Formal series and numerical integrators: some history and some new techniques

This paper provides a brief history of B-series and the associated Butcher group and presents the new theory of word series and extended word series. B-series (Hairer and Wanner 1976) are formal series of functions parameterized by rooted trees. They greatly simplify the study of Runge-Kutta schemes and other numerical integrators. We examine the problems that led to the introduction of B-series and survey a number of more recent developments, including applications outside numerical mathematics. Word series (series of functions parameterized by words from an alphabet) provide in some cases a very convenient alternative to B-series. Associated with word series is a group G of coefficients with a composition rule simpler than the corresponding rule in the Butcher group. From a more mathematical point of view, integrators, like Runge-Kutta schemes, that are affine equivariant are represented by elements of the Butcher group, integrators that are equivariant with respect to arbitrary changes of variables are represented by elements of the word group G.Comment: arXiv admin note: text overlap with arXiv:1502.0552

Absorption Time of the Moran Process

The Moran process models the spread of mutations in populations on graphs. We investigate the absorption time of the process, which is the time taken for a mutation introduced at a randomly chosen vertex to either spread to the whole population, or to become extinct. It is known that the expected absorption time for an advantageous mutation is O(n^4) on an n-vertex undirected graph, which allows the behaviour of the process on undirected graphs to be analysed using the Markov chain Monte Carlo method. We show that this does not extend to directed graphs by exhibiting an infinite family of directed graphs for which the expected absorption time is exponential in the number of vertices. However, for regular directed graphs, we show that the expected absorption time is Omega(n log n) and O(n^2). We exhibit families of graphs matching these bounds and give improved bounds for other families of graphs, based on isoperimetric number. Our results are obtained via stochastic dominations which we demonstrate by establishing a coupling in a related continuous-time model. The coupling also implies several natural domination results regarding the fixation probability of the original (discrete-time) process, resolving a conjecture of Shakarian, Roos and Johnson.Comment: minor change

Mapping film exhibition in Scotland before permanent cinemas

First paragraph: A central idea in current film historiography is that “the experience of cinema does not exist outside the experience of space” (Allen “The place” 16). This is reflected in the wealth of studies about the places of exhibition which has transformed the once text-centred perspective of film studies, its sources and methodologies, by understanding cinema-going in localized terms, as part of the fabric of everyday life.&nbsp

On the Value of Pedagogical Assets

[EN] University education is facing new strategical changes that will lead to deep structural changes. Course organization is evolving and the organizational decisions have an economical impact. We propose a method to measure the present value of a pedagogical asset under a return rate. We apply the method to three courses in the Computer Science curricula taught at the Facultat d’Informatica de Barcelona of the Universitat Politècnica de Catalunya, Barcelona Tech.  A large, compulsory, first year course (PRO1), a medium size undergraduate course (ALG) and a small specialized master course (AGT). Our results highlight that the present value gets higher values as a function of the size of the course and it goes in a negative relationship with respect  to the level of computer support involved in their teaching.This worok was partially supported by MINECO and FEDER funds under grant TIN2017-86727-C2-1-R (GRAMM) and AGAUR under grant 2017SGR-786. M. Serna was also partially funded by MINECO under grant MDM-2014-044 (BGSMath).http://ocs.editorial.upv.es/index.php/HEAD/HEAD18Gabarro, J.; Serna, M. (2018). On the Value of Pedagogical Assets. Editorial Universitat Politècnica de València. 425-432. https://doi.org/10.4995/HEAD18.2018.8007OCS42543