The Navier-Stokes regularity problem

There is currently no proof guaranteeing that, given a smooth initial condition, the three-dimensional Navier–Stokes equations have a unique solution that exists for all positive times. This paper reviews the key rigorous results concerning the existence and uniqueness of solutions for this model. In particular, the link between the regularity of solutions and their uniqueness is highlighted

Sport-Related Injuries in Canadian Interuniversity Athletics: A Descriptive Epidemiologic Analysis of Knee Injuries, 2014-2017

The purpose of this study was to identify and establish the extent of sport-related injury in Canadian university varsity athletics, focusing on knee injury and significant sport injury. Of the SRIs reported, 20% were significant in nature and 23.2% were knee injuries. A larger percentage of knee SRIs were significant compared non-knee SRIs. Those suffering a knee injury were 4.5 times more likely to suffer a significant injury than those afflicted with non-knee injuries. Mens volleyball athletes are two times more likely to suffer a knee injury, once controlled for sport group. Once controlled for sport, mens hockey athletes are 2.3 times more likely to suffer a significant injury than any other sport (Table 4). Mens basketball has the highest rate of injury (3.32 per athlete) during this reporting period and football has the highest rate of significant knee injury at 0.27 per athlete (Table 1)

Parametrization of global attractors experimental observations and turbulence

This paper is concerned with rigorous results in the theory of turbulence and fluid flow. While derived from the abstract theory of attractors in infinite-dimensional dynamical systems, they shed some light on the conventional heuristic theories of turbulence, and can be used to justify a well-known experimental method. Two results are discussed here in detail, both based on parametrization of the attractor. The first shows that any two fluid flows can be distinguished by a sufficient number of point observations of the velocity. This allows one to connect rigorously the dimension of the attractor with the Landau–Lifschitz ‘number of degrees of freedom’, and hence to obtain estimates on the ‘minimum length scale of the flow’ using bounds on this dimension. While for two-dimensional flows the rigorous estimate agrees with the heuristic approach, there is still a gap between rigorous results in the three-dimensional case and the Kolmogorov theory. Secondly, the problem of using experiments to reconstruct the dynamics of a flow is considered. The standard way of doing this is to take a number of repeated observations, and appeal to the Takens time-delay embedding theorem to guarantee that one can indeed follow the dynamics ‘faithfully’. However, this result relies on restrictive conditions that do not hold for spatially extended systems: an extension is given here that validates this important experimental technique for use in the study of turbulence. Although the abstract results underlying this paper have been presented elsewhere, making them specific to the Navier–Stokes equations provides answers to problems particular to fluid dynamics, and motivates further questions that would not arise from within the abstract theory itself

Botswana as a Role Model for Country Success

I argue that the economic success of Botswana can be explained by the historical development of its institutions which is related to the trajectory of the Tswana states over the past 200 years. These institutions created a much more stable and accountable government than elsewhere in Africa after independence with the desire and incentive to adopt good economic policies. There are two main lessons from this experience. The first is how successful an African economy can become using simple orthodox well-understood policies. The second is that successful development in Africa will be helped by a focus on the development of state institutions. Though Botswana inherited different institutions from elsewhere, it also built on these, in particular trying to create a national identity and to continually modernize and adapt institutions. There are many lessons for other African countries from these policy choices.governance, patrimonialism, state formation

Elites and Institutional Persistence

Particular sets of institutions, once they become established in a society, have a strong tendency to persist. In this paper I argue that understanding how elites form and reproduce is key to understanding the persistence of institutions over time. I illustrate this idea with a simple political economy theory of institutions and through examples from Liberia, the US, South Africa and Germany I show how elites influence institutions. To change institutions requires having an understanding of how reforms influence the preferences, capabilities and strategies of elites.elites, political economy, persistence of institutions

Evolution of the cluster abundance in non-Gaussian models

We carry out N-body simulations of several non-Gaussian structure formation models, including Peebles' isocurvature cold dark matter model, cosmic string models, and a model with primordial voids. We compare the evolution of the cluster mass function in these simulations with that predicted by a modified version of the Press-Schechter formalism. We find that the Press-Schechter formula can accurately fit the cluster evolution over a wide range of redshifts for all of the models considered, with typical errors in the mass function of less than 25%, considerably smaller than the amount by which predictions for different models may differ. This work demonstrates that the Press-Schechter formalism can be used to place strong model independent constraints on non-Gaussianity in the universe.Comment: 11 pages, 12 postscipt figure

Embedding Properties of sets with finite box-counting dimension

In this paper we study the regularity of embeddings of finite--dimensional subsets of Banach spaces into Euclidean spaces. In 1999, Hunt and Kaloshin [Nonlinearity 12 1263-1275] introduced the thickness exponent and proved an embedding theorem for subsets of Hilbert spaces with finite box--counting dimension. In 2009, Robinson [Nonlinearity 22 711-728] defined the dual thickness and extended the result to subsets of Banach spaces. Here we prove a similar result for subsets of Banach spaces, using the thickness rather than the dual thickness. We also study the relation between the box-counting dimension and these two thickness exponents for some particular subsets of $\ell_{p}$.Comment: Submitted, Referres comments addresse