if it is too easy for you, don't take it! You'll be here for years, and there will be more challenging stuff later. I'm taking classes just to get the credits, at least mostly. Professors, don't fall all over yourself trying to prove that you know more about the subject than we do. If we didn't accept that as a first principle, we wouldn't sign up for your class! Also, don't dismiss legitimate questions about what your arbitrary notation means (not my question, even), especially when you don't define it, and when you choose to use non-standard symbols!
Anyway, that is the report from the front lines. All quiet on the western front.
I know I am complaining about things, but this is probably the last time I will do it, and that's what blogs are for, anyway. The worst part is that there is one guy who is in all my classes and constantly complains to me about how the classes are too easy and that "graduate students shouldn't be allowed to take these classes because they are too easy." Maybe a direct quote. I realize I am complaining about the same thing, sort of, but I don't really mind them being too easy. I am more annoyed by having to do homework, but it's not like it takes a huge amount of time. I'm more more annoyed by students (and professors!) who think classes are beneath them. Students, if it is too easy for you, don't take it! You'll be here for years, and there will be more challenging stuff later. I'm taking classes just to get the credits, at least mostly. Professors, don't fall all over yourself trying to prove that you know more about the subject than we do. If we didn't accept that as a first principle, we wouldn't sign up for your class! Also, don't dismiss legitimate questions about what your arbitrary notation means (not my question, even), especially when you don't define it, and when you choose to use non-standard symbols!
Anyway, that is the report from the front lines. All quiet on the western front.
Tuesday, September 7, 2010
Break *Updated*
I've added a couple new things I've watched.
As if working about 10 hours a week over the summer wasn't leisurely enough, I'm now officially on break for a couple of weeks, so I have plenty of time to use my roommate's netflix account to watch tons of movies, as well as waste time on the internet. So, I've been catching up on some movies that aren't exactly new releases but that I felt like I should see, sort of.
Aliens - I am just finishing this and it is awesome. It's incredible how much more real it looks (because of the use of actual sets and props and actors) than Avatar, and how much better James Cameron used to be.
The Insider - This is kind of awesome, as well. It's sort of a different role for Al Pacino, but he still gets to yell a lot, which he is very well suited to.
The Rock - Good and dumb fun. Reminds me a lot of ConAir, but somehow not as good for not having ridiculous villain characters.
Spy Game - Robert Redford and Brad Pitt. Current and future heartthrobs team up to make a cool movie about spies. Good stuff.
The IT Crowd - not a movie, but a British TV show I had heard something about. It's kind of funny, not amazing or anything, but very British.
Wall Street - I liked this movie, though it seemed rather hamfisted in its morality. I'm guessing it just hasn't aged well, but seeing Martin Sheen play Charlie Sheen's dad is actually pretty cool. There's not really much to it, just a sort of simple morality play, but not a bad one by any means.
Raising Arizona - The first real Coen Brothers' movie. It is good, I think. I was exhausted for some reason when we were watching it, so I fell asleep for the second half, but I'll go back and finish it. I am a big fan of these guys' movies, though, so if you don't like them (The Big Lebowski, O Brother, Where Art Though, No Country for Old Men,...) you probably won't like this one, either.
Moon - This is an excellently composed science fiction movie, not huge in scope or even particularly innovative. Anyone with experience at all with SF or just mystery-type plots in general should be able to see what's coming about a parsec away (haha), but that doesn't mean it's not a good watch.
The Office (UK) - I'm a big fan of the US version of The Office, and consider it far and away better than the original version, but I hadn't seen the last few episodes or the Christmas special that acts as an epilogue to the series, although it is actually about the length of a whole season (or series, if you prefer the Brit term). They are both good shows, and they really do complement each other. There's an excellent essay comparing the two versions that is linked to in the Wikipedia article, which I won't bother looking up. I agree with the author that the British version is much darker and that Michael Scott is much more likable than David Brent, but that both versions bring something to the show. I completely disagree that Dwight is unbearable, however, and feel he is much MUCH funnier than his British equivalent, Garreth. Anyway, see the series, read the article, enjoy.
Batman: Mystery of the Batwoman (or something like that) - I can't resist DC animated stuff. This is basically a movie follow up to Batman: The New Adventures, which itself was a follow up series to The Adventures of Batman and Robin, or Batman: The Animated series, whichever title you prefer. It's quality, as everything from the Timmverse is, but I have a preference for the earlier style, as opposed to the simpler more "cartoonish" style of TNBA, but that's just me. The story is ok, but definitely not as good as some other ones. It was annoying to me that they sort of brought up Batgirl and then just forgot about her. Also, Robin didn't do anything but fly the Batwing, which seems particularly out of character, but I suppose it is to give more screen time to the titular character. Ah, well.
Let the Right One In - This is a Norwegian (I think) vampire movie about a junior high (again, I think) boy and his new friend who turns out to be a vampire. It's generally thought of as a horror movie, but it works much better as just a drama with a vampire. It's surprisingly sweet and understated despite people getting bitten to death by a 12 year old. Very good, but obviously not for everyone. Probably unnecessary word of warning: lots of extremely pale people.
McDonald's Breakfast - Not a movie, but always satisfies. Their coffee is so good since they started pretending to be a low rent Starbucks in the morning. This is supposedly an intentional move on their part to capitalize on the recession and people not being able to afford $5 coffee any more. I stopped getting sugar in my coffee a long while back now, and I told the girl behind the counter I was "on a diet," to which she mentioned that I was, indeed, also ordering a McGriddle. Haha, but I can't help it; the McGriddle is probably one of the most brilliant fast food items ever created, even if it has been overshadowed by that sodium bomb known as the Double Down.
Also, this site has a wonderful series of articles mocking and destroying the stupid arguments made by the tea parties, and I feel people should read it just because it puts into words so succinctly what is wrong with these people.
Since I've already established this blog as left-wing extremist by the measuring stick of a seemingly increasingly conservative (not coincidentally, also increasingly stupid and inching steadily towards collapse) society, here is another link for you about income disparity. Enjoy.
As if working about 10 hours a week over the summer wasn't leisurely enough, I'm now officially on break for a couple of weeks, so I have plenty of time to use my roommate's netflix account to watch tons of movies, as well as waste time on the internet. So, I've been catching up on some movies that aren't exactly new releases but that I felt like I should see, sort of.
Aliens - I am just finishing this and it is awesome. It's incredible how much more real it looks (because of the use of actual sets and props and actors) than Avatar, and how much better James Cameron used to be.
The Insider - This is kind of awesome, as well. It's sort of a different role for Al Pacino, but he still gets to yell a lot, which he is very well suited to.
The Rock - Good and dumb fun. Reminds me a lot of ConAir, but somehow not as good for not having ridiculous villain characters.
Spy Game - Robert Redford and Brad Pitt. Current and future heartthrobs team up to make a cool movie about spies. Good stuff.
The IT Crowd - not a movie, but a British TV show I had heard something about. It's kind of funny, not amazing or anything, but very British.
Wall Street - I liked this movie, though it seemed rather hamfisted in its morality. I'm guessing it just hasn't aged well, but seeing Martin Sheen play Charlie Sheen's dad is actually pretty cool. There's not really much to it, just a sort of simple morality play, but not a bad one by any means.
Raising Arizona - The first real Coen Brothers' movie. It is good, I think. I was exhausted for some reason when we were watching it, so I fell asleep for the second half, but I'll go back and finish it. I am a big fan of these guys' movies, though, so if you don't like them (The Big Lebowski, O Brother, Where Art Though, No Country for Old Men,...) you probably won't like this one, either.
Moon - This is an excellently composed science fiction movie, not huge in scope or even particularly innovative. Anyone with experience at all with SF or just mystery-type plots in general should be able to see what's coming about a parsec away (haha), but that doesn't mean it's not a good watch.
The Office (UK) - I'm a big fan of the US version of The Office, and consider it far and away better than the original version, but I hadn't seen the last few episodes or the Christmas special that acts as an epilogue to the series, although it is actually about the length of a whole season (or series, if you prefer the Brit term). They are both good shows, and they really do complement each other. There's an excellent essay comparing the two versions that is linked to in the Wikipedia article, which I won't bother looking up. I agree with the author that the British version is much darker and that Michael Scott is much more likable than David Brent, but that both versions bring something to the show. I completely disagree that Dwight is unbearable, however, and feel he is much MUCH funnier than his British equivalent, Garreth. Anyway, see the series, read the article, enjoy.
