This set is due Feb. 8 at the beginning of lecture. Of course, let me know if more time is needed or anything like that.

0. During lecture I have sometimes skipped some arguments or not given as much detail as you may have wanted. If there was a result that in particular required of you some effort to complete in detail, please state it here and show me how you filled in the gaps left in lecture. Also, if there is a result for which you do not see how to fill in the details, let me know as well, as I may have overlooked something and it may be worth going back over it in class.

1. Give an example of a bounded set for which

does not exist.

2. Compute .

3. From the book, solve exercises 1.1.3, 1.1.5, 1.1.6, and 1.1.15.

[To get you started on 1.1.3: First verify in that assigns value 0 to any point. For this, use monotonicity and translation invariance, arguing first that for any . Then find that in terms of , and use this to find for any box with rational coordinates. Use this to compute for any box, and conclude by analyzing arbitrary elementary sets.

Note we essentially solved 1.1.15 in class, but under the assumption that 1.1.6 holds.]

4. From the book, solve Exercises 1.1.7-10. Make sure to explain in 1.1.9 why Tao’s definition of compact convex polytopes coincides with what should be our intuitive definition. Please also verify that convex polytopes are indeed convex.

(For a nice argument verifying that indeed , at least for even values of , see the paper “On the volumes of balls” by Blass and Schanuel, available here.)

5. From the book, solve exercise 1.1.11.

(If you are not comfortable with linear algebra beyond size , at least argue in the plane and in .)

6. From the book, solve exercise 1.1.13.

7. From the book, solve exercise 1.1.17.

43.614000-116.202000

Advertisements

Like this:

LikeLoading...

Related

This entry was posted on Wednesday, January 25th, 2012 at 1:08 pm and is filed under 515: Analysis II. You can follow any responses to this entry through the RSS 2.0 feed.
You can leave a response, or trackback from your own site.

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, […]

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 relevant search term is ethnomathematics. There are several journals devoted to this topic (for instance, Revista latinoamericana de etnomatemática). Browsing them (if you have access to MathSciNet, the relevant MSC class is 01A70) and looking at their references should help you get started. Another place to look for this is in journals of history of mat […]

Some of the comments in the previous answers make a subtle mistake, and I think it may be worth clarifying some issues. I am assuming the standard sort of set theory in what follows. Cantor's diagonal theorem (mentioned in some of the answers) gives us that for any set $X$, $|X|

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: […]

(I have added a missing prime in the hint on question 3.) Thanks to Tara for noticing it.