Relative entropy methods for hyperbolic and diffusive limits

We review the relative entropy method in the context of hyperbolic and diffusive relaxation limits of entropy solutions for various hyperbolic models. The main example consists of the convergence from multidimensional compressible Euler equations with friction to the porous medium equation \cite{LT12}. With small modifications, the arguments used in that case can be adapted to the study of the diffusive limit from the Euler-Poisson system with friction to the Keller-Segel system \cite{LT13}. In addition, the $p$--system with friction and the system of viscoelasticity with memory are then reviewed, again in the case of diffusive limits \cite{LT12}. Finally, the method of relative entropy is described for the multidimensional stress relaxation model converging to elastodynamics \cite[Section 3.2]{LT06}, one of the first examples of application of the method to hyperbolic relaxation limits

Exploring dialogic approaches in teaching and learning: a study in a rural Kenyan community

In Kenya, there is a need to investigate the pedagogies that are currently applied to primary schools. This is important, because educators and researchers question the predominant use of teacher-centred pedagogies in Kenyan schools. Although pedagogical reforms have been slow in Kenya, the current government has stressed the importance of developing an education system that meets the needs of the students, a system that is also globally competitive. The use of dialogic pedagogies in classroom learning has been seen as valuable since students can actively engage (Mercer, 2008) and can question issues that relate to them (Freire, 1993; Ladson-Billings, 1994). These engagements have the potential to prepare students better for their lives outside of school. This study is a mixture of methods under an ethnographic approach, through which I have aimed to obtain Kenyan teachers’ insider interpretations of their setting and practice. As Tabulawa suggests (2013), insiders’ voices and engagement are critical to the progress of pedagogical development in Africa. Therefore, I have explored with Kenyan teachers the current pedagogies of teaching and learning. I used four pedagogical spaces as lenses to help to determine how dialogic pedagogies can be applicable. The four pedagogic ‘spaces’ in the study are: interaction spaces, physical spaces, cultural spaces, and policy spaces. Two rural schools in Central Kenya were involved in data collection, plus a local community church. The study revealed teachers’ everyday practices and provided information on key areas that would help pedagogical development. Additionally, the study indicated the need for pedagogical reforms that value the local context, meet Kenyan students’ needs and engage teachers in the process. A comprehensive scrutiny of the findings led to the recommendation of the ‘Harambee’ approach, which suggests how dialogic pedagogies could be employed in the Kenyan education

Relative entropy in diffusive relaxation

We establish convergence in the diffusive limit from entropy weak solutions of the equations of compressible gas dynamics with friction to the porous media equation away from vacuum. The result is based on a Lyapunov type of functional provided by a calculation of the relative entropy. The relative entropy method is also employed to establish convergence from entropic weak solutions of viscoelasticity with memory to the system of viscoelasticity of the rate-type

Intangible Capital and Economic Growth

Published macroeconomic data traditionally exclude most intangible investment from measured GDP. This situation is beginning to change, but our estimates suggest that as much as $800 billion is still excluded from U.S. published data (as of 2003), and that this leads to the exclusion of more than$3 trillion of business intangible capital stock. To assess the importance of this omission, we add capital to the standard sources-of-growth framework used by the BLS, and find that the inclusion of our list of intangible assets makes a significant difference in the observed patterns of U.S. economic growth. The rate of change of output per worker increases more rapidly when intangibles are counted as capital, and capital deepening becomes the unambiguously dominant source of growth in labor productivity. The role of multifactor productivity is correspondingly diminished, and labor's income share is found to have decreased significantly over the last 50 years.

Indexation Rules, Risk Aversion, and Imperfect Information

Nominal wage adjustment is modeled as resulting from bargaining between a risk neutral firm and a risk averse worker, in an environment where the rate of inflation is a random variable. Risk aversion makes for endogenous indexation arrangements, which deliver partial indexation as they exploit imperfect inflation indices; risk aversion also generates a positive correlation between indexation and inflation variance. The model suggests a distinction between complete vs incomplete inflation adjustment on the one hand, and perfect vs imperfect adjustment on the other hand

Understanding when Dynamics-Invariant Data Augmentations Benefit Model-Free Reinforcement Learning Updates

Recently, data augmentation (DA) has emerged as a method for leveraging domain knowledge to inexpensively generate additional data in reinforcement learning (RL) tasks, often yielding substantial improvements in data efficiency. While prior work has demonstrated the utility of incorporating augmented data directly into model-free RL updates, it is not well-understood when a particular DA strategy will improve data efficiency. In this paper, we seek to identify general aspects of DA responsible for observed learning improvements. Our study focuses on sparse-reward tasks with dynamics-invariant data augmentation functions, serving as an initial step towards a more general understanding of DA and its integration into RL training. Experimentally, we isolate three relevant aspects of DA: state-action coverage, reward density, and the number of augmented transitions generated per update (the augmented replay ratio). From our experiments, we draw two conclusions: (1) increasing state-action coverage often has a much greater impact on data efficiency than increasing reward density, and (2) decreasing the augmented replay ratio substantially improves data efficiency. In fact, certain tasks in our empirical study are solvable only when the replay ratio is sufficiently low

On-Policy Policy Gradient Reinforcement Learning Without On-Policy Sampling

On-policy reinforcement learning (RL) algorithms perform policy updates using i.i.d. trajectories collected by the current policy. However, after observing only a finite number of trajectories, on-policy sampling may produce data that fails to match the expected on-policy data distribution. This sampling error leads to noisy updates and data inefficient on-policy learning. Recent work in the policy evaluation setting has shown that non-i.i.d., off-policy sampling can produce data with lower sampling error than on-policy sampling can produce. Motivated by this observation, we introduce an adaptive, off-policy sampling method to improve the data efficiency of on-policy policy gradient algorithms. Our method, Proximal Robust On-Policy Sampling (PROPS), reduces sampling error by collecting data with a behavior policy that increases the probability of sampling actions that are under-sampled with respect to the current policy. Rather than discarding data from old policies -- as is commonly done in on-policy algorithms -- PROPS uses data collection to adjust the distribution of previously collected data to be approximately on-policy. We empirically evaluate PROPS on both continuous-action MuJoCo benchmark tasks as well as discrete-action tasks and demonstrate that (1) PROPS decreases sampling error throughout training and (2) improves the data efficiency of on-policy policy gradient algorithms. Our work improves the RL community's understanding of a nuance in the on-policy vs off-policy dichotomy: on-policy learning requires on-policy data, not on-policy sampling

HST emission line images of the Orion HII region: proper motions and possible variability

Using HST emission line images of the Orion Nebula, separated by 7 years in epoch, we have obtained evidence of localized temporal variability of both density and temperature during this period. Applying a digital filter to reduce high frequency noise, we used images in Hα and [OIII] to quantify separately the variability in these two parameters. We detected fractional temperature variations of order 0.4% on scales of 2′10−2 pc. The same images yielded proper motion information; using cross-correlation to optimize the accuracy of the differential measurements we produced velocity field maps across the nebula, with vectors ranging up to ~130kms−1 across the line of sight. It is notable that in zones of rapid proper motion we find by far the largest density variations, as would be expected. It is much easier to quantify the temperature variations, on the other hand, in zones with low or zero detectable proper motion (see the other figure here), though these temperature variations appear across the whole face of the nebul