Batman: Mystery of the Batwoman (or something like that) - I can't resist DC animated stuff. This is basically a movie follow up to Batman: The New Adventures, which itself was a follow up series to The Adventures of Batman and Robin, or Batman: The Animated series, whichever title you prefer. It's quality, as everything from the Timmverse is, but I have a preference for the earlier style, as opposed to the simpler more "cartoonish" style of TNBA, but that's just me. The story is ok, but definitely not as good as some other ones. It was annoying to me that they sort of brought up Batgirl and then just forgot about her. Also, Robin didn't do anything but fly the Batwing, which seems particularly out of character, but I suppose it is to give more screen time to the titular character. Ah, well.
Let the Right One In - This is a Norwegian (I think) vampire movie about a junior high (again, I think) boy and his new friend who turns out to be a vampire. It's generally thought of as a horror movie, but it works much better as just a drama with a vampire. It's surprisingly sweet and understated despite people getting bitten to death by a 12 year old. Very good, but obviously not for everyone. Probably unnecessary word of warning: lots of extremely pale people.
McDonald's Breakfast - Not a movie, but always satisfies. Their coffee is so good since they started pretending to be a low rent Starbucks in the morning. This is supposedly an intentional move on their part to capitalize on the recession and people not being able to afford $5 coffee any more. I stopped getting sugar in my coffee a long while back now, and I told the girl behind the counter I was "on a diet," to which she mentioned that I was, indeed, also ordering a McGriddle. Haha, but I can't help it; the McGriddle is probably one of the most brilliant fast food items ever created, even if it has been overshadowed by that sodium bomb known as the Double Down.
Also, this site has a wonderful series of articles mocking and destroying the stupid arguments made by the tea parties, and I feel people should read it just because it puts into words so succinctly what is wrong with these people.
Since I've already established this blog as left-wing extremist by the measuring stick of a seemingly increasingly conservative (not coincidentally, also increasingly stupid and inching steadily towards collapse) society, here is another link for you about income disparity. Enjoy.
Monday, August 30, 2010
Friday, August 20, 2010
Short Review
Since nobody else appears to be updating, I guess I will have to take the wheel. I saw Inception recently, and I have to say it pretty much lives up to the (ridiculous) hype. It's pretty exciting and the multiple layers keep you interested. The rules are all explained without delving into the mechanics of shared dreaming, which would have been a nightmare (ha ha) to sit through. Instead, we get walked through the rules nicely as the story progresses, just enough to let us understand and not enough to bore us. Good acting and all that. Despite what you may have heard, the story isn't confusing at all; it's very straightforward and easy to follow as long as you can keep track of dreams within dreams (there are only three levels, so it's not like you'll forget what is going on). The ambiguous ending is well done even if it is obvious; if you know the premise of the movie, in fact, you can probably guess the ending without seeing it. Anyway, check it out if you somehow haven't by now.
Also, if you aren't watching the new episodes of Futurama, then you are missing out. They're even better now than before they went off the air.
Also, if you aren't watching the new episodes of Futurama, then you are missing out. They're even better now than before they went off the air.
Friday, August 13, 2010
Tragedy!
Cathy is ending! What will the world do without her?
The teeth of great lions are broken; the lion perishes for lack of prey, and the cubs of the lioness are scattered!
Please take Funky Winkerbean instead!
The teeth of great lions are broken; the lion perishes for lack of prey, and the cubs of the lioness are scattered!
Please take Funky Winkerbean instead!
Monday, August 9, 2010
Camden Yards
It's been a bit since my last update, so how about a short one. Over the weekend, I went down with two of my roommates to Baltimore, where we met up with Eric to take in an Orioles vs. White Sox game. Camden Yards, or rather, Oriole Park at Camden Yards, is a nice ball park, pretty large and very clean. It also has nice views of the city, if you are facing away from the field, with no walls to obscure your view, at least at the upper levels. I should have some pictures, but I have not received them as of now. Anyway, the White Sox, who are in a division race, beat the O's, who are only in a race in the purely mathematical sense of it is theoretically possible that they win something like 90% of their remaining games and all other teams in their division do horribly from here on out. So, Eric was happy with that. Baltimore fans seem pleasant, knowing that their team is pretty bad, and didn't mind sitting with a St. Louis fan, a White Sox fan, a Phillies fan, and a Yankees (boo!) fan. Afterwards, we enjoyed some Chicago style pizza near the park. I know that's weird. Next stop, NY, NY!
Wednesday, August 4, 2010
Cheesesteak
I finally had a Cheesesteak from the legendary Pat's. I don't have a picture of the "wiz wit" that I ate because I took it on my phone and I don't know how to transfer the pictures. Anyway, it was awesome, but if you go, be sure to follow their ordering instructions. They have them posted on a sign. You need to have your money ready when you order or they will start yelling at you. I can read, but apparently some people can't, so I got to witness that first hand. The reason for their strictness is the absurdly long lines they have, and thus the need to rush everyone through. Anyway, it was awesome, and I suggest it to anybody who loves great (but terrible for you) food. Maybe Geno's next time...
Saturday, July 3, 2010
I Lament the Coming of 3D
Let me just say that Toy Story 3 is awesome, much cooler than Amtrak, which doesn't run sufficient numbers of trains over the July 4 weekend for me to get down to Virginia. I saw Toy Story 3 a little while back and I was very pleased, movie magic and all that. It provided a nice bit of closure to a trilogy that was good through and through, and had some of the best laughs of the series. An instant classic, yada yada.
I had to see it in 3D, however, which annoyed me right off the bat, since 3D tickets are more expensive, and movie tickets in general are too expensive, since you can't know going in if the movie is even going to be worth watching. There are hints, of course. If it is directed by M. Night whatshisface or stars Gerard Butler, you are probably better off saving your money. But that was not the case, and the movie turned out to be worth the price of admission, or at least what the price of admission would have been if it weren't in 3D and the theater didn't change management to some jerks who don't give student discounts when they think they can get more money out of you (any time they are playing anything you would want to see).
But my main quibble is that 3D doesn't work for me, at least not very well, and a lot of the time, just makes it look like things on screen are surrounded by halos of light or that there are two of them. When it does work, it doesn't really add anything to the movie as such, besides sometimes a headache. So, long story short, I wish 3D would die, but it probably won't because people have always appreciated flashy crap more than a good story, and we all know which of those thing Hollywood is better at churning out, anyway.
In closing, a joke from one of my roommates:
What's the difference between a teabag and England?
A teabag stays in the cup longer. Ha ha, take that, English people.
I had to see it in 3D, however, which annoyed me right off the bat, since 3D tickets are more expensive, and movie tickets in general are too expensive, since you can't know going in if the movie is even going to be worth watching. There are hints, of course. If it is directed by M. Night whatshisface or stars Gerard Butler, you are probably better off saving your money. But that was not the case, and the movie turned out to be worth the price of admission, or at least what the price of admission would have been if it weren't in 3D and the theater didn't change management to some jerks who don't give student discounts when they think they can get more money out of you (any time they are playing anything you would want to see).
But my main quibble is that 3D doesn't work for me, at least not very well, and a lot of the time, just makes it look like things on screen are surrounded by halos of light or that there are two of them. When it does work, it doesn't really add anything to the movie as such, besides sometimes a headache. So, long story short, I wish 3D would die, but it probably won't because people have always appreciated flashy crap more than a good story, and we all know which of those thing Hollywood is better at churning out, anyway.
In closing, a joke from one of my roommates:
What's the difference between a teabag and England?
A teabag stays in the cup longer. Ha ha, take that, English people.
Monday, June 28, 2010
World Cup Update
I actually watched a game. It is still uninteresting, but at least when you are watching at a bar with a bunch of other people cheering for the same team, some of whom are dressed like Uncle Sam, etc., there is a good atmosphere. I guess I will just "root for" glorious Nippon now, although they will probably also lose.
Friday, June 11, 2010
World Cup
I guess the world cup is starting (has started?), so I will post my thoughts on that. Here they are:
It is boring.
