32,025 research outputs found
Random Matrix Models, Double-Time Painlev\'e Equations, and Wireless Relaying
This paper gives an in-depth study of a multiple-antenna wireless
communication scenario in which a weak signal received at an intermediate relay
station is amplified and then forwarded to the final destination. The key
quantity determining system performance is the statistical properties of the
signal-to-noise ratio (SNR) \gamma\ at the destination. Under certain
assumptions on the encoding structure, recent work has characterized the SNR
distribution through its moment generating function, in terms of a certain
Hankel determinant generated via a deformed Laguerre weight. Here, we employ
two different methods to describe the Hankel determinant. First, we make use of
ladder operators satisfied by orthogonal polynomials to give an exact
characterization in terms of a "double-time" Painlev\'e differential equation,
which reduces to Painlev\'e V under certain limits. Second, we employ Dyson's
Coulomb Fluid method to derive a closed form approximation for the Hankel
determinant. The two characterizations are used to derive closed-form
expressions for the cumulants of \gamma, and to compute performance quantities
of engineering interest.Comment: 72 pages, 6 figures; Minor typos corrected; Two additional lemmas
added in Appendix
Specific "scientific" data structures, and their processing
Programming physicists use, as all programmers, arrays, lists, tuples,
records, etc., and this requires some change in their thought patterns while
converting their formulae into some code, since the "data structures" operated
upon, while elaborating some theory and its consequences, are rather: power
series and Pad\'e approximants, differential forms and other instances of
differential algebras, functionals (for the variational calculus), trajectories
(solutions of differential equations), Young diagrams and Feynman graphs, etc.
Such data is often used in a [semi-]numerical setting, not necessarily
"symbolic", appropriate for the computer algebra packages. Modules adapted to
such data may be "just libraries", but often they become specific, embedded
sub-languages, typically mapped into object-oriented frameworks, with
overloaded mathematical operations. Here we present a functional approach to
this philosophy. We show how the usage of Haskell datatypes and - fundamental
for our tutorial - the application of lazy evaluation makes it possible to
operate upon such data (in particular: the "infinite" sequences) in a natural
and comfortable manner.Comment: In Proceedings DSL 2011, arXiv:1109.032
Trusting Computations: a Mechanized Proof from Partial Differential Equations to Actual Program
Computer programs may go wrong due to exceptional behaviors, out-of-bound
array accesses, or simply coding errors. Thus, they cannot be blindly trusted.
Scientific computing programs make no exception in that respect, and even bring
specific accuracy issues due to their massive use of floating-point
computations. Yet, it is uncommon to guarantee their correctness. Indeed, we
had to extend existing methods and tools for proving the correct behavior of
programs to verify an existing numerical analysis program. This C program
implements the second-order centered finite difference explicit scheme for
solving the 1D wave equation. In fact, we have gone much further as we have
mechanically verified the convergence of the numerical scheme in order to get a
complete formal proof covering all aspects from partial differential equations
to actual numerical results. To the best of our knowledge, this is the first
time such a comprehensive proof is achieved.Comment: N° RR-8197 (2012). arXiv admin note: text overlap with
arXiv:1112.179
The collocation and meshless methods for differential equations in R(2)
In recent years, meshless methods have become popular ones to solve differential equations. In this thesis, we aim at solving differential equations by using Radial Basis Functions, collocation methods and fundamental solutions (MFS). These methods are meshless, easy to understand, and even easier to implement
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