Stationary solutions to the Falk system on shape memory alloys

We study the Falk model system describing martensitic phase transitions in shape memory alloys. Its physically closed stationary state is formulated as a nonlinear eigenvalue problem with a non‐local term. Then, some results on existence, stability, and bifurcation of the solution are proven. In particular, we prove the existence of dynamically stable nontrivial stationary solutions. Copyright © 2009 John Wiley & Sons, Ltd.     

Weak solutions for Falk's model of shape memory alloys

In this paper we discuss the system of two partial differential equations governing the dynamics of phase transitions in shape memory alloys. We consider the one‐dimensional model proposed by Falk, in which a term containing a fourth‐derivative appears. The main purpose is to show the uniqueness for weak solutions of the problem by using the approximate dual equations for the system without growth condition for the free energy function. Copyright © 2000 John Wiley & Sons, Ltd.     

A-priori and a-posteriori error analysis for a family of Reissner–Mindlin plate elements

A family of plate elements introduced by Falk and Tu [25] is considered. A new stability and a-priori error analysis is given. In addition, an a-posteriori error estimate is proved. The analysis is confirmed by numerical benchmark computations. AMS subject classification (2000)  65F20     

Weak solutions for the Falk model system of shape memory alloys in energy class

We show the unique global existence of energy class solutions for the Falk model system of shape memory alloys under the general non‐linearity as well as considered in Aiki (Math. Meth. Appl. Sci. 2000; 23 : 299). Our main tools of the proofs are the Strichartz type estimate for the Boussinesq type equation and the maximal regularity estimate for the heat equation. Copyright © 2005 John Wiley & Sons, Ltd.     

A Berry-Esseen theorem for the kernel quantile estimator with application to studying the deficiency of quantile estimators

A Berry-Esseen bound is established for the kernel quantile estimator under various conditions. The results improve an earlier result of Falk (1985,Ann. Statist.,13, 428–433) and rely on the local smoothness of the quantile function. This new Berry-Esseen bound is applied to studying the deficiency of the sample quantile estimator with respect to the kernel quantile estimator. A new result is obtained which is an extension of that in Falk (1985).     

Interior estimates for a low order finite element method for the Reissner–Mindlin plate model

Interior error estimates are obtained for a low order finite element introduced by Arnold and Falk for the Reissner–Mindlin plates. It is proved that the approximation error of the finite element solution in the interior domain is bounded above by two parts: one measures the local approximability of the exact solution by the finite element space and the other the global approximability of the finite element method. As an application, we show that for the soft simply supported plate, the Arnold–Falk element still achieves an almost optimal convergence rate in the energy norm away from the boundary layer, even though optimal order convergence cannot hold globally due to the boundary layer. Numerical results are given which support our conclusion. This revised version was published online in June 2006 with corrections to the Cover Date.     

On the complex Falk–Langemeyer method

Numerical Algorithms - A new algorithm for the simultaneous diagonalization of two complex Hermitian matrices is derived. It is a proper generalization of the known Falk–Langemeyer algorithm...     

A ROBUST DISCRETIZATION OF THE REISSNER-MINDLIN PLATE WITH ARBITRARY POLYNOMIAL DEGREE

A numerical scheme for the Reissner-Mindlin plate model is proposed.The method is based on a discrete Helmholtz decomposition and can be viewed as a generalization of the nonconforming finite element scheme of Arnold and Falk[SIAM J.Numer.Anal.,26(6):1276-1290,1989].The two unknowns in the discrete formulation are the in-plane rotations and the gradient of the vertical displacement.The decomposition of the discrete shear variable leads to equivalence with the usual Stokes system with penalty term plus two Poisson equations and the proposed method is equivalent to a stabilized discretization of the Stokes system that generalizes the Mini element.The method is proved to satisfy a best-approximation result which is robust with respect to the thickness parameter t.     

