Gradient flows as a selection procedure for equilibria of nonconvex energies

For atomistic material models, global minimization gives the wrong qualitative behavior; a theory of equilibrium solutions needs to be defined in different terms. In this paper, a concept based on gradient flow evolutions, to describe local minimization for simple atomistic models based on the Lennard–Jones potential, is presented. As an application of this technique, it is shown that an atomistic gradient flow evolution converges to a gradient flow of a continuum energy as the spacing between the atoms tends to zero. In addition, the convergence of the resulting equilibria is investigated in the case of elastic deformation and a simple damaged state

Durable goods monopoly with stochastic costs

I study the problem of a durable goods monopolist who lacks commitment power and whose marginal cost of production varies stochastically over time. I show that a monopolist with stochastic costs usually serves the different types of consumers at different times and charges them different prices. When the distribution of consumer valuations is discrete, the monopolist exercises market power and there is inefficient delay. When there is a continuum of types, the monopolist cannot extract rents and the market outcome is efficient

The role of the patch test in 2D atomistic-to-continuum coupling methods

For a general class of atomistic-to-continuum coupling methods, coupling multi-body interatomic potentials with a P1-finite element discretisation of Cauchy--Born nonlinear elasticity, this paper adresses the question whether patch test consistency (or, absence of ghost forces) implies a first-order error estimate. In two dimensions it is shown that this is indeed true under the following additional technical assumptions: (i) an energy consistency condition, (ii) locality of the interface correction, (iii) volumetric scaling of the interface correction, and (iv) connectedness of the atomistic region. The extent to which these assumptions are necessary is discussed in detail.Comment: Version 2: correction of some minor mistakes, added discussion of multiple connected atomistic region, minor improvements of styl

A theory of political gridlock

This paper studies how electoral incentives influence the outcomes of political negotiations. It considers a game between two political parties that have to bargain over which policy to implement. While bargaining, the parties' popularity varies over time. Changes in popularity are partly exogenous and partly driven by the parties' actions. There is an election scheduled at a future date and the party with more popularity at the election date wins the vote. Electoral incentives can have substantial effects on bargaining outcomes. Periods of gridlock may arise when the election is close and parties have similar levels of popularity

A-posteriori existence in adaptive computations

This short note demonstrates that it is not necessary to assume the existence of exact solutions in an a-posteriori error analysis. If the residual of a stable numerical solution is sufficiently small there exists a nearby exact solution for which an a-posteriori error estimate holds.\ud \ud We first develop the idea in an abstract Banach space setting and then demonstrate some further practical details at the nonlinear Laplace equation.\ud \ud The author acknowledges the financial support received from the European research project HPRN-CT-2002-00284: New Materials, Adaptive Systems and their Nonlinearities. Modelling, Control and Numerical Simulation, and the kind hospitality of Carlo Lovadina (University of Pavia)

Continuum Limit of a One-Dimensional Atomistic Energy Based on Local Minimization

For atomistic energies, global minimization gives the wrong qualitative behaviour and therefore continuum limits should be formulated in terms of local minimization. In this paper, a possible process is suggested, to describe local minimization for a simple one-dimensional problem with body and surface energy. It is shown that an atomistic gradient flow evolution converges to a continuum gradient flow as the spacing between the atomis tends to zero. In addition, the convergence of local minimizers is investigated, in the case of both elastic deformation and fracture