Normal forms approach to diffusion near hyperbolic equilibria

We consider the exit problem for small white noise perturbation of a smooth dynamical system on the plane in the neighborhood of a hyperbolic critical point. We show that if the distribution of the initial condition has a scaling limit then the exit distribution and exit time also have a joint scaling limit as the noise intensity goes to zero. The limiting law is computed explicitly. The result completes the theory of noisy heteroclinic networks in two dimensions. The analysis is based on normal forms theory.Comment: 21 page

Regularity of invariant densities for 1D systems with random switching

© 2015 IOP Publishing Ltd & London Mathematical Society.This is a detailed analysis of invariant measures for one-dimensional dynamical systems with random switching. In particular, we prove the smoothness of the invariant densities away from critical points and describe the asymptotics of the invariant densities at critical points

Language Models as Knowledge Bases?

Recent progress in pretraining language models on large textual corpora led to a surge of improvements for downstream NLP tasks. Whilst learning linguistic knowledge, these models may also be storing relational knowledge present in the training data, and may be able to answer queries structured as "fill-in-the-blank" cloze statements. Language models have many advantages over structured knowledge bases: they require no schema engineering, allow practitioners to query about an open class of relations, are easy to extend to more data, and require no human supervision to train. We present an in-depth analysis of the relational knowledge already present (without fine-tuning) in a wide range of state-of-the-art pretrained language models. We find that (i) without fine-tuning, BERT contains relational knowledge competitive with traditional NLP methods that have some access to oracle knowledge, (ii) BERT also does remarkably well on open-domain question answering against a supervised baseline, and (iii) certain types of factual knowledge are learned much more readily than others by standard language model pretraining approaches. The surprisingly strong ability of these models to recall factual knowledge without any fine-tuning demonstrates their potential as unsupervised open-domain QA systems. The code to reproduce our analysis is available at https://github.com/facebookresearch/LAMA

The optimal sink and the best source in a Markov chain

It is well known that the distributions of hitting times in Markov chains are quite irregular, unless the limit as time tends to infinity is considered. We show that nevertheless for a typical finite irreducible Markov chain and for nondegenerate initial distributions the tails of the distributions of the hitting times for the states of a Markov chain can be ordered, i.e., they do not overlap after a certain finite moment of time. If one considers instead each state of a Markov chain as a source rather than a sink then again the states can generically be ordered according to their efficiency. The mechanisms underlying these two orderings are essentially different though.Comment: 12 pages, 1 figur

Invariant densities for dynamical systems with random switching

We consider a non-autonomous ordinary differential equation on a smooth manifold, with right-hand side that randomly switches between the elements of a finite family of smooth vector fields. For the resulting random dynamical system, we show that H\"ormander type hypoellipticity conditions are sufficient for uniqueness and absolute continuity of an invariant measure.Comment: 16 pages; we replaced our original article to point out and close a gap in the discussion of the Lorenz system in Section 7 (see Remark 2); this gap is only present in the journal version of this article --- it wasn't present in the previous arxiv versio

Geodesics and the competition interface for the corner growth model

We study the directed last-passage percolation model on the planar square lattice with nearest-neighbor steps and general i.i.d. weights on the vertices, out- side of the class of exactly solvable models. Stationary cocycles are constructed for this percolation model from queueing fixed points. These cocycles serve as bound- ary conditions for stationary last-passage percolation, solve variational formulas that characterize limit shapes, and yield existence of Busemann functions in directions where the shape has some regularity. In a sequel to this paper the cocycles are used to prove results about semi-infinite geodesics and the competition interface

Stationary cocycles and Busemann functions for the corner growth model

We study the directed last-passage percolation model on the planar square lattice with nearest-neighbor steps and general i.i.d. weights on the vertices, out- side of the class of exactly solvable models. Stationary cocycles are constructed for this percolation model from queueing fixed points. These cocycles serve as bound- ary conditions for stationary last-passage percolation, solve variational formulas that characterize limit shapes, and yield existence of Busemann functions in directions where the shape has some regularity. In a sequel to this paper the cocycles are used to prove results about semi-infinite geodesics and the competition interface

The Body Dances: Carnival Dance and Organization

Building on the work of Pierre Bourdieu and Maurice Merleau-Ponty we seek to open up traditional categories of thought surrounding the relation `body-organization' and elicit a thought experiment: What happens if we move the body from the periphery to the centre? We pass the interlocking theoretical concepts of object-body/subject-body and habitus through the theoretically constructed empirical case of `carnival dance' in order to re-evaluate such key organizational concepts as knowledge and learning. In doing so, we connect with an emerging body of literature on `sensible knowledge'; knowledge that is produced and preserved within bodily practices. The investigation of habitual appropriation in carnival dance also allows us to make links between repetition and experimentation, and reflect on the mechanism through which the principles of social organization, whilst internalized and experienced as natural, are embodied so that humans are capable of spontaneously generating an infinite array of appropriate actions. This perspective on social and organizational life, where change and permanence are intricately interwoven, contrasts sharply with the dominant view in organization studies which juxtaposes change/ creativity and stability

Re-storying and visualizing the changing entrepreneurial identities of Bill Gates and Richard Branson.

The storytelling in textual and visual re-constructions of Bill Gates and Richard Branson by their organizations produces entrepreneurial identities bound into particular social power-knowledge relations. Our purpose is to examine how these organizations, and their critics, mobilize storytelling in acts of re-storying (enlivening) or re-narrating (branding a monologic) practices using Internet technologies to invite viewers to frame the world of entrepreneurship. We use visual discourse and storytelling methods to analyze how Microsoft and Virgin Group use various kinds of entrepreneurial images and textual narratives to re-narrate and produce particular brands of capitalism. These organizations' scoptic regimes of representation are contested in counter-visualizing and counterstory practices of external stakeholders. We suggest that the image and textual practices of storytelling have changed as both entrepreneurs court philanthropic and social entrepreneur identity markers. Our contribution to entrepreneurial identity is to apply double and multiple narrations, the appropriation of another's narrative words (or images) into another's narrative, and relate such storytelling moves to visuality