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Generic transient memory formation in disordered systems with noise
Out-of-equilibrium disordered systems may form memories of external driving
in a remarkable fashion. The system "remembers" multiple values from a series
of training inputs yet "forgets" nearly all of them at long times despite the
inputs being continually repeated. Here, learning and forgetting are
inseparable aspects of a single process. The memory loss may be prevented by
the addition of noise. We identify a class of systems with this behavior,
giving as an example a model of non-brownian suspensions under cyclic shear.Comment: 4 pages, 3 figure
On the degree two entry of a Gorenstein -vector and a conjecture of Stanley
In this note we establish a (non-trivial) lower bound on the degree two entry
of a Gorenstein -vector of any given socle degree and any
codimension .
In particular, when , that is for Gorenstein -vectors of the form
, our lower bound allows us to prove a conjecture of Stanley
on the order of magnitude of the minimum value, say , that may
assume. In fact, we show that
In general, we wonder whether our lower bound is sharp for all integers
and .Comment: A few minor changes. To appear in Proc. of the AM
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