312 research outputs found
On the existence of a solution to a spectral estimation problem \emph{\`a la} Byrnes-Georgiou-Lindquist
A parametric spectral estimation problem in the style of Byrnes, Georgiou,
and Lindquist was posed in \cite{FPZ-10}, but the existence of a solution was
only proved in a special case. Based on their results, we show that a solution
indeed exists given an arbitrary matrix-valued prior density. The main tool in
our proof is the topological degree theory.Comment: 6 pages of two-column draft, accepted for publication in IEEE-TA
An Extension Theorem with an Application to Formal Tree Series
A grove theory is a Lawvere algebraic theory T for which each hom-set T(n,p) is a commutative monoid; composition on the right distributes over all finite sums: (\sum f_i) . h = \sum f_i . h. A matrix theory is a grove theory in which composition on the left and right distributes over finite sums. A matrix theory M is isomorphic to a theory of all matrices over the semiring S = M(1,1). Examples of grove theories are theories of (bisimulation equivalence classes of) synchronization trees, and theories of formal tree series over a semiring S . Our main theorem states that if T is a grove theory which has a matrix subtheory M which is an iteration theory, then, under certain conditions, the fixed point operation on M can be extended in exactly one way to a fixed point operation on T such that T is an iteration theory. A second theorem is a Kleene-type result. Assume that T is an iteration grove theory and M is a sub iteration grove theory of T which is a matrix theory. For a given collection Sigma of scalar morphisms in T we describe the smallest sub iteration grove theory of T containing all the morphisms in M union Sigma
Symbolic Algorithms for Language Equivalence and Kleene Algebra with Tests
We first propose algorithms for checking language equivalence of finite
automata over a large alphabet. We use symbolic automata, where the transition
function is compactly represented using a (multi-terminal) binary decision
diagrams (BDD). The key idea consists in computing a bisimulation by exploring
reachable pairs symbolically, so as to avoid redundancies. This idea can be
combined with already existing optimisations, and we show in particular a nice
integration with the disjoint sets forest data-structure from Hopcroft and
Karp's standard algorithm. Then we consider Kleene algebra with tests (KAT), an
algebraic theory that can be used for verification in various domains ranging
from compiler optimisation to network programming analysis. This theory is
decidable by reduction to language equivalence of automata on guarded strings,
a particular kind of automata that have exponentially large alphabets. We
propose several methods allowing to construct symbolic automata out of KAT
expressions, based either on Brzozowski's derivatives or standard automata
constructions. All in all, this results in efficient algorithms for deciding
equivalence of KAT expressions
Multi-channel Bethe-Salpeter equation
A general form of multi-channel Bethe-Salpeter equation is considered. In
contradistinction to the hitherto applied approaches, our coupled system of
equations leads to the simultaneous solutions for all relativistic four-point
Green functions (elastic and inelastic) appearing in a given theory. A set of
relations which may be helpful in approximate treatments is given. An example
of extracting useful information from the equations is discussed: we consider
the most general trilinear coupling of N different scalar fields and obtain -
in the ladder approximation - closed expressions for the Regge trajectories and
their couplings to different channels in the vicinity of l = -1. Sum rules and
an example containing non-obvious symmetry are discussed.Comment: 16 pages. Extended version published in JHEP. Uses JHEP.cls
(included
Nonlinear behavior in a piezoelectric resonator: a method of analysis
Theories used for understanding nonlinear behavior of piezoelectric resonators are usually only valid for a given range of amplitudes. Thus, important discrepancies can sometimes be observed between theory and experiment. In this work, a simplified model of the resonator is assumed in order to extend the analysis of nonlinear behavior to any kind of nonlinear function, without a significant increase of mathematical complexity. Nevertheless, nonlinearities are considered to be weak enough to be taken as perturbations. An asymptotic method is used to obtain the first and second order perturbations of the response to an harmonic excitation applied to the system, and each one is separated into Fourier series. Nonlinearity is described by two functions-/spl Phi/, (S,D,S/spl dot/,D/spl dot/) and /spl Psi/ (S,D,S/spl dot/,D/spl dot/)-that must be added to the constitutive equations that give T and E as functions of S and D. These functions can be split into their symmetrical and antisymmetrical parts, which have different incidence over the perturbation terms. In order to simplify the problem, no mechanical excitation is considered, the electrical one is taken as strictly harmonic, and the current rather than the e.m.f. is taken as initial data. As an application example, this method is applied in order to find the second harmonic generation for a particular kind of nonlinearity.Peer ReviewedPostprint (published version
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