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Optimal linear stability condition for scalar differential equations with distributed delay
Linear scalar differential equations with distributed delays appear in the
study of the local stability of nonlinear differential equations with feedback,
which are common in biology and physics. Negative feedback loops tend to
promote oscillations around steady states, and their stability depends on the
particular shape of the delay distribution. Since in applications the mean
delay is often the only reliable information available about the distribution,
it is desirable to find conditions for stability that are independent from the
shape of the distribution. We show here that for a given mean delay, the linear
equation with distributed delay is asymptotically stable if the associated
differential equation with a discrete delay is asymptotically stable. We
illustrate this criterion on a compartment model of hematopoietic cell dynamics
to obtain sufficient conditions for stability
Small eta-N scattering lengths favour eta-d and eta-alpha states
Unstable states of the eta meson and the 3He nucleus predicted using the time
delay method were found to be in agreement with a recent claim of eta-mesic 3He
states made by the TAPS collaboration. Here, we extend this method to a
speculative study of the unstable states occurring in the eta-d and eta-4He
elastic scattering. The T-matrix for eta-4He scattering is evaluated within the
Finite Rank Approximation (FRA) of few body equations. For the evaluation of
time delay in the eta-d case, we use a parameterization of an existing Faddeev
calculation and compare the results with those obtained from FRA. With an eta-N
scattering length, fm, we find an eta-d unstable
bound state around -16 MeV, within the Faddeev calculation. A similar state
within the FRA is found for a low value of , namely, fm. The existence of an eta-4He unstable bound state close to
threshold is hinted by fm, but is ruled out by
large scattering lengths.Comment: 21 pages, LaTex, 7 Figure
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