Unified theory for finite Markov chains

We provide a unified framework to compute the stationary distribution of any finite irreducible Markov chain or equivalently of any irreducible random walk on a finite semigroup $S$. Our methods use geometric finite semigroup theory via the Karnofsky-Rhodes and the McCammond expansions of finite semigroups with specified generators; this does not involve any linear algebra. The original Tsetlin library is obtained by applying the expansions to $P(n)$, the set of all subsets of an $n$ element set. Our set-up generalizes previous groundbreaking work involving left-regular bands (or $\mathscr{R}$-trivial bands) by Brown and Diaconis, extensions to $\mathscr{R}$-trivial semigroups by Ayyer, Steinberg, Thi\'ery and the second author, and important recent work by Chung and Graham. The Karnofsky-Rhodes expansion of the right Cayley graph of $S$ in terms of generators yields again a right Cayley graph. The McCammond expansion provides normal forms for elements in the expanded $S$. Using our previous results with Silva based on work by Berstel, Perrin, Reutenauer, we construct (infinite) semaphore codes on which we can define Markov chains. These semaphore codes can be lumped using geometric semigroup theory. Using normal forms and associated Kleene expressions, they yield formulas for the stationary distribution of the finite Markov chain of the expanded $S$ and the original $S$. Analyzing the normal forms also provides an estimate on the mixing time.Comment: 29 pages, 12 figures; v2: Section 3.2 added, references added, revision of introduction, title change; v3: typos fixed and clarifications adde

Placing large group relations into pedestrian dynamics: psychological crowds in counterflow

Understanding influences on pedestrian movement is important to accurately simulate crowd behaviour, yet little research has explored the psychological factors that influence interactions between large groups in counterflow scenarios. Research from social psychology has demonstrated that social identities can influence the micro-level pedestrian movement of a psychological crowd, yet this has not been extended to explore behaviour when two large psychological groups are co-present. This study investigates how the presence of large groups with different social identities can affect pedestrian behaviour when walking in counterflow. Participants (N = 54) were divided into two groups and primed to have identities as either ‘team A’ or ‘team B’. The trajectories of all participants were tracked to compare the movement of team A when walking alone to when walking in counterflow with team B, based on their i) speed of movement and distance walked, and ii) proximity between participants. In comparison to walking alone, the presence of another group influenced team A to collectively self-organise to reduce their speed and distance walked in order to walk closely together with ingroup members. We discuss the importance of incorporating social identities into pedestrian group dynamics for empirically validated simulations of counterflow scenarios

Organic pig production – With particular reference to Danish production conditions

While there is a markedly growth in organically farmed land and in the organic food sector as such in Europe and North America, the organic pig sector has not developed correspondingly in most countries. This weaker development seems most likely to be due to difficulties for pig producers to comply with the organic standards, which impose comparatively more pronounced changes in the way of production than e.g. in ruminant production systems. Pigs should have access to roughage and to grazing in the summer period although finishers can be kept in barns if access to an outdoor rum. A common way to comply with these regulations is to have sows on grassland all year round, but to rear the finishers in stables. Outdoor piglet production can be very efficient but careful measures needs to be taken to avoid environmental risks. Ringing of sows to avoid rooting on the grassland is a matter of concern. The regulations for housing include a considerable larger area per pig than in conventional production. This support good production and health results in the finisher production, and the regulation stating no tail docking does not seem to impose problems. However, such stables are often very expensive to establish and outdoor rearing of finishers throughout the growing period should be considered. Recent results indicate that this is a viable option. The ban on use of synthetic amino acids and GMO products for feeding represent a challenge in the feeding of finishers. There is a risk regarding carcass quality when soybean meal and amino acid are substituted with more oil rich ingredients for protein supply. On the other hand inclusion of new protein sources in the diet may results in particular meat quality characteristics including positive effects. It is concluded that there is a need in organic pig production to produce “new” products, which differ from the conventional pork products in order to comply with consumer expectations

Random walks on semaphore codes and delay de Bruijn semigroups

We develop a new approach to random walks on de Bruijn graphs over the alphabet $A$ through right congruences on $A^k$, defined using the natural right action of $A^+$. A major role is played by special right congruences, which correspond to semaphore codes and allow an easier computation of the hitting time. We show how right congruences can be approximated by special right congruences.Comment: 34 pages; 10 figures; as requested by the journal, the previous version of this paper was divided into two; this version contains Sections 1-8 of version 1; Sections 9-12 will appear as a separate paper with extra material adde

The semaphore codes attached to a Turing machine via resets and their various limits

We introduce semaphore codes associated to a Turing machine via resets. Semaphore codes provide an approximation theory for resets. In this paper we generalize the set-up of our previous paper "Random walks on semaphore codes and delay de Bruijn semigroups" to the infinite case by taking the profinite limit of $k$-resets to obtain $(-\omega)$-resets. We mention how this opens new avenues to attack the P versus NP problem.Comment: 28 pages; Sections 3-6 appeared in a previous version of arXiv:1509.03383 as Sections 9-12 (the split of the previous paper was suggested by the journal); Sections 1-2 and 7 are ne

CoRe: A way to build pedagogical content knowledge for beginning teachers

Research has shown that one of the factors which enables teachers to be effective is their rich pedagogical content knowledge (PCK). Beginning teachers need support to develop this PCK and recent research in the field has proposed a conceptual tool known as “content representations”, or CoRes, as a model for doing this. The study reported here brought together science and technology experts in content and pedagogy, early career secondary teachers, and researchers to design a CoRe to assist development of teacher PCK. The study then researched the early career teachers’ use of the CoRe in their planning and delivery of a unit in their classrooms to examine the effect of the CoRe on teaching and learning, and on the development of the teachers’ PCK

Children’s Creativity Lab : creating a ‘pen of the future’

This work is funded by EPSRC and SICSA.Technology is changing the way we acquire new skills and proficiencies and handwriting is no exception to this. However, while some technological advancements exist in this area, the question of how we can digitally enhance the process of learning handwriting remains under-explored. Being immersed in this process on an everyday basis, we believe that school aged children can provide valuable ideas and insights into the design of future writing tools for learners developing their (hand)writing skills. As end-users of the proposed technology, we explore including children in a form of informed participatory design during a creativity lab where we invited 12 children, aged 11–12, to put themselves into the shoes of a product designers and create a Pen of the Future using prototyping materials. In this paper we describe our methodology and discuss the design ideas that children came up with and how these may inform the design of future writing tools.Postprin

Groundwater animals

Groundwater animals are adapted to live in environments with no light and limited nutrients, They can provide insights into fundamental questions of evolution, ecology and biodiversity. They also have an important role to play in informing the reconstruction of past changes in geomorphology and climate, and can be used for characterising aquifers. The BGS is undertaking a systematic survey of selected areas and lithologies in the UK where groundwater animals have not been investigated. This is important because little is known about groundwater ecosystems in the UK despite the unique contribution to biodiversity made by these animals. Groundwater organisms are also thought to provide ecosystem services by means of their role in nutrient cycling and natural remediation of pollutants in the subsurface. They may also be useful indicators of human impacts on groundwaters