2,924 research outputs found
Phononic Rogue Waves
We present a theoretical study of extreme events occurring in phononic
lattices. In particular, we focus on the formation of rogue or freak waves,
which are characterized by their localization in both spatial and temporal
domains. We consider two examples. The first one is the prototypical nonlinear
mass-spring system in the form of a homogeneous Fermi-Pasta-Ulam-Tsingou (FPUT)
lattice with a polynomial potential. By deriving an approximation based on the
nonlinear Schroedinger (NLS) equation, we are able to initialize the FPUT model
using a suitably transformed Peregrine soliton solution of the NLS, obtaining
dynamics that resembles a rogue wave on the FPUT lattice. We also show that
Gaussian initial data can lead to dynamics featuring rogue wave for
sufficiently wide Gaussians. The second example is a diatomic granular crystal
exhibiting rogue wave like dynamics, which we also obtain through an NLS
reduction and numerical simulations. The granular crystal (a chain of particles
that interact elastically) is a widely studied system that lends itself to
experimental studies. This study serves to illustrate the potential of such
dynamical lattices towards the experimental observation of acoustic rogue
waves.Comment: 9 pages, 4 figure
Noise Sensitivity of Boolean Functions and Applications to Percolation
It is shown that a large class of events in a product probability space are
highly sensitive to noise, in the sense that with high probability, the
configuration with an arbitrary small percent of random errors gives almost no
prediction whether the event occurs. On the other hand, weighted majority
functions are shown to be noise-stable. Several necessary and sufficient
conditions for noise sensitivity and stability are given.
Consider, for example, bond percolation on an by grid. A
configuration is a function that assigns to every edge the value 0 or 1. Let
be a random configuration, selected according to the uniform measure.
A crossing is a path that joins the left and right sides of the rectangle, and
consists entirely of edges with . By duality, the probability
for having a crossing is 1/2. Fix an . For each edge , let
with probability , and
with probability , independently of the
other edges. Let be the probability for having a crossing in
, conditioned on . Then for all sufficiently large,
.Comment: To appear in Inst. Hautes Etudes Sci. Publ. Mat
On the Sensitivity Conjecture
The sensitivity of a Boolean function f:{0,1}^n -> {0,1} is the maximal number of neighbors a point in the Boolean hypercube has with different f-value. Roughly speaking, the block sensitivity allows to flip a set of bits (called a block) rather than just one bit, in order to change the value of f. The sensitivity conjecture, posed by Nisan and Szegedy (CC, 1994), states that the block sensitivity, bs(f), is at most polynomial in the sensitivity, s(f), for any Boolean function f. A positive answer to the conjecture will have many consequences, as the block sensitivity is polynomially related to many other complexity measures such as the certificate complexity, the decision tree complexity and the degree. The conjecture is far from being understood, as there is an exponential gap between the known upper and lower bounds relating bs(f) and s(f).
We continue a line of work started by Kenyon and Kutin (Inf. Comput., 2004), studying the l-block sensitivity, bs_l(f), where l bounds the size of sensitive blocks. While for bs_2(f) the picture is well understood with almost matching upper and lower bounds, for bs_3(f) it is not. We show that any development in understanding bs_3(f) in terms of s(f) will have great implications on the original question. Namely, we show that either bs(f) is at most sub-exponential in s(f) (which improves the state of the art upper bounds) or that bs_3(f) >= s(f){3-epsilon} for some Boolean functions (which improves the state of the art separations).
We generalize the question of bs(f) versus s(f) to bounded functions f:{0,1}^n -> [0,1] and show an analog result to that of Kenyon and Kutin: bs_l(f) = O(s(f))^l. Surprisingly, in this case, the bounds are close to being tight. In particular, we construct a bounded function f:{0,1}^n -> [0, 1] with bs(f) n/log(n) and s(f) = O(log(n)), a clear counterexample to the sensitivity conjecture for bounded functions.
Finally, we give a new super-quadratic separation between sensitivity and decision tree complexity by constructing Boolean functions with DT(f) >= s(f)^{2.115}. Prior to this work, only quadratic separations, DT(f) = s(f)^2, were known
SU(2) lattice gluon propagators at finite temperatures in the deep infrared region and Gribov copy effects
We study numerically the SU(2) Landau gauge transverse and longitudinal gluon
propagators at non-zero temperatures T both in confinement and deconfinement
phases. The special attention is paid to the Gribov copy effects in the
IR-region. Applying powerful gauge fixing algorithm we find that the Gribov
copy effects for the transverse propagator D_T(p) are very strong in the
infrared, while the longitudinal propagator D_L(p) shows very weak (if any)
Gribov copy dependence. The value D_T(0) tends to decrease with growing lattice
size; however, D_T(0) is non-zero in the infinite volume limit, in disagreement
with the suggestion made in [1]. We show that in the infrared region D_T(p) is
not consistent with the pole-type formula not only in the deconfinement phase
but also for T < T_c. We introduce new definition of the magnetic infrared mass
scale ('magnetic screening mass') m_M. The electric mass m_E has been
determined from the momentum space longitudinal gluon propagator. We study also
the (finite) volume and temperature dependence of the propagators as well as
discretization errors.Comment: 11 pages, 14 figures, 3 tables. Few minor change
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