## Milica Andelic

## Some remarks on the second largest eigenvalue of a graph

For a given real number k we give several sufficient and/or necessary conditions for a graph to have the second largest eigenvalue less than or equal to k. Some applications are included.

Rui Borges Lopes

## The location-routing problem: variants, methods, and promising applications

Location-routing is a branch of locational analysis that takes into account distribution aspects. This talk will focus on these problems, where its general structure will be briefly introduced, and main variants in the literature presented. As interest in the area has been growing, current methods will be discussed and an overview of best exact and heuristic methods will be provided (with emphasis on the capacitated location-routing problem). Finally, several promising applications are discussed, and a decision support tool for supporting the aforementioned problems will be shown.

Alexander Plakhov

## Besicovitch's magic method and problems of minimal resistance

A flow of point particles falls vertically downward on a square table of size 1m x 1m. It is allowed to put inclined mirrors on the table; the height of each mirror is 1cm at most. The inclination of the reflecting part of each mirror is 45°. (

incident flow. We solve this problem and some related problems of Newtonian aerodynamics. The idea of the main part of construction is inspired by Besicovitch’s solution of the Kakeya problem: what is the minimum area of a plane region in which a unit line segment can be rotated continuously through 360°.

*A mirror is a**domain in R³ with lateral surface being a cylindrical one with vertical generating line,**bounded below by the plane of the table and above by a plane with the slope 45°*.) When a particle hits a mirror, it is reflected elastically and then moves freely in a horizontal direction. The mirrors do not impede the motion of particles; that is, the reflected particles do not meet mirrors anymore. It is required to find a configuration of mirrors that ensures redirecting (from vertical to horizontal direction) at least 99% of theincident flow. We solve this problem and some related problems of Newtonian aerodynamics. The idea of the main part of construction is inspired by Besicovitch’s solution of the Kakeya problem: what is the minimum area of a plane region in which a unit line segment can be rotated continuously through 360°.