List of expository papers
October 8, 2010
Here is a list of my expository articles. These papers explain things that are already well-known, but at the most introductory level possible. The content is far from original, and sometimes the only innovation is an elaboration of some main ideas a more veteran reader would have recognized without further comment:
Convolutions and the Weierstrass Approximation Theorem – intended for an undergraduate math major audience; experience with epsilon-delta arguments (elementary real analysis) is almost necessary.
Kakeya sets: Unit Length Line Segments In Every Direction In The x-y Plane Cleverly Arranged To Take Up Zero Area – intended for an undergraduate audience. Strong technical and intuitive understanding of “calculus 1” material (integration) is required. Elementary real analysis is useful in places but not required to get the main idea.
Euler’s Formula and the Fundamental Theorem of Algebra – To understand how complex polynomials work, you must first understand what it looks like when you multiply two complex numbers. The deep but easy-to-use identity called Euler’s Formula allows us to do this. One common proof of the FToA follows from Euler’s Formula without the need for extra machinery.
Euler’s Formula and the Fundamental Theorem of Algebra (rough sketch of another popular proof) – At the moment, there is only an animation of what happens when you plug in a circle of z-values into a polynomial as you increase the radius. This is one version of the proof, but the reason the animation looks the way it does is Euler’s Formula. I am reasonably happy with the .pdf I have written, but the topological proof hinted at by the animation could warrant an additional essay. I am not sure whether such an article should be a .pdf (with no animation) or take the form of a web page.
More On Vertical Tangent Lines – Intended for someone who at least has a solid understanding of undergraduate calculus, especially formal definitions and epsilon-delta proofs. Also essential is a solid understanding of the more theoretical applications of the Mean Value Theorem that are sometimes included in problem sets. This paper isn’t too over the top, but someone new to the material should maybe read slowly, as the Mean Value Theorem can tend to be deceptively subtle – there is always more to it than meets the eye.
Non-standard Analysis (NSA) – If you’re good with epsilon-delta arguments and constructing the reals as Cauchy sequences of rationals, then hopefully this slide show is self-explanatory without need for a lecture. Also contains some spoofs. Contains a simple proof of Tychonoff’s product theorem using Robinson’s equivalent definition of compactness (proof of equivalence omitted). This should demonstrate that NSA might have some potential for making some proofs easier in the case where a proposition depends in a very deep way on the Axiom of Choice.