During my vacation time, I found a couple days for an old project I’ve been wanting to complete for a while now:

Euler’s Formula and the Fundamental Theorem of Algebra

EDIT: V. 1.2 is up now.

For a few years now, I have wished that the Fundamental Theorem of Algebra had been written up at a much lower level, and that someone had given me such a write-up at, say, age 16. I am not sure whether this first draft is a success or not, but I have attempted to write what sort of thing I had in mind, complete with remarks that are included in undergraduate lectures, but not always present in undergraduate textbooks. The audience for this is supposed to be relatively general, though familiarity with Taylor Series would be an advantage.

The interplay between Euler’s Formula and the FToA is strong. Although students are often introduced to both in high school, the FToA is never proved (but Gauss sure was smart! He proved it five different ways! Wish you were him, don’t you?), and Euler’s Formula is reduced to the status of a bit of numerological trivia I like to call Bieber’s Theorem.

It’s not longish because it’s that hard. It’s long because I’m attempting to make it that easy. Though I suppose there’s always a little bit more to it than I thought that there was. Depending on the audience, a paper 2-3x as lengthy can often take a quarter as long to read, depending on how useful the expanded explanations are.

The theorem has a few other proofs, one of which in particular I want to sketch in a later draft. The proof presented in the first draft is much closer to a fully rigorous complete proof than the others. Some content has been left out, but it is content that is easier to fill in details for in a self-contained sort of way, when I get around to it.

I found this while fishing files out of an old computer the other week, and I figured you should have it. It’s all rather silly, but here’s my writeup of the secretary problem from 2004 or so. There’s a funny identity for partial sums of the harmonic series at the end, though, and it’s pretty bizarre. But it’s true.

It could probably have done just as well without the self-deprecating weirdness about girls being pretty and wishing they’d like me, but that’s where my head was at the time, and it is what it is.

There’s a discussion on Wikipedia as well.

Update

August 15, 2009

Convolutions and the Weierstrass approximation theorem has been updated and expanded. Now it also covers the “other” Weierstrass approximation theorem, taking a one-page tour of selected topics from the first chapter of a harmonic analysis text along the way for motivation. Also, a solution to the heat equation in one dimension has been briefly provided without proof in the remarks section.

So now you all should know what a convolution is and have some idea that they’re important.

In other news, a summary of the noodle paper is under review at Comptes Rendus Mathematique. If they choose to publish it, I’d like to link it here and say a few words about it having to do with some recent thoughts of mine concerning the lacunary circular operator, as well as possible connections to Furstenburg sets and incidence geometry. In particular, a comparison of the solution to the joints problem with the estimate taken from the multilinear Kakeya conjecture indicates to me that there is some reason to think that maybe incidence geometry won’t provide the answer to the sharpness of the LloglogL bound on the lacunary circular operator. I’d like to say something at length, but I want to share the paper with you first.

I have a new short paper which proves the Weierstrass approximation theorem in a manner quite similar to how it is done in Rudin’s Principles of Mathematical Analysis, but with much more motivation for the technique. In particular, the underlying idea, the convolution (by an approximate identity) is given center stage rather than relegated to the status of a magic trick. If you have read and vaguely understood the details of Rudin’s proof but are mystified as to how anyone could have thought of that or what the main idea is, then I hope that this paper can help you.

This paper is probably a bit advanced for a level 2 paper, but in principle, the prerequisites aren’t too intense, and I have some hope that an undergrad could get something from this without having taken Advanced Calc first – maybe a little help from an advanced friend or a professor might be necessary in this case.

Click here for the paper.

Edit: Version 2-0, with some introductory harmonic analysis and Weierstrass for trig polynomials. (8/15/09)

Hmmmm, apparently a little bit is already known about Buffon’s noodle, after all. I will have to look at this more closely.

Buffon’s Noodle

April 7, 2009

I almost forgot to mention the paper that myself and my advisor, Alexander Volberg, have out. Here it is.

In my post about Kakeya sets in R^2, I talked about the Buffon needle probability of a certain square Cantor set. The square Cantor set you end up with at the “end” of the construction is an abstraction, and not something one can draw a complete picture of by hand, as is the case with its corresponding Kakeya set. However, by a fact called “continuity of measures”, we know that partial constructions get successively closer to having zero needle probability/area (in the Cantor/Kakeya side of the picture, respectively), in fact converging to zero in the limit as n goes to infinity. One may then ask how quickly this convergence happens, so that, for example, one may actually construct by hand an actual set with footlong line segments in every direction whose total area is, say, less than .002 square feet.

