A pair trade is a portfolio consisting of a long position in one asset and a short position in another, and it is a widely used investment strategy in the financial industry. Recently, Ekström, Lindberg, and Tysk studied the problem of optimally closing a pair trading strategy when the difference of the two assets is modelled by an Ornstein-Uhlenbeck process. In the present work the model is generalized to also include jumps. More precisely, we assume that the difference between the assets is an Ornstein-Uhlenbeck type process, driven by a Lévy process of finite activity. We prove a necessary condition for optimality (a so-called verification theorem), which takes the form of a free boundary problem for an integro-differential equation. We analyze a finite element method for this problem and prove rigorous error estimates, which are used to draw conclusions from numerical simulations. In particular, we present strong evidence for the existence and uniqueness of an optimal solution.

Most three dimensional constitutive relations that have been developed to describe the behavior of bodies are correlated against one dimensional and two dimensional experiments. What is usually lost sight of is the fact that infinity of such three dimensional models may be able to explain these experiments that are lower dimensional. Recently, the notion of maximization of the rate of entropy production has been used to obtain constitutive relations based on the choice of the stored energy and rate of entropy production, etc. In this paper we show different choices for the manner in which the body stores energy and dissipates energy and satisfies the requirement of maximization of the rate of entropy production that can all describe the same experimental data. All of these three dimensional models, in one dimension, reduce to the model proposed by Burgers to describe the viscoelastic behavior of bodies.

The framework for shape and topology sensitivity analysis in geometrical domains with cracks is established for elastic bodies in two spatial dimensions. The equilibrium problem for the elastic body with cracks is considered. Inequality type boundary conditions are prescribed at the crack faces providing a non-penetration between the crack faces. Modelling of such problems in two spatial dimensions is presented with all necessary details for further applications in shape optimization in structural mechanics. In the paper, general results on the shape and topology sensitivity analysis of this problem are provided. The results are of interest of their own. In particular, the existence of the shape and topological derivatives of the energy functional is obtained. The results presented in the paper can be used for numerical solution of shape optimization and inverse problems in structural mechanics.

The paper is devoted to the analysis of the discontinuous Galerkin finite element method (DGFEM) applied to the space semidiscretization of a nonlinear nonstationary convection-diffusion problem with mixed Dirichlet-Neumann boundary conditions. General nonconforming meshes are used and the NIPG, IIPG and SIPG versions of the discretization of diffusion terms are considered. The main attention is paid to the impact of the Neumann boundary condition prescribed on a part of the boundary on the truncation error in the approximation of the nonlinear convective terms. The estimate of this error allows to analyse the error estimate of the method. The results obtained represent the completion and extension of the analysis from V. Dolejsi, M. Feistauer, Numer. Funct. Anal. Optim. 26 (2005), 349-383, where the truncation error in the approximation of the nonlinear convection terms was proved only in the case when the Dirichlet boundary condition on the whole boundary of the computational domain was considered.

We consider the singular boundary value problem $$({t^n}u't))' + {t^n}f(t,u(t)) = 0,{\rm{ }}\mathop {\lim }\limits_{t \to 0 + } {t^n}u'(t) = 0,{\rm{ }}{a_0}u(1) + {a_1}u'(1 - ) = A,$$ where f(t, x) is a given continuous function defined on the set (0, 1]×(0,∞) which can have a time singularity at t = 0 and a space singularity at x = 0. Moreover, n ∈ ℕ, n ⩾ >2, and a 0, a 1, A are real constants such that a 0 ∈ (0,1), whereas a 1,A ∈ [0,∞). The main aim of this paper is to discuss the existence of solutions to the above problem and apply the general results to cover certain classes of singular problems arising in the theory of shallow membrane caps, where we are especially interested in characterizing positive solutions. We illustrate the analytical findings by numerical simulations based on polynomial collocation.

The nonlinear integro-differential system associated with the penetration of a magnetic field into a substance is considered. The asymptotic behavior as t → ∞ of solutions for two initial-boundary value problems are studied. The problem with non-zero conditions on one side of the lateral boundary is discussed. The problem with homogeneous boundary conditions is studied too. The rates of convergence are given. Results presented show the difference between stabilization characters of solutions of these two cases.

We employ the active set strategy which was proposed by Facchinei for solving large scale bound constrained optimization problems. As the special structure of the bound constrained problem, a simple rule is used for updating the multipliers. Numerical results show that the active set identification strategy is practical and efficient.

In this paper, the evolution equations with nonlinear term describing the resonance interaction between the long wave and the short wave are studied. The semi-discrete and fully discrete Crank-Nicholson Fourier spectral schemes are given. An energy estimation method is used to obtain error estimates for the approximate solutions. The numerical results obtained are compared with exact solution and found to be in good agreement.

The generalized FGM distribution and related copulas are used as bivariate models for the distribution of spheroidal characteristics. It is shown that this model is suitable for the study of extremes of the 3D spheroidal particles observed in terms of their random planar sections.

