The formation of liquid drop patterns on solid surfaces is a fundamental process for both industry and nature. Now, a team of scientists including Oxford Mathematician Andreas Münch and colleagues from the Weierstrass Institute in Berlin, and the University of Saarbrücken can explain exactly how it happens.
In this collaboration with researchers from the Umeå University and the University of Zurich, Renaud Lambiotte from Oxford Mathematics explores the use of higher-order networks to analyse complex data.
We’re all familiar with liquid droplets moving under gravity (especially if you live somewhere as rainy as Oxford). However, emerging applications such as lab-on-a-chip technologies require precise control of extremely small droplets; on these scales, the forces associated with surface tension become dominant over gravity, and it is therefore not practical to rely on the weight of the drops for motion.
The concept of equilibrium is one of the most central ideas in economics. It is one of the core assumptions in the vast majority of economic models, including models used by policymakers on issues ranging from monetary policy to climate change, trade policy and the minimum wage. But is it a good assumption?
Differential equations arising in physics and elsewhere often describe the evolution in time of quantities which also depend on other (typically spatial) variables. Well known examples of such evolution equations include the heat equation and the wave equation.
Oxford Mathematician Carla Groenland talks about her and Oxford colleagues' work on graph reconstruction.
A graph $G$ consists of a set of vertices $V(G)$ and a set of edges $E(G)$ which may connect two (distinct) vertices. (There are no self-loops or multiple edges.)
Oxford Mathematician Erik Panzer talks about his and colleagues' work on devising an algorithm to compute Kontsevich's star-product formula explicitly, solving a problem open for more than 20 years.
"The transition from classical mechanics to quantum mechanics is marked by the introduction of non-commutativity. For example, let us consider the case of a particle moving on the real line.
From commutative classical mechanics...
When mathematicians solve a differential equation, they are usually converting unbounded operators (such as differentiation) which are represented in the equation into bounded operators (such as integration) which represent the solutions. It is rarely possible to give a solution explicitly, but general theory can often show whether a solution exists, whether it is unique, and what properties it has. For this, one often needs to apply suitable (bounded) functions $f$ to unbounded operators $A$ and obtain bounded operators $f(A)$ with good properties. This is t