5,884 research outputs found
Algebraic number theory and code design for Rayleigh fading channels
Algebraic number theory is having an increasing impact in code design for many different coding applications, such as single antenna fading channels and more recently, MIMO systems.
Extended work has been done on single antenna fading channels, and algebraic lattice codes have been proven to be an effective tool. The general framework has been settled in the last ten years and many explicit code constructions based on algebraic number theory are now available.
The aim of this work is to provide both an overview on algebraic lattice code designs for Rayleigh fading channels, as well as a tutorial introduction to algebraic number theory. The basic facts of this mathematical field will be illustrated by many examples and by the use of a computer algebra freeware in order to make it more accessible
to a large audience
Quantum Random Self-Modifiable Computation
Among the fundamental questions in computer science, at least two have a deep
impact on mathematics. What can computation compute? How many steps does a
computation require to solve an instance of the 3-SAT problem? Our work
addresses the first question, by introducing a new model called the ex-machine.
The ex-machine executes Turing machine instructions and two special types of
instructions. Quantum random instructions are physically realizable with a
quantum random number generator. Meta instructions can add new states and add
new instructions to the ex-machine. A countable set of ex-machines is
constructed, each with a finite number of states and instructions; each
ex-machine can compute a Turing incomputable language, whenever the quantum
randomness measurements behave like unbiased Bernoulli trials. In 1936, Alan
Turing posed the halting problem for Turing machines and proved that this
problem is unsolvable for Turing machines. Consider an enumeration E_a(i) =
(M_i, T_i) of all Turing machines M_i and initial tapes T_i. Does there exist
an ex-machine X that has at least one evolutionary path X --> X_1 --> X_2 --> .
. . --> X_m, so at the mth stage ex-machine X_m can correctly determine for 0
<= i <= m whether M_i's execution on tape T_i eventually halts? We demonstrate
an ex-machine Q(x) that has one such evolutionary path. The existence of this
evolutionary path suggests that David Hilbert was not misguided to propose in
1900 that mathematicians search for finite processes to help construct
mathematical proofs. Our refinement is that we cannot use a fixed computer
program that behaves according to a fixed set of mechanical rules. We must
pursue methods that exploit randomness and self-modification so that the
complexity of the program can increase as it computes.Comment: 50 pages, 3 figure
Intertwining wavelets or Multiresolution analysis on graphs through random forests
We propose a new method for performing multiscale analysis of functions
defined on the vertices of a finite connected weighted graph. Our approach
relies on a random spanning forest to downsample the set of vertices, and on
approximate solutions of Markov intertwining relation to provide a subgraph
structure and a filter bank leading to a wavelet basis of the set of functions.
Our construction involves two parameters q and q'. The first one controls the
mean number of kept vertices in the downsampling, while the second one is a
tuning parameter between space localization and frequency localization. We
provide an explicit reconstruction formula, bounds on the reconstruction
operator norm and on the error in the intertwining relation, and a Jackson-like
inequality. These bounds lead to recommend a way to choose the parameters q and
q'. We illustrate the method by numerical experiments.Comment: 39 pages, 12 figure
Limits of spiked random matrices II
The top eigenvalues of rank spiked real Wishart matrices and additively
perturbed Gaussian orthogonal ensembles are known to exhibit a phase transition
in the large size limit. We show that they have limiting distributions for
near-critical perturbations, fully resolving the conjecture of Baik, Ben Arous
and P\'{e}ch\'{e} [Duke Math. J. (2006) 133 205-235]. The starting point is a
new -diagonal form that is algebraically natural to the problem; for
both models it converges to a certain random Schr\"{o}dinger operator on the
half-line with matrix-valued potential. The perturbation determines
the boundary condition and the low-lying eigenvalues describe the limit,
jointly as the perturbation varies in a fixed subspace. We treat the real,
complex and quaternion () cases simultaneously. We further
characterize the limit laws in terms of a diffusion related to Dyson's Brownian
motion, or alternatively a linear parabolic PDE; here appears simply as
a parameter. At , the PDE appears to reconcile with known Painlev\'{e}
formulas for these -parameter deformations of the GUE Tracy-Widom law.Comment: Published at http://dx.doi.org/10.1214/15-AOP1033 in the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
The Bounded Confidence Model Of Opinion Dynamics
The bounded confidence model of opinion dynamics, introduced by Deffuant et
al, is a stochastic model for the evolution of continuous-valued opinions
within a finite group of peers. We prove that, as time goes to infinity, the
opinions evolve globally into a random set of clusters too far apart to
interact, and thereafter all opinions in every cluster converge to their
barycenter. We then prove a mean-field limit result, propagation of chaos: as
the number of peers goes to infinity in adequately started systems and time is
rescaled accordingly, the opinion processes converge to i.i.d. nonlinear Markov
(or McKean-Vlasov) processes; the limit opinion processes evolves as if under
the influence of opinions drawn from its own instantaneous law, which are the
unique solution of a nonlinear integro-differential equation of Kac type. This
implies that the (random) empirical distribution processes converges to this
(deterministic) solution. We then prove that, as time goes to infinity, this
solution converges to a law concentrated on isolated opinions too far apart to
interact, and identify sufficient conditions for the limit not to depend on the
initial condition, and to be concentrated at a single opinion. Finally, we
prove that if the equation has an initial condition with a density, then its
solution has a density at all times, develop a numerical scheme for the
corresponding functional equation, and show numerically that bifurcations may
occur.Comment: 43 pages, 7 figure
Matrix geometric approach for random walks: stability condition and equilibrium distribution
In this paper, we analyse a sub-class of two-dimensional homogeneous nearest
neighbour (simple) random walk restricted on the lattice using the matrix
geometric approach. In particular, we first present an alternative approach for
the calculation of the stability condition, extending the result of Neuts drift
conditions [30] and connecting it with the result of Fayolle et al. which is
based on Lyapunov functions [13]. Furthermore, we consider the sub-class of
random walks with equilibrium distributions given as series of product-forms
and, for this class of random walks, we calculate the eigenvalues and the
corresponding eigenvectors of the infinite matrix appearing in the
matrix geometric approach. This result is obtained by connecting and extending
three existing approaches available for such an analysis: the matrix geometric
approach, the compensation approach and the boundary value problem method. In
this paper, we also present the spectral properties of the infinite matrix
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