Semiclassical Dynamics with Exponentially Small Error Estimates

We construct approximate solutions to the time--dependent Schr\"odinger equation $i \hbar (\partial \psi)/(\partial t) = - (\hbar^2)/2 \Delta \psi + V \psi$ for small values of $\hbar$. If $V$ satisfies appropriate analyticity and growth hypotheses and $|t|\le T$, these solutions agree with exact solutions up to errors whose norms are bounded by $C \exp{-\gamma/\hbar}$, for some $C$ and $\gamma>0$. Under more restrictive hypotheses, we prove that for sufficiently small $T', |t|\le T' |\log(\hbar)|$ implies the norms of the errors are bounded by $C' \exp{-\gamma'/\hbar^{\sigma}}$, for some $C', \gamma'>0$, and $\sigma>0$

A Time-Dependent Born-Oppenheimer Approximation with Exponentially Small Error Estimates

We present the construction of an exponentially accurate time-dependent Born-Oppenheimer approximation for molecular quantum mechanics. We study molecular systems whose electron masses are held fixed and whose nuclear masses are proportional to $\epsilon^{-4}$, where $\epsilon$ is a small expansion parameter. By optimal truncation of an asymptotic expansion, we construct approximate solutions to the time-dependent Schr\"odinger equation that agree with exact normalized solutions up to errors whose norms are bounded by \ds C \exp(-\gamma/\epsilon^2), for some C and $\gamma>0$

Exponentially Accurate Semiclassical Dynamics: Propagation, Localization, Ehrenfest Times, Scattering and More General States

We prove six theorems concerning exponentially accurate semiclassical quantum mechanics. Two of these theorems are known results, but have new proofs. Under appropriate hypotheses, they conclude that the exact and approximate dynamics of an initially localized wave packet agree up to exponentially small errors in $\hbar$ for finite times and for Ehrenfest times. Two other theorems state that for such times the wave packets are localized near a classical orbit up to exponentially small errors. The fifth theorem deals with infinite times and states an exponentially accurate scattering result. The sixth theorem provides extensions of the other five by allowing more general initial conditions

Trade and the (Dis)Incentive to Reform Labor Markets: The Case of Reform in the European Union

In a closed economy general equilibrium model, Hopenhayn and Rogerson (1993) find large welfare gains to removing firing restrictions. We explore the extent to which international trade alters this result. When economies trade, labor market policies in one country spill over to other countries through their effect on the terms of trade. A key finding in the open economy is that the share of the welfare gains from domestic labor market reform exported substantially exceeds the share of goods exported. In our baseline case, 105 percent of the welfare gains are exported even though the domestic economy only exports 30 percent of its goods. Thus, with international trade a country receives little to no benefit, and possibly even loses, from unilaterally reforming its labor market. A coordinated elimination of firing taxes yields considerable benefits. We find the welfare gains to the U.K. from labor market reform by its continental trading partners of 0.21 percent of steady state consumption. This insight provides some explanation for recent efforts toward labor market reform in the European Union.Firing Costs, International Trade, Labor Market Reform

Qualitative Data Analysis Challenges in Co-Designing Educational Technology Systems for Refugee Children

There is a growing interest in the potential for technology to facilitate emergency education of refugee children. But designing in this space requires knowledge of the displaced population and the contextual dynamics surrounding it. Design should therefore be informed by both existing research across relevant disciplines, and from those who are on the ground facing the problem in real life. This paper describes a process that is based on literature from emergency education, student engagement and motivation, educational technology, and participatory design. We describe how this process was implemented leading to the design of a digital learning space for children living in a refugee camp in Greece. The challenge of data analysis is critical, as the qualitative data in the process is elicited from activities of various natures and thus moving from qualitative data to designs is a critical challenge that we are looking to cover for our process to be complete and applicable. We discuss some of the challenges that can be expected in such context

Exponentially Accurate Semiclassical Tunneling Wave Functions in One Dimension

We study the time behavior of wave functions involved in tunneling through a smooth potential barrier in one dimension in the semiclassical limit. We determine the leading order component of the wave function that tunnels. It is exponentially small in $1/\hbar$. For a wide variety of incoming wave packets, the leading order tunneling component is Gaussian for sufficiently small $\hbar$. We prove this for both the large time asymptotics and for moderately large values of the time variable

Reconciling Pyroclastic Flow and Surge: the Multiphase Physics of Pyroclastic Density Currents.

Two end-member types of pyroclastic density current are commonly recognized: pyroclastic surges are dilute currents in which particles are carried in turbulent suspension and pyroclastic flows are highly concentrated flows. We provide scaling relations that unify these end-members and derive a segregation mechanism into basal concentrated flow and overriding dilute cloud based on the Stokes number (ST), the Stability factor (ET) and the Dense-Dilute condition (DD). We recognize five types of particle behaviors within a fluid eddy as a function of ST and ET : (1) particles sediment from the eddy, (2) particles are preferentially settled out during the downward motion of the eddy, but can be carried during its upward motion, (3) particles concentrate on the periphery of the eddy, (4) particles settling can be delayed or “fast-tracked” as a function of the eddy spatial distribution, and (5) particles remain homogeneously distributed within the eddy. We extend these concepts to a fully turbulent flow by using a prototype of kinetic energy distribution within a full eddy spectrum and demonstrate that the presence of different particle sizes leads to the density stratification of the current. This stratification may favor particle interactions in the basal part of the flow and DD determines whether the flow is dense or dilute. Using only intrinsic characteristics of the current, our model explains the discontinuous features between pyroclastic flows and surges while conserving the concept of a continuous spectrum of density currents

Tightly-Coupled Multiprocessing for a Global Illumination Algorithm

{dret | elf} @ dgp.toronto.edu A prevailing trend in computer graphics is the demand for increasingly realistic global illumination models and algorithms. Despite the fact that the computational power of uniprocessors is increasing, it is clear that much greater computational power is required to achieve satisfactory throughput. The obvious next step is to employ parallel processing. The advent of affordable, tightly-coupled multiprocessors makes such an approach widely available for the first time. We propose a tightly-coupled parallel decomposition of FIAT, a global illumination algorithm, based on space subdivision and power balancing, that we have recently developed. This algorithm is somewhat ambitious, and severely strains existing uniprocessor environments. We discuss techniques for reducing memory contention and maximising parallelism. We also present empirical data on the actual performance of our parallel solution. Since the model of parallel computation that we have employed is likely to persist for quite some time, our techniques are applicable to other algorithms based on space subdivision. 1