9 research outputs found
Ramsey numbers of bounded degree trees versus general graphs
For every and , we prove that there exists a constant
such that the following holds. For every graph with
and every tree with at least vertices and maximum
degree at most , the Ramsey number is
, where is the size of a smallest colour
class across all proper -colourings of . This is tight up to the value of
, and confirms a conjecture of Balla, Pokrovskiy, and Sudakov
Global and local
The global/local contrast is ubiquitous in mathematics. This paper explains it with straightforward examples. It is possible to build a circular staircase that is rising at any point (locally) but impossible to build one that rises at all points and comes back to where it started (a global restriction). Differential equations describe the local structure of a process; their solution describes the global structure that results. The interplay between global and local structure is one of the great themes of mathematics, but rarely discussed explicitly
Scientiļ¬c uncertainty and decision making
It is important to have an adequate model of uncertainty, since decisions must be
made before the uncertainty can be resolved. For instance, ļ¬ood defenses must be
designed before we know the future distribution of ļ¬ood events. It is standardly
assumed that probability theory oļ¬ers the best model of uncertain information. I
think there are reasons to be sceptical of this claim.
I criticise some arguments for the claim that probability theory is the only
adequate model of uncertainty. In particular I critique Dutch book arguments,
representation theorems, and accuracy based arguments.
Then I put forward my preferred model: imprecise probabilities. These are sets
of probability measures. I oļ¬er several motivations for this model of uncertain
belief, and suggest a number of interpretations of the framework. I also defend
the model against some criticisms, including the so-called problem of dilation.
I apply this framework to decision problems in the abstract. I discuss some
decision rules from the literature including Leviās E-admissibility and the more
permissive rule favoured by Walley, among others. I then point towards some
applications to climate decisions. My conclusions are largely negative: decision
making under such severe uncertainty is inevitably diļ¬cult.
I ļ¬nish with a case study of scientiļ¬c uncertainty. Climate modellers attempt
to oļ¬er probabilistic forecasts of future climate change. There is reason to be
sceptical that the model probabilities oļ¬ered really do reļ¬ect the chances of future
climate change, at least at regional scales and long lead times. Indeed, scientiļ¬c
uncertainty is multi-dimensional, and diļ¬cult to quantify. I argue that probability
theory is not an adequate representation of the kinds of severe uncertainty that
arise in some areas in science. I claim that this requires that we look for a better
framework for modelling uncertaint
Fuelling the zero-emissions road freight of the future: routing of mobile fuellers
The future of zero-emissions road freight is closely tied to the sufficient availability of new and clean fuel options such as electricity and Hydrogen. In goods distribution using Electric Commercial Vehicles (ECVs) and Hydrogen Fuel Cell Vehicles (HFCVs) a major challenge in the transition period would pertain to their limited autonomy and scarce and unevenly distributed refuelling stations. One viable solution to facilitate and speed up the adoption of ECVs/HFCVs by logistics, however, is to get the fuel to the point where it is needed (instead of diverting the route of delivery vehicles to refuelling stations) using "Mobile Fuellers (MFs)". These are mobile battery swapping/recharging vans or mobile Hydrogen fuellers that can travel to a running ECV/HFCV to provide the fuel they require to complete their delivery routes at a rendezvous time and space. In this presentation, new vehicle routing models will be presented for a third party company that provides MF services. In the proposed problem variant, the MF provider company receives routing plans of multiple customer companies and has to design routes for a fleet of capacitated MFs that have to synchronise their routes with the running vehicles to deliver the required amount of fuel on-the-fly. This presentation will discuss and compare several mathematical models based on different business models and collaborative logistics scenarios