Velocity of the fourth sound in liquid helium II via extended thermodynamics

This work continues a study begun in previous works, where, using Extended Thermodynamics, a monofluid model of liquid helium II is formulated. The wave propagation in bulk liquid helium II is studied in the hypothesis that the thermal dilatation is not zero. The propagation of fourth sound, studied previously neglecting both the thermal dilatation and finite volume of the powder, is studied without these simplified hypotheses: a scattering correction n is introduced to take into account the porosity. The model is more general than the standard two-fluid model because it allows that a small amount of entropy is associated with helium when it flows through a very thin capillary or a porous medium. A comparison with experimental data is performed. From experimental values for velocities and attenuations of the two sounds in bulk liquid helium, the model provides the velocity of fourth sound in a porous medium. These values are determined at various temperatures and pressures and compared with fourth sound measurements in a packed powder.     

Numerical simulation of the performance of a snow fence with airfoil snow plates by FVM

The aim of this work is to analyze the efficiency of a snow fence with airfoil snow plates to avoid the snowdrift formation, to improve visibility and to prevent blowing snow disasters on highways and railways. In order to attain this objective, it is necessary to solve particle transport equations along with the turbulent fluid flow equations since there are two phases: solid phase (snow particles) and fluid phase (air). In the first place, the turbulent flow is modelled by solving the Reynolds-averaged Navier-Stokes (RANS) equations for incompressible viscous flows through the finite volume method (FVM) and then, once the flow velocity field has been determined, representative particles are tracked using the Lagrangian approach. Within the particle transport models, we have used a particle transport model termed as Lagrangian particle tracking model, where particulates are tracked through the flow in a Lagrangian way. The full particulate phase is modelled by just a sample of about 15,000 individual particles. The tracking is carried out by forming a set of ordinary differential equations in time for each particle, consisting of equations for position and velocity. These equations are then integrated using a simple integration method to calculate the behaviour of the particles as they traverse the flow domain. Finally, the conclusions of this work are exposed.     

Local and global existence criteria for capillary surfaces in wedges

This paper addresses a conjecture of Concus and Finn [Capillary Wedges Revisited, SIAM J. Math. Anal., in press] on conditions for local existence of solutions of the zero-gravity capillarity equation at a boundary protruding corner pointP of prescribed opening 2. Geometrically, surfaces of constant mean curvatureH are sought as graphs which meet vertical walls over the boundary in prescribed angles, which are locally constant except for a possible jump discontinuity atP. The conjecture is settled more or less completely in the affirmative, depending on whetherH is to be prescribed. The proof proceeds through a global existence theorem for moon domains, which seems of independent interest.     

Complete convergence and Cesàro summation for i.i.d. random variables

Summary Various results generalizing summation methods for divergent series of real numbers to analogous results for independent, identically distributed random variables have appeared during the last two decades. The main result of this paper provides necessary and sufficient conditions for the complete convergence of the Cesàro means of i.i.d random variables.     

Adaptive pseudospectral solution of a diffuse interface model

A diffuse interface type model, using an energy-based variational formulation with a free energy that is a function of the density and its gradients is presented. All of the boundary terms are retained and related to external surface forces, which can be of particular interest when considering the fluid–fluid–solid region. The numerical solution of these types of problems can be troublesome if a thin transition layer is desired. Here, Chebyshev pseudospectral methods with mesh adaptation for the solution of diffuse interface type problems are studied. A mesh adaptation algorithm based in the equidistribution principle following a continuation process is derived. In order to achieve high precision for problems exhibiting thin transition layers, a modified version of the arc-length monitor function is proposed which yields a sufficiently smooth coordinate transformation. At every step of the continuation process, a fixed number of iterations is implemented, so that the equidistribution equations are not solved completely at each step, which saves a considerable amount of computational effort. Numerical results for the static phase field model exhibiting thin transition layers are presented.     

