175, 275 -Homework 9 and suggestions for next week

Homework 9 is due Tuesday, November 11, at the beginning of lecture. The usual considerations apply.

In 175 we will try to cover this week until section 8.4 at least, but probably we won’t get there until next week. The key section here is 8.2; make sure you understand the notions discussed in 8.2 before going further. If you want to read ahead from 8.4, continue with sections 8.5 and 8.6; the difference between conditional and absolute convergence is very important here.

In 275 we will cover from section 13.4 on, and the goal is to reach 13.8, which probably won’t happen until next week or even the one after if things do not go well. Besides these topics, I will discuss the `mean value property’ of harmonic functions.

Homework 9:

175: Do not use the solutions manual for any of these problems.

Section 8.1. Exercises 86, 88, 127. Also, the following exercise:

Starting with a given , define the subsequent terms of a sequence by setting . Determine whether the sequence converges, and if it does, find its limit. More precisely: You must indicate for which values of the sequence diverges, and for which it converges, and for those that converges, you must identify the limit, which may again depend on . You may want to try studying the sequence with different initial values of (choose a large range of possible values) to get a feeling for what is going on.

Section 8.2. Exercises 14, 22, 38, 40 (do not use a calculator for this one; you can use that if necessary), 64-68, 71.

Section 8.3. Exercises 26, 35, 41, 43, 44.

There are 19 problems in total. Turn in at least 10. The others (at most 9) will be due November 18 together with a few additional exercises for that week. I suggest you start working on these problems early, as some may be a bit longer than usual.

(1) Patrick Dehornoy gave a nice talk at the Séminaire Bourbaki explaining Hugh Woodin's approach. It omits many technical details, so you may want to look at it before looking again at the Notices papers. I think looking at those slides and then at the Notices articles gives a reasonable picture of what the approach is and what kind of problems remain […]

It is not possible to provide an explicit expression for a non-linear solution. The reason is that (it is a folklore result that) an additive $f:{\mathbb R}\to{\mathbb R}$ is linear iff it is measurable. (This result can be found in a variety of places, it is a standard exercise in measure theory books. As of this writing, there is a short proof here (Intern […]

I learned of this problem through Su Gao, who heard of it years ago while a post-doc at Caltech. David Gale introduced this game in the 70s, I believe. I am only aware of two references in print: Richard K. Guy. Unsolved problems in combinatorial games. In Games of No Chance, (R. J. Nowakowski ed.) MSRI Publications 29, Cambridge University Press, 1996, pp. […]

Let $C$ be the standard Cantor middle-third set. As a consequence of the Baire category theorem, there are numbers $r$ such that $C+r$ consists solely of irrational numbers, see here. What would be an explicit example of a number $r$ with this property? Short of an explicit example, are there any references addressing this question? A natural approach would […]

Suppose $M$ is an inner model (of $\mathsf{ZF}$) with the same reals as $V$, and let $A\subseteq \mathbb R$ be a set of reals in $M$. Suppose further that $A$ is determined in $M$. Under these assumptions, $A$ is also determined in $V$. The point is that since winning strategies are coded by reals, and any possible run of the game for $A$ is coded by a real, […]

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I will show that for any positive integers $n,\ell,k$ there is an $M$ so large that for all positive integers $i$, if $i/M\le \ell$, then the difference $$ \left(\frac iM\right)^n-\left(\frac{i-1}M\right)^n $$ is less than $1/k$. Let's prove this first, and then argue that the result follows from it. Note that $$ (i+1)^n-i^n=\sum_{k=0}^{n-1}\binom nk i^ […]

I think it is cleaner to argue without induction. If $n$ is a positive integer and $n\ge 8$, then $7n$ is both less than $n^2$ and a multiple of $n$, so at most $n^2-n$ and therefore $7n+1$ is at most $n^2-n+1

Let PRA be the theory of Primitive recursive arithmetic. This is a subtheory of PA, and it suffices to prove the incompleteness theorem. It is perhaps not the easiest theory to work with, but the point is that a proof of incompleteness can be carried out in a significantly weaker system than the theories to which incompleteness actually applies. It is someti […]