I don't get why soccer fans are so insecure about their sport. Yes, Americans don't like it. It's not that we don't understand it, at least not in any cases I've ever seen. We just don't like it. We have other sports that are just more interesting. We have seen it and fail to see why you would get so excited about 88 minutes of low-speed passing and maybe two minutes of anything meaningful.
Only soccer fans don't get this. I like hockey and of course was crazy excited about the Stanley Cup this year. I watched almost every game the Flyers played and it was awesome, but I get that some people don't like it. I don't try to get other people to like it; I am satisfied with watching it. Soccer fans always try to convince you that just because nations are competing with nations now instead of teams within a nation, that the game has somehow become less mind-numbing. It hasn't. That's it.
It is boring.
I don't get why soccer fans are so insecure about their sport. Yes, Americans don't like it. It's not that we don't understand it, at least not in any cases I've ever seen. We just don't like it. We have other sports that are just more interesting. We have seen it and fail to see why you would get so excited about 88 minutes of low-speed passing and maybe two minutes of anything meaningful.
Only soccer fans don't get this. I like hockey and of course was crazy excited about the Stanley Cup this year. I watched almost every game the Flyers played and it was awesome, but I get that some people don't like it. I don't try to get other people to like it; I am satisfied with watching it. Soccer fans always try to convince you that just because nations are competing with nations now instead of teams within a nation, that the game has somehow become less mind-numbing. It hasn't. That's it.
Sunday, May 30, 2010
Monday, May 24, 2010
Graphs and Donuts
I think I wrote a couple posts ago that I would post something about some stuff that I am fooling around with in a mental sense, so I will do that.
I came across a book about Topological Graph theory and since that sounded good and weird and like I wouldn't need to find a bunch of inequalities to do it, I started reading it, and came upon a problem involving planar graphs. A little explanation, though not anything rigorous:
First you have to know what a graph is. It is essentially a set of points connected by line segments. The points are called vertices, and the segments are called edges. Technically, graphs are sets which contain two sets, an edge set and a vertex set, where the vertex set is basically single numbers (generally integers), and edges are pairs of numbers, where both of the integers in the pair appear in the vertex set. But for our purposes, they are just dots with lines connecting them.
For most purposes, simple graphs will suffice, which are graphs where only one edge can appear between any two vertices, and no edge can connect a vertex to itself. If you are inclined to chemistry, you can think of them as molecule diagrams where no double bonds are allowed, but any number of bonds can be made between atoms. Actually, there is an interesting problem in graph theory of counting isomers of carbon chains of arbitrary length, but I digress.
The next thing you have to understand is what is meant by planar graphs. A planar graph is a graph that can be drawn in the plane without any edges intersecting. The analyst in my office would complain about that definition, but my tensor professor would love it because he insists that "R^2 is not a plane; a plane is a plane. R^2 is a set of pairs of numbers. What is a plane? I can't tell you, but I know what it is." That isn't an exact quote, but it is close. He also says things like, "There's no such thing as integration by parts. There is only the product rule." (^_^)
I will now give examples, like a good textbook:
This graph is K4 (the 4 is supposed to be a subscript, but blogs...). That is shorthand for the complete graph on four vertices, meaning that it has every edge that can exist on four vertices. Exercise: how many edges does Kn (the complete graph on n vertices) have?
Is K4 planar? One is tempted to say "no," since those two edges in the box intersect, and that's a no-no according to our definition. However, what if we draw it like this:
Now we have the same graph drawn in the plane, with no edges intersecting, so the answer above should actually be "yes." Technically, these graphs are not the same, they are only "isomorphic," but all that means is that if you treat graphs as they should be treated, as sets, and make a renaming function (i.e., an isomorphism) between the sets, all the relationships are preserved.
A better question is what graphs are (not) planar? A dude named Kuratowski essentially answered that question all at once and very neatly. He said (and proved) that all the non-planar graphs contain a copy of K5 or K3,3 (the bipartite graph on 3 and 3 vertices; a picture will follow). Technically, they just have to contain a subdivision of one of these two graphs. That is, if you take one of those graphs and add vertices on already established edges, you also can't draw that on the plane, but the important point is that we can tell, in a certain sense, very easily if a graph can be drawn in the plane.
K5
K3,3
The second one is called the complete bipartite graph because it has two subsets of vertices {1,2,3} and {4,5,6} that have no edges inside them, but all the other edges are there.
So, how do we know that things are planar/not planar? To show that something is planar, it's easy enough to just draw it in the plane, though in practice this could be staggeringly difficult for large graphs. To show something is non-planar, you can't just show one non-planar representation; you have to show that no planar representation is possible. Here's a bare-bones proof that K5 is non-planar, using the concept that that previous planar representation of K4 is "unique" up to bending, stretching and renaming, none of which affects planarity.
Now try putting the fifth vertex in any of the regions we've created and see if you can draw an edge to all the other vertices. So K5 isn't planar! I'll leave a proof of K3,3 up to you, but it can be done in much the same way.
Now, a mind that likes puzzles might wonder what is so special about the plane? What about other shapes, like a ball? Well, that question has been asked, but not totally answered. From now on I'm going to call "shapes" "surfaces" because that is the actual term. I'm not going to give you a real definition of a surface, but it essentially boils down to adding loops and crosscaps to the sphere.
The first part of the question is are there "planar" graphs and "non-planar" graphs for other surfaces, and the first answer is yes. In fact, some dude showed that for any surfaces, there is a finite set like {K5, K3,3} that generates all the non-embeddable graphs. However, except for the sphere, nobody knows what they are, so that is what I am messing with, but it is pretty complex. Let's consider the sphere:
There's our good buddy K4 drawn on the surface of a sphere. It shouldn't come as a surprise that anything we can embed on a plane we can embed on a sphere, since a sphere is what's called "locally Euclidean;" it looks like a plane as long as you only look in a little disk. But is that drawing still unique, as it was in the plane?
Oh, ho, ho! What if we wrap the last edge around the back? Well, it turns out that that doesn't change anything at all, because wrapping around the back is the same as stretching back and back and back. So, well, the sphere isn't too interesting. It's the same as a plane, but we like to use a sphere to judge from because it's what's called "compact." Don't worry about that much.
So what's the next simplest thing? A doughnut, or donut to normal people. Mathematicians of course call it a torus because we are contrarians by nature. Let's try drawing K4 again:
Try and draw the fifth vertex and you'll run into the same problem as before. But does this mean that K5 is non-embeddable on the torus? You should know by now that that isn't the case. What if we start by drawing the box in more natural ways?
Now try to draw that fifth vertex and you'll find that not only can you draw K5, but you can draw it in more than one way! So a donut has a little more freedom than a ball.
So, I'm not really sure which graphs can and can't be embedded yet, but I've managed to embed K5 but not K6 on the torus, as well as K3,4. I don't know if I embedded K4,4 yet, as I have tons of scrap paper with drawings on them now. Anyway, that's what I've been up to.
I came across a book about Topological Graph theory and since that sounded good and weird and like I wouldn't need to find a bunch of inequalities to do it, I started reading it, and came upon a problem involving planar graphs. A little explanation, though not anything rigorous:
First you have to know what a graph is. It is essentially a set of points connected by line segments. The points are called vertices, and the segments are called edges. Technically, graphs are sets which contain two sets, an edge set and a vertex set, where the vertex set is basically single numbers (generally integers), and edges are pairs of numbers, where both of the integers in the pair appear in the vertex set. But for our purposes, they are just dots with lines connecting them.
For most purposes, simple graphs will suffice, which are graphs where only one edge can appear between any two vertices, and no edge can connect a vertex to itself. If you are inclined to chemistry, you can think of them as molecule diagrams where no double bonds are allowed, but any number of bonds can be made between atoms. Actually, there is an interesting problem in graph theory of counting isomers of carbon chains of arbitrary length, but I digress.
