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Archive for April, 2012

A student I’m tutoring was working unsuccessfully on the following problem from the 2011 Mandelbrot Competition:

Let a, b be positive integers such that \log_a b = (\log 23)(\log_6 7) + \log_2 3 + \log_6 7. Find the minimum value of ab.

After some tinkering, I concluded that the problem as stated has no solution. I am now almost certain it was printed incorrectly: \log 23 should be replaced by \log_2 3, and then we can solve the problem as follows:

\log_a b + 1 = (\log_2 3 + 1)(\log_6 7 + 1) = \log_2 6 \log_6 42 = \log_2 42.

It follows that \log_a b = \log_2 21. Since a, b are positive integers we must have a \ge 2, and then it follows that the smallest solution occurs when a = 2, b = 21. But what I’d like to discuss, briefly, is the argument showing that the misprinted problem has no solution.

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The Yoneda lemma I

For two categories C, D let D^C denote the functor category, whose objects are functors C \to D and whose morphisms are natural transformations. For C a locally small category, the Yoneda embedding is the functor C \to \text{Set}^{C^{op}} sending an object x \in C to the contravariant functor \text{Hom}(-, x) and sending a morphism x \to y to the natural transformation \text{Hom}(-, x) \to \text{Hom}(-, y) given by composition. The goal of the next few posts is to discuss some standard properties of this embedding and try to gain some intuition about it.

Below, whenever we talk about the Yoneda lemma we implicitly restrict our attention to locally small categories.

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