Study of the linear ablation growth rate for the quasi isobaric model of Euler equations with thermal conductivity

In this paper, we study a linear system related to the 2d system of Euler equations with thermal conduction in the quasi-isobaric approximation of Kull-Anisimov [14]. This model is used for the study of the ablation front instability, which appears in the problem of inertial confinement fusion. This physical system contains a mixing region, in which the density of the gaz varies quickly, and one denotes by L0 an associated characteristic length. The system of equations is linearized around a stationary solution, and each perturbed quantity is written using the normal modes method. The resulting linear system is not self-adjoint, of order 5, with coefficients depending on x and on physical parameters $\alpha, \beta$. We calculate Evans function associated with this linear system, using rigorous constructions of decreasing at $\pm \infty$ solutions of systems of ODE. We prove that for $\alpha$ small, there is no bounded solution of the linearized system.Comment: Indiana University Mathematical Journal (2007) in pres

The linear and non linear Rayleigh-Taylor instability for the quasi isobaric profile

We study the stability of the system of the Euler equation in the neighborhood of a stationary profile associated with the quasi isobaric model in a gravity field. This stationary profile is not bounded below, hence the operator is not coercive. We use this linear result to deduce a nonlinear resul

Numerical Eulerian method for linearized gas dynamics in the high frequency regime

We study the propagation of an acoustic wave in amoving fluid in the high frequency regime. We calculate a high-frequency approximation of the solution of this problemusing an Eulerianmethod. The model retained is a linearized Euler system around a mean fluid flow. For any regular mean flow, we derive a conservative transport equation for the geometrical optics approximation. We introduce the stretching matrix corresponding to this system, from which we deduce the geometrical spreading, key tool for computing the geometrical optics approximation. Finally, we construct and implement a numerical scheme in the Eulerian framework for the eikonal equation. This Eulerian formulation applies also for the transport equation on the stretching matrix

The Erpenbeck High Frequency Instability Theorem for Zeldovitch–von Neumann–Döring Detonations

Abstract The rigorous study of spectral stability for strong detonations was begun by Erpenbeck (Phys. Fluids 5:604–614 1962). Working with the Zeldovitch–von Neumann–Döring (ZND) model (more precisely, Erpenbeck worked with an extension of ZND to general chemistry and thermodynamics), which assumes a finite reaction rate but ignores effects such as viscosity corresponding to second order derivatives, he used a normal mode analysis to define a stability function {V(\tau,\epsilon)}$whose zeros in {\mathfrak{R}\tau > 0}$ correspond to multidimensional perturbations of a steady detonation profile that grow exponentially in time. Later in a remarkable paper (Erpenbeck in Phys. Fluids 9:1293–1306, 1966; Stability of detonations for disturbances of small transverse wavelength, 1965) he provided strong evidence, by a combination of formal and rigorous arguments, that for certain classes of steady ZND profiles, unstable zeros of V exist for perturbations of sufficiently large transverse wavenumber {\epsilon}$, even when the von Neumann shock, regarded as a gas dynamical shock, is uniformly stable in the sense defined (nearly 20 years later) by Majda. In spite of a great deal of later numerical work devoted to computing the zeros of {V(\tau,\epsilon)}$ , the paper (Erpenbeck in Phys. Fluids 9:1293–1306, 1966) remains one of the few works we know of [another is Erpenbeck (Phys. Fluids 7:684–696, 1964), which considers perturbations for which the ratio of longitudinal over transverse components approaches ∞] that presents a detailed and convincing theoretical argument for detecting them. The analysis in Erpenbeck (Phys. Fluids 9:1293–1306, 1966) points the way toward, but does not constitute, a mathematical proof that such unstable zeros exist. In this paper we identify the mathematical issues left unresolved in Erpenbeck (Phys. Fluids 9:1293–1306, 1966) and provide proofs, together with certain simplifications and extensions, of the main conclusions about stability and instability of detonations contained in that paper. The main mathematical problem, and our principal focus here, is to determine the precise asymptotic behavior as {\epsilon\to\infty}$of solutions to a linear system of ODEs in x, depending on {\epsilon}$ and a complex frequency τ as parameters, with turning points x * on the half-line [0,∞)

When Analytic Calculus Cracks AdaBoost Code

The principle of boosting in supervised learning involves combining multiple weak classifiers to obtain a stronger classifier. AdaBoost has the reputation to be a perfect example of this approach. We have previously shown that AdaBoost is not truly an optimization algorithm. This paper shows that AdaBoost is an algorithm in name only, as the resulting combination of weak classifiers can be explicitly calculated using a truth table. This study is carried out by considering a problem with two classes and is illustrated by the particular case of three binary classifiers and presents results in comparison with those from the implementation of AdaBoost algorithm of the Python library scikit-learn.Comment: 8 pages, 1 figur

Time Transfer functions as a way to validate light propagation solutions for space astrometry

Given the extreme accuracy of modern space astrometry, a precise relativistic modeling of observations is required. Concerning light propagation, the standard procedure is the solution of the null-geodesic equations. However, another approach based on the Time Transfer Functions (TTF) has demonstrated its capability to give access to key quantities such as the time of flight of a light signal between two point-events and the tangent vector to its null-geodesic in a weak gravitational field using an integral-based method. The availability of several models, formulated in different and independent ways, must not be considered like an oversized relativistic toolbox. Quite the contrary, they are needed as validation to put future experimental results on solid ground. The objective of this work is then twofold. First, we build the time of flight and tangent vectors in a closed form within the TTF formalism giving the case of a time dependent metric. Second, we show how to use this new approach to obtain a comparison of the TTF with two existing modelings, namely GREM and RAMOD. In this way, we evidentiate the mutual consistency of the three models, opening the basis for further links between all the approaches, which is mandatory for the interpretation of future space missions data. This will be illustrated through two recognized cases: a static gravitational field and a system of monopoles in uniform motion.Comment: 16 pages, submitted to CQ