ChapterZero

When are the levels sets of an arbitrary function the boundary of some convex set? I.e. if I give you the level set , what kind of conditions on and guarantee this?

I’ve been spending a lot of time looking at the equation lately. From Mathematica’s implicit plot, it sure looks like an ellipse (locally). So I’m wondering what kind of results you can get about level sets.

Thinking out loud, for one dimensional level sets in 2d, one condition that looks like it might be necessary and sufficient if you assume the level set is a smooth manifold is that the gradient on the level set has to be a one to one function onto the unit circle. I say this because it looks like you could stretch the unit circle onto the boundary of an arbitrary convex set by just growing it at different rates in the direction of its outward normal.

On a related note, are there higher dimensional analogs to the Jordan Curve Theorem: if it’s hard to show that a simple closed curve separates the plane into two disjoint regions, is it harder to show compact submanifolds do the same to space? I’ve not actually seen the proof of the JCT, so I don’t know if the arguments used are intrinsic to the plane, or more generally applicable.

Forget Sun Tzu’s Art of War, and Musashi’s Book of Five Rings. Just give me Machiavelli’s The Prince! The clear-headed, cold-hearted, ruthless, unsentimental practicality of his advice leaves me breathless and yearning for more. I’m not sure what the reason is, but I know that the same part of my brain fires off when I read Ender’s Game and the sequels about Bean, Vinge’s masterpieces ‘A Deepness in the Sky’ and ‘A Fire upon the Deep’, and when I watch certain episodes of Xena (especially the one where she takes on an entire army single-handedly), or when I watch Cellular, or the last Die Hard movie.

There’s something beautiful about doing whatever the f*ck it takes to get where you want to be: setting an impossible goal, and then redefining reality to achieve it. As I see it, this requires an objectification of others, a gaining of a sense of yourself as a chessmaster. Call me a sociopath, but I’d like to achieve this state someday.

Until then, I’ll get my inspiration by reading about people like Jason Itzler.

This is the end of the third week of CDS202, the geometry course based on MTA, being taught by Marsden. So far the class hasn’t been exactly what I’d hoped for: the lectures are superficial to the point of being useless, while the assignments cover more material than I can absorb in a week.

The first week, which focused on topology, covered the entirety of Chapter 1– not so taxing if you’ve seen topology before (I didn’t do have to do much reading), but I imagine the students with a less structured mathematical background had it hard. The second week covered multilinear algebra, functional analysis (banach and hilbert spaces, the three important theorems of functional analysis, and other miscellany), and differentiation in normed spaces, including taylor’s theorem and the converse, the implicit and inverse function theorems, and the calculus of variations in function spaces. Did I mention there are only two lectures a week? Consequently, Marsden doesn’t even have time to state all the theorems, much less give anything beyond a very broad stroke overview of the material. The third week covered manifolds, tangent bundles, vector bundles, submersions, immersions, tranversality, and Sard’s and Smale’s theorems.

So this course is insane. Fun, in a desperate not to fall behind way, but insane.

Here’s a challenging question: Let . Show that the mapping given by is onto.

It came up during the course of my first CDS202 assignment ( I both posed and solved it, on my way to proving something else). Give it a try, and if there’s interest I can give hints, or post my solution.

OK, not so hard… sue me.

My officemate asked me a Matlab question that’s worthy of consideration: what is a fast method for locating a submatrix within a matrix? In his case, we were dealing with two dimensional matrices, but to make it more challenging, attempt it with matrices of arbitrary dimension.

Some history: I coded a solution to this problem during my undergraduate research, to use in registering biomedical volumes. When I started working on the project, there were some reference subvolumes that had been extracted from various volumes, and exactly where they’d been extracted from had been lost. The data sets were relatively large, so the code had to be fast … too bad I can’t recall my solution off the top of my head.

It’d be intereresting to compare the performance of various solutions. I’ll try to recreate mine.

Here’s something new and exciting: let be a stationary Gaussian r.v.. Find the expected number of zero crossings in a unit interval. I’ll give you a hint: it can be expressed in terms of the second and zeroth moments of the signal’s spectrum. And I have no idea how to be about it

If you get that one, try to show that the expected number of extrema in a unit interval can be expressed in terms of the fourth and zeroth spectral moments.

I’ve started making a list of exotic locations that I’m going to visit when I have a job to take some time off from. This is one of my new hobbies

First up is Lake Baikal in Siberia, the largest lake in the world by volume. It is located over a young continental rift, so is more than a mile deep, and every year gets deeper. The water has incredible clarity– so much so that you can see down for over a 100 meters in the interior of the lake. Lake Baikal is also one of the most biodiverse lakes: 80% of its 1550 species of wildlife can be found nowhere else in the world. The nerpa, one of the three freshwater species of seals, as well as the most pleasant and sociable of seals, is found only at Lake Baikal — this strange fact has yet to be explained.

Sounds like a beautiful place, and makes you reconsider all the rumors about Siberia being a forbidding, desolate place.

Here’s a trick question: how many years would it take for Lake Baikal to be drained by the Niagra river, if it had no other inlets and outlets?

From the komplex plane:

A mathematician, native Texan, once was asked in his class: “What is mathematics good for?”

He replied: “This question makes me sick. Like when you show somebody the Grand Canyon for the first time, and he asks you `What’s is good for?’ What would you do? Why, you would kick the guy off the cliff”.

but to be fair, there’s the other side:

A mathematician decides he wants to learn more about practical problems. He sees a seminar with a nice title: “The Theory of Gears.” So he goes. The speaker stands up and begins, “The theory of gears with a real number of teeth is well known …”