I’m going to attempt to classify chords from first principles, forgetting about the restrictions imposed by existing terminology. Chords are essentially a partial harmonic series. Therefore, they can be indicated as a series of ascending integers indicating ratios of frequencies of elements in the chord, such as 1:2 (octave), 2:3 (fifth), 4:5:6 (major triad), and so on. This is for pure tone combinations. Real instruments contain overtones in each note, so the total effect is more complicated (or collapsed, depending on the view). We will just deal with pure tones for now.
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An elementary, nice lemma relating to the optimization of multivariable functions says that the smallest “big thing” is still bigger than the biggest “small thing”, in other words,

\(\min_x \max_y f(x,y) \ge \max_y \min_x f(x,y)\).
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It’s not like I play piano (I don’t), but I’ve never been convinced by most of Lang Lang’s performances. Too raw. Sure, one can argue it’s his own interpretation but it really is a bad one if it doesn’t make sense… I mean Beethoven is not video game music, which seems to be Lang Lang’s self professed straightjacket… Barenboim explains it well and I must say I agree with most of his criticisms:

fakalin pointed me to this product from MSR. I was actually sort of aware of this during my stint at MSR, via overheard hallway conversations, but didn’t know it was going to be released as a product.

http://research.microsoft.com/en-us/um/redmond/projects/songsmith/

So I downloaded it to see what’s up. It has been called the reverse Karaoke program. It has only been released a month and it appears there are already a handful of parodies of well known songs. There is one that turned a rap by Eminem into bluegrass (stupid vulgar song, but anyway):

http://www.youtube.com/watch?v=2ScVTBg2vxk
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Wow, how do you exactly train a lizard?

Here is a problem on complexity, or alternatively, convergence.

If an integer \(n\) is reduced by the largest square less than it each time until it is terminated at zero, show that the number of steps taken is at most order \(\log \log n\).

More precisely, let \(f(n) = n – {\left\lfloor{\sqrt{n}}\right\rfloor}^2\) and \(g(n)=k\) be the minimum number of steps until \(f^k(n)=0\). Prove that there exists some fixed \(A\) such that \(g(n)< A \log \log n\) for all \(n\ge 3\).
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Cloud computing is taking off. That’s like the first sentence of some recent “introduction” mumbo jumbo I wrote for some paper. There are of course different models of this.

One is to use all services that Google provides, which are entirely built on web applications. I don’t believe this is the right model.
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I suppose good penmanship is the basis of good calligraphy, since calligraphy is mainly the addition of (variable) brush width to the structure of the characters. This bulk structure is really the key and it is particularly difficult to get correctly without muscle memory. That’s why they tell you to trace character books over and over.

However, there is a way to figure this matter of structure from first principles (and perhaps generate a more unique style as a result), albeit with the tradeoff that you cannot be quick, you must be careful.
(Read the article)

Got it installed and seems like a clean update on Vista. Somebody must have cracked the whip on simplicity, since nearly everything involving user interaction got simpler. Since it is mostly feature extensions on Vista, it is quite stable.

Some less noticed changes:
* IE8 now runs all tabs and windows in separate processes, so there is no longer a distinction between tabs and windows. There is also (finally) a Mozilla style jump-highlight in-page search. There is a convenient “In Private” mode that leaves behind nothing, but it is kind of stupid in that it doesn’t sandbox in cookies to delete them afterwards but in fact doesn’t appear to store them at all, breaking some sites… or maybe it’s just a bug. There are also these “accelerators” to web services (like smart tags on crack), not that useful in my opinion.
* English ink input in continuous mode now displays recognitions in-place, rather than in typeface underneath.
* Services for Unix (the POSIX subsystem) is much much improved and is actually usable for compilation.
* Monad (or PowerShell), which got dropped from Vista, is in. Very nice.
* Desktop backgrounds now come in sets of images, rather than one image.
* Yet another new directory structure for user home directory. The “data” folders in the home directory like Pictures, Music, Movies, Documents are now symbolically separated into a “Libraries” indexing structure (kind of like in WMP), and apparently you can create multiple libraries. Not sure if this is implemented cleanly enough, but intersting.

That’s about it.

In my undergraduate EM class, I didn’t particularly pay attention to this part of the course, because it wasn’t on the test. I ended up never knowing what the heck the Smith chart is supposed to be — always thought it was some kind of polar to rectangular complex number conversion chart. Today through random browsing I found this simply excellent explanation:

http://www.maxim-ic.com/appnotes.cfm/an_pk/742/

Turns out it is not quite what I thought, and it is pretty neat. It does convert between two complex numbers, but the relationship has nothing to do with rectangular to polar. It’s the real and imaginary grid lines of normalized load impedance (the circles) layered on top of the real and imaginary grid lines of normalized reflection coefficient (the straight lines). Normalized load impedance and normalized reflection coefficient are functions of each other, so the Smith chart is used to convert between them. Very nice!