We consider a model for transient conductive-radiative heat transfer in grey materials. Since the domain contains an enclosed cavity, nonlocal radiation boundary conditions for the conductive heat-flux are taken into account. We generalize known existence and uniqueness results to the practically relevant case of lower integrable heat-sources, and of nonsmooth interfaces. We obtain energy estimates that involve only the L (p) norm of the heat sources for exponents p close to one. Such estimates are important for the investigation of models in which the heat equation is coupled to Maxwell's equations or to the Navier-Stokes equations (dissipative heating), with many applications such as crystal growth.

Using the critical point theory and the method of lower and upper solutions, we present a new approach to obtain the existence of solutions to a p-Laplacian impulsive problem. As applications, we get unbounded sequences of solutions and sequences of arbitrarily small positive solutions of the p-Laplacian impulsive problem.

We consider a mathematical model of nutrient-autotroph-herbivore interaction with nutrient recycling from both autotroph and herbivore. Local and global stability criteria of the model are studied in terms of system parameters. Next we incorporate the time required for recycling of nutrient from herbivore as a constant discrete time delay. The resulting DDE model is analyzed regarding stability and bifurcation aspects. Finally, we assume the recycling delay in the oscillatory form to model the daily variation in nutrient recycling and deduce the stability criteria of the variable delay model. A comparison of the variable delay model with the constant delay one is performed to unearth the biological relevance of oscillating delay in some real world ecological situations. Numerical simulations are done in support of analytical results.

NURBS (Non-Uniform Rational B-Splines) belong to special approximation curves and surfaces which are described by control points with weights and B-spline basis functions. They are often used in modern areas of computer graphics as free-form modelling, modelling of processes. In literature, NURBS surfaces are often called tensor product surfaces. In this article we try to explain the relationship between the classic algebraic point of view and the practical geometrical application on NURBS.

The purpose of this paper is to study the existence of periodic solutions for the non-autonomous second order Hamiltonian system {(u) over double dot(t) = del F(t, u(t)), a.e.t is an element of [0, T], u(0) - u(T) = (u) over dot(0) - (u) over dot(T) = 0.

We investigate diverse separation properties of two convex polyhedral sets for the case when there are parameters in one row of the constraint matrix. In particular, we deal with the existence, description and stability properties of the separating hyperplanes of such convex polyhedral sets. We present several examples carried out on PC. We are also interested in supporting separation (separating hyperplanes support both the convex polyhedral sets at given faces) and permanent separation (a hyperplane separates the convex polyhedral sets for all feasible parameters). Finally, we show how the developed theory is applicable in multiobjective linear programming.

The present paper studies the following constrained vector optimization problem: $$\mathop {\min }\limits_C f(x),g(x) \in - K,h(x) = 0$$ , where f: ℝ n → ℝ m , g: ℝ n → ℝ p are locally Lipschitz functions, h: ℝ n → ℝ q is C 1 function, and C ⊂ ℝ m and K ⊂ ℝ p are closed convex cones. Two types of solutions are important for the consideration, namely w-minimizers (weakly efficient points) and i-minimizers (isolated minimizers of order 1). In terms of the Dini directional derivative first-order necessary conditions for a point x 0 to be a w-minimizer and first-order sufficient conditions for x 0 to be an i-minimizer are obtained. Their effectiveness is illustrated on an example. A comparison with some known results is done.

We consider a quasilinear parabolic problem with time dependent coefficients oscillating rapidly in the space variable. The existence and uniqueness results are proved by using Rothe’s method combined with the technique of two-scale convergence.Moreover, we derive a concrete homogenization algorithm for giving a unique and computable approximation of the solution.

Stochastic homogenization (with multiple fine scales) is studied for a class of nonlinear monotone eigenvalue problems. More specifically, we are interested in the asymptotic behaviour of a sequence of realizations of the form -div(a(T-1(x/epsilon(1))omega(1), T-2(x/epsilon(2))omega(2), del u(epsilon)(omega))) = lambda C-omega(epsilon)(u(epsilon)(omega)). It is shown, under certain structure assumptions on the random map a(omega(1), omega(2), xi), that the sequence {lambda(omega,k)(epsilon), u(epsilon)(omega,k)} of kth eigenpairs converges to the kth eigenpair {lambda(k), u(k)} of the homogenized eigenvalue problem -div(b(del u)) = lambda(C) over bar (u). For the case of p-Laplacian type maps we characterize b explicitly.

A non-linear semi-coercive beam problem is solved in this article. Suitable numerical methods are presented and their uniform convergence properties with respect to the finite element discretization parameter are proved here. The methods are based on the minimization of the total energy functional, where the descent directions of the functional are searched by solving the linear problems with a beam on bilateral elastic "springs". The influence of external loads on the convergence properties is also investigated. The effectiveness of the algorithms is illustrated on numerical examples.

We consider a quasistatic contact problem for an electro-viscoelastic body. The contact is frictional and bilateral with a moving rigid foundation which results in the wear of the contacting surface. The damage of the material caused by elastic deformation is taken into account, its evolution is described by an inclusion of parabolic type. We present a weak formulation for the model and establish existence and uniqueness results. The proofs are based on classical results for elliptic variational inequalities, parabolic inequalities and fixed point arguments.