A defect-based a posteriori error estimate for Krylov subspace approximations to the matrix exponential is introduced. This error estimate constitutes an upper norm bound on the error and can be computed during the construction of the Krylov subspace with nearly no computational effort. The matrix exponential function itself can be understood as a time propagation with restarts. In practice, we are interested in finding time steps for which the error of the Krylov subspace approximation is smaller than a given tolerance. Finding correct time steps is a simple task with our error estimate. Apart from step size control, the upper error bound can be used on the fly to test if the dimension of the Krylov subspace is already sufficiently large to solve the problem in a single time step with the required accuracy.

During this seminar, we will present a new mathematical model describing the transport of nitric oxide (NO) in a realistic geometrical representation of the lungs. Nitric oxide (NO) is naturally produced in the bronchial region of the lungs. It is a physiological molecule that has antimicrobial properties and allows the relaxation of muscles. It is well known that the measurement of the molar fraction of NO in the exhaled air, the so-called FeNO, allows a monitoring of asthmatic patients, since the production of this molecule in the lungs is increased in case of inflammation. However, recent clinical studies have shown that the amount of NO in the exhaled air can also be affected by « non-inflammatory » processes, such as the action of a bronchodilator or a respiratory physiotherapy session for a patient with cystic fibrosis. Using our new model, we will highlight the complex interplay between different transport phenomena in the lungs. More specifically, we will show why changes taking place in the deepest part of the lungs are expected to impact the FeNO. This gives a new light on the clinical studies mentioned below, allowing to confer a new role to the NO for the management of various pulmonary pathologies.

The last Fridays@2 of the year will be the Prelims Preparation Lecture aimed at first-year undergraduates. Richard Earl and Vicky Neale will highlight some key points to be aware of as you prepare for exams, thinking both about exam technique and revision strategy, and a student will offer some tips from their personal experience.  This will complement the Friday@2 event in Week 2, on Managing exam anxiety.  As part of the Prelims Preparation session, we'll look through two past exam questions, giving tips on how to structure a good answer.  You'll find that most helpful if you've worked through the questions yourself beforehand, so this is advance notice so that you can slot the questions into your timetable for the next few days.  They are both from 2013, one is Q5 from Maths I (on the Groups and Group Actions course), and the other is Q3 from Maths IV (on the Dynamics course).  You can access these, and a large collection of other past Prelims exam questions, via the archive.

This week’s Fridays@2 session, led by Dr Richard Earl and Dr Vicky Neale, is intended to provide advice on exam preparation and how to approach the Part A and Part B exams.

This session is aimed at second years and third years who will be sitting exams this term. Next week’s Fridays@2 will be for first years and will look at preparing for Prelims papers.
 

This workshop will focus on the main causes of exam stress, anxiety and panic and look at practical strategies to manage and overcome these issues. We will also review strategies to best support exam preparation.

Dr Ruth Collins is a Chartered Psychologist who specialises in the management of anxiety and panic. She is also a trained mindfulness teacher and an associate of the Oxford Mindfulness Centre.

This year sees the 100th anniversary of Emmy Noether receiving her Habilitation and thus becoming the first women to be granted the right to teach and lecture at a university in Prussia (now Germany).  Noether shaped modern algebra and her influence was felt in many other fields including topology.

We will start by exploring what algebraic topology is, how the subject was shaped by algebra (under the influence of Noether), before considering some current challenges and applications.

We will discuss the similarity and difference between deterministic and stochastic NLS. Different notions (or possible formulations) of local solutions will also be discussed. We will also present a global well posedness result for stochastic mass critical NLS. Joint work with Weijun Xu (Oxford)

We present some regularity results for the gradient of solutions to very degenerate equations, which exhibit a great lack of ellipticity.
In particular we show that local weak solutions of the orthotropic p−harmonic equation are locally Lipschitz, for every $p\geq 2$ and in every dimension.
The results presented in this talk have been obtained in collaboration with Pierre Bousquet (Toulouse), Lorenzo Brasco (Ferrara) and Anna Verde (Napoli).
 

Black holes are predicted by Einstein's theory of general relativity, and now we have ample observational evidence for their existence. However theoretically there are many unanswered questions about how black holes come into being. In this talk, with tools from hyperbolic PDE, quasilinear elliptic equations and geometric analysis, we will prove that, through a nonlinear focusing effect, initially low-amplitude and diffused gravitational waves can give birth to a trapped (black hole) region in our universe. This result extends the 2008 Christodoulou’s monumental work and it also proves a conjecture of Ashtekar on black-hole thermodynamics