The next thing you have to understand is what is meant by planar graphs. A planar graph is a graph that can be drawn in the plane without any edges intersecting. The analyst in my office would complain about that definition, but my tensor professor would love it because he insists that "R^2 is not a plane; a plane is a plane. R^2 is a set of pairs of numbers. What is a plane? I can't tell you, but I know what it is." That isn't an exact quote, but it is close. He also says things like, "There's no such thing as integration by parts. There is only the product rule." (^_^)
I will now give examples, like a good textbook:
This graph is K4 (the 4 is supposed to be a subscript, but blogs...). That is shorthand for the complete graph on four vertices, meaning that it has every edge that can exist on four vertices. Exercise: how many edges does Kn (the complete graph on n vertices) have?
Is K4 planar? One is tempted to say "no," since those two edges in the box intersect, and that's a no-no according to our definition. However, what if we draw it like this:
Now we have the same graph drawn in the plane, with no edges intersecting, so the answer above should actually be "yes." Technically, these graphs are not the same, they are only "isomorphic," but all that means is that if you treat graphs as they should be treated, as sets, and make a renaming function (i.e., an isomorphism) between the sets, all the relationships are preserved.
A better question is what graphs are (not) planar? A dude named Kuratowski essentially answered that question all at once and very neatly. He said (and proved) that all the non-planar graphs contain a copy of K5 or K3,3 (the bipartite graph on 3 and 3 vertices; a picture will follow). Technically, they just have to contain a subdivision of one of these two graphs. That is, if you take one of those graphs and add vertices on already established edges, you also can't draw that on the plane, but the important point is that we can tell, in a certain sense, very easily if a graph can be drawn in the plane.
K5
K3,3
The second one is called the complete bipartite graph because it has two subsets of vertices {1,2,3} and {4,5,6} that have no edges inside them, but all the other edges are there.
So, how do we know that things are planar/not planar? To show that something is planar, it's easy enough to just draw it in the plane, though in practice this could be staggeringly difficult for large graphs. To show something is non-planar, you can't just show one non-planar representation; you have to show that no planar representation is possible. Here's a bare-bones proof that K5 is non-planar, using the concept that that previous planar representation of K4 is "unique" up to bending, stretching and renaming, none of which affects planarity.
Now try putting the fifth vertex in any of the regions we've created and see if you can draw an edge to all the other vertices. So K5 isn't planar! I'll leave a proof of K3,3 up to you, but it can be done in much the same way.
Now, a mind that likes puzzles might wonder what is so special about the plane? What about other shapes, like a ball? Well, that question has been asked, but not totally answered. From now on I'm going to call "shapes" "surfaces" because that is the actual term. I'm not going to give you a real definition of a surface, but it essentially boils down to adding loops and crosscaps to the sphere.
The first part of the question is are there "planar" graphs and "non-planar" graphs for other surfaces, and the first answer is yes. In fact, some dude showed that for any surfaces, there is a finite set like {K5, K3,3} that generates all the non-embeddable graphs. However, except for the sphere, nobody knows what they are, so that is what I am messing with, but it is pretty complex. Let's consider the sphere:
There's our good buddy K4 drawn on the surface of a sphere. It shouldn't come as a surprise that anything we can embed on a plane we can embed on a sphere, since a sphere is what's called "locally Euclidean;" it looks like a plane as long as you only look in a little disk. But is that drawing still unique, as it was in the plane?
Oh, ho, ho! What if we wrap the last edge around the back? Well, it turns out that that doesn't change anything at all, because wrapping around the back is the same as stretching back and back and back. So, well, the sphere isn't too interesting. It's the same as a plane, but we like to use a sphere to judge from because it's what's called "compact." Don't worry about that much.
So what's the next simplest thing? A doughnut, or donut to normal people. Mathematicians of course call it a torus because we are contrarians by nature. Let's try drawing K4 again:
Try and draw the fifth vertex and you'll run into the same problem as before. But does this mean that K5 is non-embeddable on the torus? You should know by now that that isn't the case. What if we start by drawing the box in more natural ways?
Now try to draw that fifth vertex and you'll find that not only can you draw K5, but you can draw it in more than one way! So a donut has a little more freedom than a ball.
So, I'm not really sure which graphs can and can't be embedded yet, but I've managed to embed K5 but not K6 on the torus, as well as K3,4. I don't know if I embedded K4,4 yet, as I have tons of scrap paper with drawings on them now. Anyway, that's what I've been up to.
Now that Lost has supposedly ended (I did not watch it, but it's all over the internet), I'm expecting a bevy of posts from the Losties soon, but maybe not. It seems the consensus is that the finale was lame and didn't answer any interesting questions (again, I don't know which those are).
For my part, I just watched Superman: Doomsday, which was pretty good. In a way, it is disappointing because it's just a little story. When DC killed Superman, it was a big deal (like even non-comic nerds bought those issues). Then there is a long saga of four dudes claiming to be Superman taking over and us finding out who they all are, and eventually it segues into Hal Jordan, the Green Lantern, becoming Parallax and all this junk. Anyway, it's a huge deal, not just for Metropolis, but for people in general. In this movie, we don't get to see anything but Metropolis, and that is almost distracting, since I couldn't help thinking that when a somewhat evil Superman shows up and starts laying down the law, it seems unlikely to me that somebody like Batman would just let it happen and not try to Kryptonite the guy. There are just so many things that could happen, but we get a little story about Lex Luthor and then Superman just unceremoniously comes back to life and we get a battle Royale, which would be way more entertaining if it was with Captain Marvel or something. Ah, just limitations on movies.
Anyway, there's not much going on here. It's just muggy and gross. That's it for me.
For my part, I just watched Superman: Doomsday, which was pretty good. In a way, it is disappointing because it's just a little story. When DC killed Superman, it was a big deal (like even non-comic nerds bought those issues). Then there is a long saga of four dudes claiming to be Superman taking over and us finding out who they all are, and eventually it segues into Hal Jordan, the Green Lantern, becoming Parallax and all this junk. Anyway, it's a huge deal, not just for Metropolis, but for people in general. In this movie, we don't get to see anything but Metropolis, and that is almost distracting, since I couldn't help thinking that when a somewhat evil Superman shows up and starts laying down the law, it seems unlikely to me that somebody like Batman would just let it happen and not try to Kryptonite the guy. There are just so many things that could happen, but we get a little story about Lex Luthor and then Superman just unceremoniously comes back to life and we get a battle Royale, which would be way more entertaining if it was with Captain Marvel or something. Ah, just limitations on movies.
Anyway, there's not much going on here. It's just muggy and gross. That's it for me.
Thursday, May 20, 2010
Half-Post 350?
350 isn't an important number, is it? I don't know; I'm not too good with numbers. Maybe that's why lately I've been fascinated with embeddings of graphs on surfaces. Maybe I will post about that when I have time to draw some pictures.
In other news, I think Holiday picked up a hit the other day, even if we lost. At least the Reds lost, too.
In other news, I think Holiday picked up a hit the other day, even if we lost. At least the Reds lost, too.
Friday, May 7, 2010
Lol-iday
Matt Holliday needs to learn to hit a pitch. I went to two of those games and he was awful. Pujols was barely better. If we can't be Kendrick, then there is something wrong with the club. Hopefully we can pick up a few wins against the Buccos.
Also, can we not pick up a shortstop who bats above .200?
Also, can we not pick up a shortstop who bats above .200?
Sunday, May 2, 2010
野球
On Friday I went with a couple dudes to watch the Mets absolutely destroy the Phillies. Too bad, but I can't say I care that much. At least it's not the Yankees doing the winning. It was pretty crazy, though, since the Mets are the Phillies biggest rivals and this was their first meeting of the season. The stadium was packed and the parking lots (all the stadiums are right next to each other, so the parking lots can all be used at once) were full of people tailgating. So, that was cool. Alright, that's it.
Wednesday, April 14, 2010
Hi Emily!
I know you probably don't read this blog, but if you do read it, I just want you to know that I am writing you a letter; I've just got no ideas.
Thursday, March 25, 2010
Too Many Mornings
In an effort to balance out that last post, I'll talk about something that is probably of marginally more interest to people but still sort of galvanizing: Bob Dylan.
He's famous among his fans or rock fans or music fans or something for recording many (at times very) different versions of his songs, and I've had one of those songs that keeps popping up in his repertoire over the decades stuck in my head, so I thought I'd use that as an example. Here's the original, from The Times They Are A-Changin' (1964). It's got a nice simple melancholy to it, which is pretty much how most of the album feels. It's probably only his second best acoustic album, after The Freewheelin' Bob Dylan, but still leagues better than Another Side, which is pretty weak in my opinion. This is probably one of the best tracks from the album, which says something for an album that contains THE LONESOME DEATH OF HATTIE CARROLL - a favorite track of IWU roommates.
