Mental Illness

There has been a lot of talk about Eastern vs Western worldview differences, and I was curious as to whether mental illness had a lower frequency in the East. My hypothesis was that people whose worldview involved meditation and introspection as opposed to a worldview almost entirely based on external things would make a significant difference.

There are two immediate problems. Due to the worldview differences, the documented cases of mental illness in the East is probably significantly lower than the actual cases. The other major problem is that these worldviews are not very separate anymore. Any study in today’s globalized world would be tainted, since there are very few people who have exclusively one or the other view.

So I started web researching to see what would come up. It seems as if a study such as this may not have ever been done, but there has been lots of social studies to see how conceptions of mental illness are difference. So I altered my search to see if meditation in particular had been studied in relation to mental illness. This proved challenging as well, in that many non-scientific sites came up as personal testimony from someone that claimed to have cured an illness/mental illness through meditation.

There were far too many sites for me to try to sort through if any had what I was looking for considering this is just a random passing question I had. So briefly let’s look at some causes and see if these causes are less likely to occur in someone with an introspective worldview.

Cause 1: Neurotransmitter imbalance. I don’t really want to comment on this one without scientific support, but I am sure that there are scientific studies of people seriously altering their brain chemistry. Now if they had any control over neurotransmitters is another question. It doesn’t seem too unreasonable that meditation could help prevent this.

Cause 2: Heredity. Now according to it is pretty random whether or not you inherit a mental illness, and it usually takes high stress, abuse, or trauma to trigger it. This seems to suggest to me that it is not genetic (though they claim it is), and that it is more of a mimic syndrome. You see someone in your family with a mental illness your whole life, then something traumatic happens and you develop those same symptoms. If this is the case, then meditation can definitely help empower someone to overcome the sense that you must develop these symptoms.

Cause 3: Infections/Brain injury. There doesn’t seem to be much you can do about an actually physical accident that occurs. There are without a doubt people who claim to have healed themselves through meditation, but this is far less common and also we are looking at prevention not healing. So can meditation help prevent an accident from occurring, probably not.

Cause 4: Poor nutrition/toxins. I do have to say that this is where Eastern culture and not just meditation can probably help. Many Eastern cultures such as Buddhism, Taoism, etc advocate a vegetarian or semi-vegetarian all-natural diet. The West seems obsessed with highly processed food and also animal products that have been produced in mass quantity. This mass quantity usually means that chemicals were used to enhance the amount. So I’d say the Eastern vs Western can be a prevention here, but meditation alone probably won’t do much.

I’ll keep digging randomly to see what I turn up and maybe post on this topic again in a week or two if I find anything new. Note: this is not a scientific or documented study. It was all opinion. No information here should be used as if it were fact.


Cat’s Eyes

I’ve been running a very unscientific experiment for awhile now. It has produced some very interesting results that I can’t seem to explain. So every time my cat comes up to me and wants to jump on my lap (several times a day for at least a month I’ve been running this), I try something new. I either look near her, or don’t look at her at all, or look at her eyes, or some variation. The point was that I noticed she seemed to only make the jump when after we made eye contact.

Recently I’ve tried to be very specific about the different between looking near or at a different part of her body vs making eye contact. It may be my imagination as hardcore scientists would probably tell me, but I’m am quite convinced that she understands the difference between eye contact and non-eye contact.

Here is why I don’t understand this. Even if we make a huge leap and grant full out consciousness to cats (probably an overkill assumption), I still can’t explain it. Humans understand eye contact because they are aware that that is where sight comes from. I can see of no scenario in which a cat can come to the realization that eyes are where sight and recognition come from.

The only explanation I can come up with is that she is getting up there in cat years, and so has developed a conditioned response through years of experience that eye contact means that I am paying attention to her. As a good Pavlovian, we could say that she doesn’t have to have any idea what the cause is or why it works, it is merely a conditioned reaction.

Anyone else that might know something about animal behavior want to take a crack at this one?

