Nothing is more fun than some grungy analysis – except perhaps grungy analysis with a very cheeky solution.

For , define the function by

Show that, for any , converges as , and identify the limit.

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# Coffee-Break Problem, 30/11/20

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# Coffee-Break Problem, 05/10/20

# Coffee-Break Problem, 28/09/20

Nothing is more fun than some grungy analysis – except perhaps grungy analysis with a very cheeky solution.

For , define the function by

Show that, for any , converges as , and identify the limit.

Let be independent random variables in , with a density proportional to .

Describe the law of , conditioned on , when is very large.

How does your answer depend on ?

*The behaviour in the case is the starting point for the theory of subexponential distributions. What other distributions can you find that behave similarly?*

Consider hitting keys on a keyboard (a-z), uniformly and independently of previous keystrokes. How long, on average, before you type “abcd”? What about “abab” or “abba”?

Let be a bounded, open set, and for all , let be the open set given by removing all points distance at most from .

Suppose is connected. Must be connected for sufficiently small ?

Consider the following model for fruit fly populations: every fruit fly dies after a time , and is then replaced by two identical fruit flies. These fruit flies then reproduce in the same way, identically to their parent (and all other flies).

Fruit flies are small, but make up for this smallness by a large population. Writing for the number of fruit flies at time , let us consider a regime in which – so that , representing the total mass of flies, is a macroscopic object.

Suppose that the initial number of flies is distributed according to a distribution, for some small , and we want to know the probability that for some fixed . Show that, no matter how small is, there is a certain (exponentially small) probability of this happening:

What happens if we take the limits in the other order?

A rather nice mechanics problem I saw back in the day.

Consider a pendulum: a ball of mass , at the end of a light, inextensible string of length . The pendulum swings, making maximum angle to the vertical which we suppose to be small.

To first order in , what is the period of the oscillations?

Consider a super-linear Ornstein-Uhlenbeck Process given by the stochastic differential equation . Why does this SDE have global solutions, even though the coefficients aren’t Lipschitz?

Let be two solutions, driven by the same Brownian motion, where have all moments finite. Considering for a well-chosen , how fast do the coupled processes diverge?

Suppose that I am running a speed-meeting session, where I ‘randomly’ assign participants into pairs. This is repeated over a number of rounds, hoping that each participant meets several others.

On average, how many people are assigned the same partner in two consecutive rounds? If I run rounds, how many new people (on average) does each participant meet?

Let be the the product probability space given by , for an uncountable ordinal , and let be a filtration with .

On this probability space, let be the space of all -bounded martingales , equipped with the distance . Define also as martingales with some Hölder-regularity:

Show that is meagre in the sense of Baire.

Write down axioms for the theory of algebraically closed fields.

Can you do so in a way that every axiom is of the form , for some quantifier-free statement ? That is, so that the statments don’t involve any further uses of or ?