A mean-field game economic growth model

Here, we examine a mean-field game (MFG) that models the economic growth of a population of non-cooperative rational agents. In this MFG, agents are described by two state variables - the capital and consumer goods they own. Each agent seeks to maximize their utility by taking into account statistical data of the total population. The individual actions drive the evolution of the players, and a market-clearing condition determines the relative price of capital and consumer goods. We study the existence and uniqueness of optimal strategies of the agents and develop numerical methods to compute these strategies and the equilibrium price

Quantum optimal transport and weak topologies

Several extensions of the classical optimal transport distances to the quantum setting have been proposed. In this paper, we investigate the pseudometrics introduced by Golse, Mouhot and Paul in [Commun Math Phys 343:165-205, 2016] and by Golse and Paul in [Arch Ration Mech Anal 223:57-94, 2017]. These pseudometrics serve as a quantum analogue of the Monge--Kantorovich--Wasserstein distances of order $2$ on the phase space. We prove that they are comparable to negative Sobolev norms up to a small term in the semiclassical approximation, which can be expressed using the Wigner--Yanase Skew information. This enables us to improve the known results in the context of the mean-field and semiclassical limits by requiring less regularity on the initial data.Comment: 25 page

Optimal Semiclassical Regularity of Projection Operators and Strong Weyl Law

Projection operators arise naturally as one particle density operators associated to Slater determinants in fields such as quantum mechanics and the study of determinantal processes. In the context of the semiclassical approximation of quantum mechanics, projection operators can be seen as the analogue of characteristic functions of subsets of the phase space, which are discontinuous functions. We prove that projection operators indeed converge to characteristic functions of the phase space and that in terms of quantum Sobolev spaces, they exhibit the same maximal regularity as characteristic functions. This can be interpreted as a semiclassical asymptotic on the size of commutators in Schatten norms. Our study answers a question raised in [J. Chong, L. Lafleche, C. Saffirio, arXiv:2103.10946 [math.AP]] about the possibility of having projection operators as initial data, and also gives a strong convergence result for the Weyl law.Comment: 18 pages, 2 figures. v2: context on the semiclassical mean-field limit added, estimate of Remark 2.2 improved, bibliography update

We visit the cemetery where you might be buried

These poems explore the death of a spouse through metaphorical explorations of the sensual and spiritual dimensions of grief

We reminisce about our first winter solstice

These three poems explore the death of a spouse through poems that address the sensual and spiritual losses

Design and Monitoring of Earth Embankments over Permafrost

As the northern regions of Canada are developed, there is an increasing need to protect the fragile ecology as well as to maximize usage of local construction materials. The construction of earth dykes to retain liquid wastes is a common requirement in municipal and industrial developments. Frozen core earthfill dykes provide an effective technique to cut off seepage in cold permafrost areas (Sayles, 1984). The seepage of water through an unfrozen overburden or fractured bedrock foundation can occur and accelerate the thermal deterioration of an earth embankment. The development of the active layer during the summer reduces the dam\u27s ability to retain water if the freeboard is inadequate. Several earthfill dams were built at the Lupin mine near Contwoyto Lake in the Canadian Arctic to form a mine tailings pond. Even though design forecasts indicated 9 m high structures would remain frozen after impoundment of the reservoir, very few case histories were found to support the design. Several earth dams have been monitored since 1982. Initially, ground temperature measurements were taken with thermistor strings in short boreholes. More recently, deep boreholes were instrumented with thermistor strings and the ground probing radar has been used to confirm and locate unfrozen zones within the dams. Specifically, the performance of three dams is reviewed here. The first dam, the base case, was built over virgin cold permafrost. The complete dam section froze during the first winter after construction. Part of the second dam was built over a 5 m deep talik associated with a seasonal creek and possibly a fault zone. The talik is apparently mostly refrozen and continuing to cool, however, geophysical surveys indicate a possible unfrozen remnant. The third dam was built across the reservoir after impoundment and during the winter. The internal nature of that dam and its thermal behaviour are quite different from the above two. The thermal regime of the dams and underlying foundation has changed considerably over the five years following construction. The results of the ground temperature and radar profiles are compared for various seasons to reconstruct the transient thermal regime at uninstrumented sections. The findings are significant for the design and monitoring of future water retaining structures in the North

From many-body quantum dynamics to the Hartree-Fock and Vlasov equations with singular potentials

We obtain the combined mean-field and semiclassical limit from the $N$-body Schr\"{o}dinger equation for fermions interacting via singular potentials. To obtain the result, we first prove the uniformity in Planck's constant $h$ propagation of regularity for solutions to the Hartree\unicode{x2013}Fock equation with singular pair interaction potentials of the form $\pm |x-y|^{-a}$, including the Coulomb and gravitational interactions. In the context of mixed states, we use these regularity properties to obtain quantitative estimates on the distance between solutions to the Schr\"{o}dinger equation and solutions to the Hartree\unicode{x2013}Fock and Vlasov equations in Schatten norms. For $a\in(0,1/2)$, we obtain local-in-time results when $N^{-1/2} \ll h \leq N^{-1/3}$. In particular, it leads to the derivation of the Vlasov equation with singular potentials. For $a\in[1/2,1]$, our results hold only on a small time scale, or with an $N$-dependent cutoff.Comment: 75 pages; introduction improved and some errors in the propagation of regularity part correcte