Film Analysis

Today I saw a great film, yesterday I saw a horrible film. The great film was The Hawk is Dying (THID) and the horrible film was Ellie Parker (EP). It is rare that I see such a contrast so close together, so I’ve decided to analyze it. I haven’t analyzed any films yet on here (in my memory). Here is an interesting tidbit about why I’ve chosen these; both are independent. This means that to the average box office movie-goer, both of these films would probably be written off as “unbearable,” “plotless,” or “arty.” To the average independent film-goer, both of these would be considered huge successes. This gives me great opportunity to explore at a deeper level than just genre basing my opinions.

My typical analysis of whether a film is bad, OK, good, or great goes through many layers. So let’s lay out a bunch of parts of a film first. There is: acting, script, editing, directing, and cinematography as major categories (of course there are others, but these catch my attention first). To pass the first level, a film has to show proficiency in each of these areas. THID definitely passes round one in each of these categories. EP had some slight issues with acting. It was superficial at times. I’ll get to directing later, but this was definitely very immature directing, as well. Overall, both pass round one, but EP shows some weaknesses that could pose problems in later rounds.

Round two: originality vs predictability. In my mind, you either follow the formula or you break out of it. Now these are both independent films, so by their very nature they don’t follow the formula or else they would have funding from a major studio. Here is the interesting thing, though. THID decided to take a rather mundane idea (taming a hawk) and pull out a very original and intense film. EP decided to take the originality route from the ground up. This is sort cliche in my opinion. It is much more impressive to take something that is common place and make it original than to just toss together all sorts of randomness to make it original. Whatever the case may be, both pass round two.

Round three: balancing the creative and technical. This is the eternal struggle in creating art. You must know and be able to follow the rules in order to break them. A musician must have incredible technical proficiency that comes from training scales, arpeggios, and etudes with the relentless click of a metronome before they can play expressively, bending the tempo and tone to fit the mood. A film can easily fall prey to being too technical and dry or too artistic and without any technique. The two go hand-in-hand. THID blends this beautifully. The technique is often awe inspiring. Angles, lighting, and acting are pitch perfect to fit the scene. Every shot is thought about. EP goes over-the-top. It is what I often call “young director’s syndrome.” A director that is overflowing with ideas often lets too many of them spill over into the film. EP is filled with rapid editing showing a tour de force in both camera work and editing. It serves no purpose, though. It is a mindless show of flash. It lacks the artistic side. This can be explained perfectly, though. This was Scott Coffey’s first directing experience. He had been an actor in several Lynch films. For an aspiring director, it is clear that he was trying to imitate Lynch. The thing he didn’t realize is that Lynch does things with a great vision and masterful artistry. The technical display by Lynch is vacuous like what Coffey did. THID is clearly superior in this round.

Finally, I fall to overall purpose. Now I am a big supporter of art for art’s sake (meaning a work of art doesn’t need a purpose), but that is part of the process. I determine if there was a purpose or not. If not, did it accomplish that? If so, how well and accessible was it? THID also masterfully pulled this off. Nothing was handed to the watcher. This is important to me. If the purpose is handed to me, then I feel like the director and writer didn’t trust me (a common problem with major pictures). I like that I had to work and figure things out. Overall, nothing was beyond the watcher’s capabilities, though. EP on the other hand had another young director’s faux pas. It confused its purpose. It posed as a truly pure art film without a purpose, while at the same time it had one. This is a common mistake. Young directors want arty directors to respect them, so they think if their film has purpose then they will be written off as not serious or commercial or something. What they don’t realize is that trying to come across as something your not ruins the film.

Overall, THID was excellent. It had great technical proficiency balanced with the perfect amount of artistry to keep the watcher interested. It took the common place and made it original. EP tried to be too much. It was like freshmen writing class. You have an overambitious student take a grand topic and write a short essay on it. They try to cover everything. The true art of writing comes when you can take a very small idea and develop it in a long work. The artist that tries too much ends up digging their own grave. Yes, I was taught this by my freshmen writing professor when she took me aside and harshly told me that I was trying to write a lifetime’s work into a 10 page essay. Narrow the focus. Use the tools that suit the purpose. Be ambitious, but don’t let every idea that pops into your head appear in your work. This is the difference between an experienced director and a new one.

