## Evgeny Lakshtanov

## On some constructive and explicit solutions in inverse scattering theory

First I will discuss some constructive methods of potential/obstacle reconstruction and will give a short review of Interior Transmission Eigenvalues. These eigenvalues play the same role in transmission scattering problems as Laplacian spectrum in the scattering by an obstacle. Finally I will present some our recent explicit formula for potential reconstruction.

Results mentioned above were obtained in collaboration with R.Novikov, B.Vainber.

Results mentioned above were obtained in collaboration with R.Novikov, B.Vainber.

Rui Borges Lopes

## Location-arc routing problem: Formulation and heuristic approaches

Location-routing is a branch of locational analysis that takes into account distribution aspects. Within these problems it is easy to consider scenarios where the demand is located on the edges of a network, referred in the literature as location-arc routing problems (LARP). Examples of such scenarios include locating facilities for postal delivery, garbage collection, road maintenance, winter gritting and street sweeping.

This talk will address the LARP, where a formulation and some heuristic approaches were recently put forward. Regarding the heuristic approaches new constructive and improvement methods are presented and used within different metaheuristic frameworks. New test instances are also proposed and used to compare the heuristic methods.

This talk will address the LARP, where a formulation and some heuristic approaches were recently put forward. Regarding the heuristic approaches new constructive and improvement methods are presented and used within different metaheuristic frameworks. New test instances are also proposed and used to compare the heuristic methods.

Rute Lemos

## Some results concerning the C-determinantal range

If A,C are square complex matrices, the C-determinantal range of
A is a subset of the complex plane intimately connected, for
normal matrices, with
Marcus-Oliveira Conjecture [3,4]
This set
can be considered as a variation of
the C-numerical range of A and
a certain parallelism exists between some of their properties. We survey some of them, present crucial differences and derive some consequences from the elliptical range theorem. We also consider the additive Frobenius endomorphisms of the determinantal range. Further, we revisit and improve two known results when C is Hermitian.
The first one concerns a condition stated in [1] that in general is not true. A correct criteria for the C-numerical range to be real is presented. The other is concerning the case when the C-determinantal range is a line segment [2,Theorem 3.3].

This is a joint work with A. Guterman and G. Soares.

References

This is a joint work with A. Guterman and G. Soares.

References

- [1] C.-K. Li, C-numerical ranges and C-numerical radii,
- [2] C.-K. Li, Y.-T. Poon and N.-S. Sze, Ranks and determinants of the sum of matrices from unitary orbits,
- [3] M. Marcus, Derivations, Plücker relations and the numerical range,
- [4] G. N. de Oliveira, Normal matrices (research problem),

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