# Coffee-Break Problem, 30/11/20

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

For $n\ge 1$, define the function $f_n: [0,\infty)\to \mathbb{R}$ by

$f_n(t)=\sum_{k=1}^n \binom{n}{k} (-1)^k \left(1-\frac{k}{n}\right)^{\lceil tn\log n\rceil }.$

Show that, for any $t\neq 1$, $f_n(t)$ converges as $n\to \infty$, and identify the limit.

# Coffee-Break Problem, 23/11/20

Let $X_1, X_2....X_m$ be independent random variables in $\mathbb{R}_+$, with a density proportional to $\propto \exp(-x^\alpha), \alpha>0$.

Describe the law of $(X_1/R, ...., X_m/R)$, conditioned on $\sum_{i=1}^m X_i \ge R$, when $R$ is very large.

How does your answer depend on $\alpha$?

The behaviour in the case $\alpha<1$ is the starting point for the theory of subexponential distributions. What other distributions can you find that behave similarly?

# Coffee-Break Probem, 16/11/20

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”?

# Coffee-Break Problem, 09/11/20

Let $U \subset \mathbb{R}^2$ be a bounded, open set, and for all $\epsilon>0$, let $U_\epsilon$ be the open set given by removing all points distance at most $\epsilon$ from $U^\mathrm{c}$.

Suppose $U$ is connected. Must $U_\epsilon$ be connected for sufficiently small $\epsilon>0$?

# Coffee-Break Problem, 02/11/20

Consider the following model for fruit fly populations: every fruit fly dies after a time $T \sim \text{Exponential}(\lambda)$, 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 $X_t$ for the number of fruit flies at time $t$, let us consider a regime in which $X_t=\mathcal{O}(N)$ – so that $Y_t=X_t/N$, representing the total mass of flies, is a macroscopic object.

Suppose that the initial number of flies $X_0$ is distributed according to a $\text{Poisson}(N\delta)$ distribution, for some small $\delta>0$, and we want to know the probability that $X_t/N >x$ for some fixed $t>0, x>0$. Show that, no matter how small $\delta >0$ is, there is a certain (exponentially small) probability of this happening:

$\liminf_{\delta\to 0}\left[\liminf_{N\to \infty}\frac{1}{N} \log \mathbb{P}\left(X_t /N >x\right)\right]>-\infty.$

What happens if we take the limits $N\to \infty, \delta \to 0$ in the other order?

# Coffee-Break Problem, 26/10/20

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

Consider a pendulum: a ball of mass $m$, at the end of a light, inextensible string of length $L$. The pendulum swings, making maximum angle to the vertical $\theta_0$ which we suppose to be small.

To first order in $\theta_0$, what is the period of the oscillations?

# Coffee-Break Problem, 19/10/20

Consider a super-linear Ornstein-Uhlenbeck Process given by the stochastic differential equation $dX_t=-|X_t|^\gamma X_t dt + \frac{1}{2}dB_t, \quad \gamma>0$. Why does this SDE have global solutions, even though the coefficients aren’t Lipschitz?

Let $X_t, Y_t$ be two solutions, driven by the same Brownian motion, where $X_0, Y_0$ have all moments finite. Considering $(1+|X_t|^p+|Y_t|^p)|X_t-Y_t|^2$ for a well-chosen $p>0$, how fast do the coupled processes diverge?

# Coffee-Break Problem, 12/10/20

Suppose that I am running a speed-meeting session, where I ‘randomly’ assign $2n$ 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 $m$ rounds, how many new people (on average) does each participant meet?

# Coffee-Break Problem, 05/10/20

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be the the product probability space given by $(\{0,1\}, \{\emptyset,\{0\},\{1\},\{0,1\}\}, \frac{1}{2}\delta_0+ \frac{1}{2}\delta_1)^{\otimes \alpha}$, for an uncountable ordinal $\alpha$, and let $(\mathcal{F}_t)_{0\le t\le 1}$ be a filtration with $\mathcal{F}_1=\mathcal{F}$.

On this probability space, let $\mathcal{M}$ be the space of all $L^2$-bounded martingales $(M_t)_{0\le t\le 1}$, equipped with the distance $d(M,N):=(\mathbb{E}|M_1-N_1|^2)^{1/2}$. Define also $\mathcal{R}$ as martingales with some Hölder-regularity:

$\mathcal{R}=\big\{M\in \mathcal{M}: \text{ for some nonempty open interval }I\subset [0,1]\text{ and }\beta>0, \mathbb{P}(M \text{ is }\beta\text{-H\"older continuous on }I)>0\big\}.$

Show that $\mathcal{R} \subset \mathcal{M}$ is meagre in the sense of Baire.

# Coffee-Break Problem, 28/09/20

Write down axioms for the theory of algebraically closed fields.

Can you do so in a way that every axiom is of the form $\forall x_1 .... \forall x_n \hspace{0.1cm} \phi(x_1,....,x_n)$, for some quantifier-free statement $\phi$? That is, so that the statments $\phi$ don’t involve any further uses of $\forall$ or $\exists$?