Hope you now have a better idea of how I critique films 🙂

On Parenting

Quick update. I decided to check out what randomly linked from last post. It turns out one of them is to a parenting blog and that children should be taught strict manners. Clearly, I’m in general opposed to this (of course, children shouldn’t go around terrorizing, but teaching them to blindly follow rules that have no justification is a scary thought as well).

All I have to say is that posting some of my views in the comments was not a good idea. Lots of scary moms started yelling at me that I had no idea what I was talking about. How dare I criticize when I don’t have any children. I find this slightly ironic considering the post. It seems to me quite impolite to dismiss someone’s opinion because they approach it from outside your little group. I actually find that the opinions I value most come from outside the group. They have a fresh perspective I haven’t thought about before.

Also, this is a scary attitude to have. The “You can only have a valid opinion if you have an invested interest” attitude. Essentially they are like when a tobacco company does research on the health effect of smoking. Of course, when you don’t go outside the group all your data is going to fit your predetermined conclusion. They seem to think that parents that raise their kids to be polite turn out polite and parents that don’t instill some sort of military-like rigid manners will turn out to be these horrible people. In my experience there seems to be very little correlation.

I wanted to write: a child grows up, and the character that person becomes is more than the sum of its parts…erm, I mean, is more than the sum of one part: parenting. There are: brain chemistry, early psychology developed while at school and with friends, hormone alterations, genetics, mass media. That is just a tiny list, too. How could a parent possibly have that much impact with so much out of their hands? I’m not saying that parenting isn’t important. It is one of the parts, but it shouldn’t be considered the sole factor in how a child turns out as these mothers seem to think.

I just had to get that out there. I couldn’t do it on their site. Is this really that unreasonable?

Ethics of Manners

This has bothered me from my earliest years. When I was little, I never understood the seemingly pointless rules people had to follow called “manners.” I was going to do some research before writing this to make sure I’m not way off base. I also wanted it to be well-researched so that it would be taken seriously. Oh well, I’m more of an impromptu type of person.

I’ve had several experiences in the past couple of weeks that has brought this back into my mind. I tried to read Lynn Truss’ Talk to the Hand: The Utter Bloody Rudeness of the World Today, or Six Good Reasons to Stay Home and Bolt the Door. I became infuriated at how she was confusing and oversimplifying ethical issues to make it sound like the problem was with people’s lack of manners. In fact, I admit I never finished it due to this frustration. The second was with my whole rant on Sam Harris who basically is claiming that upholding manners is causing lots of unnecessary suffering. Let’s be rude! These two things started allowing me to notice manners in the world more.

In the beginning there was the word and according to the definitions are:

2a. “the prevailing customs, ways of living, and habits of a people, class, period.”

3. “a person’s outward bearing; way of speaking to and treating others”

I think that in 2a we immediately see a clash of interest for me. You should always have good reasons to do things. Doing something because it is the way it has always been done is not a good reason. This is why I hate that culturally imposed behavior is so hard to break. Sometimes it is not good. Slavery was socially acceptable at one time. If people hadn’t been “utter bloody rude,” then the practice would still be going on today. OK. So anyone with philosophical training is going to call a red herring on me right about now. I’m examining the definition of manners and I bring up slavery. All I’m trying to say is that according to 2a, manners are a social norm, and in the past social norms have been seen to be unethical. I’ll build the case later that manners are precisely this.

One semi-irrelevant thing to think about is that if manners are a form of ethical conduct, then we have a case of cultural ethical relativism, since every culture has different sets of manners. In fact, some sets of manners are in direct conflict with each other.

Evidence for manners being unethical: 1. they are a form of lying, 2. they are useless and waste people’s time (and hence are ironically rude), 3. they allow people to practice unethical behavior behind an acceptable name.