Analysis of a nonlinear degenerating PDE system for phase transitions in thermoviscoelastic materials

We address the analysis of a nonlinear and degenerating PDE system, proposed by M. Frémond for modelling phase transitions in viscoelastic materials subject to thermal effects. The system features an internal energy balance equation, governing the evolution of the absolute temperature ?, an evolution equation for the phase change parameter χ, and a stress-strain relation for the displacement variable u. The main novelty of the model is that the equations for χ and u are coupled in such a way as to take into account the fact that the properties of the viscous and of the elastic parts influence the phase transition phenomenon in different ways. However, this brings about an elliptic degeneracy in the equation for u which needs to be carefully handled.In this paper, we first prove a local (in time) well-posedness result for (a suitable initial-boundary value problem for) the above mentioned PDE system, in the (spatially) three-dimensional setting. Secondly, we restrict to the one-dimensional case, in which, for the same initial-boundary value problem, we indeed obtain a global well-posedness theorem.     

A new proof of regularity for¶two-shaded image segmentations

In this paper, we consider minimizing the Mumford-Shah functional over two-valued functions in the plane, which is equivalent to minimizing over characteristic functions. Existence of minimizers is straightforward and we show that any minimizing set is essentially open, has a boundary with finitely many connected components, and each component is C ¹. The relatively quick proof does not rely on quasi-minimal surface or varifold theory, but on uniform estimates on the regularity of the boundary. Received: 4 February 1997 / Revised version: 4 February 1998     

Matching behaviour is asymptotically normal

Ak-matching in a graphG is a set ofk edges, no two of which have a vertex in common. The number of these inG is writtenp(G, k). Using an idea due to L. H. Harper, we establish a condition under which these numbers are approximately normally distributed. We show that our condition is satisfied ifn=|V(G)| is large compared to the maximum degree Δ of a vertex inG(i.e. Δ=o(n)) orG is a large complete graph. One corollary of these results is that the number of points fixed by a randomly chosen involution in the symmetric groupS is asymptotically normally distributed.     

A note on the de La Vallée Poussin criterion for uniform integrability

In this paper, we present new versions of the classical de La Vallée Poussin criterion for uniform integrability. Our results concern the uniform integrability of a continuous function relative to a sequence of distribution functions. We apply our results to obtain a result on the convergence of a sequence of integrals which we illustrate with an example.     

Hypersurfaces with constant mean curvature and prescribed area

Summary We consider—in the setting of geometric measure theory—hypersurfacesT (of codimension one) with prescribed boundaryB in Euclideann+1 space which maximize volume (i.e.T together with a fixed hypersurfaceT ₀ encloses oriented volume) subject to a mass constraint. We prove existence and optimal regularity of solutionsT of such variational problems and we show that, on the regular part of its support,T is a classical hypersurface of constant mean curvature. We also prove that the solutionsT become more and more spherical as the valuem of the mass constraint approaches ∞. This work was done at the Centre for Mathematics and its Applications at the Australian National University, Canberra while the author was a visiting member This article was processed by the author using the LAT_EX style filecljour1 from Springer-Verlag.     

Time dependent critical fluctuations of a one dimensional local mean field model

Summary One-dimensional stochastic Ising systems with a local mean field interaction (Kac potential) are investigated. It is shown that near the critical temperature of the equilibrium (Gibbs) distribution the time dependent process admits a scaling limit given by a nonlinear stochastic PDE. The initial conditions of this approximation theorem are then verified for equilibrium states when the temperature goes to its critical value in a suitable way. Earlier results of Bertini-Presutti-Rüdiger-Saada are improved, the proof is based on an energy inequality obtained by coupling the Glauber dynamics to its voter type, linear approximation.     

A presentation of the double Pieri algebra

The double Pieri algebra, constructed by Howe, Lee, and the author in [10], [12], encodes information on the decomposition of Pieri type tensor products of irreducible representations for complex classical groups. In this paper we study its finite presentation in terms of generators and relations, and then prove that it admits the structure of a cluster algebra. We also study its ${\mathfrak{sl}}_2$-module structure.     

