A theory of elasticity with spatial distribution of matter. Three-dimensional complex structure

References [1 and 2] consider a theory of elasticity with spatial distribution of matter for a medium having simple structure and for a one-dimensional medium having complex structure. In the present article the general case of a three-dimensional medium with complex structure is examined. The general scheme of the one-dimensional case [2] is retained; chief attention is directed toward the specific character of the three-dimensional problem. The original micro-model is a complex crystal lattice [3]. In Section 1 this model is generalized to the case of a continuous distribution of matter. The displacements of the mass centers of the unit cells and the micro-strains of the cells are introduced as the kinematic variables. The force variables are the micro-moments. The transition to an exact continuous representation is carried out, and the equations of an elastic medium of complex structure with spatial distribution of matter are derived. The operators corresponding to the continuous theory are expressed in terms of the original microparameters. It is shown that the well known conditions of symmetry of the tensor of elastic constants, which are usually interpreted as the condition of absence of initial stresses [3 and 4], are consequences of the invariance of the elastic energy under translation and rotation. In Section 2 some special models are examined, and the equations of a medium are obtained for the approximation of weak dispersion of matter. These equations contain as a special case the equations of linear nonsymmetric elasticity (couple-stress theory) [5 to 7]. However, in the latter it turns out that the orders of approximation are inconsistent in the various equations from the point of view of the theory of spatial distribution.

In Section 3 the equations of a medium having complex structure are transformed in the acoustic range into equations, one of which contains only a single kinematic variable (the displacement of the mass centers) and the others of which are explicitly solvable for the remaining kinematic variables. The first equation of this set coincides in form with the equation for a medium with simple structure, but differs from it by the presence of a timewise dispersion which is unrelated to energy dissipation. Expressions are written for the energy density, and it is shown that it is possible to introduce a symmetric stress tensor, as in the case of a simple structure.     

Uncertainty Quantification for Two-Phase Flow in Heterogeneous Porous Media

The simulation of flow and transport in porous media such as aquifers often involve dealing with complex heterogeneities. They are characterized by varying hydrogeological properties which differ strongly from the adjacent medium and often lead to significant changes of the flow behavior. However detailed information about the location of such heterogeneities is not always known. The deterministic models thus need to be extended stochastically to quantify uncertainties. As mathematical model we use the capillarity-free fractional flow formulation for two immiscible and incompressible fluid phases in a two-dimensional and partitioned domain. To cope with the randomly located heterogeneity interfaces we employ a stochastic Galerkin (SG) method [4]. The physical space of this system then is modelled by a central upwind finite volume scheme [5] in combination with mixed finite elements [7]. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the structure of kripke models of heyting arithmetic

Since in Heyting Arithmetic (HA) all atomic formulas are decidable, a Kripke model for HA may be regarded classically as a collection of classical structures for the language of arithmetic, partially ordered by the submodel relation. The obvious question is then: are these classical structures models of Peano Arithmetic (PA)? And dually: if a collection of models of PA, partially ordered by the submodel relation, is regarded as a Kripke model, is it a model of HA? Some partial answers to these questions were obtained in [6], [3], [1] and [2]. Here we present some results in the same direction, announced in [7]. In particular, it is proved that the classical structures at the nodes of a Kripke model of HA must be models of IΔ₁ (PA^- with induction for provably Δ₁ formulas) and that the relation between these classical structures must be that of a Δ₁-elementary submodel. MSC: 03F30, 03F55.     

Interacting Fisher–Wright diffusions in a catalytic medium

We study the longtime behaviour of interacting systems in a randomly fluctuating (space–time) medium and focus on models from population genetics. There are two prototypes of spatial models in population genetics: spatial branching processes and interacting Fisher–Wright diffusions. Quite a bit is known on spatial branching processes where the local branching rate is proportional to a random environment (catalytic medium). Here we introduce a model of interacting Fisher–Wright diffusions where the local resampling rate (or genetic drift) is proportional to a catalytic medium. For a particular choice of the medium, we investigate the longtime behaviour in the case of nearest neighbour migration on the d-dimensional lattice. While in classical homogeneous systems the longtime behaviour exhibits a dichotomy along the transience/recurrence properties of the migration, now a more complicated behaviour arises. It turns out that resampling models in catalytic media show phenomena that are new even compared with branching in catalytic medium. Received: 15 November 1999 / Revised version: 16 June 2000 / Published online: 6 April 2001     

The GARCH(1,1)-M model: results for the densities of the variance and the mean

This paper starts from the GARCH(1,1)-M model of Bollerslev [Generalized autoregressive conditional heteroskedasticity, Journal of Econometrics 31 (1986) 307–327], and investigates the limit diffusion form as it is presented in Nelson [ARCH models as diffusion approximations, Journal of Econometrics 45 (1990) 7–38]. The distribution for the conditional variance process is derived, and in the limit for t going to infinity is shown to coincide with the stationary distribution given in Nelson [ARCH models as diffusion approximations, Journal of Econometrics 45 (1990) 7–38]. In addition it is shown how the distribution for the complete model can be arrived at; explicit calculations are given in case the conditional variance is a martingale.     