1. According to definition 3 (I should probably map this out visually for people who are no good at trying to follow where all these numbers are, but that would be polite and thus evidence against my case), manners are a person’s “outward bearing.” There is very explicit intonation there that this is not what the person is truly thinking. Let’s get real. Manners teach us to lie with dignity and in a socially acceptable way. You hate someone’s hair. Manners tells us to not go up to that person and say, “I hate your hair.” We’ll come back to this in 3 (not definition, but evidence 3).

2. Manners are rude. I may have accidentally constructed a zen koan on this one. I think it is rude to waste people’s time. This is probably the general consensus. Well, picture yourself in this common situation. You are passing someone you know. You have nothing to say to them. Manners says that you should be polite and make at least a little small talk. This accomplishes nothing. In fact, usually there are lies exchanged (see 1) such as, “How are you?” “Fine.” It could be the worst day of your life. You will probably say, “fine.” (see the play Wit by Margaret Edson). A few minutes later, you have exchanged absolutely no useful information, and everyone’s time has been wasted. Hmm…seems rude to have wasted that person’s time. What ever happened to that bit of manners that says, “If you have nothing useful to say, don’t say anything at all.” Erm…that’s not quite right, but a little altering of the truth can be polite we’ve already established.

3. This is a bit more serious, so I’ll drop the lighthearted tone. Now I’m talking about respect and human rights. This is where the slavery example comes back. There is a fine line between respecting a culture’s practices and allowing violations of human rights to occur. An example from all over the news recently (I think last week) was that a (the) gay Anglican bishop was not invited to the national conference. There is only one ordained gay Anglican bishop, because it is still technically against policy. The conference was holding debates about whether to allow gay people to be ordained. Don’t you think the only person with first hand experience should be allowed to state an opinion on the issue? So you might not like this example, but it was recent and it could be any of the hundreds of current examples of inequality being practiced somewhere.

We can always find a “proper manners” or respect argument to hide behind. We say that that is their culture, their belief, their faith, and if someone doesn’t like it then they shouldn’t practice that religion. What people don’t realize is that their manners are not saying what they think. If we allow a group of people to say that one type of person is better or worse than another (at a fundamental level), then this is not confined to the group. This is sending a message to what is now a globalized world that this type of practice is acceptable. The time for politeness is over. We can continue to use our excuse that we are being respectful to someone’s beliefs, but it is unethical to hide behind this cultural norm. Good manners are the cause of a lot of needless suffering and inequality.

Conclusion: the practice of good manners is unethical.

Measure Decomposition Theorems

Well, I’ve been mostly posting comments around on other people’s blogs and not really getting around to my own. I’m giving up on NCG for now. It seems that the stuff I already know I’m reading, and I’m skipping the stuff that will take effort to sort through. This seems pointless, especially with Analysis prelims coming up. That’s why things may take a turn in that direction over the next couple of weeks. I do still have at least one major ethical issue I want to sort out, though.

So. What is the most confusing part of measure theory? To me it is the fact that there are tons of ways to decompose your measure. In fact, I usually can’t remember which one is named what, and when to use which one. This post is an attempt to sort out which one is which, and what to look out for when you want to use them.

Jordan Decomposition: Any real measure \mu on a \sigma-algebra can be expressed in terms of two positive measures, called the positive and negative variations (\mu^+ and \mu^-) by \mu=\mu^+-\mu^-. This allows us to examine the total variation more easily, since |\mu|=\mu^+ + \mu^-. Also, it is quite simple to prove the existence and uniqueness since we can write \mu^+=\frac{1}{2}(|\mu| + \mu) and \mu^-=\frac{1}{2}(|\mu|-\mu).

Jordan decomposition seems to be used when you can prove something for positive measures and need to extend it to all measures. Since J decomp gets you any measure in terms of positive measures, this eases the process. The other main use is when you invoke the uniqueness along with the next decomp theorem.

Hahn Decomposition: This is different from all the rest. It is not a decomposition of the measure, but of the measure space. It says: Let \mu be a real measure on a \sigma-algebra \mathfrak{M} in a set X. Then there exist sets A and B in \mathfrak{M} such that X=A\cup B, A\cap B=\emptyset and such that the positive and negative variations \mu^+, \mu^- satisfy \mu^+(E)=\mu(A\cap E) and \mu^-(E)=-\mu(B\cap E), for any E\in\mathfrak{M}.