A novel PDE based image restoration: Convection-diffusion equation for image denoising

In this paper we present a convection-diffusion equation for processing image denoising, edge preservation and compression. We compare it with a popular nonlinear diffusion model which has been widely implemented in image denoising for Gaussian white noise. Here we show that this convection-diffusion model effectively removes noise, especially for the mixture of Gaussian and salt-and-pepper noises. We propose the modified streamline diffusion method [Y. Shih, H.C. Elman, Modified streamline diffusion schemes for convection-diffusion problems, Comput. Methods Appl. Mech. Eng, 1998.] for the discretization of this convection-diffusion model to prevent internal layers because of the discontinuities while using the coarsening algorithm for the image compression. Numerical experiments have shown that our convection-diffusion model for removing both Gaussian and salt-and-pepper noises, efficiently and reliably preserves edges quite satisfactorily.     

Large deviations for the Boussinesq equations under random influences

A Boussinesq model for the Bénard convection under random influences is considered as a system of stochastic partial differential equations. This is a coupled system of stochastic Navier–Stokes equations and the transport equation for temperature. Large deviations are proved, using a weak convergence approach based on a variational representation of functionals of infinite-dimensional Brownian motion.     

Kinetic limits for a class of interacting particle systems

Summary We study one dimensional particle systems in which particles travel as independent random walks and collide stochastically. The collision rates are chosen so that each particle experiences finitely many collisions per unit time. We establish the kinetic limit and derive the discrete Boltzmann equation for the macroscopic particle density.     

Quadrature rule-based bounds for functions of adjacency matrices

Bounds for entries of matrix functions based on Gauss-type quadrature rules are applied to adjacency matrices associated with graphs. This technique allows to develop inexpensive and accurate upper and lower bounds for certain quantities (Estrada index, subgraph centrality, communicability) that describe properties of networks.     

A two-sample test for comparison of long memory parameters

We construct a two-sample test for comparison of long memory parameters based on ratios of two rescaled variance (V/S) statistics studied in Giraitis et al. [L. Giraitis, R. Leipus, A. Philippe, A test for stationarity versus trends and unit roots for a wide class of dependent errors, Econometric Theory 21 (2006) 989-1029]. The two samples have the same length and can be mutually independent or dependent. In the latter case, the test statistic is modified to make it asymptotically free of the long-run correlation coefficient between the samples. To diminish the sensitivity of the test on the choice of the bandwidth parameter, an adaptive formula for the bandwidth parameter is derived using the asymptotic expansion in Abadir et al. [K. Abadir, W. Distaso, L. Giraitis, Two estimators of the long-run variance: beyond short memory, Journal of Econometrics 150 (2009) 56-70]. A simulation study shows that the above choice of bandwidth leads to a good size of our comparison test for most values of fractional and ARMA parameters of the simulated series.     

Implicit time discretization for the mean curvature flow equation

In this paper we apply the method of implicit time discretization to the mean curvature flow equation including outer forces. In the framework ofBV-functions we construct discrete solutions iteratively by minimizing a suitable energy-functional in each time step. Employing geometric and variational arguments we show an energy estimate which assures compactness of the discrete solutions. An additional convergence condition excludes a loss of area in the limit. Thus existence of solutions to the continuous problem can be derived. We append a brief discussion of the related Mullins-Sekerka equation.This work was supported by the Deutsche Forschungsgemeinschaft through Sonderforschungsbereich 256, Bonn     

Stability for hypersurfaces of constant mean curvature with free boundary

The partitioning problem for a smooth convex bodyB ³ consists in to study, among surfaces which divideB in two pieces of prescribed volume, those which are critical points of the area functional.We study stable solutions of the above problem: we obtain several topological and geometrical restrictions for this kind of surfaces. In the case thatB is a Euclidean ball we obtain stronger results.Antonio Ros is partially supported by DGICYT grant PB91-0731 and Enaldo Vergasta is partially supported by CNPq grant 202326/91-8.     

Well-posedness of a diffusive gyro-kinetic model

We study a finite Larmor radius model used to describe the ions distribution function in the core of a tokamak plasma, that consists in a gyro-kinetic transport equation coupled with an electro-neutrality equation. Since the last equation does not provide enough regularity on the electric potential, we introduce a simple linear collision operator adapted to the finite Larmor radius approximation. We next study the two-dimensional dynamics in the direction perpendicular to the magnetic field. Thanks to the smoothing effects of the collision and the gyro-average operators, we prove the global existence of solutions, as well as short time uniqueness and stability.