The present paper represents a continuation of Sofonea and Matei's paper (Sofonea, M. and Matei, A. (2011) History-dependent quasivariational inequalities arising in contact mechanics. Eur. J. Appl. Math. 22, 471–491). There a new class of variational inequalities involving history-dependent operators was considered, an abstract existence and uniqueness result was proved and it was completed with a regularity result. Moreover, these results were used in the analysis of various frictional and frictionless models of contact. In this current paper we present a penalization method in the study of such inequalities. We start with an example which motivates our study; it concerns a mathematical model which describes the quasistatic contact between a viscoelastic body and a foundation; the material's behaviour is modelled with a constitutive law with long memory, the contact is frictionless and is modelled with a multivalued normal compliance condition and unilateral constraint. Then we introduce the abstract variational inequalities together with their penalizations. We prove the unique solvability of the penalized problems and the convergence of their solutions to the solution of the original problem, as the penalization parameter converges to zero. Finally, we turn back to our contact model, apply our abstract results in the study of this problem and provide their mechanical interpretation.

With financial modelling requiring a better understanding of model risk, it is helpful to be able to vary assumptions about underlying probability distributions in an efficient manner, preferably without the noise induced by resampling distributions managed by Monte Carlo methods. This paper presents differential equations and solution methods for the functions of the form Q(x) = F−1(G(x)), where F and G are cumulative distribution functions. Such functions allow the direct recycling of Monte Carlo samples from one distribution into samples from another. The method may be developed analytically for certain special cases, and illuminate the idea that it is a more precise form of the traditional Cornish–Fisher expansion. In this manner the model risk of distributional risk may be assessed free of the Monte Carlo noise associated with resampling. The method may also be regarded as providing both analytical and numerical bases for doing more precise Cornish–Fisher transformations. Examples are given of equations for converting normal samples to Student t, and converting exponential to normal. In the case of the normal distribution, the change of variables employed allows the sampling to take place to good accuracy based on a single rational approximation over a very wide range of sample space. The avoidance of branching statements is of use in optimal graphics processing unit (GPU) computations as it avoids the effect of branch divergence. We give a branch-free normal quantile that offers performance improvements in a GPU environment while retaining the best precision characteristics of well-known methods. We also offer models with low probability branch divergence. Comparisons of new and existing forms are made on Nvidia GeForce GTX Titan and Tesla C2050 GPUs. We argue that in both single- and double-precisions, the change-of-variables approach offers the most GPU-optimal Gaussian quantile yet, working faster than the Cuda 5.5 built-in function.

This paper analyses a canonical class of one-dimensional superlinear indefinite boundary value problems of great interest in population dynamics under non-homogeneous boundary conditions; the main bifurcation parameter in our analysis is the amplitude of the superlinear term. Essentially, it continues the analysis of López-Gómez et al. (López-Gómez, J., Tellini, A. & Zanolin, F. (2014) High multiplicity and complexity of the bifurcation diagrams of large solutions for a class of superlinear indefinite problems. Comm. Pure Appl. Anal. 13(1), 1–73) with empty overlapping, by computing the bifurcation diagrams of positive steady states of the model and by proving analytically a number of significant features, which have been observed from the numerical experiments carried out here. The numerics of this paper, besides being very challenging from the mathematical point of view, are imperative from the point of view of population dynamics, in order to ascertain the dimensions of the unstable manifolds of the multiple equilibria of the problem, which measure their degree of instability. From that point of view, our results establish that under facilitative effects in competitive media, the harsher the environmental conditions, the richer the dynamics of the species, in the sense discussed in Section 1.

In this work we investigate limiting values of the lift and drag coefficients of profiles in the Helmholtz–Kirchhoff (infinite cavity) flow. The coefficients are based on the wetted arc length of profile surfaces. The problem is to find global minimum and maximum values of the drag coefficient CD under a given lift coefficient CL. We reduce the problem to a constrained problem of calculus of variations and solve it analytically. In so doing we do not only determine extremals but also strictly prove that these extremals realize global extrema. The proofs are based on non-trivial application of Jensen's inequality. The solution of the problem allows us to construct the domain of possible variations of coefficients CL and CD and define maximum and minimum values of the lift-to-drag ratios CL/CD for a given CL.

We present the first macroscopical model for charge transport in compound semiconductors to make use of analytic ellipsoidal approximations for the energy dispersion relationships in the neighbours of the lowest minima of the conduction bands. The model considers the main scattering mechanisms charges undergo in polar semiconductors, that is the acoustic, polar optical, intervalley non-polar optical phonon interactions and the ionized impurity scattering. Simulations are shown for the cases of bulk 4H and 6H-SiC.