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Linear Recurrences 201
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While working through the details of the last post I discovered a beautiful result. It is by no means novel but it renders clear the relationship (which I neglected previously) between linear recurrences, ODEs and indeed anything which is ‘sequence like’.
Linear Recurrences 101
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This post is something of a PSA rather than original work. It’s a rather standard result and I haven’t offered anything particularly novel, but I think it should be more widely known.
Making Change is Complex
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Abstract
The change making problem is a computer science classic. However, dynamic programming can only take you so far, and in the limit we are left only with analysis. In this post we determine the aymptotics of the number of ways to make change with prescribed denominations using methods of complex analysis inspired by the Hardy Littlewood Circle Method. We will make use of generating functions, though no prior knowledge of them will be required. Familiarity with asymptotic notation and complex analysis, particularly the Laurent Series and the Residue Theorem, will be assumed.
An Invitation to Cooking Steak I: First Principles
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This one was requested by some friends and is something I’ve been thinking about putting together anyway. It occurs to me that I have way too much to say so I’m going to split it into several chapters.
Races Within Races
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Abstract
Somewhat recently a number of my friends participated in Sydney’s City2Surf, a 14km run from the CBD to Bondi. These friends also regularly participate in ‘Park Runs’ most Saturday mornings, 5km runs held in parks throughout the city. Ben is one of these friends. Ben takes his running quite seriously. Amazingly, his City2Surf performance was so stellar that he actually set a new 5km record! In fact, his total 14km pace almost rivalled his previous personal best 5km pace. This prompted the following question, if one runs a long race at a certain pace, is it necessarily the case that for any given length shorter than that of the race, there exists a segment of the long race of the shorter length which was run at the same pace or better? Amongst all races, what is the slowest of the fastest short races contained in a long race? How much wood would a wood chucker chuck if a wood chucker could chuck wood? Join me on a mathematically low tech journey to ask and answer two of these inane questions.
A Proof of Euclid’s Theorem
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This is my proof of Euclid’s Theorem. There are many like it, but this one is mine.
Euler’s Theorem via Necklaces II
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Abstract
In the previous post a combinatorial proof of Fermat’s little theorem (FLT) was given, with the claim that it would be extended to a proof of the (almost) more general Euler’s Theorem. In this post we make good on that promise,
Euler’s Theorem via Necklaces I
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Abstract
You may be aware of a cute proof of Fermat’s Little Theorem that exploits rotational symmetries of necklaces. It’s also possible you haven’t heard of Fermat’s Little Theorem (not to be confused with his last!), in which case I have the pleasure of acquainting you with both the theorem and the aforementioned proof. Fermat’s Little Theorem (from now on FLT) can be seen as a special case of the more general Euler’s Theorem (plus an edge case). One might ask if the combinatorial proof of FLT can be lifted to a proof of Euler’s Theorem.