662,467 research outputs found
Necessary Conditions for K/2 Degrees of Freedom
Stotz et al., 2016, reported a sufficient (injectivity) condition for each
user in a K-user single-antenna constant interference channel to achieve 1/2
degree of freedom. The present paper proves that this condition is necessary as
well and hence provides an equivalence characterization of interference channel
matrices allowing full degrees of freedom
Interference alignment for the MIMO interference channel
We study vector space interference alignment for the MIMO interference
channel with no time or frequency diversity, and no symbol extensions. We prove
both necessary and sufficient conditions for alignment. In particular, we
characterize the feasibility of alignment for the symmetric three-user channel
where all users transmit along d dimensions, all transmitters have M antennas
and all receivers have N antennas, as well as feasibility of alignment for the
fully symmetric (M=N) channel with an arbitrary number of users.
An implication of our results is that the total degrees of freedom available
in a K-user interference channel, using only spatial diversity from the
multiple antennas, is at most 2. This is in sharp contrast to the K/2 degrees
of freedom shown to be possible by Cadambe and Jafar with arbitrarily large
time or frequency diversity.
Moving beyond the question of feasibility, we additionally discuss
computation of the number of solutions using Schubert calculus in cases where
there are a finite number of solutions.Comment: 16 pages, 7 figures, final submitted versio
Darboux points and integrability of homogeneous Hamiltonian systems with three and more degrees of freedom
We consider natural complex Hamiltonian systems with degrees of freedom
given by a Hamiltonian function which is a sum of the standard kinetic energy
and a homogeneous polynomial potential of degree . The well known
Morales-Ramis theorem gives the strongest known necessary conditions for the
Liouville integrability of such systems. It states that for each there
exists an explicitly known infinite set \scM_k\subset\Q such that if the
system is integrable, then all eigenvalues of the Hessian matrix V''(\vd)
calculated at a non-zero \vd\in\C^n satisfying V'(\vd)=\vd, belong to
\scM_k. The aim of this paper is, among others, to sharpen this result. Under
certain genericity assumption concerning we prove the following fact. For
each and there exists a finite set \scI_{n,k}\subset\scM_k such that
if the system is integrable, then all eigenvalues of the Hessian matrix
V''(\vd) belong to \scI_{n,k}. We give an algorithm which allows to find
sets \scI_{n,k}. We applied this results for the case and we found
all integrable potentials satisfying the genericity assumption. Among them
several are new and they are integrable in a highly non-trivial way. We found
three potentials for which the additional first integrals are of degree 4 and 6
with respect to the momenta.Comment: 54 pages, 1 figur
Isolated Horizons and Black Hole Entropy in Loop Quantum Gravity
We review the black hole entropy calculation in the framework of Loop Quantum
Gravity based on the quasi-local definition of a black hole encoded in the
isolated horizon formalism. We show, by means of the covariant phase space
framework, the appearance in the conserved symplectic structure of a boundary
term corresponding to a Chern-Simons theory on the horizon and present its
quantization both in the U(1) gauge fixed version and in the fully SU(2)
invariant one. We then describe the boundary degrees of freedom counting
techniques developed for an infinite value of the Chern-Simons level case and,
less rigorously, for the case of a finite value. This allows us to perform a
comparison between the U(1) and SU(2) approaches and provide a state of the art
analysis of their common features and different implications for the entropy
calculations. In particular, we comment on different points of view regarding
the nature of the horizon degrees of freedom and the role played by the
Barbero-Immirzi parameter. We conclude by presenting some of the most recent
results concerning possible observational tests for theory
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