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

April 12, 2012

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.

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305 – A brief update on n(3)

April 12, 2012

This continues the previous post on A lower bound for n(3).

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 n(k), the length of the longest possible sequence x_1,x_2,\dots,x_n from \{1,2,\dots,k\} with the property that for no i<j\le n/2, the sequence x_i,x_{i+1},\dots,x_{2i} is a subsequence of x_j,x_{j+1},\dots,x_{2j}.

Friedman says that “A good upper bound for n(3) is work in progress”, and states (without proof):

Theorem. n(3)\le A_k(k), where k=A_5(5).

Here, A_1,A_2,\dots are the functions of the Ackermann hierarchy (as defined in the previous post).

He also indicates a much larger lower bound for n(4). We need some notation first: Let A(m)=A_m(m). Use exponential notation to denote composition, so A^3(n)=A(A(A(n))).

Theorem. Let m=A(187196). Then n(4)>A^m(1).

He also states a result relating the rate of growth of the function n(\cdot) 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

April 11, 2012

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

Recall that if G is a group and a,b\in G, then their commutator is defined as


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

Note that, since any permutation has the same parity as its inverse, any commutator in S_n is even. This means that G'\le A_n.

The following short program is Sage allows us to verify that, for 3\le i\le 6, every element of (S_i)' is actually a commutator. The program generates a list of the commutators of S_i, 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 {}|A_i|, so (S_i)'=A_i in these 4 cases:

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305 – Cube moves

April 11, 2012

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:

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305 – Homework V

April 9, 2012

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.

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