Noncommutative phase spaces are generated and classified in the framework of centrally extended anisotropic planar kinematical Lie groups as well as in the framework of noncentrally extended planar absolute time Lie groups. Through these constructions the coordinates of the phase spaces do not commute due to the presence of naturally introduced fields giving rise to minimal couplings. By symplectic realizations methods, physical interpretations of generators coming from the obtained structures are given.
This paper deals with the construction of noncommutative phase spaces as coadjoint orbits of noncentral extensions of Galilei and Para-Galilei groups in two-dimensional space. The noncommutativity is due to the presence of a dual magnetic field B * in the Galilei case and of a magnetic field B in the Para-Galilei case.
In epidemiology, the effective reproduction number $R_{e}$
R
e
is used to characterize the growth rate of an epidemic outbreak. If $R_{e} >1$
R
e
>
1
, the epidemic worsens, and if $R_{e}< 1$
R
e
<
1
, then it subsides and eventually dies out. In this paper, we investigate properties of $R_{e}$
R
e
for a modified SEIR model of COVID-19 in the city of Houston, TX USA, in which the population is divided into low-risk and high-risk subpopulations. The response of $R_{e}$
R
e
to two types of control measures (testing and distancing) applied to the two different subpopulations is characterized. A nonlinear cost model is used for control measures, to include the effects of diminishing returns. Lowest-cost control combinations for reducing instantaneous $R_{e}$
R
e
to a given value are computed. We propose three types of heuristic strategies for mitigating COVID-19 that are targeted at reducing $R_{e}$
R
e
, and we exhibit the tradeoffs between strategy implementation costs and number of deaths. We also consider two variants of each type of strategy: basic strategies, which consider only the effects of controls on $R_{e}$
R
e
, without regard to subpopulation; and high-risk prioritizing strategies, which maximize control of the high-risk subpopulation. Results showed that of the three heuristic strategy types, the most cost-effective involved setting a target value for $R_{e}$
R
e
and applying sufficient controls to attain that target value. This heuristic led to strategies that begin with strict distancing of the entire population, later followed by increased testing. Strategies that maximize control on high-risk individuals were less cost-effective than basic strategies that emphasize reduction of the rate of spreading of the disease. The model shows that delaying the start of control measures past a certain point greatly worsens strategy outcomes. We conclude that the effective reproduction can be a valuable real-time indicator in determining cost-effective control strategies.
Let (M, g) be a non-compact and complete Riemannian manifold with minimal horospheres and infinite injectivity radius. In this paper we prove that bounded functions on (M, g) satisfying the mean-value property are constant. We thus extend a result of Ranjan and Shah ['Harmonic manifolds with minimal horospheres', J. Geom. Anal. 12(4) (2002), 683-694] where they proved a similar result for bounded harmonic functions on harmonic manifolds with minimal horospheres.2000 Mathematics subject classification: 53C21, 53C25.
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