Things to note. This is not unique! Also, you get as a quick corollary that since the positive and negative variations are concentrated on disjoint sets, they are mutually singular. The Hahn decomp is usually invoked in conjunction with the J decomp, as in, “Let ____ be the J decomposition and A, B be the respective Hahn decomp.” These two together get you that the J decomp is minimal. In other words, if \mu=\lambda_1 - \lambda_2, where \lambda_1 and \lambda_2 are positive measures, then \lambda_1\geq \mu^+ and \lambda_2\geq\mu^-.

Lebesgue Decomposition: Let \mu be a positive \sigma-finite measure and let \lambda be a complex measure on the same sigma algebra, then there is a unique pair of complex measures \lambda_a and \lambda_s such that \lambda=\lambda_a + \lambda_s and \lambda_a \ll \mu and \lambda_s \perp \mu. Also, if \lambda is positive and finite, then so are the two parts of the decomp. Caution: the measure \mu MUST be sigma finite. This theorem says that given any complex measure and any sigma finite measure, you can decompose the complex one into two unique parts that are absolutely continuous with respect to and mutually singular with the sigma finite measure respectively.

The major use of this is when you want to invoke the Radon-Nikodym theorem to get an integral representation of your measure. The Radon-Nikodym theorem only works if your measure is absolutely continuous with respect to the other. Luckily, with Lebesgue decomposition you can always apply R-D to at least a part of the measure.

Polar Decomposition: Let \mu be a complex measure on a sigma algebra. Then there is a measurable function h such that |h(x)|=1 for all x and such that d\mu=hd|\mu|. Note that the name “polar” is in reference to the polar form of writing a complex number as the product of its absolutely value and a number of absolute value 1. I’m not entirely sure I’ve ever used this. I guess the main place that it seems useful is when working with the integral representation of the measure. If you need to manipulate with the total variation, then this gives you how to put it into the integral representation.

Those seem to be the big ones. This is quite possibly the most useful math post I’ve made. I didn’t go into too much depth, but hopefully if someone is struggling with the differences between these, or trying to get a vague idea of when to use them, this post will help. I suppose I could have elaborated a little by proving the simple claims and showing counterexamples for the “cautions.” This would have given a feel for using them. Oh well.

Ben Gibbard

I’m about to speak blasphemy. So close your eyes and don’t read if you live in Seattle and are a fan of indie music. This got a vote, so I’ll post about it. I’ll return to NCG next time. My outline is to discuss why Death Cab used to be a good band, what went wrong, and why their latest album is pretty horrible.

In the days of old Death Cab did some very good things. First, musically it was interesting. They are a “pop” band, and I think that is a fair label. Now I’m talking “pop” as a style and not as something that is “popular.” This means that it has pretty standard song format, something along the lines of ABABCA. The chords flowed it in a traditional sense. Any band that does only those two things I would consider bad, though.

They took this pop formula and altered it. They weren’t as repetitive as standard pop bands. They altered chords, or didn’t resolve their 9ths or suspensions. There was interesting texture. The instrumentation was nonstandard at times. There was always something that made it worth going back to. Also, the melody was rarely your typical boring melody that you catch onto after one listen. It was just solid, creative, original music making.

Lyrically, I found Death Cab to be quite successful as well. They didn’t moan about the standard boring trivial pop ideas. They had lyrics that required interpretation. Sometimes being quite poetic. Some examples are definitely needed:

Burn it down till the embers smoke on the ground
And start new when your heart is an empty room
With walls of the deepest blue

You may tire of me as our December sun is setting because I’m not who I used to be
No longer easy on the eyes but these wrinkles masterfully disguise
The youthful boy below who turned your way and saw
Something he was not looking for: both a beginning and an end

this is the moment that you know
that you told you loved her but you don’t.
you touch her skin and then you think
that she is beautiful but she don’t mean a thing to me.

They are harsh and poetic, beautiful and about real issues.

