Modeling several competitive leaders and followers acting in an electricity market leads to coupled systems of mathematical programs with equilibrium constraints, called equilibrium problems with equilibrium constraints (EPECs). We consider a simplified model for competition in electricity markets under uncertainty of demand in an electricity network as a (stochastic) multi-leader-follower game. First order necessary conditions are developed for the corresponding stochastic EPEC based on a result of Outrata. For applying the general result an explicit representation of the co-derivative of the normal cone mapping to a polyhedron is derived. Then the co-derivative formula is used for verifying constraint qualifications and for identifying M-stationary solutions of the stochastic EPEC if the demand is represented by a finite number of scenarios.

The paper presents the theory of the discontinuous Galerkin finite element method for the space-time discretization of a linear nonstationary convection-diffusion-reaction initial-boundary value problem. The discontinuous Galerkin method is applied separately in space and time using, in general, different nonconforming space grids on different time levels and different polynomial degrees p and q in space and time discretization, respectively. In the space discretization the nonsymmetric interior and boundary penalty approximation of diffusion terms is used. The paper is concerned with the proof of error estimates in “L 2(L 2)”-and “ $$\sqrt \varepsilon L^2 (H^1 )$$ ”-norms, where ɛ ⩾ 0 is the diffusion coefficient. Using special interpolation theorems for the space as well as time discretization, we find that under some assumptions on the shape regularity of the meshes and a certain regularity of the exact solution, the errors are of order O(h p + τ q ). The estimates hold true even in the hyperbolic case when ɛ = 0.

We consider a tree-shaped network of vibrating elastic strings, with feedback acting on the root of the tree. Using the d’Alembert representation formula, we show that the input-output map is bounded, i.e. this system is a well-posed system in the sense of G. Weiss (Trans. Am. Math. Soc. 342 (1994), 827–854). As a consequence we prove that the strings networks are not exponentially stable in the energy space. Moreover, we give explicit polynomial decay estimates valid for regular initial data.

In this paper we study set-valued optimization problems with equilibrium constraints (SOPECs) described by parametric generalized equations in the form 0 ∈ G(x) + Q(x), where both G and Q are set-valued mappings between infinite-dimensional spaces. Such models particularly arise from certain optimization-related problems governed by set-valued variational inequalities and first-order optimality conditions in nondifferentiable programming. We establish general results on the existence of optimal solutions under appropriate assumptions of the Palais-Smale type and then derive necessary conditions for optimality in the models under consideration by using advanced tools of variational analysis and generalized differentiation.

We consider a special class of optimization problems that we call Mathematical Programs with Vanishing Constraints, MPVC for short, which serves as a unified framework for several applications in structural and topology optimization. Since an MPVC most often violates stronger standard constraint qualification, first-order necessary optimality conditions, weaker than the standard KKT-conditions, were recently investigated in depth. This paper enlarges the set of optimality criteria by stating first-order sufficient and second-order necessary and sufficient optimality conditions for MPVCs.

The paper is devoted to verification of accuracy of approximate solutions obtained in computer simulations. This problem is strongly related to a posteriori error estimates, giving computable bounds for computational errors and detecting zones in the solution domain where such errors are too large and certain mesh refinements should be performed. A mathematical model consisting of a linear elliptic (reaction-diffusion) equation with a mixed Dirichlet/Neumann/Robin boundary condition is considered in this work. On the base of this model, we present simple technologies for straightforward constructing computable upper and lower bounds for the error, which is understood as the difference between the exact solution of the model and its approximation measured in the corresponding energy norm. The estimates obtained are completely independent of the numerical technique used to obtain approximate solutions and are “flexible” in the sense that they can be, in principle, made as close to the true error as the resources of the used computer allow.

The main focus in this paper is on homogenization of the parabolic problem ∂ t uɛ − ∇ · (a(x/ɛ,t/ɛ,t/ɛ r )∇u ɛ ) = f. Under certain assumptions on a, there exists a G-limit b, which we characterize by means of multiscale techniques for r > 0, r ≠ 1. Also, an interpretation of asymptotic expansions in the context of two-scale convergence is made.

The solvability of the contact problem is proved provided the plate is simply supported. The singular memory material is assumed. This makes it possible to get a priori estimates important for the strong convergence of gradients of velocities of solutions to the penalized problem.

