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.

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, […]

Yes. This is obvious if there are no such cardinals. (I assume that the natural numbers of the universe of sets are the true natural numbers. Otherwise, the answer is no, and there is not much else to do.) Assume now that there are such cardinals, and that "large cardinal axiom" is something reasonable (so, provably in $\mathsf{ZFC}$, the relevant […]

Please send an email to mathrev@ams.org, explaining the issue. (This is our all-purpose email address; any mistakes you discover, not just regarding references, you can let us know there.) Give us some time, I promise we'll get to it. However, if it seems as if the request somehow fell through the cracks, you can always contact one of your friendly edit […]

The characterization mentioned by Mohammad in his answer really dates back to Lev Bukovský in the early 70s, and, as Ralf and Fabiana recognize in their note, has nothing to do with $L$ or with reals (in their note, they indicate that after proving their result, they realized they had essentially rediscovered Bukovský's theorem). See MR0332477 (48 #1080 […]

For $\lambda$ a scalar, let $[\lambda]$ denote the $1\times 1$ matrix whose sole entry is $\lambda$. Note that for any column vectors $a,b$, we have that $a^\top b=[a\cdot b]$ and $a[\lambda]=\lambda a$. The matrix at hand has the form $A=vw^\top$. For any $u$, we have that $$Au=(vw^\top)u=v(w^\top u)=v[w\cdot u]=(w\cdot u)v.\tag1$$ This means that there are […]

That you can list $K $ does not mean you can list its complement. Perhaps the thing to note to build your intuition is that the program is not listing the elements of $K $ in increasing order. Indeed, maybe program 20 halts on input 20 but only does it after several million steps, while program 19 doesn't halt on input 19 and program 21 halts on input 2 […]

A reasonable follow-up question is whether there are some natural algebraic properties that the class of cardinals satisfies (provably in $\mathsf{ZF}$ or in $\mathsf{ZF}$ together with a weak axiom of choice). This is a natural problem and was investigated by Tarski in the 1940s, see MR0029954 (10,686f). Tarski, Alfred. Cardinal Algebras. With an Appendix: […]

Yes. Lev Bukovský proved a very general theorem that deals precisely with this problem: MR0332477 (48 #10804). Characterization of generic extensions of models of set theory, Fundamenta Mathematica 83 (1973), pp. 35–46. Bukovský characterizes when, for a given regular cardinal $\lambda$, $V$ is a $\lambda$-cc generic extension of a given inner model $W$. For […]

This is a question with a long history. As I mentioned in a comment, I think the best reference to get started on these matters is MR0373902 (51 #10102). Marek, W.; Srebrny, M. Gaps in the constructible universe. Ann. Math. Logic 6 (1973/74), 359–394. The paper does not require knowledge of fine structure, it is directly concerned with the question, provides […]