A Posteriori Error Estimates for Finite Element Exterior Calculus: The de Rham Complex

Finite element exterior calculus (FEEC) has been developed over the past decade as a framework for constructing and analyzing stable and accurate numerical methods for partial differential equations by employing differential complexes. The recent work of Arnold, Falk, and Winther includes a well-developed theory of finite element methods for Hodge–Laplace problems, including a priori error estimates. In this work we focus on developing a posteriori error estimates in which the computational error is bounded by some computable functional of the discrete solution and problem data. More precisely, we prove a posteriori error estimates of a residual type for Arnold–Falk–Winther mixed finite element methods for Hodge–de Rham–Laplace problems. While a number of previous works consider a posteriori error estimation for Maxwell’s equations and mixed formulations of the scalar Laplacian, the approach we take is distinguished by a unified treatment of the various Hodge–Laplace problems arising in the de Rham complex, consistent use of the language and analytical framework of differential forms, and the development of a posteriori error estimates for harmonic forms and the effects of their approximation on the resulting numerical method for the Hodge–Laplacian.     

Efficient estimators and LAN in canonical bivariate POT models

Bivariate generalized Pareto distributions (GPs) with uniform margins are introduced and elementary properties such as peaks-over-threshold (POT) stability are discussed. A unified parameterization with parameter ?∈[0,1] of the GPs is provided by their canonical parameterization. We derive efficient estimators of ? and of the dependence function of the GP in various models and establish local asymptotic normality (LAN) of the loglikelihood function of a 2×2 table sorting of the observations. From this result we can deduce that the estimator of ? suggested by Falk and Reiss (2001, Statist. Probab. Lett. 52, 233-242) is not efficient, whereas a modification actually is.     

A stability and numerical study of the solutions of a Timoshenko system with distributed delay

In this work, we analyze the stability of the semigroup associated with a Timoshenko beam model with distributed delay in the rotation angle equation. We show that the type of stability resulting from the semigroup is directly related to some model coefficients, which constitute the velocities of the system's component equations. In the case of stability of the polynomial type, we prove that rate obtained is optimal. We conclude the work performing a numerical study of the solutions and their energies, associated to discrete system.     

Stability analysis of a reduced model of the lac operon under impulsive and switching control

The study of the stability is an important motif in biological systems. Stability plays a significant role in some of the basic processes of life. In this paper, we provide a theoretical method for analyzing the exponential stability of a reduced model of the lac operon under the impulsive and switching control. Under the hybrid control, the reduced model becomes a nonlinear hybrid impulsive and switching system, and we get the sufficient conditions of its exponential stability. Finally, we give examples to illustrate the effectiveness of our method.     

Some notes on multivariate generalized Pareto distributions

The investigation of multivariate generalized Pareto distributions (GPDs) has begun only recently and there are slightly varying definitions of GPDs available. In this article we investigate the one from Section 5.1 of Falk et al. [Laws of Small Numbers: Extremes and Rare Events, second ed., Birkhäuser, Basel, 2004], which does not differ in the area of interest from those of other authors. We first give an interpretation of the case of independence in terms of the peaks-over-threshold approach. This case is also used in dimension d=3 by Falk et al. [Laws of Small Numbers: Extremes and Rare Events, second ed., Birkhäuser, Basel, 2004] as a counterexample to show that GP functions are not necessarily distribution functions on their entire support. We generalize this counterexample to an arbitrary dimension d≥3 and demonstrate also that other GP functions show this behavior. Finally we show that different GPDs can lead to the same conditional probability measure in the area of interest.     