The paper studies a new class of robust regression estimators based on the two-step least weighted squares (2S-LWS) estimator which employs data-adaptive weights determined from the empirical distribution or quantile functions of regression residuals obtained from an initial robust fit. Just like many existing two-step robust methods, the proposed 2S-LWS estimator preserves robust properties of the initial robust estimate. However, contrary to the existing methods, the first-order asymptotic behavior of 2S-LWS is fully independent of the initial estimate under mild conditions. We propose data-adaptive weighting schemes that perform well both in the cross-section and time-series data and prove the asymptotic normality and efficiency of the resulting procedure. A simulation study documents these theoretical properties in finite samples.

An introduction to the worst scenario method is given. We start with an example and a general abstract scheme. An analysis of the method both on the continuous and approximate levels is discussed. We show a possible incorporation of the method into the fuzzy set theory. Finally, we present a survey of applications published during the last decade.

We prove some optimal regularity results for minimizers of the integral functional ∫ f(x, u, Du) dx belonging to the class K ≔ {u ∈ W 1,p (Ω): u ⩾ ψ, where ψ is a fixed function, under standard growth conditions of p-type, i.e. $$L^{ - 1} \left| z \right|^p \leqslant f(x,s,z) \leqslant L(1 + \left| z \right|^p ).$$ .

Over the past fifty years, finite element methods for the approximation of solutions of partial differential equations (PDEs) have become a powerful and reliable tool. Theoretically, these methods are not restricted to PDEs formulated on physical domains up to dimension three. Although at present there does not seem to be a very high practical demand for finite element methods that use higher dimensional simplicial partitions, there are some advantages in studying the methods independent of the dimension. For instance, it provides additional insights into the structure and essence of proofs of results in one, two and three dimensions. In this survey paper we review some recent progress in this direction.

In this paper Rothe's classical method is extended so that it can be used to solve some linear parabolic boundary value problems in non-cylindrical domains. The corresponding existence and uniqueness theorems are proved and some further results and generalizations are discussed and applied.

In this paper we consider proper cycles of indefinite integral quadratic forms F = (a, b, c) with discriminant Δ. We prove that the proper cycles of F can be obtained using their consecutive right neighbors R i(F) for i ⩾ 0. We also derive explicit relations in the cycle and proper cycle of F when the length l of the cycle of F is odd, using the transformations τ(F) = (−a, b, −c) and ϰ(F) = (−c, b, −a).

In this work we derive a pair of nonlinear eigenvalue problems corresponding to the one-band effective Hamiltonian accounting for the spin-orbit interaction governing the electronic states of a quantum dot. We show that the pair of nonlinear problems allows for the minmax characterization of its eigenvalues under certain conditions which are satisfied for our example of a cylindrical quantum dot and the common InAs/GaAs heterojunction. Exploiting the minmax property we devise an efficient iterative projection method simultaneously handling the pair of nonlinear problems and thereby saving about 25% of the computation time as compared to the Nonlinear Arnoldi method applied to each of the problems separately.

This paper studies the existence of solutions to the singular boundary value problem $$\left\{ \begin{gathered} - u'' = g(t,u) + (h,u),t \in (0,1), \hfill \\ u(0) = 0 = u(1), \hfill \\ \end{gathered} \right.$$ , where g: (0, 1) × (0, ∞) → ℝ and h: (0, 1) × [0, ∞) → [0, ∞) are continuous. So our nonlinearity may be singular at t = 0, 1 and u = 0 and, moreover, may change sign. The approach is based on an approximation method together with the theory of upper and lower solutions.

We study a 2D model of the orientation distribution of fibres in a paper machine headbox. The goal is to control the orientation of fibres at the outlet by shape variations. The mathematical formulation leads to an optimization problem with control in coefficients of a linear convection-diffusion equation as the state problem. Existence of solutions both to the state and the optimization problem is analyzed and sensitivity analysis is performed. Further, discretization is done and a numerical example is shown.

This paper is devoted to the problem of existence of a solution for a non-resonant, non-linear generalized multi-point boundary value problem on the interval [0, 1]. The existence of a solution is obtained using topological degree and some a priori estimates for functions satisfying the boundary conditions specified in the problem.

Simple rank statistics are used to test that two samples come from the same distribution. Šidák’s E-test (Apl. Mat. 22 (1977), 166–175) is based on the number of observations from one sample that exceed all observations from the other sample. A similar test statistic is defined in Ann. Inst. Stat. Math. 52 (1970), 255–266. We study asymptotic behavior of the moments of both statistics.

In one if his paper Luo transformed the problem of sum-fuzzy rationality into artificial learning procedure and gave an algorithm which used the learning rule of perception. This paper extends the Luo method for finding a sum-fuzzy implementation of a choice function and offers an algorithm based on the artificial learning procedure with fixed fraction. We also present a concrete example which uses this algorithm.