In this talk, we present recent mathematical results in Maxwell variational inequalities with technological applications in high-temperature superconductivity and EM-shielding. The established well-posedness opens a way to analyze the corresponding numerical approximations, optimal design and inverse problems, which are mainly complicated by the hyperbolic Maxwell structure and the non-smooth character. Results in these directions are presented.

We consider a sequence of linear-quadratic optimization problems defined in an abstract functional framework. Each problem is accompanied by the constraint of reaching a given target with some precision. We show that problems are well posed and we characterise their solutions. The main result provides the conditions under which these solutions converge to the minimizer of the limit problem. The theory can be applied to a wide range of problems: elliptic, parabolic, control ones etc. In the talk, the following examples will be presented: 1. optimal controllability of the heat equation 2. optimal solvability of the Poisson equation.

Given a set of departments and the pairwise transport weights between them, the Single-Row Facility Layout Problem (SRFLP) asks for an arrangement of the departments on one side of a path such that the weighted sum of the distances between the departments is minimized. The best integer linear programming approach for the SRFLP is based on the betweenness model (Amaral, 2009).
In this talk we present new classes of inequalities for the associated betweenness polytope and show their validity using combinatorial arguments. Using a cutting plane approach allows us to solve more and larger instances of the SRFLP to optimality than the lower bounding approach of Amaral. In the second part of the talk we consider the Multi-Bay Facility Layout Problem and Facility Layouts with Pier Type Patterns which are special Combined Cell Layout Problems (CCLP). Here the path structure is more complicated. We solve these problems exactly by enumerating over all assignments of the departments to the cells and solving several CCLP with fixed-cell assignment using the betweenness model in each of the subproblems. We show how to reduce the number of distinguishable cell assignments significantly by merging two cells of type single-row.

Copulas and doubly stochastic measures are mostly well known in the statistical community due to Sklar's famous theorem saying that copulas are the link between multivariate distribution functions and their marginals. Copulas, however, exhibit various interesting and surprising mathematical aspects going far beyond statistics. The talk will discuss some of those aspects, in particular some interrelations with functional analysis, geometry, number theory, and fractals.

Positivity-preserving schemes are of high importance and gained more and more attention in recent years. However, they often result in nonlinear iteration schemes even when applied to a linear test problem y′ = Ay. Hence, a stability analysis becomes more complex. Moreover, the scheme often has to preserve additionally other properties of the analytic solution such as steady states and linear invariants. As a result, steady states of the test problem become non-hyperbolic fixed points of the method. In general, the stability analysis thereby turns into a case by case study.
In this talk, we present sufficient conditions for stability as well as local convergence, and hence, overcome this case by case study for a large class of steady state preserving schemes to which reasonable positivity-preserving schemes belong.
To illustrate the theoretical results, we consider so-called modified Patankar-type schemes that are positive and conservative numerical methods when applied to a positive and conservative production-destruction system. Applying the main theorem we are able to define stability functions which are the basis for the stability analysis. Finally, numerical experiments are presented to confirm the theoretical results.