Discrete compactness property for quadrilateral finite element spaces

The main purpose of the present article is to prove the discrete compactness property for Arnold‐Boffi‐Falk spaces of any order. Results of numerical experiments confirming the theory are also reported. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

On global stability of the intra-host dynamics of malaria and the immune system

In this paper we consider an intra-host model for the dynamics of malaria. The model describes the dynamics of the blood stage malaria parasites and their interaction with host cells, in particular red blood cells (RBC) and immune effectors. We establish the equilibrium points of the system and analyze their stability using the theory of competitive systems, compound matrices and stability of periodic orbits. We established that the disease-free equilibrium is globally stable if and only if the basic reproduction number satisfies R₀?1 and the parasite will be cleared out of the host. If R₀>1, a unique endemic equilibrium is globally stable and the parasites persist at the endemic steady state. In the presence of the immune response, the numerical analysis of the model shows that the endemic equilibrium is unstable.     

On the dynamical behavior of three species food web model

In this paper, a mathematical model consisting of two preys one predator with Beddington–DeAngelis functional response is proposed and analyzed. The local stability analysis of the system is carried out. The necessary and sufficient conditions for the persistence of three species food web model are obtained. For the biologically reasonable range of parameter values, the global dynamics of the system has been investigated numerically. Number of bifurcation diagrams has been obtained; Lyapunov exponents have been computed for different attractor sets. It is observed that the model has different types of attractors including chaos.     

On the p-version approximation in the boundary element method for a variational inequality of the second kind modelling unilateral contact and given friction

In this paper we complement recent work of Maischak and Stephan on adaptive hp-versions of the BEM for unilateral Signorini problems, respectively on FEM-BEM coupling in its h-version for a nonlinear transmission problem modelling Coulomb friction contact. Here we focus on the boundary element method in its p-version to treat a scalar variational inequality of the second kind that models unilateral contact and Coulomb friction in elasticity together. This leads to a nonconforming discretization scheme. In contrast to the work cited above and to a related paper of Guediri on a boundary variational inequality of the second kind modelling friction we take the quadrature error of the friction functional into account of the error analysis. At first without any regularity assumptions, we prove convergence of the BEM Galerkin approximation in the energy norm. Then under mild regularity assumptions, we establish an a priori error estimate that is based on a novel Céa–Falk lemma for abstract variational inequalities of the second kind.     

Robust network design in telecommunications under polytope demand uncertainty

We consider a model for robust network design in telecommunications, in which we minimize the cost of the maximum mismatch between supply and demand. In the present study, the demand is uncertain and takes its values in a polytope defined by constraints. This problem is hardly tractable, so we limit ourselves to computing lower bounds (by a column-generation mechanism) and upper bounds (using an algorithm due to Falk and Soland for maximizing a separable convex function over a polytope). The experimental gap obtained turns out to be large, and this seems to be mainly due to poor upper bounds. Two possible solutions are suggested for further research aimed at improving them: dc optimization (to minimize the difference of two convex functions) and AARC modeling (affinely adjustable robust counterpart).     

The inf-sup condition and error estimates for the Arnold-Falk plate bending element

Summary We prove a discrete inf-sup condition for a mixed plate element introduced by Arnold and Falk.By using this condition, we obtain optimal error estimates for the shear strain which are valid uniformly in the plate thickness.This research was performed while the author was visiting the Center for Applied Mathematics, Purdue University, West Lafayette, IN 47907, USA     

Testing the independence of maxima: from bivariate vectors to spatial extreme fields

Characterizing the behaviour of multivariate or spatial extreme values is of fundamental interest to understand how extreme events tend to occur. In this paper we propose to test for the asymptotic independence of bivariate maxima vectors. Our test statistic is derived from a madogram, a notion classically used in geostatistics to capture spatial structures. The test can be applied to bivariate vectors, and a generalization to the spatial context is proposed. For bivariate vectors, a comparison to the test by Falk and Michel (Ann Inst Stat Math 58:261–290, 2006) is conducted through a simulation study. In the spatial case, special attention is paid to pairwise dependence. A multiple test procedure is designed to determine at which lag asymptotic independence takes place. This new procedure is based on the bootstrap distribution of the number of times the null hypothesis is rejected. It is then tested on maxima of three classical spatial models and finally applied to two climate datasets.