Folk fans everywhere were saddened when Dylan abandoned this sound and started hanging out with the guys who would become The Band, but he didn't forget this number and mixed it up to fit their "thin, wild mercury" sound. You have to fast forward a bit to get to the song. I would have thought this would have been an easier version to find, since it is on The Bootleg Series, Vol. 4, but I guess not.
After the whole electric thing, he went back to recording acoustic stuff, famously making a short but sweet country album called Nashville Skyline, a favorite of Dan's if I recall correctly. While recording for that album, I believe, he ran into Johnny Cash, and they recorded duets of some of each of their songs as well as some others. This is one of them. It's got a very simple Johnny Cash sound to it, with Dylan using his country crooner voice. Not my favorite, but not terrible.
Just a few years later, we get a collaboration with another famous Dylan friend, George Harrison. I believe this was recorded around the time of New Morning, but I'm not sure. I kind of like it, but I've got a soft spot for that album. Overall it's not that good a version and seems highly out of character for the lyrics, but it's worth a listen, at least.
A few years later we get to painted face, cowboy hat wearing, going a bit crazy Rolling Thunder Revue Dylan, which is one of my favorite eras. This version is from Hard Rain, a live album much maligned for being listless and not as good as the first half of his tour with the RTR. I think that is part of the charm of it; the RTR was a big, energetic last push by Bob before he seems to have lost it all for a decade or so, and here we get to see Dylan rough around the edges, basically yelling into mic, breathing new life into the song.
Quite a far cry from this. Leather pants?
Alright, well, that's about it. You can dig up some more versions of it on YouTube if you want.
He's famous among his fans or rock fans or music fans or something for recording many (at times very) different versions of his songs, and I've had one of those songs that keeps popping up in his repertoire over the decades stuck in my head, so I thought I'd use that as an example. Here's the original, from The Times They Are A-Changin' (1964). It's got a nice simple melancholy to it, which is pretty much how most of the album feels. It's probably only his second best acoustic album, after The Freewheelin' Bob Dylan, but still leagues better than Another Side, which is pretty weak in my opinion. This is probably one of the best tracks from the album, which says something for an album that contains THE LONESOME DEATH OF HATTIE CARROLL - a favorite track of IWU roommates.
Folk fans everywhere were saddened when Dylan abandoned this sound and started hanging out with the guys who would become The Band, but he didn't forget this number and mixed it up to fit their "thin, wild mercury" sound. You have to fast forward a bit to get to the song. I would have thought this would have been an easier version to find, since it is on The Bootleg Series, Vol. 4, but I guess not.
After the whole electric thing, he went back to recording acoustic stuff, famously making a short but sweet country album called Nashville Skyline, a favorite of Dan's if I recall correctly. While recording for that album, I believe, he ran into Johnny Cash, and they recorded duets of some of each of their songs as well as some others. This is one of them. It's got a very simple Johnny Cash sound to it, with Dylan using his country crooner voice. Not my favorite, but not terrible.
Just a few years later, we get a collaboration with another famous Dylan friend, George Harrison. I believe this was recorded around the time of New Morning, but I'm not sure. I kind of like it, but I've got a soft spot for that album. Overall it's not that good a version and seems highly out of character for the lyrics, but it's worth a listen, at least.
A few years later we get to painted face, cowboy hat wearing, going a bit crazy Rolling Thunder Revue Dylan, which is one of my favorite eras. This version is from Hard Rain, a live album much maligned for being listless and not as good as the first half of his tour with the RTR. I think that is part of the charm of it; the RTR was a big, energetic last push by Bob before he seems to have lost it all for a decade or so, and here we get to see Dylan rough around the edges, basically yelling into mic, breathing new life into the song.
Quite a far cry from this. Leather pants?
Alright, well, that's about it. You can dig up some more versions of it on YouTube if you want.
Wednesday, March 24, 2010
I Review a Movie I Have Not Seen
The Blind Side is terrible. My roommates are in love with it. I was so bored within the first fifteen minutes of it that I gave up and went to surf the internet instead. How did this movie succeed? It seems like it's just some sort of Hallmark Channel movie of the week. Does it make white people feel good about themselves, thinking that in the same situation they would do the same thing when they no doubt wouldn't? I don't get it.
Monday, March 22, 2010
Hidden Post
I had a new post, but it was rather lengthy, and due to extenuating circumstances, it didn't get done until after my last couple short posts. So, I put it up today, but I guess blogger puts things in order by when you start them, not when you finish them, so you'll have to scroll down if you want to get to it. WARNING: LOTS OF MATH.
Saturday, March 20, 2010
Finally Done
To slow this blog's descent into nothingness, I thought I'd post an announcement that I finally finished my responsibilities for this quarter with a few hours of grading exams. It's very frustrating having to deal with people who lie to your face about turning things in that they didn't and people who cheat just cleverly enough not to get caught, and even worse, people who do both. Ah, well, good luck getting into med school with a C in what's basically a high school math course.
The weather was fantastic today, so after spending hours in a gray windowless office, I took the opportunity to read the internet and promptly fall asleep. Maybe I'll come up with something to put up to match Dan's bizarre short story-like post. Recombobulation indeed.
Update: Since apparently Sarah is in Milwaukee, she should be careful:
The weather was fantastic today, so after spending hours in a gray windowless office, I took the opportunity to read the internet and promptly fall asleep. Maybe I'll come up with something to put up to match Dan's bizarre short story-like post. Recombobulation indeed.
Update: Since apparently Sarah is in Milwaukee, she should be careful:
Tuesday, March 2, 2010
No Posts
I know I haven't posted anything in forever. I am working on that, but it is nearing the end of the quarter so it is unlikely I will have adequate time to type a bunch of stuff up because I am busy typing up other stuff when I do finally manage to figure it out. Also, grading. I just finished a marathon session of it and that was only half of the problems on half the exams. Fortunately I am only responsible for half the problems. Peace!!!!!!!
Friday, February 26, 2010
New Post
I'd had this update sitting on my hard drive for a few weeks, incomplete, before today, but I just got around to finishing it because of being sick and finals and whatnot keeping me out of the blogosphere. Now I'm on break, so "enjoy."
I've talked about topologies before, so if you are interested in reading my ramblings but don't know what it is, you can find it in some old post. Or you can just use Wikipedia like a normal person.
Anyway, many useful topologies are defined by a function called a metric, which just measures distance between two points. A metric d is defined as having three properties:
1) d(x,y) >= 0 and d(x,y) = 0 iff x = y (if we relax this last condition, it is a pseudo-metric
2) d(x,y) = d(y,x)
3) d(x,z) <= d(x,y) + d(y,z)
You can see that distance in the normal sense meets all of these conditions. In fact, property 3) is called the triangle inequality and you have to use it all the time in analysis.
The way that a metric induces a topology is pretty straightforward. You just define an "open ball" as the set of all points less than some distance from a point, which you can call the center of the ball. Picturing this in the Euclidean 3-space, known to non math nerds as just 3 dimensions in the usual sense, means spheres of some radius, not including any points on the surface. From here, you just say that a set is open if every point in the set has an open ball containing it which is entirely contained in the set. In Euclidean space, this again translates into anything that is missing its boundary. I'm not going to define boundary for you, although it is a rigorously defined thing in general topology, but it should be clear in Euclidean space what that means. For example, in Euclidean 1-space, heretofore known as the real line, the boundary of the interval [a,b) is {a,b}. So, this set is not open, since it contains some of its boundary. More rigorously, open balls in 1-space just mean open intervals, so if you try to put an open ball around a, it will contain things to the left of a, which can't be in the original interval. Then that set can't be open. It's not closed, either, but I won't get into that.
You'll note that the way we defined this topology does indeed give us a topology:
1) The empty set has no points, so it vacuously meets the condition to be open. Obviously, any open ball of a point will be contained in the entire space, so the entire space is also open.
