We consider abstract second order evolution equations with unboundedfeedback with delay. Existence results are obtained under somerealistic assumptions. Sufficient and explicit conditions are derived that guarantee the exponential or polynomial stability. Some new examples that enter into our abstract framework are presented.

DiPerna's and Majda's generalization of Young measures is used to describe oscillations and concentrations in sequences of maps $\{u_k\}_{k\in{\mathbb N}} \subset L^p(\Omega;{\mathbb R}^m)$ satisfying a linear differential constraint ${\mathcal A}u_k=0$ . Applications to sequential weak lower semicontinuity of integral functionals on ${\mathcal A}$ -free sequences and to weak continuity of determinants are given. In particular, we state necessary and sufficient conditions for weak* convergence of det $\nabla\varphi_k\stackrel{*}{\rightharpoonup}{\rm det}\nabla\varphi$ in measures on the closure of $\Omega\subset{\mathbb R}^n$ if $\varphi_k\rightharpoonup\varphi$ in $W^{1,n}(\Omega;{\mathbb R}^n)$ . This convergence holds, for example, under Dirichlet boundary conditions. Further, we formulate a Biting-like lemmaprecisely stating which subsets $\Omega_j\subset \Omega$ must be removed to obtain weak lower semicontinuity of $u\mapsto\int_{\Omega\setminus\Omega_j} v(u(x))\,{\rm d}x$ along $\{u_k\}\subset L^p(\Omega;{\mathbb R}^m)\cap{\rm ker}\ {\mathcal A}$ . Specifically, $\Omega_j$ are arbitrarily thin “boundary layers”.

The left-invariant sub-Riemannian problem on the group of motions of a plane is considered. Sub-Riemannian geodesics are parameterized by Jacobi's functions. Discrete symmetries of the problem generated by reflections of pendulum are described. The corresponding Maxwell points are characterized, on this basis an upper bound on the cut time is obtained.

We consider the magnetic induction equation for the evolution of a magnetic field in a plasma where the velocity is given. The aim is to design a numerical scheme which also handles the divergence constraint in a suitable manner. We design and analyze an upwind scheme based on the symmetrized version of the equations in the non-conservative form. The scheme is shown to converge to a weak solution of the equations. Furthermore, the discrete divergence produced by the scheme is shown to be bounded. We report several numerical experiments that show that the stable upwind scheme of this paper is robust.

The atomistic to continuum interface for quasicontinuum energiesexhibits nonzero forces under uniform strain that have beencalled ghost forces.In this paper,we prove for a linearization of a one-dimensional quasicontinuum energyaround a uniform strainthat the effect of the ghost forces on the displacementnearly cancels and has a small effect on the error away from the interface.We give optimal order error estimatesthat show that the quasicontinuum displacementconverges to the atomistic displacement at the rate O(h)in the discrete $\ell^\infty$ andw 1,1 norms where h is the interatomic spacing.We also give a proof that the error in the displacement gradientdecays away from the interface to O(h) at distance O(h|logh|)in the atomistic region and distance O(h) in the continuum region.Our work gives an explicit and simplified form for the decay of the effect of theatomistic to continuum coupling error in terms of a general underlying interatomic potential and gives the estimates described above in the discrete $\ell^\infty$ and w1,p norms.

A new Schwarz method for nonlinear systems is presented, constituting the multiplicative variant of a straightforward additive scheme. Local convergence can be guaranteed under suitable assumptions.The scheme is applied to nonlinear acoustic-structure interaction problems. Numerical examples validate the theoretical results. Further improvements are discussed by means of introducing overlapping subdomains and employing an inexact strategy for the local solvers.

We state a superconvergence result for the lowest order Raviart-Thomas approximation of eigenvalue problems. It is known that a similar superconvergence result holds for the mixed approximation of Laplace problem; here we introduce a new proof, since the one given for the source problem cannot be generalized in a straightforward way to the eigenvalue problem. Numerical experiments confirm thesuperconvergence property and suggest that it also holds for the lowest orderBrezzi-Douglas-Marini approximation.

In this article, we analyze the stability of various numerical schemes for differential models of viscoelastic fluids. More precisely, we consider the prototypical Oldroyd-B model, for which a free energy dissipation holds, and we show under which assumptions such a dissipation is also satisfied for the numerical scheme. Among the numerical schemes we analyze, we consider some discretizations based on the log-formulation of the Oldroyd-B system proposed by Fattal and Kupferman in [J. Non-Newtonian Fluid Mech.123 (2004) 281–285], for which solutions in some benchmark problems have been obtained beyond the limiting Weissenberg numbers for the standard scheme (see [Hulsen et al.J. Non-Newtonian Fluid Mech. 127 (2005) 27–39]). Our analysis gives some tracks to understand these numerical observations.

