# Coffee-Break Problem, 21/09/20

I haven’t had a mechanics problem here for a while, so here’s something `elementary’ to amuse you.

I suspend a 1kg bag of sand above a bowl, which rests on pair of scales. At time $t=0$, I puncture the bag; sand then flows at a constant rate into the bowl.

Sketch the (qualitative behaviour of) the output of the scales.

# Coffee-Break Problem, 14/09/20

Construct a communtative, unital Banach Algebra $A$ and a closed, unital subalgebra $B\subset A$, along with an element $x\in B$ which is invertible in $A$ but not in $B$.

# Coffee-Break Problem, 07/09/20

Let $\mathcal{X} \subset L^\infty([0,1])$ be the set of all measurable functions $f: [0,1]\rightarrow \{0,1\}$. Consider the weak-star topology $\tau$ on $L^\infty([0,1])$ induced by the functions $\phi_g(f):=\int_0^1 fg dx$, where $g$ runs over $L^1([0,1])$.

With respect to this topology, what is the closure $\overline{\mathcal{X}}$?

# Coffee-Break Problem, 31/08/20

Fix an integer $x\ge 2$ which is not a power of $10$. For $n\ge 1$, let $p_n$ be the proportion

$p_n := \frac{\#\{1\le m\le n: x^m \text{ has first decimal digit }1\}}{n}.$

Show that $p_n$ converges as $n\rightarrow \infty$, and identify the limit.

# Coffee-Break Problem, 24/08/20

Consider noughts-and-crosses on a $3\times 3\times 3$ grid. There are multiple possible interpretations of the rules: we can choose any of

A: No gravity, so that any cell can be played at any time;
B: Gravity, but no rows: any cell can be played so long as all the cells under it have been played;
C: Rows: the first horizontal layer must be completed before the second layer can be started, and the second layer must be completed before the third layer can be started;

and either of

L: Long Diagonals allowed for a victory condition, or
NL: No Long Diagonals.

With each possible set of rules and perfect play, who wins?

# Coffee-Break Problem, 17/08/20

Walking from Athens to Thebes, I am stopped by a Sphinx who has a grudge against pure mathematicians, and does not wish to allow me to pass.

The Sphinx proposes the following game: they think of a number, and I can ask yes/no questions; I win by finding the Sphinx’s number. Unknown to me, the Sphinx never actually chooses a number, and makes up the answers as they go along; being a perfect logician, these answers are chosen consistently so that I can never catch them in a lie.

Prove that the Sphinx has a winning strategy: the answers can be chosen so that the game never terminates.

What if the Sphinx is constrained to choose all the yes/no answers before I start my questions?

For bonus points: relate your solution to this previous problem.

# Coffee-Break Problem, 10/08/20

Consider a network of queues, each with one server. At the $i^\text{th}$ queue, the first customer in the queue is served with rate $\mu_i$, and moves to another queue $j$ with probability $p_{ij}$, or leaves the system with probability $1-\sum_j p_{ij}$.

Customers enter the network in queue $1$, and arrive opportunistically: when the system is non-empty, customers arrive $\lambda$, but if the system is already empty, they arrive at some other rate $\lambda'$.

Fixing $\lambda$, show that the proportion of customers arriving when the system is in a given state does not depend on $\lambda'$.

# Coffee-Break Problem, 03/08/20.

A lovely little problem, putting the ‘fun’ in ‘functional analysis’.

Let $V\subset C([0,1])$ be a vector subspace such that the norms $\|f\|_\infty:=\sup_x |f(x)|$ and $\|f\|_2:=(\int_0^1 |f(x)|^2 dx)^{1/2}$ induce the same topology on $V$.

Show that $\text{dim}(V)<\infty$.

# Coffee-Break Problem, 27/07/20

Today’s problem is not particularly deep, but made me smile when I first realised it.

Let $G=(V,E)$ be a (potentially infinite) graph, and let $\mathcal{F}=\{E'\subset E: (V,E') \text{ contains no cycles}\}$ be the space of forests – that is, subgraphs of $G$ made of unions of trees.

Equip $\mathcal{F}$ with the topology whose basic open sets are $U=\{E'\in \mathcal{F}: e_1\in E', ....e_n\in E', e_{n+1}\not \in E',...., e_{n+m}\not \in E'\}$, for $n, m\ge 0$ and $e_1, ...e_{n+m} \in E$. Show that this topology is compact and Hausdorff, and if $E$ is countable, then it admits a metric.

The significance is that compact and Hausdorff spaces have very nice properties when we look at the space of probability measures – for instance, given a sequence $\mu_n, n\ge 1$ of Borel probability measures on $\mathcal{F}$, we can always pass to a weakly convergent subsequence.

# Coffee-Break Problem, 20/06/20

Just in case you thought Brownian motion was a nice, regular object, here’s a process with even worse regularity.

Construct a continuous martingale $X_t, 0\le t\le 1$ whose sample paths are locally constant $dt$ almost everywhere, and are not $\alpha$ -Hölder continuous, for any $\alpha>0$.