We show that the basic reproduction number of an SIS patch model with standard incidence is either strictly decreasing and strictly convex with respect to the diffusion coefficient of infected subpopulation if the patch reproduction numbers of at least two patches in isolation are distinct or constant otherwise. Biologically, it means that fast diffusion of the infected people reduces the risk of infection. This completely solves and generalizes a conjecture by Allen et al. (SIAM J Appl Math, 67: 1283-1309, 2007. Furthermore, a substantially improved lower bound on the multipatch reproduction number, a generalized monotone result on the spectral bound the Jacobian matrix of the model system at the disease-free equilibrium, and the limiting endemic equilibrium are obtained. The approach and results can be applied to a class of epidemic patch models where only one class of infected compartments migrate between patches and one transmission route is involved.AMS subject classifications. 91D25, 34D20, 92D30, 34D05, 15B48, 15A42.
Let G be a complex connected simple algebraic group with a fixed real form σ. Let G(R) = G σ be the corresponding group of real points. This paper reports a finiteness theorem for the classification of irreducible unitary Harish-Chandra modules of G(R) (up to equivalence) having non-vanishing Dirac cohomology. Moreover, we study the distribution of the spin norm along Vogan pencils for certain G(R), with particular attention paid to the unitarily small convex hull introduced by Salamanca-Riba and Vogan.2010 Mathematics Subject Classification. Primary 22E46.
Let
g
\mathfrak {g}
be any finite-dimensional complex simple Lie algebra. In this paper, we show that the spin norm increases strictly along any pencil once it goes beyond the u-small convex hull of
g
\mathfrak {g}
.
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