Franklin Pezzuti Dyer

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Welcome to my "hit list"! This is a list of math problems/puzzles I've come up with that I haven't been able to solve, and which particularly bother me - usually because it seems like there should be a nice solution. If you happen to solve any of them, for the love of Gauss, please send it to me and unburden my troubled mind! (I'll give you a shout-out.)

  1. Given a polynomial $P$ and an arbitrary $z\in\mathbb C$, is there a closed-form formula for the value of where $P^{\circ n}$ denotes n-fold composition of $P$. For first degree polyomials, this is easy - the limit always equals zero (when it exists). What about quadratics? Can you even find the value of the limit for some particular value of $z$ for which it equals and $\ne 0$?

  2. Can you find a closed-form evaluation of the series for some $x\in (-1,1)$ and $\alpha\notin\mathbb Q$? (For $\alpha\in\mathbb Q$, I know how to evaluate this series, but it's a good exercise.)

  3. For what $\alpha_1, \alpha_2, \beta_1, \beta_2 \in\mathbb R$ does there exist a continuous, bounded function $f:\mathbb R\to\mathbb R$ such that ? Or, even better, what are necessary and sufficient conditions on $\alpha_1,...,\alpha_n,\beta_1,...,\beta_n\in\mathbb R$ for the existence of a continuous function $f:\mathbb R\to\mathbb R$ satisfying ?

  4. What is the largest possible determinant of a matrix whose entries are only zeroes and ones?

  5. (This is more of an open-ended question.) In general, the notion of a monotonic function is usually only defined on a domain and codomain consisting of ordered sets. However, we may extend this notion to functions $f:\mathrm{X}\to\mathrm{Y}$ between inner product vector spaces $\mathrm{X}, \mathrm{Y}$ by saying that $f$ is monotone if and only if for all $x_1,x_2,x_3,x_4\in\mathrm{X}$. It is an exercise to prove that this is equivalent to the standard notion of monotonicity when $\mathrm{X},\mathrm{Y}$ are ordered fields with their inner product defined equal to their product. Using this extended definition, what results about monotonic functions on ordered sets can we extend to arbitrary inner product vector spaces? For instance, must a monotonic function be continuous except at countably many points? What other results can we extend?

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