Front Propagation Problems with Nonlocal Terms, II

We study the propagation of hyper surfaces Σ_t of ^N satisfying the equation V = h(x, Ω_t), where V is the normal velocity of Σ_t at x, Ω_t is the interior of Σ_t, and h is a given evolution law. We prove the existence of generalized solutions and an inclusion principle for these generalized solutions. Copyright 2001 Academic Press.Dans cet article nous étudions certains problèmes de propagation d'hypersurfaces Σ_t de ^N suivant des lois de la forme: V = h(x, Ω_t), où V est la vitesse normale à Σ_t en x, Ω_t est l'intérieur de Σ_t et h est une loi d'évolution. On définit une notion de solutions généralisées, et on montre l'existence de telles solutions. De plus on démontre un principe d'inclusion pour ces solutions.     

High order shear horizontal modes for minimum remnant thickness

Thickness mapping in aging structures suffering from corrosion is challenging especially when the structure is only partially accessible. In plates the high order shear horizontal guided wave modes all have a cutoff frequency thickness product below which they cannot propagate. This property is potentially attractive to estimate the minimum remnant thickness between two transducers. When using a source and a sensor array it is possible to control the number of modes being excited and the size of the region interrogated by the technique. Finite element simulations were used to show that by exciting multiple guided wave modes simultaneously and identifying which modes are received by a sensor array it is possible to estimate the minimum remaining thickness along the propagation path. Initial experimental results showed excellent agreement with the finite element simulations when the plate is uniform and with a thickness reduction between the source and the sensor arrays the minimum remnant thickness was underestimated by approximately 20%.     

Index growth of hypersurfaces with constant mean curvature

In this paper we give the precise index growth for the embedded hypersurfaces of revolution with constant mean curvature (cmc) 1 in (Delaunay unduloids). When n=3, using the asymptotics result of Korevaar, Kusner and Solomon, we derive an explicit asymptotic index growth rate for finite topology cmc 1 surfaces with properly embedded ends. Similar results are obtained for hypersurfaces with cmc bigger than 1 in hyperbolic space. Received: 6 July 2000; in final form: 10 September 2000 / Published online: 25 June 2001     

A combined 14N/27Al nuclear magnetic resonance and powder X-ray diffraction study of impurity phases in beta-sialon ceramics

Beta-sialons are ceramic phases occurring in the SiO(2)-Si(3)N(4)-AlN-Al(2)O(3) system. A series of samples with differing compositions has been investigated by magic-angle spinning nuclear magnetic resonance (NMR) spectroscopy and powder X-ray diffraction (XRD). Although the constituent nitrogen nuclei occupy positions of low symmetry in the beta-sialon structure, 14N NMR spectra could be recorded for the samples examined. The origin of the 14N signal could be traced to the presence of an aluminium nitride (AlN) impurity phase with the help of 27Al NMR and XRD results. Similarly, the existence of Al(2)O(3) grains could be readily detected for a number of samples. Thus, the combination of 14N and 27Al NMR is shown to be an especially effective tool in identifying and characterizing impurity phases in sialon ceramics, complementing the results obtained from standard XRD analysis.     

A Note on the Morozov Principle via Lagrange Duality

Considering a general linear ill-posed equation, we explore the duality arising from the requirement that the discrepancy should take a given value based on the estimation of the noise level, as is notably the case when using the Morozov principle. We show that, under reasonable assumptions, the dual function is smooth, and that its maximization points out the appropriate value of Tikhonov’s regularization parameter. The numerical relevance of our approach is established by means of an illustrative example from nonparametric instrumental regression, a standard problem in statistics.     

Thermal fluctuations in pinned elastic systems: field theory of rare events and droplets

Using the functional renormalization group (FRG) we study the thermal fluctuations of elastic objects (displacement field u, internal dimension d) pinned by a random potential at low temperature T, as prototypes for glasses. A challenge is how the field theory can describe both typical (minimum energy T = 0) configurations, as well as thermal averages which, at any non-zero T as in the phenomenological droplet picture, are dominated by rare degeneracies between low lying minima. We show that this occurs through an essentially non-perturbative thermal boundary layer (TBL) in the (running) effective action Γ [u] at T > 0 for which we find a consistent scaling ansatz to all orders. The TBL describes how temperature smoothes the singularities of the T = 0 theory and contains the physics of rare thermal excitations (droplets). The formal structure of this TBL, which involves all cumulants of the coarse grained disorder, is first explored around d = 4 using a one-loop Wilson RG. Next, a more systematic exact RG (ERG) method is employed, and first tested on d = 0 models where it can be pushed quite far. There we obtain precise relations between TBL quantities and droplet probabilities (those are constrained by exact identities which are then checked against recent exact results). Our analysis is then extended to higher d, where we illustrate how the TBL scaling remains consistent to all orders in the ERG and how droplet picture results can be retrieved. Since correlations are determined deep in the TBL (by derivatives of Γ [u] at u = 0), it remains to be understood (in any d) how they can be retrieved (as u = 0⁺ limits in the non-analytic T = 0 effective action), i.e., how to recover a T = 0 critical theory. This formidable “matching problem” is solved in detail for d = 0, N = 1 by studying the (partial) TBL structure of higher cumulants when points are brought together. We thereby obtain the β-function at T = 0, all ambiguities removed, displayed here up to four loops. A discussion of the d > 4 case and an exact solution at large d are also provided.     

A note on bochner type theorems for complete manifolds

In this note we give an extension of Bochner's vanishing theorem to complete manifolds; this generalizes earlier results of J. Dodziuk, K.D. Elworthy and S. Rosenberg. This research was partially done under the E.E.C. Contract # SC 1-0105-C “GADGET” at the C.N.R.S. U.R.A. 188     

Dynamically orthogonal field equations for continuous stochastic dynamical systems

In this work we derive an exact, closed set of evolution equations for general continuous stochastic fields described by a Stochastic Partial Differential Equation (SPDE). By hypothesizing a decomposition of the solution field into a mean and stochastic dynamical component, we derive a system of field equations consisting of a Partial Differential Equation (PDE) for the mean field, a family of PDEs for the orthonormal basis that describe the stochastic subspace where the stochasticity ‘lives’ as well as a system of Stochastic Differential Equations that defines how the stochasticity evolves in the time varying stochastic subspace. These new evolution equations are derived directly from the original SPDE, using nothing more than a dynamically orthogonal condition on the representation of the solution. If additional restrictions are assumed on the form of the representation, we recover both the Proper Orthogonal Decomposition equations and the generalized Polynomial Chaos equations. We apply this novel methodology to two cases of two-dimensional viscous fluid flows described by the Navier–Stokes equations and we compare our results with Monte Carlo simulations.     

Variable-range hopping and quantum creep in one dimension

We study the quantum nonlinear response to an applied electric field E of a one-dimensional pinned charge-density wave or Luttinger liquid in the presence of disorder. From an explicit construction of low-lying metastable states and of bounce instanton solutions between them, we demonstrate quantum creep v=e(-c/E(1/2)) as well as a sharp crossover at E=E(*) towards a linear response form consistent with variable-range hopping arguments, but dependent only on electronic degrees of freedom.     

Classroom note: Using Platonic bodies to generate a random sample from a population having two characteristics linked with a predetermined coefficient of correlation

This note describes a simple method of generating a random sample of N pairs (U _i ,?W _i ) from a population whose elements have two characteristics U  and W associated with a known coefficient of correlation. Although the method described is extremely advantageous when Platonic bodies are used, it can be generalized to include any discrete uniform distribution.