High-order numerical solution of the nonlinear Helmholtz equation with axial symmetry

The nonlinear Helmholtz (NLH) equation models the propagation of intense laser beams in a Kerr medium. The NLH takes into account the effects of nonparaxiality and backward scattering that are neglected in the more common nonlinear Schrödinger model. In [G. Fibich, S. Tsynkov, High-order two-way artificial boundary conditions for nonlinear wave propagation with backscattering, J. Comput. Phys., 171 (2001) 632–677] and [G. Fibich, S. Tsynkov, Numerical solution of the nonlinear Helmholtz equation using nonorthogonal expansions, J. Comput. Phys., 210 (2005) 183–224], a novel high-order numerical method for solving the NLH was introduced and implemented in the case of a two-dimensional Cartesian geometry. The NLH was solved iteratively, using the separation of variables and a special nonlocal two-way artificial boundary condition applied to the resulting decoupled linear systems. In the current paper, we propose a major improvement to the previous method. Instead of using LU decomposition after the separation of variables, we employ an efficient summation rule that evaluates convolution with the discrete Green's function. We also extend the method to a three-dimensional setting with cylindrical symmetry, under both Dirichlet and Sommerfeld-type transverse boundary conditions.     

Numerical solution of electromagnetic scattering from a large partly covered cavity

The paper focuses on the numerical study of electromagnetic scattering from two-dimensional (2D) large partly covered cavities, which is described by the Helmholtz equation with a nonlocal boundary condition on the aperture. The classical five-point finite difference method is applied for the discretization of the Helmholtz equation and a linear approximation is used for the nonlocal boundary condition. We prove the existence and uniqueness of the numerical solution when the medium in the cavity is y-direction layered or the number of the mesh points on the aperture is large enough. The fast algorithm proposed in Bao and Sun (2005) [2] for open cavity models is extended to solving the partly covered cavity problem with (vertically) layered media. A preconditioned Krylov subspace method is proposed to solve the partly covered cavity problem with a general medium, in which a layered medium model is used as a preconditioner of the general model. Numerical results for several types of partly covered cavities with different wave numbers are reported and compared with those by ILU-type preconditioning algorithms. Our numerical experiments show that the proposed preconditioning algorithm is more efficient for partly covered cavity problems, particularly with large wave numbers.     

On the singularity of multivariate skew-symmetric models

In recent years, the skew-normal models introduced by Azzalini (1985) [1]-and their multivariate generalizations from Azzalini and Dalla Valle (1996) [4]-have enjoyed an amazing success, although an important literature has reported that they exhibit, in the vicinity of symmetry, singular Fisher information matrices and stationary points in the profile log-likelihood function for skewness, with the usual unpleasant consequences for inference. It has been shown (DiCiccio and Monti (2004) [23], DiCiccio and Monti (2009) [24] and Gómez et al. (2007) [25]) that these singularities, in some specific parametric extensions of skew-normal models (such as the classes of skew-t or skew-exponential power distributions), appear at skew-normal distributions only. Yet, an important question remains open: in broader semiparametric models of skewed distributions (such as the general skew-symmetric and skew-elliptical ones), which symmetric kernels lead to such singularities? The present paper provides an answer to this question. In very general (possibly multivariate) skew-symmetric models, we characterize, for each possible value of the rank of Fisher information matrices, the class of symmetric kernels achieving the corresponding rank. Our results show that, for strictly multivariate skew-symmetric models, not only Gaussian kernels yield singular Fisher information matrices. In contrast, we prove that systematic stationary points in the profile log-likelihood functions are obtained for (multi)normal kernels only. Finally, we also discuss the implications of such singularities on inference.     

Some remarks on the two-level DEA model

The two-level DEA model was introduced to increase the discriminational power of Data Envelopment Analysis (DEA) models. This nonlinear model was presented by Meng et al. (2008) [3], and then converted into a linear model by Kao (2008) [4].In this paper two subjects will be discussed: first, we show that the two-level DEA model is a special case of DEA models where weight restrictions are applied. Then, we express that the nonlinear model is equivalent to the conventional DEA model.     

