515 – Caratheodory’s characterization of measurability (Homework 3)

This set is due Friday, April 27.

The goal of these problems is to prove Carathéodory‘s theorem that “extracts” a measure from any outer measure. In particular, when applied to Lebesgue outer measure, this construction recovers Lebesgue measure.

305 – A brief update on n(3)

This continues the previous post on A lower bound for .

Only recently I was made aware of a note dated November 22, 2001, posted on Harvey Friedman‘s page, “Lecture notes on enormous integers”. In section 8, Friedman recalls the definition of the function , the length of the longest possible sequence from with the property that for no , the sequence is a subsequence of .

Friedman says that “A good upper bound for is work in progress”, and states (without proof):

Theorem. , where .

Here, are the functions of the Ackermann hierarchy (as defined in the previous post).

He also indicates a much larger lower bound for . We need some notation first: Let . Use exponential notation to denote composition, so .

Theorem. Let . Then .

He also states a result relating the rate of growth of the function to what logicians call subsystems of first-order arithmetic. A good reference for this topic is the book Metamathematics of First-order Arithmetic, by Hájek and Pudlák, available through Project Euclid.

There is a recent question on MathOverflow on this general topic.

305 – Derived subgroups of symmetric groups

One of the problems in the last homework set is to study the derived group of the symmetric group .

Recall that if is a group and , then their commutator is defined as

.

The derived group is the subgroup of generated by the commutators.

Note that, since any permutation has the same parity as its inverse, any commutator in is even. This means that .

The following short program is Sage allows us to verify that, for , every element of is actually a commutator. The program generates a list of the commutators of , then verifies that this list is closed under products and inverses (so it is a group). It also lists the size of this group. Note that the size is precisely , so in these 4 cases:

305 – Cube moves

Here is a small catalogue of moves of the Rubik’s cube. Appropriately combining them and their natural analogues under rotations or reflections, allow us to solve Rubik’s cube starting from any (legal) position. I show the effect the moves have when applied to the solved cube.

But, first, some relevant links:

• Rubik’s Cube World  Records. Here is the video to the current record.
• Lego Rubik’s Cube Solver. And the current mechanical record, the CubeStormer II.
• Thomas Rokicki, “Twenty-two moves suffice for Rubik’s Cube®“. Math. Intelligencer, vol 32 (1), (2010), 33–40.
• Exactly 20 moves are required: “God’s number is 20“. Moves are counted here in the Half-turn metric, where any turn on any face by any angle is one move.

305 – Homework V

This is the last homework set of the term. It is due Friday, April 27, 2012, at the beginning of lecture, but I am fine collecting it during dead week, if that works better.