2) If a point is in a union of open sets, it's in one of them, so it's got one of these open balls, which as a subset of one of the sets, is a subset of the union, so a union of open sets is open.
3) If you intersect two open sets and choose a point in their intersection, then there is an open ball centered at that point corresponding to each of the two sets. Just choose the minimum of those two radii and you've got yourself the open ball you wanted.
Anyway, I was just dragging my feet on undergrad stuff until now. It's time to step it up to measure theory, but not really. I'm just going to glide over all the annoying parts of trying to set up integration the Lebesgue way because it doesn't matter for what I want to get to.
Intuitively, how "big" is the interval [0,1]? It should have length 1-0 = 1, right? How about the interval (a,b)? If you said b-a, you know how to generalize, but aren't Stieltjes (which just means you are normal). Anyway, how about the interval (a,b]? It should still be b-a, right? All we added was one point, and that point should be infinitely small in a certain geometric sense.
Now, how about a union of intervals, like, say, [0,1] U [2,3]? It should just be 2, to my mind since it's just two (disjoint) intervals of length 1. And how about [a,b]U[c,d], assuming c>b; that is those intervals are disjoint? If you said b-a+d-c, congrats. So I think we're clear on how to "measure" intervals, and I'll let you work out for yourself how to do it if you want to union a countable number of intervals.
But, how do you measure something that's weird looking? For example, how big is the set of integers? Well, it should work out to be 0, since if you think about it, they don't really take up any space on the line. They're just like inch marks on an infinitely long ruler. To cut to the chase, the way that Lebesgue thought to do this was to put intervals (which we know how to measure) around sets, and call the measure of the set the infimum of the intervals we can put around it. The infimum of an ordered set is just its greatest lower bound for my intro analysis students out there. So, revisiting the integers question above with this new definition of measure makes it obvious because we can certainly cover the integers with a bunch of very small intervals, arbitrarily small, in fact. That's a sort of baby analysis problem for you, so I won't bother working out all the details without TeX handy.
Alright, so I jumped around a bit, going from metrics to measures, which seem from the names like they should be the same thing, but aren't. Now I'm going to tell you how to induce a metric from a measure, which as you recall, we used to induce a topology. Stick with me through all this terminology.
Some of the properties of a measure make it very attractive as a candidate to induce a topology not on say, the real line itself, but rather on the set of its subsets, called its power set, since a measure measures sets, not points. How would we do that? Intuitively, sets are close together if they overlap quite a bit and far apart if they don't. Less intuitively but still pretty clear, what we are really concerned with there is the parts of sets that DON'T overlap. For example, [0,2] and [1,3] have as their intersection [1,2], but so do [-100,2] and [1, 100] but this second set of measures seems more far apart than the first one. So what we want to measure is the symmetric difference of two sets, (A/B)U(B/A).
This seems to work out nicely, as (A/A)U(A/A) is empty, so it has measure 0. Furthermore, using the symmetric difference makes our would-be metric symmetric. You can check the triangle inequality for yourself, but you'll note that there is a little problem with our definition.
What is the "distance" between [0,1]U{2} and [0,1]? These are not the same sets (not the same points in a metric way of thinking), but their symmetric difference is {2}, which has measure zero, so according to our "metric" these points (sets to a measure way of thinking) are the same. That's a peculiarity.
So what do we do? What mathematicians always do in this kind of situation. Mod out.
What I mean is, these sets aren't equal, but the "metric" tells us that they are, so let's just say that they are and work from there. More precisely, let's define an equivalence relation R by saying that two sets are equivalent if their symmetric difference is 0. Now we have a new space, the set of equivalence classes mod R of [measurable]* sets of real numbers. Using the our "metric" based on the symmetric difference of two sets now gives an actual metric. So the question is, what are open sets in our induced topology? What are closed sets? Compact? Etc., Etc.
Have fun with that for a while, I'm on break.
*I say measurable because it turns out that not all sets are measurable (in the Lebesgue sense). What kind of sets aren't measurable? I don't know, but I can tell you what kind of sets are: Borel sets. These are the kind of sets that you get by performing countable set operations (unions, intersections, complements) on intervals. So, you are going to have to work hard to find a set that isn't measurable, but they are out there. In fact, because there are non-measurable sets in Euclidean 3-space, you can take apart the surface of a sphere and rotate the parts around without stretching them or anything and reassemble them into two spheres of the same size as the original sphere. I know this makes no sense, but it's called the Banach-Tarski paradox and it's one of the coolest results out there.
I've talked about topologies before, so if you are interested in reading my ramblings but don't know what it is, you can find it in some old post. Or you can just use Wikipedia like a normal person.
Anyway, many useful topologies are defined by a function called a metric, which just measures distance between two points. A metric d is defined as having three properties:
1) d(x,y) >= 0 and d(x,y) = 0 iff x = y (if we relax this last condition, it is a pseudo-metric
2) d(x,y) = d(y,x)
3) d(x,z) <= d(x,y) + d(y,z)
You can see that distance in the normal sense meets all of these conditions. In fact, property 3) is called the triangle inequality and you have to use it all the time in analysis.
The way that a metric induces a topology is pretty straightforward. You just define an "open ball" as the set of all points less than some distance from a point, which you can call the center of the ball. Picturing this in the Euclidean 3-space, known to non math nerds as just 3 dimensions in the usual sense, means spheres of some radius, not including any points on the surface. From here, you just say that a set is open if every point in the set has an open ball containing it which is entirely contained in the set. In Euclidean space, this again translates into anything that is missing its boundary. I'm not going to define boundary for you, although it is a rigorously defined thing in general topology, but it should be clear in Euclidean space what that means. For example, in Euclidean 1-space, heretofore known as the real line, the boundary of the interval [a,b) is {a,b}. So, this set is not open, since it contains some of its boundary. More rigorously, open balls in 1-space just mean open intervals, so if you try to put an open ball around a, it will contain things to the left of a, which can't be in the original interval. Then that set can't be open. It's not closed, either, but I won't get into that.
You'll note that the way we defined this topology does indeed give us a topology:
1) The empty set has no points, so it vacuously meets the condition to be open. Obviously, any open ball of a point will be contained in the entire space, so the entire space is also open.
2) If a point is in a union of open sets, it's in one of them, so it's got one of these open balls, which as a subset of one of the sets, is a subset of the union, so a union of open sets is open.
3) If you intersect two open sets and choose a point in their intersection, then there is an open ball centered at that point corresponding to each of the two sets. Just choose the minimum of those two radii and you've got yourself the open ball you wanted.
Anyway, I was just dragging my feet on undergrad stuff until now. It's time to step it up to measure theory, but not really. I'm just going to glide over all the annoying parts of trying to set up integration the Lebesgue way because it doesn't matter for what I want to get to.
Intuitively, how "big" is the interval [0,1]? It should have length 1-0 = 1, right? How about the interval (a,b)? If you said b-a, you know how to generalize, but aren't Stieltjes (which just means you are normal). Anyway, how about the interval (a,b]? It should still be b-a, right? All we added was one point, and that point should be infinitely small in a certain geometric sense.
Now, how about a union of intervals, like, say, [0,1] U [2,3]? It should just be 2, to my mind since it's just two (disjoint) intervals of length 1. And how about [a,b]U[c,d], assuming c>b; that is those intervals are disjoint? If you said b-a+d-c, congrats. So I think we're clear on how to "measure" intervals, and I'll let you work out for yourself how to do it if you want to union a countable number of intervals.
But, how do you measure something that's weird looking? For example, how big is the set of integers? Well, it should work out to be 0, since if you think about it, they don't really take up any space on the line. They're just like inch marks on an infinitely long ruler. To cut to the chase, the way that Lebesgue thought to do this was to put intervals (which we know how to measure) around sets, and call the measure of the set the infimum of the intervals we can put around it. The infimum of an ordered set is just its greatest lower bound for my intro analysis students out there. So, revisiting the integers question above with this new definition of measure makes it obvious because we can certainly cover the integers with a bunch of very small intervals, arbitrarily small, in fact. That's a sort of baby analysis problem for you, so I won't bother working out all the details without TeX handy.
Alright, so I jumped around a bit, going from metrics to measures, which seem from the names like they should be the same thing, but aren't. Now I'm going to tell you how to induce a metric from a measure, which as you recall, we used to induce a topology. Stick with me through all this terminology.