A conceptual numerical strategy for rate-independent processes in the energetic formulation is proposed and its convergence is proved under various rather mild data qualifications. The novelty is that we obtain convergence ofsubsequences of space-time discretizations even in case where the limitproblem does not have a unique solution and we need noadditional assumptions on higher regularity of the limit solution.The variety of general perspectives thus obtained is illustrated on several specific examples: plasticity with isotropic hardening, damage, debonding, magnetostriction, and two models of martensitic transformation in shape-memory alloys.

This note introduces quasi-local-balance for discrete-time Markov chains with absorbing states. From quasi-local-balance product-form quasi-stationary distributions are derived by analogy with product-form stationary distributions for Markov chains that satisfy local balance.

A substance carried convectively through the liver by the blood undergoes two successive enzymatic transformations. The resulting concentrations of the three forms of the substance are determined, as functions of position along the blood flow in the steady state, by coupled ordinary differential equations of the first order on a finite interval. The densities along the blood flow of the activities of the two (immobile) transforming enzymes are described by two non-negative and normalised functions of position.

The problem, suggested by recent experimental results, is to choose these two functions so as to minimise the concentration of the once-transformed (intermediate) form of the substance at one boundary (the liver outlet). That minimisation is particularly significant biologically when the intermediate form is toxic and the second transformation renders it harmless. In this problem of optimal control (exerted perhaps by natural selection), the classical approach through Euler's equations is inapplicable because of the constraints on the two density functions. Moreover, when the enzyme kinetics and hence the differential equations are non-linear, the functional to be minimised is not obtainable explicitly. Instead it appears, after some manipulation of the coupled equations, as the terminal boundary value of the solution of a non-linear Volterra integral equation, which involves the two density functions (one explicitly and one implicitly) as control variables.

Appropriate existence, uniqueness and boundedness results are obtained for the solution of this integral equation, and the problem is then solved rigorously for one class of non-linearities (including saturation kinetics). Some unanswered questions are posed for another class (including substrate-inhibition kinetics).

We carry out a study of the peristaltic motion of an incompressible micropolar fluid in a two-dimensional channel. The effects of viscoelastic wall properties and micropolar fluid parameters on the flow are investigated using the equations of the fluid as well as of the deformable boundaries. A perturbation technique is used to determine flow characteristics. The velocity profile is presented and discussed briefly. We find the critical values of the parameters involving wall characteristics, which cause mean flow reversal.

A mathematical model is presented in which the long Jump is treated as the motion of a projectile under gravity with slight drag. The first two terms of a perturbation solution are obtained and are shown to be more accurate than earlier approximate analytical solutions. Results from the perturbation analysis are just as accurate as results from various numerical schemes, and require far less computer time.

The model is modified to include the observation that a long-jumper's centre of mass is forward of his feet at take-off and behind his feet on landing.

The modified model is used to determine the take-off angle for the current world long jump record, resulting in several interesting observations for athletic coaches.

It has been known for some time that if a certain non-degeneracy condition is satisfied then the successive solution estimates x(r) produced by barrier function techniques lie on a smooth trajectory. Accordingly, extrapolation methods can be used to calculate x(0). In this paper we analyse the situation further treating the special case of the log barrier function. If the non-degeneracy assumption is not satisfied then the approach to x(0) is like r½ rather than like r which would be expected in the non-degenerate case. A measure of sensitivity is introduced which becomes large when the non- degeneracy assumption is close to violation, and it is shown that this sensitivity measure is related to the growth of dix/dri with respect to i for fixed r small enough on the solution trajectory. With this information it is possible to analyse the extrapolation procedure and to predict the number of stages of extrapolation which are useful.

In this paper we study the best constant in the Sobolev trace embedding H1 (Ω) →Lq(∂Ω) in a bounded smooth domain for 1 < q < 2+ = 2(N - 1)/(N - 2), that is, critical or subcritical q. First, we consider a domain with periodically distributed holes inside which we impose that the involved functions vanish. There exists a critical size of the holes for which the limit problem has an extra term. For sizes larger than critical the best trace constant diverges to infinity and for sizes smaller than critical it converges to the best constant in the domain without holes. Also, we study the problem with the holes located on the boundary of the domain. In this case another critical exists and its extra term appears on the boundary.

The reflection-transmission properties of water waves obliquely incident upon a vortex sheet in water of finite depth are studied. The problem is reduced to that of solving two integral equation. An accurate Galerkin solution is obtained which supports the use of the “variational method” in water wave problems that has recently been questioned by Kirby and Dalyrmple.

A heuristic methodology for the identification of a circuit passing through all the vertices only once in a graph is presented. The procedure is based upon defining a normal form of a matrix and then transforming the adjacency matrix into its normal form. For a class of graphs known to be Hamiltonian, it is conjectured that this methodology will identify circuits in a small number of steps and in many cases merely by observation.