On to the new album. First off, it is far far too repetitive. I could barely listen to the album five times. I had the whole thing learned on the second listen. On the first listen some of the structure and melody and lyrics were so cliche and unoriginal that I could actually sing along without having listened to it before. I should be a little more specific.

Bixby Canyon Bridge: To be fair, this is one of the better songs. The lyrics aren’t too straightforward and the song structure is nontraditional. Still, they have this problem with repetition that I’ll get to on the next song.

I Will Possess Your Heart: Possibly the most repetitive song in history? Such a great idea, too. Too bad. Now I’m not against repetition if done properly. I think this was an attempt to recreate Transatlanticism, the problem is that along with the length and repetition of Transatlanticism there was a major overall direction. The repetition didn’t matter because it also kept changing and stayed interesting as it made its journey. This song has no direction. It builds for several minutes, then comes back and never really gets to where it was going. It’s too bad, since that bass line is great and could have been utilized more successfully if used sparingly. I won’t even go there with the lyrics. I’ll just say “I will possess your heart”?!?!

No Sunlight: I made it through this song maybe twice before I decided that if I ever heard the words no and sunlight next to each other I would probably punch whoever said them in the face. Possibly the most uninteresting song lyrically ever produced. Look up the lyrics. You should never see something like:

No sunlight, no sunlight.
No sunlight, no sunlight. (At all)
No sunlight, no sunlight.
No sunlight, no sunlight.
No sunlight, anymore…

That (At all) truly changes things up…gag!

Cath…: Definitely the closest to their old self. Quite interesting chord blocking. Nice shift up in the overall traditional structure. Not too repetitive. Lyrically it is OK: “As the flashbulbs burst she holds a smile. Like someone would hold a crying child.” Thank goodness that word finally creeped in, “like.” We like to call that a simile. Wow. You decided to use a poetic device? It only took a half hour into the album. I’ll listen to this several more times. Yay.

Talking Bird: No comment. See bad comments above.

You Can Do Better Than Me: Lyrically this album has been bad so far, but it is mostly due to the straightforward writing. No room for interpretation and about pretty simple things. Here we hit an all time low. “You can do better than me, but I can’t do better than you.” Why listen to Death Cab when you could just open up the Oxford English Dictionary of cliche and out would pop this song?

Grapevine Fire: 6/8 time. Really changing it up now…

Your New Twin Sized Bed: OED of cliche. “You look so defeated lying there in your new twin size bed.”

Long Division: See negative comments above.

Pity and Fear: The most original song musically on the album. They use almost a Mediterranean scale structure. I think I’ve been overemphasizing lyrics and not the nonoriginality of the songs so far. So far they have been ungodly boring and repetitive. The repetitiveness continues in this, but I think it works just because of the interesting chord structure and nonstandard form of the song. Yay number 2.

The Ice is Getting Thinner: OED of cliche. “There’s little we can say and even less than we can do,
to stop the ice from getting thinner under me and you.”

So overall rating: Somewhere less than 5 out of 10. Come on guys. If we wanted to hear unoriginal songs with unoriginal lyrics we would just turn on the radio.

Lebesgue Points

Just a quick detour. I’ve found a new reason to dislike analysis. I’m trying to learn Radon-Nikodym derivatives (i.e. an attempt to take derivatives in a general measure theory sense and maintain the Fundamental Theorem of Calculus for the Lebesgue integral), and Rudin uses the approach of Lebesgue Points. Since I’ve never learned this before, I’m not sure if the other methods are easier, but this is certainly proving to be rough. Apparently we are supposed to be familiar with random facts about LPs, even though this is the very first time the definition is given. So here are the random unproven statements about Lebesgue points that I’ve encountered and my proofs to go along with them. I don’t think all of these are what Rudin had in mind, since my proofs are far more complicated than one could probably just think through.

Definition: Let f\in L^1(\mathbb{R}^k), then x is a Lebesgue point of f if \displaystyle \lim_{r\to 0}\frac{1}{m(B_r)}\int_{B_r}|f(y)-f(x)|dm(y)=0. Where m is Lebesgue measure, and that B notation is the open ball centered at x of radius r. Yeah. Not the simplest definition to be assuming knowledge of.