Some of the properties of a measure make it very attractive as a candidate to induce a topology not on say, the real line itself, but rather on the set of its subsets, called its power set, since a measure measures sets, not points. How would we do that? Intuitively, sets are close together if they overlap quite a bit and far apart if they don't. Less intuitively but still pretty clear, what we are really concerned with there is the parts of sets that DON'T overlap. For example, [0,2] and [1,3] have as their intersection [1,2], but so do [-100,2] and [1, 100] but this second set of measures seems more far apart than the first one. So what we want to measure is the symmetric difference of two sets, (A/B)U(B/A).
This seems to work out nicely, as (A/A)U(A/A) is empty, so it has measure 0. Furthermore, using the symmetric difference makes our would-be metric symmetric. You can check the triangle inequality for yourself, but you'll note that there is a little problem with our definition.
What is the "distance" between [0,1]U{2} and [0,1]? These are not the same sets (not the same points in a metric way of thinking), but their symmetric difference is {2}, which has measure zero, so according to our "metric" these points (sets to a measure way of thinking) are the same. That's a peculiarity.
So what do we do? What mathematicians always do in this kind of situation. Mod out.
What I mean is, these sets aren't equal, but the "metric" tells us that they are, so let's just say that they are and work from there. More precisely, let's define an equivalence relation R by saying that two sets are equivalent if their symmetric difference is 0. Now we have a new space, the set of equivalence classes mod R of [measurable]* sets of real numbers. Using the our "metric" based on the symmetric difference of two sets now gives an actual metric. So the question is, what are open sets in our induced topology? What are closed sets? Compact? Etc., Etc.
Have fun with that for a while, I'm on break.
*I say measurable because it turns out that not all sets are measurable (in the Lebesgue sense). What kind of sets aren't measurable? I don't know, but I can tell you what kind of sets are: Borel sets. These are the kind of sets that you get by performing countable set operations (unions, intersections, complements) on intervals. So, you are going to have to work hard to find a set that isn't measurable, but they are out there. In fact, because there are non-measurable sets in Euclidean 3-space, you can take apart the surface of a sphere and rotate the parts around without stretching them or anything and reassemble them into two spheres of the same size as the original sphere. I know this makes no sense, but it's called the Banach-Tarski paradox and it's one of the coolest results out there.
Friday, February 12, 2010
Converge Slow, Homie
Since a string of digits recently asked me to mention something about math on here, I'll bring up a problem I am working on that should be easier than it is.
Hopefully, we are all familiar with convergence in the numerical sense, but if not, I'll try to hand-wave at it so that even an eighth grader can understand it. I've been told a good teacher can explain anything so that an eighth grader can understand it. It seems like an arbitrary line to me, but maybe a good enough one. Perhaps that is when people start displaying abstract thinking ability.
So, we can start with a sequence. A sequence is just a special kind of function, and for our purposes, we'll stick to sequences of real numbers. As to what a real number is, it's just about any kind of number you can think of that doesn't involve i somewhere. So whole numbers, 0, fractions, even irrational stuff, like 2^(1/2) or pi.
That said, a sequence is just a function of natural numbers, so something like
1, 2, 4, 9, 16, ... you can see how this sequence "goes to infinity," in that it just keeps getting bigger (I am purposefully being vague about this concept). On the other hand, the sequence s(n) = 1/n, that is
1, 1/2, 1/3, 1/4, ... doesn't keep getting bigger; it keeps getting smaller. However, it doesn't "go to negative infinity." In fact it demonstrates the central idea of calculus, which is convergence. In particular, it is said to converge to 0, or that the limit as n approaches infinity of s(n) is 0. What do I mean by that? I mean that we can think of this sequence as approximating 0, as if I didn't know what 0 was, but I was guessing at it, and each time I guessed, my guess got closer. The sequence is said to converge to a number if it approximates that number to any error. More formally,
A sequence s(n) is said to converge to a number L if and only if for all E > 0, there exists an index N such that if n > N, |s(n) -L| < E.
If you think about it, it just says that the sequence gets as close to L as we would like and stays at least that close; that we can approximate L infinitely well with a big enough term of s(n).
A proof of convergence usually goes like this: Given E > 0 (we actually use epsilon, usually)
|s(n) - L |
(bunch of algebra with inequalities)
< E for n such and such
That is, we usually set E and find an expression for N in terms of E that suffices. In the simple case above, you just set N = 1/E and you're good.
Sometimes it isn't so easy, and that is what I'm dealing with at the moment. The sequence I'm looking at is
S(n) = 1, 1/2, (1/2)(3/4), (1/2)(3/4)(5/6), ...
And so on. The denominators are just the product of the even numbers and the numerators are the product of the odds, always smaller. Each factor is less than 1, so each term is smaller than the last term, but that's not enough to show that it converges to 0. One idea might be a comparison test.
That is, if I can show that for any positive integer k, there's a positive integer N such that S(N) < 1/k, I can just compare it with the previous limit problem and say the limits must be the same. [It is easy enough to show that a sequence of positive numbers cannot converge to a negative number, so the new sequence must be "squeezed" between the old 1/n sequence and 0.]
Just working out some terms of the sequence explicitly, I've found that the limit must be less than .15, and I'm convinced that it is actually 0; that I can somehow show it is squeezed down by 1/n if we look far enough along the sequence. The problem is that this convergence is very slow. You'll note that each subsequent factor is bigger than the last, in fact, the last factor converges to 1. However, they still are less than one, so they make each term decrease, just by less and less. It is rather annoying and making it hard for me to find the right expression or technique.
Anyway, it is just part of a somewhat bigger problem related to a theorem of Tauber.
Hopefully, we are all familiar with convergence in the numerical sense, but if not, I'll try to hand-wave at it so that even an eighth grader can understand it. I've been told a good teacher can explain anything so that an eighth grader can understand it. It seems like an arbitrary line to me, but maybe a good enough one. Perhaps that is when people start displaying abstract thinking ability.
So, we can start with a sequence. A sequence is just a special kind of function, and for our purposes, we'll stick to sequences of real numbers. As to what a real number is, it's just about any kind of number you can think of that doesn't involve i somewhere. So whole numbers, 0, fractions, even irrational stuff, like 2^(1/2) or pi.
That said, a sequence is just a function of natural numbers, so something like
1, 2, 4, 9, 16, ... you can see how this sequence "goes to infinity," in that it just keeps getting bigger (I am purposefully being vague about this concept). On the other hand, the sequence s(n) = 1/n, that is
1, 1/2, 1/3, 1/4, ... doesn't keep getting bigger; it keeps getting smaller. However, it doesn't "go to negative infinity." In fact it demonstrates the central idea of calculus, which is convergence. In particular, it is said to converge to 0, or that the limit as n approaches infinity of s(n) is 0. What do I mean by that? I mean that we can think of this sequence as approximating 0, as if I didn't know what 0 was, but I was guessing at it, and each time I guessed, my guess got closer. The sequence is said to converge to a number if it approximates that number to any error. More formally,
A sequence s(n) is said to converge to a number L if and only if for all E > 0, there exists an index N such that if n > N, |s(n) -L| < E.
If you think about it, it just says that the sequence gets as close to L as we would like and stays at least that close; that we can approximate L infinitely well with a big enough term of s(n).
A proof of convergence usually goes like this: Given E > 0 (we actually use epsilon, usually)
|s(n) - L |
(bunch of algebra with inequalities)
< E for n such and such
That is, we usually set E and find an expression for N in terms of E that suffices. In the simple case above, you just set N = 1/E and you're good.
Sometimes it isn't so easy, and that is what I'm dealing with at the moment. The sequence I'm looking at is
S(n) = 1, 1/2, (1/2)(3/4), (1/2)(3/4)(5/6), ...
And so on. The denominators are just the product of the even numbers and the numerators are the product of the odds, always smaller. Each factor is less than 1, so each term is smaller than the last term, but that's not enough to show that it converges to 0. One idea might be a comparison test.
That is, if I can show that for any positive integer k, there's a positive integer N such that S(N) < 1/k, I can just compare it with the previous limit problem and say the limits must be the same. [It is easy enough to show that a sequence of positive numbers cannot converge to a negative number, so the new sequence must be "squeezed" between the old 1/n sequence and 0.]
