7,274 research outputs found
Echo State Condition at the Critical Point
Recurrent networks with transfer functions that fulfill the Lipschitz
continuity with K=1 may be echo state networks if certain limitations on the
recurrent connectivity are applied. It has been shown that it is sufficient if
the largest singular value of the recurrent connectivity is smaller than 1. The
main achievement of this paper is a proof under which conditions the network is
an echo state network even if the largest singular value is one. It turns out
that in this critical case the exact shape of the transfer function plays a
decisive role in determining whether the network still fulfills the echo state
condition. In addition, several examples with one neuron networks are outlined
to illustrate effects of critical connectivity. Moreover, within the manuscript
a mathematical definition for a critical echo state network is suggested
The Osculating cone to special Brill-Noether Loci
In this paper, we describe the osculating cone to Brill-Noether loci
at smooth isolated points of for a smooth
canonically embedded curve of even genus . In particular, we show
that the canonical curve is a component of the osculating cone. The proof
is based on techniques introduced by George Kempf in 1986.Comment: 18 pages, v3: to appear in Collectanea Mathematic
The propagation of Elastic Waves in Granular Solid Hydrodynamics
The anisotropic, stress-dependent velocity of elastic waves in glass beads --
as observed by Y. Khidas and X. Jia, see [Phys. Rev. E, 81:021303, Feb. 2010]
-- is shown to be well accounted for by ``granular solid hydrodynamics,'' a
broad-range macroscopic theory of granular behavior. As the theory makes no
reference to fabric anisotropy, the influence of which on sound is in doubt.Comment: 4 pages, 1 fi
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