Modulated phases in magnetic models frustrated by long-range interactions

We study an Ising model in one dimension with short range ferromagnetic and long range (power law) antiferromagnetic interactions. We show that the zero temperature phase diagram in a (longitudinal) field H involves a sequence of up and down domains whose size varies continuously with H, between -H_c and H_c which represent the edge of the ferromagnetic up and down phases. The implications of long range interaction in many body systems are discussed.Comment: 5 pages, 3 figure

Almost direct summands

In this short note, we point out how some new cases of Hochster's direct summand conjecture can be deduced from fundamental theorems in p-adic Hodge theory due to Faltings. The cases tackled include the ones when the ramification of the map being considered lies entirely in characteristic p.Comment: 5 pages, comments welcome

Between Sense and Sensibility: Declarative narrativisation of mental models as a basis and benchmark for visuo-spatial cognition and computation focussed collaborative cognitive systems

What lies between `\emph{sensing}' and `\emph{sensibility}'? In other words, what kind of cognitive processes mediate sensing capability, and the formation of sensible impressions ---e.g., abstractions, analogies, hypotheses and theory formation, beliefs and their revision, argument formation--- in domain-specific problem solving, or in regular activities of everyday living, working and simply going around in the environment? How can knowledge and reasoning about such capabilities, as exhibited by humans in particular problem contexts, be used as a model and benchmark for the development of collaborative cognitive (interaction) systems concerned with human assistance, assurance, and empowerment? We pose these questions in the context of a range of assistive technologies concerned with \emph{visuo-spatial perception and cognition} tasks encompassing aspects such as commonsense, creativity, and the application of specialist domain knowledge and problem-solving thought processes. Assistive technologies being considered include: (a) human activity interpretation; (b) high-level cognitive rovotics; (c) people-centred creative design in domains such as architecture & digital media creation, and (d) qualitative analyses geographic information systems. Computational narratives not only provide a rich cognitive basis, but they also serve as a benchmark of functional performance in our development of computational cognitive assistance systems. We posit that computational narrativisation pertaining to space, actions, and change provides a useful model of \emph{visual} and \emph{spatio-temporal thinking} within a wide-range of problem-solving tasks and application areas where collaborative cognitive systems could serve an assistive and empowering function.Comment: 5 pages, research statement summarising recent publication