A specialized branch & bound & cut for Single-Allocation Ordered Median Hub Location problems

The Single-Allocation Ordered Median Hub Location problem is a recent hub model introduced by Puerto et al. (2011) [32] that provides a unifying analysis of the class of hub location models. Indeed, considering ordered objective functions in hub location models is a powerful tool in modeling classic and alternative location paradigms, that can be applied with success to a large variety of problems providing new distribution patterns induced by the different users’ roles within the supply chain network. In this paper, we present a new formulation for the Single-Allocation Ordered Median Hub Location problem and a branch-and-bound-and-cut (B&B&Cut) based algorithm to solve optimally this model. A simple illustrative example is discussed to demonstrate the technique, and then a battery of test problems with data taken from the AP library are solved. The paper concludes that the proposed B&B&Cut approach performs well for small to medium sized problems.     

Dynamics of the discrete Seno population model: Combined effects of harvest timing and intensity on population stability

The paper contains new results on the impact of harvesting times and intensities on the stability properties of Seno population models. It is proved that sufficiently high harvest intensities are stabilizing for any harvesting time in the sense that they create a positive equilibrium that attracts all positive solutions. Moreover, in the special case that the nonlinearity in the Seno model is a Ricker function, we derive a global stability result independent of timing and valid for low to medium harvesting efforts. The proof is based on a characterization of those harvesting intensities which guarantee a negative Schwarzian derivative for all harvesting times. Finally, we rigorously show that timing can be stabilizing as well as destabilizing by itself. In particular, a recent conjecture formulated by Cid et al. (2014) [1] is shown to be false.     

Effects of variable viscosity in a third grade fluid with porous medium: An analytic solution

This study extends the analysis of ref. [Hayat T, Ellahi R, Asghar S. The influence of variable viscosity and viscous dissipation on the non-Newtonian flow: An analytic solution, Commun Nonlinear Sci Numer Simul 2007;12:300–313] in a porous medium by employing modified Darcy’s law. Beside this Reynolds and Vogels models of temperature dependent viscosity are considered. The problem is solved using homotopy analysis method (HAM). Expressions of velocity and temperature profiles are constructed analytically and explained with the help of graphs.     

On the theory of dynamic thermoelasticity : PMM, vol. 42, no. 6, 1978, pp. 1093–1098

Use of the causality principle as radiation condition in dynamical problems of thermoelasticity is proposed. It follows from an analysis of the fundamental mathematical models describing the thermoelastic behavior of a continuous medium and used in the solution of specific problems, that some will yield physically unrealizable solutions. To eliminate the ambiguity in the solution which occurs, an approach is possible which has an explicit physical meaning and is based on the causality principle [1, 2]; it is required that the time source not yield a response earlier than the time of starting up of the source. Different kinds of radiation conditions of the Sommerfeld type are known in thermoelasticity problems [3 – 6].

To extract the unique solution in dynamical thermoelasticity problems, it is proposed in this paper to use the causality principle, which is equivalent to the requirement of analyticity of the solution in the upper half of the complex frequency plane; there are studied the analytic properties of the solutions of the fundamental boundary value problems for the models used most often for thermoelastic media, and there are made deductions about their physical realizability.     

Assessing the predictive effectiveness of the variety seeking and reinforcement models

Prediction of customer choice behaviour has been a big challenge for marketing researchers. They have adopted various models to represent customers purchase patterns. Some researchers considered simple zero–order models. Others proposed higher–order models to represent explicitly customers tendency to seek [variety] or [reinforcement] as they make repetitive choices. Nevertheless, the question [Which model has the highest probability of representing some future data?] still prevails. The objective of this paper is to address this question. We assess the predictive effectiveness of the well–known customer choice models. In particular, we compare the predictive ability of the [dynamic attribute satiation] (DAS) model due to McAlister (Journal of Consumer Research, 91, pp. 141–150, 1982) with that of the well–known stochastic variety seeking and reinforcement behaviour models. We found that the stochastic [beta binomial] model has the best predictive effectiveness on both simulated and real purchase data. Using simulations, we also assessed the effectiveness of the stochastic models in representing various complex choice processes generated by the DAS. The beta binomial model mimicked the DAS processes the best. In this research we also propose, for the first time, a stochastic choice rule for the DAS model.     

