46,501 research outputs found
Kochen-Specker Vectors
We give a constructive and exhaustive definition of Kochen-Specker (KS)
vectors in a Hilbert space of any dimension as well as of all the remaining
vectors of the space. KS vectors are elements of any set of orthonormal states,
i.e., vectors in n-dim Hilbert space, H^n, n>3 to which it is impossible to
assign 1s and 0s in such a way that no two mutually orthogonal vectors from the
set are both assigned 1 and that not all mutually orthogonal vectors are
assigned 0. Our constructive definition of such KS vectors is based on
algorithms that generate MMP diagrams corresponding to blocks of orthogonal
vectors in R^n, on algorithms that single out those diagrams on which algebraic
0-1 states cannot be defined, and on algorithms that solve nonlinear equations
describing the orthogonalities of the vectors by means of statistically
polynomially complex interval analysis and self-teaching programs. The
algorithms are limited neither by the number of dimensions nor by the number of
vectors. To demonstrate the power of the algorithms, all 4-dim KS vector
systems containing up to 24 vectors were generated and described, all 3-dim
vector systems containing up to 30 vectors were scanned, and several general
properties of KS vectors were found.Comment: 19 pages, 6 figures, title changed, introduction thoroughly
rewritten, n-dim rotation of KS vectors defined, original Kochen-Specker 192
(117) vector system translated into MMP diagram notation with a new graphical
representation, results on Tkadlec's dual diagrams added, several other new
results added, journal version: to be published in J. Phys. A, 38 (2005). Web
page: http://m3k.grad.hr/pavici
Inferring hidden states in Langevin dynamics on large networks: Average case performance
We present average performance results for dynamical inference problems in
large networks, where a set of nodes is hidden while the time trajectories of
the others are observed. Examples of this scenario can occur in signal
transduction and gene regulation networks. We focus on the linear stochastic
dynamics of continuous variables interacting via random Gaussian couplings of
generic symmetry. We analyze the inference error, given by the variance of the
posterior distribution over hidden paths, in the thermodynamic limit and as a
function of the system parameters and the ratio {\alpha} between the number of
hidden and observed nodes. By applying Kalman filter recursions we find that
the posterior dynamics is governed by an "effective" drift that incorporates
the effect of the observations. We present two approaches for characterizing
the posterior variance that allow us to tackle, respectively, equilibrium and
nonequilibrium dynamics. The first appeals to Random Matrix Theory and reveals
average spectral properties of the inference error and typical posterior
relaxation times, the second is based on dynamical functionals and yields the
inference error as the solution of an algebraic equation.Comment: 20 pages, 5 figure
Optimal random search for a single hidden target
A single target is hidden at a location chosen from a predetermined
probability distribution. Then, a searcher must find a second probability
distribution from which random search points are sampled such that the target
is found in the minimum number of trials. Here it will be shown that if the
searcher must get very close to the target to find it, then the best search
distribution is proportional to the square root of the target distribution. For
a Gaussian target distribution, the optimum search distribution is
approximately a Gaussian with a standard deviation that varies inversely with
how close the searcher must be to the target to find it. For a network, where
the searcher randomly samples nodes and looks for the fixed target along edges,
the optimum is to either sample a node with probability proportional to the
square root of the out degree plus one or not at all.Comment: 13 pages, 5 figure
Broadcast Coded Slotted ALOHA: A Finite Frame Length Analysis
We propose an uncoordinated medium access control (MAC) protocol, called
all-to-all broadcast coded slotted ALOHA (B-CSA) for reliable all-to-all
broadcast with strict latency constraints. In B-CSA, each user acts as both
transmitter and receiver in a half-duplex mode. The half-duplex mode gives rise
to a double unequal error protection (DUEP) phenomenon: the more a user repeats
its packet, the higher the probability that this packet is decoded by other
users, but the lower the probability for this user to decode packets from
others. We analyze the performance of B-CSA over the packet erasure channel for
a finite frame length. In particular, we provide a general analysis of stopping
sets for B-CSA and derive an analytical approximation of the performance in the
error floor (EF) region, which captures the DUEP feature of B-CSA. Simulation
results reveal that the proposed approximation predicts very well the
performance of B-CSA in the EF region. Finally, we consider the application of
B-CSA to vehicular communications and compare its performance with that of
carrier sense multiple access (CSMA), the current MAC protocol in vehicular
networks. The results show that B-CSA is able to support a much larger number
of users than CSMA with the same reliability.Comment: arXiv admin note: text overlap with arXiv:1501.0338
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