Claim 1: If f is continuous at x, then x is a Lebesgue point of f (under the suitable conditions on f that will always be assumed in this post). Let f be continuous at x. Then let \varepsilon>0 and choose \delta>0 such that if |x-y|<\delta, then |f(x)-f(y)|<\varepsilon. Now whenever |x-0|<\delta, we have \displaystyle \big| \frac{1}{m(B_\delta)}\int_{B_\delta}|f(y)-f(x)|dm -0 \big| \leq \frac{1}{m(B_\delta)}\int_{B_\delta}\varepsilon dm =\frac{\varepsilon m(B_\delta)}{m(B_\delta)}=\varepsilon. i.e. The limit behaves as we would like and x is a Lebesgue point.

Claim 2: If x is a Lebesgue point of f, then \displaystyle f(x)=\lim_{r\to 0}\frac{1}{m(B_r)}\int_{m(B_r)}fdm. Now I’m not sure if it is just me, but things were just moved around, so the fishy business I’m going to pull doesn’t seem necessary. Let x be a Lebesgue point of f. Then

\displaystyle 0=\lim_{r\to 0}\frac{1}{m(B_r)}\int_{B_r}|f(y)-f(x)|dx
\displaystyle\geq \lim_{r\to 0}\big|\frac{1}{m(B_r)} \int_{B_r} f(y)dm -\frac{1}{m(B_r)}\int_{B_r} f(x)dm\big|
\displaystyle =\lim_{r\to 0}\big|\frac{1}{m(B_r)}\int_{B_r}fdm-f(x)\big|. Thus since the right side is positive and less than or equal to 0 get rid of the absolute value since it must be equal to 0 and we have \displaystyle 0=\lim_{r\to 0}\frac{1}{m(B_r)}\int_{B_r}fdm - f(x) and rearrange.

I think there was a third claim, but I can’t find it now. Also, these proofs may look rather trivial now, but when you are completely unfamiliar with the definition and properties, this is rather confusing to try to work out quickly to continue reading the proof. Hopefully this post will help future readers of Rudin when they come to this.

I guess since I’ve come this far I should probably post some bonus material just to see the point.

Interesting result 1: Almost every point of f is a Lebesgue point (still assuming appropriate conditions on f).

The point is to get to the definition of the derivative, so if for all measurable sets E, we have \mu(E)=\int_E fdm for some f, then f is called the Radon-Nikodym derivative and notationally it is usually written that d\mu=f dm (for the obvious reason that if you integrate both sides you get the first form). But that notation leads us nicely to a more familiar Leibniz-type notation: f=\frac{d\mu}{dm}. Now skipping some other interesting results, some of the meat of the theory comes out in a FTC type result

Interesting result 2: If f\in L^1(\mathbb{R}^k) and F(x)=\int_{-\infty}^x fdm, then F'(x)=f(x) at every Lebesgue point of f (and by IR 1 almost everywhere).


So this is going slower than I thought it would. I’m on 4 and I’ve basically given a definition that was incomplete. Maybe I’ll quit this endeavor if this one doesn’t go any better than the previous. I’m now going to make the assumption that if you could read NCG 2, then you were bored and so to cover more ground I’m going to make more demands on the reader.

A good example of the overview in NCG 2 is in electromagentism. Let V be a manifold. Let \mathcal{A} be the algebra of functions on V. It turns out the group of unitary elements of \mathcal{A} is the local gauge group of electromagnetism. Let D be the covariant derivative associated to the potential.

We can then express D by \mathcal{H}\stackrel{D}\to \Omega^1(V)\otimes_\mathcal{A} \mathcal{H}. Where \mathcal{H} is a \mathcal{A}-module and D as a derivation satisfies the Leibniz rule: D(f\psi)=df\otimes \psi + fD\psi where f\in\mathcal{A} and \psi\in\mathcal{H}. Now we know that each component of the potential couples equally with the Dirac spinors.