Vortices in the Maxwell‐Chern‐Simons theory

Our aim is to prove rigorously that the Chern‐Simons model of Hong, Kim, and Pac [13] and Jackiw and Weinberg [14] (the CS model) and the Abelian Higgs model of Ginzburg and Landau (the AH model, see [15]) are unified by the Maxwell‐Chern‐Simons theory introduced by Lee, Lee, and Min in [16] (MCS model). In [16] the authors give a formal argument that shows how to recover both the CS and AH models out of their theory by taking special limits for the values of the physical parameters involved. To make this argument rigorous, we consider the existence and multiplicity of periodic vortex solutions for the MCS model and analyze their asymptotic behavior as the physical parameters approach these limiting values. We show that, indeed, the given vortices approach (in a strong sense) vortices for the CS and AH models, respectively. For this purpose, we are led to analyze a system of two elliptic PDEs with exponential nonlinearities on a flat torus. © 2000 John Wiley & Sons, Inc.     

The marked empirical process to test a general AR-ARCH against an other general AR-ARCH when the random vectors are nonstationary and absolutely regular

In this Note, we study a procedure on goodness-of-fit testing for nonlinear time-series models against a large class of alternatives under nonstationarity and absolute regularity. For that, we define a marked empirical process based on residuals which converges in distribution to a Gaussian process with respect to the Skorohod topology. This method was first introduced by Stute (1997) and then widely developed by Ngatchou-Wandji (2002, 2005, 2008) [1], [2], [3] under more general conditions. Applications to general AR-ARCH models are given. To cite this article: M. Harel, E. Elharfaoui, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Comparison of gas-dynamic models for hypersonic flow past bodies

Various gas-dynamic models for describing chemically non-equilibrium flows are compared using the example of the steady flow past the blunt nose of the “Buran” [1] and “Space Shuttle” vehicles during their descent from orbit. Models of locally self-similar approximations of the Navier-Stokes equations [2], of a chemically equilibrium and non-equilibrium complete viscous shock layer (CVSL) [3] and a model of a thin viscous shock layer (TVSL) [4] are considered. In all the models the occurrence of physicochemical processes was taken into account in the same way using fixed values of the constants for the gas-phase chemical reactions (their effect has been considered in [5]). Good agreement between the results of calculations of the heat flux at the critical point is found.

Chemically non-equilibrium flows have been considered earlier using the approximate Navier-Stokes equations [6], within the framework of a TVSL [7] and a CVSL [8, 9](for more detail, see the review [10]). The TVSL and CVSL models were compared in [11] in the case of flows of a uniform gas.     

Basic equations of the theory of reinforced media

The author derives the basic equations of the theory of composite elastic media obtained by reinforcing some elastic medium with a large number of linear or planar elastic elements with high strength and deformation resistance. The argument is based on macrostructural considerations. The stress-strain state of each of the reinforcing elements is considered with allowance for interaction with the matrix material. In addition, the "smoothing" principle introduced in [1–3] is applied. This corresponds to approximating the reinforced medium with some equivalent quasi-homogeneous anisotropic medium.The case of a fibrous medium in which the reinforcing elements are rods or filaments [4] is discussed in detail. Allowance for moment effects leads to equations analogous to the equations of the Voight-Cosserat moment theory and its later generalizations. Similar equations are obtained for the case of laminated media, where the reinforcing elements are membranes or plates. On the basis of the viscoelastic analogy [7], the equations of the theory of reinforced media are extended to include the case in which the matrix and/or reinforcing materials are linear viscoelastic.Mekhanika Polimerov, Vol 1, No. 2, pp. 27–37, 1965     

On the continuum limit of the conformal matrix models

The double scaling limit of a new class of the multi-matrix models proposed in [1], which possess the W-symmetry at the discrete level, is investigated in details. These models are demonstrated to fall into the same universality class as the standard multi-matrix models. In particular, the transformation of the W-algebra at the discrete level into the continuum one of the paper [2] is proposed, the corresponding partition functions being compared. All calculations are demonstrated in full in the first non-trivial case of W⁽³⁾-constraints.This paper was supported in part by NSF grant PHY88-57200.Published in Teoreticheskaya i Matematicheskaya Fizika, Vol. 95, No. 2, pp. 317–340, May, 1993.     

On the existence of transmission eigenvalues in an inhomogeneous medium

We prove the existence of transmission eigenvalues corresponding to the inverse scattering problem for isotropic and anisotropic media for both the scalar problem and Maxwell's equations. Considering a generalized abstract eigenvalue problem, we are able to extend the ideas of Päivärinta and Sylvester [Transmission eigenvalues, SIAM J. Math. Anal. 40, (2008) pp. 783–753] to prove the existence of transmission eigenvalues for a larger class of interior transmission problems. Our analysis includes both the case of a medium with positive contrast and of a medium with negative contrast provided that the contrasts are large enough.