Dror Bar-Natan: Classes: 2003-04: Math 157 - Analysis I: | (12) |
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Assigned Tuesday September 23; due Friday October 3, 2PM,
at SS 1071

This document in PDF: HW.pdf

**Required email. ** The class photo will be on
the class' web site in a day or two and you are all required to find
it, find yourself in the photo, and send me an email message (either
using the feedback form on the class' web site or using my regular
email address) with the following information:

- Where are you in the picture? (Use the supplied grid. If your square has more than one face in it, add something like ``I'm the guy with the red hair''.)
- Your name.
- Your email address.
- Your telephone number (optional).
- Which of the last four pieces of information do you allow me to put on the web? If you don't write anything about this, I'll assume that your location in the photo, your name and your email address are public but that your phone number is to be kept confidential.

Your email is due like the rest of this assignment, on Friday October 3 at 2PM. If you aren't in the picture at all, talk to me after class and I'll take a (small) picture of you on the spot and edit it into the main picture.

**Required reading. ** All of Spivak Chapters 2 and 3.

**To be handed in. **

From Spivak Chapter 2: 1, 5.

From Spivak Chapter 3: 6, 13.

**Recommended for extra practice. **

From Spivak Chapter 2: 3, 4, 12, 22.

From Spivak Chapter 3: 1, 7, 21.

**An extra problem: ** (recommended, but do not
submit) Is there a problem with the following inductive proof that all
horses are of the same color?

We assert that in all sets with precisely horses, all horses are of the same color. For , this is obvious: it is clear that in a set with just one horse, all horses are of the same color. Now assume our assertion is true for all sets with horses, and let us be given a set with horses in it. By the inductive assumption, the first of those are of the same color and also the last of those. Hence they are all of the same color as illustrated below:

**Just for fun. **

From Spivak Chapter 2: 27, 28.

A little more on Chapter 2, Problem 22:

- We know that if and are non-negative then
. This is the same as saying that
,
which is the same as saying that the area of four by rectangles is
less than or equal to the area of a square with side . Can you
actually fit four by rectangles inside a square of side
without overlaps? It's fun and not too hard.
- We know that if , and are non-negative then
. This is the same as saying that
, which is the same as saying that the volume of 27
by by rectangular boxes is less than or equal to the volume
of a cube with side . Can you actually fit 27 such by by
rectangular boxes inside a cube of side without overlaps? This is
also fun, but quite hard. You have no chance of doing it without a physical
model. Make yourself one!
- The corresponding problem in 4D, involving 256 boxes of size
, is actually a little easier, though trickier, than the
3D problem. Can you do it?
- The corresponding problem in 5D, involving 3,125 boxes of size
, is an open problem -- meaning that
nobody knows how to solve it. Can you?

Horse picture from
http://lib.allconet.org/story_hour.htm.

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Dror Bar-Natan 2003-10-21