This is tending to be a little physics-like, so I’ll take a little diversion here. If you look at the Dirac equation it requires a collection of four objects satisfying: \gamma_0^2=I, \gamma_1^2=\gamma_2^2=\gamma_3^2=-I, and \gamma_i\gamma_j=-\gamma_j\gamma_i when i\neq j. These \gamma have 4 by 4 matrix representation that I won’t type (too cumbersome, but you can look up anywhere if you are terribly curious). Using Einstein summation convention, we get a nice compact form of the Dirac equation: i\hbar\gamma^i\partial_i\psi = mc\psi where \partial_i=\frac{\partial}{\partial x_i}.

So we have an interaction with the EM-field F^{ij} through the potential (A^0, A^1, A^2, A^3)=(\frac{1}{c}V, A_x, A_y, A_z). Just replace i\hbar \partial^i \to i\hbar\partial^i-eA^i to get the new Dirac equation \gamma_i(i\hbar\partial^i-eA^i)\psi=mc\psi where the state function \psi is a column spinor (\psi_i)_1^4\in\mathbb{C}^4.

OK. So back to what I was driving at. This means we can identify \mathcal{H} with \mathcal{A}. So if we define D by the rule D1=A\otimes 1=A along with the Leibniz rule. This leads to the operator D behaving exactly as I derived (no that wasn’t for nothing) above. After redefining some constants we get D\psi is the left side of the “new” Dirac equation.

And so now we have an electromagnetic theory for noncommutative geometry. Which is actually pretty important since like we said before, if we want the four forces in a unified theory we need them to work in a quantum/noncommutative framework. Hmm…sorry, since I feel like the vote that went for NCG was not a vote for physics examples.


As promised. I’m back to a conceptual description. What I think I’ll do is look to the post before and try to tease apart a single term into its concept.

What is a smooth manifold? Well first of all there are two parts to this “smooth” and “manifold.” We’ll start with manifold, and I’ll quickly say that smooth is a technical term so probably doesn’t mean quite what you think.

A manifold is a topological idea. It is also a “local” idea. This means that different manifolds can look radically different, but when you zoom in at a very small scale they start to look the same. Basically, manifolds are an attempt to generalize are what we generally think of as flat space. The surface of your table or a piece of graph paper gives you 2-dimensional space. It is flat and you can put nice perfect squares on it to give you a location. Likewise you can chunk out cubes of the air around you to give you three dimensional coordinates.

When we generalize we can have curved weird looking things, as long as at any given place we can zoom in really close and get something that looks like a piece of our flat nice space. This is the property of being “locally Euclidean.” Try it with your coffee cup. Zoom in really close at any chosen place. Pretend you’re an ant crawling around on the cup. If you were a small enough ant, even the curved parts are going to seem flat to you. So examples of a 1-dimensional would be a circle or curve. Examples of 2-dimensional manifolds would be the surface of your coffee cup or the surface of a sphere. Examples of 3-dimensional manifolds would be … heh. Notice how 1-dimensional manifolds have been living in 2 dimensional space and 2 dimensional manifolds have been living in 3-dimensional space. (There is a theorem about this I won’t get to that any n-dimensional manifold can live in 2n-dimensional space.)

There are technically two other properties that we’ve ignored. A manifold has to be second countable and Hausdorff. This was a triviality since the examples I gave were living in (were a subspace of) a higher dimensional space that we knew had these properties. So second countable is a pretty abstract idea. Technically it means that there is a countable basis. Or if you take the numbers 1,2,3,… you can for each number assign an open set. Taking this collection we can get all the other open sets. This is sort of meaningless since I didn’t describe a topological space… Hausdorff is easier. It means that given any two points that aren’t the same, we can zoom in close enough on each one, so that the zoom in of the one doesn’t have any of the same points as the zoom in of the other.

So you’re thinking, all those properties seem to be everywhere, how could something not satisfy those? That’s because a manifold is a generalization of our intuition of space. All the concepts of it are meant to be intuitive. It actually takes a little bit of work to find things that don’t satisfy those properties. I think those examples are for another time (along with what it means to be a smooth manifold).