69 research outputs found
On a zero speed sensitive cellular automaton
Using an unusual, yet natural invariant measure we show that there exists a
sensitive cellular automaton whose perturbations propagate at asymptotically
null speed for almost all configurations. More specifically, we prove that
Lyapunov Exponents measuring pointwise or average linear speeds of the faster
perturbations are equal to zero. We show that this implies the nullity of the
measurable entropy. The measure m we consider gives the m-expansiveness
property to the automaton. It is constructed with respect to a factor dynamical
system based on simple "counter dynamics". As a counterpart, we prove that in
the case of positively expansive automata, the perturbations move at positive
linear speed over all the configurations
On the Convergence of Ritz Pairs and Refined Ritz Vectors for Quadratic Eigenvalue Problems
For a given subspace, the Rayleigh-Ritz method projects the large quadratic
eigenvalue problem (QEP) onto it and produces a small sized dense QEP. Similar
to the Rayleigh-Ritz method for the linear eigenvalue problem, the
Rayleigh-Ritz method defines the Ritz values and the Ritz vectors of the QEP
with respect to the projection subspace. We analyze the convergence of the
method when the angle between the subspace and the desired eigenvector
converges to zero. We prove that there is a Ritz value that converges to the
desired eigenvalue unconditionally but the Ritz vector converges conditionally
and may fail to converge. To remedy the drawback of possible non-convergence of
the Ritz vector, we propose a refined Ritz vector that is mathematically
different from the Ritz vector and is proved to converge unconditionally. We
construct examples to illustrate our theory.Comment: 20 page
Krein signature and Whitham modulation theory: the sign of characteristics and the âsign characteristicâ
In classical Whitham modulation theory, the transition of the dispersionless Whitham equations from hyperbolic to elliptic is associated with a pair of nonzero purely imaginary eigenvalues coalescing and becoming a complex quartet, suggesting that a Krein signature is operational. However, there is no natural symplectic structure. Instead, we find that the operational signature is the âsign characteristicâ of real eigenvalues of Hermitian matrix pencils. Its role in classical Whitham singleâphase theory is elaborated for illustration. However, the main setting where the sign characteristic becomes important is in multiphase modulation. It is shown that a necessary condition for two coalescing characteristics to become unstable (the generalization of the hyperbolic to elliptic transition) is that the characteristics have opposite sign characteristic. For example the theory is applied to multiphase modulation of the twoâphase traveling wave solutions of coupled nonlinear Schrödinger equation
The EcoThermo project: key and innovative aspects
In this paper we present the most innovative aspects of the EC-FP7 EcoThermo project. The main aim of the project consists on innovating the technique of heat cost allocation in buildings with a centralized heating system, overcoming the heat cost allocator drawbacks for reliability, measurement reproducibility and traceability and contexts of applications. Given the complexity of the project, we will focus on its main aspects, such as the use of a virtual sensor to estimate the radiators heating power, the design of electronic valves fitted out with an energy harvesting system and the original wireless communication protocol
Computing the common zeros of two bivariate functions via BĂ©zout resultants
The common zeros of two bivariate functions can be computed by finding the common zeros of their polynomial interpolants expressed in a tensor Chebyshev basis. From here we develop a bivariate rootfinding algorithm based on the hidden variable resultant method and BĂ©zout matrices with polynomial entries. Using techniques including domain subdivision, BĂ©zoutian regularization, and local refinement we are able to reliably and accurately compute the simple common zeros of two smooth functions with polynomial interpolants of very high degree (â„ 1000). We analyze the resultant method and its conditioning by noting that the BĂ©zout matrices are matrix polynomials. Two implementations are available: one on the Matlab Central File Exchange and another in the roots command in Chebfun2 that is adapted to suit Chebfunâs methodology
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