Analysis – HW 4 – Fractals

October 29, 2013

This set is due Friday, November 15, at the beginning of lecture.

We discuss the Hausdorff metric on the collection of compact subsets of a complete metric space, and fractals obtained by contractions. (But note that not all fractals one may want to study come from this procedure.)

TeX source.

Pdf file.

[Edit, Nov. 12, 2013: Problem 4 has been changed to: Provide an example showing that, in general, if e\in\mathbb R^n and L is compact, then d_H(\{e\},L)\ne d(e,L). (There are cases where the equality holds, and it may be useful to provide some examples of this as well.)]

Office Hours

October 28, 2013

This week, I will not be able to hold office hours on Thursday (my son does not have school on Hallowe’en). Instead, they will be on Wednesday, 3-4:30 pm.

2013-08-11 11.41.23

Analysis – Increasing functions

October 25, 2013

Some of the topics discussed today in lecture came up on a question at Math.Stackexchange. I am reposting my answer here, slightly edited.

The question, by Mah Moud, was whether there can be a function whose derivative at a point is positive but the function is not increasing.

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Analysis – HW 3 – Strong measure zero

October 17, 2013

This set is due Wednesday, October 30, at the beginning of lecture.

[Edit, Oct. 30: The original version of the problem set had some mistakes, and has been replaced accordingly.]

Recall that a set A\subseteq \mathbb R is measure zero iff for all \epsilon>0 there is a sequence (I_n\mid n\in\mathbb N) of open intervals such that \displaystyle \sum_{n\in\mathbb N}\mathrm{lh}(I_n)<\epsilon and \bigcup_n I_n\supseteq A.

Similarly, X\subseteq\mathbb R is strong measure zero iff for any sequence (\epsilon_n\mid n\in\mathbb N) of positive reals, there is a sequence (I_n\mid n\in\mathbb N) of open intervals such that \mathrm{lh}(I_n)\le\epsilon_n for all n, and \bigcup_n I_n\supseteq X. The notion is due to Borel, in 1919.

In lecture we showed that the continuous image of a measure zero set does not need to be a set of measure zero, and that the sum of two measure zero sets does not need to be a measure zero set.

As mentioned in lecture, Borel conjectured that the strong measure zero sets are precisely the countable sets. This statement turned out to be independent of the usual axioms of set theory: If the continuum hypothesis is true, the conjecture is false. On the other hand, Laver showed in 1976 that the conjecture is true in some models of set theory.

TeX source.

Pdf file.

On proofs and more

October 17, 2013

This is a transcript of an exchange on Twitter on what mathematicians and others expect from proofs. (A previous exchange on a different topic is here. Twitter produces surprisingly nice results sometimes. What follows is a bit meandering, but interesting points are made.)

It began at 7:56 am – 27 Jun 13, with the twitter account of Republic of Mathematics (a website started by Gary Davis) quoting from Bill Thurston‘s great essay On proof and progress in mathematics. The quoted sentence was a short excerpt from the following:

The question is not even “How do mathematicians make progress in mathematics?”
Rather, as a more explicit (and leading) form of the question, I prefer “How do mathematicians advance human understanding of mathematics?”
This question brings to the fore something that is fundamental and pervasive: that what we are doing is finding ways for people to understand and think about mathematics.

To this, the account of The True Beauty of Math replied with “[Which is] why computer proofs [are of] little use.” Republic of Mathematics objected to this position, and quoted from an article by Sara Billey titled Computer Proofs. What is the value of computer assisted proofs?. The quoted sentence is an excerpt from:

Some mathematicians have tried to protect their egos by proposing that human proofs are superior to computer assisted proofs. They claim that we don’t learn as much from computer assisted proofs as we do from a human proof. They claim computer proofs can be difficult to verify. They claim computer proofs are less elegant. I find these complaints to be naive.

This was retweeted by Guy Longsworth, which is how I saw it.

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Analysis – HW 2 – Metric spaces

October 1, 2013

This set is due Wednesday, October 16, at the beginning of lecture.

TeX source.

Pdf file.