40,487 research outputs found
Rigidity of spherical codes
A packing of spherical caps on the surface of a sphere (that is, a spherical
code) is called rigid or jammed if it is isolated within the space of packings.
In other words, aside from applying a global isometry, the packing cannot be
deformed. In this paper, we systematically study the rigidity of spherical
codes, particularly kissing configurations. One surprise is that the kissing
configuration of the Coxeter-Todd lattice is not jammed, despite being locally
jammed (each individual cap is held in place if its neighbors are fixed); in
this respect, the Coxeter-Todd lattice is analogous to the face-centered cubic
lattice in three dimensions. By contrast, we find that many other packings have
jammed kissing configurations, including the Barnes-Wall lattice and all of the
best kissing configurations known in four through twelve dimensions. Jamming
seems to become much less common for large kissing configurations in higher
dimensions, and in particular it fails for the best kissing configurations
known in 25 through 31 dimensions. Motivated by this phenomenon, we find new
kissing configurations in these dimensions, which improve on the records set in
1982 by the laminated lattices.Comment: 39 pages, 8 figure
Susceptibility and Percolation in 2D Random Field Ising Magnets
The ground state structure of the two-dimensional random field Ising magnet
is studied using exact numerical calculations. First we show that the
ferromagnetism, which exists for small system sizes, vanishes with a large
excitation at a random field strength dependent length scale. This {\it
break-up length scale} scales exponentially with the squared random
field, . By adding an external field we then study the
susceptibility in the ground state. If , domains melt continuously and
the magnetization has a smooth behavior, independent of system size, and the
susceptibility decays as . We define a random field strength dependent
critical external field value , for the up and down spins to
form a percolation type of spanning cluster. The percolation transition is in
the standard short-range correlated percolation universality class. The mass of
the spanning cluster increases with decreasing and the critical
external field approaches zero for vanishing random field strength, implying
the critical field scaling (for Gaussian disorder) , where and .
Below the systems should percolate even when H=0. This implies that
even for H=0 above the domains can be fractal at low random fields, such
that the largest domain spans the system at low random field strength values
and its mass has the fractal dimension of standard percolation .
The structure of the spanning clusters is studied by defining {\it red
clusters}, in analogy to the ``red sites'' of ordinary site-percolation. The
size of red clusters defines an extra length scale, independent of .Comment: 17 pages, 28 figures, accepted for publication in Phys. Rev.
Quantum-fluctuation-induced repelling interaction of quantum string between walls
Quantum string, which was brought into discussion recently as a model for the
stripe phase in doped cuprates, is simulated by means of the
density-matrix-renormalization-group method. String collides with adjacent
neighbors, as it wonders, owing to quantum zero-point fluctuations. The energy
cost due to the collisions is our main concern. Embedding a quantum string
between rigid walls with separation d, we found that for sufficiently large d,
collision-induced energy cost obeys the formula \sim exp (- A d^alpha) with
alpha=0.808(1), and string's mean fluctuation width grows logarithmically \sim
log d. Those results are not understood in terms of conventional picture that
the string is `disordered,' and only the short-wave-length fluctuations
contribute to collisions. Rather, our results support a recent proposal that
owing to collisions, short-wave-length fluctuations are suppressed, but
instead, long-wave-length fluctuations become significant. This mechanism would
be responsible for stabilizing the stripe phase
Short-term Memory of Deep RNN
The extension of deep learning towards temporal data processing is gaining an
increasing research interest. In this paper we investigate the properties of
state dynamics developed in successive levels of deep recurrent neural networks
(RNNs) in terms of short-term memory abilities. Our results reveal interesting
insights that shed light on the nature of layering as a factor of RNN design.
Noticeably, higher layers in a hierarchically organized RNN architecture
results to be inherently biased towards longer memory spans even prior to
training of the recurrent connections. Moreover, in the context of Reservoir
Computing framework, our analysis also points out the benefit of a layered
recurrent organization as an efficient approach to improve the memory skills of
reservoir models.Comment: This is a pre-print (pre-review) version of the paper accepted for
presentation at the 26th European Symposium on Artificial Neural Networks,
Computational Intelligence and Machine Learning (ESANN), Bruges (Belgium),
25-27 April 201
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