I’ve wasted a lot of time on following the UK politics since 2016, so may I just remark that the Indyref vote was obviously influenced by promises to stay in EU made by “No” campaign, which was saying that the best way stay in EU for Scotland is to vote “No” in the Indyref. This has tricked the pro-EU Scots into voting “No”. This is something to be added to the statement in red above.

If the Tory govenment materialises this week, one would regret not voting SNP a lot.

]]>We stayed in the Quadrant again when we were in Auckland on Hood fellowships. That time, nothing went wrong (except for the washing machine shrinking our clothes).

I don’t think it was really anything for which the conference organisers need take responsibility. ]]>

3. I share your last belief but I think we are in a dwindling minority. For example, I never write analyze because it comes from the same Greek root as electrolysis.

]]>2. I don’t remember writing a report on the 2008 Auckland conference, and Google won’t find anything like this.

3. As I understand it, the rule is that Greek words take “ize” (so “diagonalize” is correct) but words that have entered English from French or Latin take “ise”. Living in St Andrews, I tend to take Chambers as the authority. Unfortunately Chambers gives “generalize” even though it admits the word comes from French.

It is my belief that the idiosyncratic spelling of English is worth retaining since it gives us information about word origins — but not if we get it wrong! ]]>

2) I just saw your report on the conference in Auckland Dec 2008 that I organized. Some of the blog entries there might have been better conveyed to the organizer.

3) The Oxford guide to English usage recommends -ize and not -ise for worlds like “diagonalize”. This is a battle I am getting a bit tired of fighting, but I think it is another thing you should know. ]]>

1. My professors should not be blamed for my mental handicaps. It is not their fault that I cannot learn more than three definitions a week and writing an integral sign fills me with anxiety and dread. I imagine there is a direct correlation between a student’s performance and their general impressions of the course – yet that does not mean the instructor did a bad job. In fact, the courses I learned the most from are the ones I did badly in, (except yours which was honestly a happy anomaly and a reflection of the great teaching).

2. Support and advice does not always seem like support and advice at the time. Eteri Tutberidze, the famous figure skating coach, was once asked how she can get 11 year olds to do quad jumps. She said roughly, that if you gave an 11 year old the choice between watching TV and running their program, they would watch TV while smoking a pack of cigarettes. So she doesn’t give them a choice. I wonder how much difference there was between an 11 year old and me in first year (or second or third year)? If teachers were allowed to clearly express their opinions, their support and advice score would definitely go down, but perhaps they would be offering more effective support and advice.

3. As teaching reviews impact whether a lecturer gets a promotion, there may be slight compromises over time. But why in the world should my very smart teachers be afraid of the stupid opinions of 20 somethings? Neophytes generally love to complain and are ungrateful – this is a universal and self evident truth. But I honestly have not had a single bad teacher ever at university and I learned something from all of them. I can’t expect everyone to be Richard Feynman: all I can expect is that they try their best. They all did.

It’s easy to show (for ) that , and similarly that . If we can show that , then it will follow that the sequence created by interleaving and satisfies the Fibonacci recurrence (and checking the early terms shows that it is the actual Fibonacci sequence).

So we need to show that . Let be a partition of , such that (or such that ). I define two partitions and of , as follows:

* The rule for defining is: replace with . (So becomes .

* The rule for defining is: if then simply reduce by 1. If then replace with .

We note that has first part 1 and does not, that every partition of appears as or for exactly one , and that the product of parts of is equal to the sum of those for and .

If you believe this slightly sketchy argument, then we’re done.

]]>Start with EO EO EO … EO (the number of EO units is w) and replace some pairs of adjacent monomers with dimers so as to eliminate all occurrences of the pattern OE. The initial E and the final O must remain since they help define the partition. Of the w-2 EO units in the middle, we arbitrarily replace some with dimers, d. There are 2^(w-2) ways to do this. Then to ensure that the pattern OE does not occur, we replace any occurrences of that pattern that remain with a dimer D.

So, for example, the third of the eight configurations listed in the previous comment for w=5 is obtained from EO EO EO EO EO by replacing the third pair with d: EO EO d EO EO. Then the remaining occurrences of OE are eliminated by replacing them with D: EDOdEDO. Similarly, the fifth configuration on the list is obtained from EO EO EO EO EO by replacing the second and fourth pairs with d: EO d EO d EO. No occurrence of the pattern OE remains, so nothing more needs to be done.

A final comment: we may use the same method for enumerating monomer-dimer configurations in the n odd case, but at the cost of including additional terms in the sum. So let n = 2k+1. Instead of adding a fictitious monomer at site 2k+2, we have the following series of possibilities: (1) put a monomer at site 2k+1, which will be an O. The rest is then an enumeration problem for the even length 2k. (2) Put a dimer at sites 2k, 2k+1 and a monomer at site 2k-1 (which will be an O). The rest is now an enumeration problem for the even length 2k-2. (3) Put dimers at 2k, 2k+1 and at 2k-2, 2k-1 and a monomer at 2k-3, leaving an enumeration problem for the length 2k-4. (4) An so on, including all even-length enumeration problems down to length 0, This is analogous to what we had to do using the earlier enumeration method when we wanted to handle the n even case.

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