22,357 research outputs found
Ten Conferences WORDS: Open Problems and Conjectures
In connection to the development of the field of Combinatorics on Words, we
present a list of open problems and conjectures that were stated during the ten
last meetings WORDS. We wish to continually update the present document by
adding informations concerning advances in problems solving
An identity of Chernoff bounds with an interpretation in statistical physics and applications in information theory
An identity between two versions of the Chernoff bound on the probability a
certain large deviations event, is established. This identity has an
interpretation in statistical physics, namely, an isothermal equilibrium of a
composite system that consists of multiple subsystems of particles. Several
information--theoretic application examples, where the analysis of this large
deviations probability naturally arises, are then described from the viewpoint
of this statistical mechanical interpretation. This results in several
relationships between information theory and statistical physics, which we
hope, the reader will find insightful.Comment: 29 pages, 1 figure. Submitted to IEEE Trans. on Information Theor
Pattern avoidance: themes and variations
AbstractWe review results concerning words avoiding powers, abelian powers or patterns. In addition we collect/pose a large number of open problems
Unforgeable Noise-Tolerant Quantum Tokens
The realization of devices which harness the laws of quantum mechanics
represents an exciting challenge at the interface of modern technology and
fundamental science. An exemplary paragon of the power of such quantum
primitives is the concept of "quantum money". A dishonest holder of a quantum
bank-note will invariably fail in any forging attempts; indeed, under
assumptions of ideal measurements and decoherence-free memories such security
is guaranteed by the no-cloning theorem. In any practical situation, however,
noise, decoherence and operational imperfections abound. Thus, the development
of secure "quantum money"-type primitives capable of tolerating realistic
infidelities is of both practical and fundamental importance. Here, we propose
a novel class of such protocols and demonstrate their tolerance to noise;
moreover, we prove their rigorous security by determining tight fidelity
thresholds. Our proposed protocols require only the ability to prepare, store
and measure single qubit quantum memories, making their experimental
realization accessible with current technologies.Comment: 18 pages, 5 figure
On optimum parameter modulation-estimation from a large deviations perspective
We consider the problem of jointly optimum modulation and estimation of a
real-valued random parameter, conveyed over an additive white Gaussian noise
(AWGN) channel, where the performance metric is the large deviations behavior
of the estimator, namely, the exponential decay rate (as a function of the
observation time) of the probability that the estimation error would exceed a
certain threshold. Our basic result is in providing an exact characterization
of the fastest achievable exponential decay rate, among all possible
modulator-estimator (transmitter-receiver) pairs, where the modulator is
limited only in the signal power, but not in bandwidth. This exponential rate
turns out to be given by the reliability function of the AWGN channel. We also
discuss several ways to achieve this optimum performance, and one of them is
based on quantization of the parameter, followed by optimum channel coding and
modulation, which gives rise to a separation-based transmitter, if one views
this setting from the perspective of joint source-channel coding. This is in
spite of the fact that, in general, when error exponents are considered, the
source-channel separation theorem does not hold true. We also discuss several
observations, modifications and extensions of this result in several
directions, including other channels, and the case of multidimensional
parameter vectors. One of our findings concerning the latter, is that there is
an abrupt threshold effect in the dimensionality of the parameter vector: below
a certain critical dimension, the probability of excess estimation error may
still decay exponentially, but beyond this value, it must converge to unity.Comment: 26 pages; Submitted to the IEEE Transactions on Information Theor
Boolean Dynamics with Random Couplings
This paper reviews a class of generic dissipative dynamical systems called
N-K models. In these models, the dynamics of N elements, defined as Boolean
variables, develop step by step, clocked by a discrete time variable. Each of
the N Boolean elements at a given time is given a value which depends upon K
elements in the previous time step.
We review the work of many authors on the behavior of the models, looking
particularly at the structure and lengths of their cycles, the sizes of their
basins of attraction, and the flow of information through the systems. In the
limit of infinite N, there is a phase transition between a chaotic and an
ordered phase, with a critical phase in between.
We argue that the behavior of this system depends significantly on the
topology of the network connections. If the elements are placed upon a lattice
with dimension d, the system shows correlations related to the standard
percolation or directed percolation phase transition on such a lattice. On the
other hand, a very different behavior is seen in the Kauffman net in which all
spins are equally likely to be coupled to a given spin. In this situation,
coupling loops are mostly suppressed, and the behavior of the system is much
more like that of a mean field theory.
We also describe possible applications of the models to, for example, genetic
networks, cell differentiation, evolution, democracy in social systems and
neural networks.Comment: 69 pages, 16 figures, Submitted to Springer Applied Mathematical
Sciences Serie
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