Let S_n be the square Cantor set, and Buf(S_n) be its needle probability (in papers, this is also called the Favard length at times). It is known that there are two constants c and C such that, for example, clogn/n < Buf(S_n) < C/(n^(1/7)).

(Each inequality is a separate paper; clogn/n paper is a lot easier).

Ok. Now here’s where I come in. It seems as if there should be little difference between Buffon’s needle toss and Buffon’s ring toss when the radius of the ring is large. For the purposes of the estimate given by Bateman and Volberg, there is no difference at all when the radius is large enough compared to the fineness of the set in question. Additionally, we can generalize the question further: non-circular rigid curves (“noodles”) may be tossed at the set, landing in random places and with random orientations. Then the “noodle probability” Bufnood(S_n) > clogn/n, as long as the noodles aren’t too curvy, which we quantify in the paper. The result, when applied to circular arcs, says that a radius of r>4^(n/5) is large enough for the Bateman-Volberg estimate Bufnood(S_n) > clogn/n to hold.

An improved version of our paper is in the works which says nothing more about general noodles, but contains simpler special arguments which vastly improve the situation for circles: now r=Cn is known to be sufficient. The newer paper will also be simplified, reorganized, and have more pictures. It will also briefly talk about a certain conjecture about a modified Buffon’s ring toss which would imply the sharpness of the LloglogL boundedness of the lacunary circular maximal operator. I’ll let you know if it’s ever available online.

I hope this rant was Level 2 except for the maximal operator stuff at the end. The papers linked are a moderately light Level 3, and another level 3 that has lost me a fair amount of sleep.

Also, nothing much seems to be known about upper bounds for buffon noodle probability. I am not optimistic that it will be easy to modify existing arguments, but perhaps we will try sometime.

Kakeya Sets in R^2

March 23, 2009

Kakeya sets: Unit Length Line Segments In Every Direction In The x-y Plane Cleverly Arranged To Take Up Zero Area

Abstract. A Kakeya set in the x-y plane is a set containing unit length line segments in all possible directions cleverly arranged to overlap so much that together they take up zero area. If we think of the slopes and y-intercepts as ordered pairs and make a graph of them, one such possible graph is given by a rotated copy of a 2-dimensional Cantor set. The fact that this Kakeya set has zero area corresponds exactly to the fact that this Cantor set has zero Bu ffon needle probability, which in turn is a consequence of a deep theorem of Abram Besicovitch. 1-dimensional Hausdorff measure is briefly discussed.

(Click here for the most recent version of the paper)

(Older draft, v. 3-2)

(Older draft, v. 3-1)

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I recently wrote this paper for my own benefit while preparing a talk for a student conference. It is a difficult task to condense to a 20-minute talk something that is demonstrably a 9-page undertaking.

The target audience of the talk (and hence the paper) is a mixed audience of undergraduates and grad students. Ideally, it should be understandable by a motivated High School student who is competent in plane geometry and has a pretty good idea of what an integral is. Most of the paper has been generalized elsewhere, but this has been intentionally left out for simplicity.

The paper may be updated later. I was thinking about adding a few exercises to the paper for fun. For example, look at equation (2.2). It maybe be difficult to apply this formula for general sets S, but it’s manageable for a couple of cases: S is a line segment, and S is a circle. Of course, one should figure out what set K in the plane one gets from the line segment set and find its area by more conventional means for comparison.

And to think that when I took Calc 1, I had just assumed that the integral of csc-cubed would never come up naturally!

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EDIT: There are slight variations on this problem. For example, one can ask what the situation is if one is required to be able to continuously maneuver a single line segment to take on all orientations as it moves within the planar set K, and in fact, this is closer to the the problem as originally posed by Kakeya. Terry Tao addresses this analog using only the usual tools and language of analysis that should make sense to any fluent veteran of Walter Rudin’s Principles of Mathematical Analysis.

EDIT 2: I presented the content of this paper at MSU’s 2009 Student Research Conference a couple weeks ago, and won third prize.

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