Just working out some terms of the sequence explicitly, I've found that the limit must be less than .15, and I'm convinced that it is actually 0; that I can somehow show it is squeezed down by 1/n if we look far enough along the sequence. The problem is that this convergence is very slow. You'll note that each subsequent factor is bigger than the last, in fact, the last factor converges to 1. However, they still are less than one, so they make each term decrease, just by less and less. It is rather annoying and making it hard for me to find the right expression or technique.
Anyway, it is just part of a somewhat bigger problem related to a theorem of Tauber.
Thursday, February 11, 2010
Snow Day 2: Bad Sequel Joke
I always see things with the number 2 appended to them, and then ":Electric Boogaloo" as a joke. I can't think of any specific times, but it seems like this is a hip joke to make, and every time it comes up I know it is going to happen and that annoys me. I suppose it is awesome to make fun of terrible movies from the 80's, but aren't we past that by this point? Can't we move on to making fun of terrible movies from the 90's?
Regardless, it is yet another snow day here in the city of brotherly love, which is hilarious to me because who cancels college classes, especially two days in a row? But, whatever, it has extended my weekend Wednesday - Monday, I think. I have to check if it is a holiday on Monday or not, but I never have specific work on Fridays and that pleases me in the greatest. I think I can manage a 2 day workweek. Alright, peace out!
Regardless, it is yet another snow day here in the city of brotherly love, which is hilarious to me because who cancels college classes, especially two days in a row? But, whatever, it has extended my weekend Wednesday - Monday, I think. I have to check if it is a holiday on Monday or not, but I never have specific work on Fridays and that pleases me in the greatest. I think I can manage a 2 day workweek. Alright, peace out!
Wednesday, February 10, 2010
Snow Day
Philly got hit pretty hard with a snowstorm, so today's classes were preemptively canceled, which is nice because it gives me more time to type up my homework and to generally laze about. As I'm not really doing anything, I don't really have that much to add at the moment. Just this:
The wind, it was howlin', and the snow was outrageous
We chopped through the night, and we chopped through the dawn
When he died, I was hopin' that it wasn't contagious
But I made up my mind and I had to go on
Know what that's from?
The wind, it was howlin', and the snow was outrageous
We chopped through the night, and we chopped through the dawn
When he died, I was hopin' that it wasn't contagious
But I made up my mind and I had to go on
Know what that's from?
Friday, January 29, 2010
Some Movies
Greetings internet. I have not posted in a bit because nothing really happens that is of interest. I did watch a few movies lately, though, so I will share my thoughts on them.
Avatar - If you are alive you have probably heard that the special effects are amazing but the story is not very good. I just want to reiterate that the story is essentially just cliches linked together and I was falling asleep after about the first twenty minutes. None of the characters is interesting in the slightest (don't let reviewers fool you into thinking otherwise about the typical hardass scientist played by Ripley herself), the premise is the kind of ludicrous tripe that gives science fiction a bad name, and even the supposedly vibrant world that Cameron has imagined is fairly dull. Making things glow does not amount to imagination, and it's actually really stupid because there's no reason whatsoever for all these organisms to evolve neon lights. I'd also like to say that the already obvious allegory for Native Americans is pounded into your head over and over again and is actually insulting. Io9 had a good article about how it is a white guilt fantasy. Anyway, I'm done complaining about this movie. Except that the 3D glasses hurt my head. Now I'm done.
The Hurt Locker - The critics really hyped this, and they are right. It is awesome. It's surprisingly funny in a very dark way. Anyway, you should see it.
Sherlock Holmes - Pretty dang good. Robert Downey, Jr. plays himself or Iron Man in old timey clothes, but who doesn't like that? Jude Law actually manages to seem like he's having fun for once instead of being so upset about having to act. The story is fun, with enough mystery to keep you interested between absurd fight scenes. I'm looking forward to a sequel, anyway.
Alright, that's it, go home.
Avatar - If you are alive you have probably heard that the special effects are amazing but the story is not very good. I just want to reiterate that the story is essentially just cliches linked together and I was falling asleep after about the first twenty minutes. None of the characters is interesting in the slightest (don't let reviewers fool you into thinking otherwise about the typical hardass scientist played by Ripley herself), the premise is the kind of ludicrous tripe that gives science fiction a bad name, and even the supposedly vibrant world that Cameron has imagined is fairly dull. Making things glow does not amount to imagination, and it's actually really stupid because there's no reason whatsoever for all these organisms to evolve neon lights. I'd also like to say that the already obvious allegory for Native Americans is pounded into your head over and over again and is actually insulting. Io9 had a good article about how it is a white guilt fantasy. Anyway, I'm done complaining about this movie. Except that the 3D glasses hurt my head. Now I'm done.
The Hurt Locker - The critics really hyped this, and they are right. It is awesome. It's surprisingly funny in a very dark way. Anyway, you should see it.
Sherlock Holmes - Pretty dang good. Robert Downey, Jr. plays himself or Iron Man in old timey clothes, but who doesn't like that? Jude Law actually manages to seem like he's having fun for once instead of being so upset about having to act. The story is fun, with enough mystery to keep you interested between absurd fight scenes. I'm looking forward to a sequel, anyway.
Alright, that's it, go home.
Tuesday, January 12, 2010
Japan Trip
I had previously posted that I would post later about my trip to Japan, but I haven't felt like doing that lately. It was a lot of fun, but I don't know how interesting it would be to anyone else. I don't have any pictures to go along with it because I didn't have a camera. But here are some things I did.
-Visited my 第に故郷 or second hometown. It was pretty nice to see some people again and surprise the folks at the BOE. I spent basically a whole day at a kindergarten that asked me to come as soon as they heard I was there, playing with little kids.
-Spent about a day and a half playing with the nephews/niece of the friend of Mie, who was also visiting Japan with her American husband. The friend was. Mie isn't married. I guess that was confusing, and I guess I also rock with little kids. We built a snowman even though there was only about an inch of snow on the ground. Japanese snowmen are made of only two snowballs instead of three.
-Visited Osaka castle, which was sort of funny in that I'd been there twice and neither of the Japanese people in our group had ever been there. It's a pretty decent castle with a museum in time.
-Ate some awesome okonomiyaki in Osaka.
-Ate some Osaka style ikayaki, which is cooked (grilled?) squid. I don't know if it's actually Osaka style, or if that's a thing, but that's what it said, and it was different from normal ikayaki.
-Ate kitakyuushuu style ramen in Fukuoka at a place called Ramen Stadium.
-Ate yakiramen, a Hakata specialty, from a street vendor in Fukuoka, in the same night.
-Stayed in an awesome ryokan with an onsen for a couple days.
-Nomihoudai!
-All around awesomeness.
That's about it for now.
-Visited my 第に故郷 or second hometown. It was pretty nice to see some people again and surprise the folks at the BOE. I spent basically a whole day at a kindergarten that asked me to come as soon as they heard I was there, playing with little kids.
-Spent about a day and a half playing with the nephews/niece of the friend of Mie, who was also visiting Japan with her American husband. The friend was. Mie isn't married. I guess that was confusing, and I guess I also rock with little kids. We built a snowman even though there was only about an inch of snow on the ground. Japanese snowmen are made of only two snowballs instead of three.
-Visited Osaka castle, which was sort of funny in that I'd been there twice and neither of the Japanese people in our group had ever been there. It's a pretty decent castle with a museum in time.
-Ate some awesome okonomiyaki in Osaka.
-Ate some Osaka style ikayaki, which is cooked (grilled?) squid. I don't know if it's actually Osaka style, or if that's a thing, but that's what it said, and it was different from normal ikayaki.
-Ate kitakyuushuu style ramen in Fukuoka at a place called Ramen Stadium.
-Ate yakiramen, a Hakata specialty, from a street vendor in Fukuoka, in the same night.
-Stayed in an awesome ryokan with an onsen for a couple days.
-Nomihoudai!
-All around awesomeness.
That's about it for now.
Sunday, January 3, 2010
Baby Got Back
I just got back from Japan and I already miss it, especially the Mie part of it. More updates when it isn't like two in the morning.
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