Pseudo Almost Periodic Mild Solution of Nonautonomous Impulsive Integro-Differential Equations

In this paper, we investigate the existence, uniqueness and stability of pseudo almost periodic mild solution to nonautonomous impulsive integro-differential equations in Banach space. The working tools are based on the fixed point theorems and Gronwall–Bellman inequality. To illustrate our main results, we study pseudo almost periodic solution of the heat equations with Dirichlet conditions.     

Experimental investigations of the controlled motion of a screwless underwater robot

In this paper we describe the results of experimental investigations of the motion of a screwless underwater robot controlled by rotating internal rotors. We present the results of comparison of the trajectories obtained with the results of numerical simulation using the model of an ideal fluid.     

Optimizing vehicle routes in a bakery company allowing flexibility in delivery dates

The work addressed in this paper is motivated from a real problem proposed to the authors by a bakery company in Northern Spain. The objective is to minimize the total distance travelled for the daily routes over the week. In order to reduce this total distance, some flexibility in the dates of delivery is introduced. A mixed-integer linear model for the problem is formulated. In addition, a two-phase method based in GRASP and path-relinking metaheuristic strategies is proposed. Computational experiments show that the method performs very well, obtaining high-quality solutions in short computational times. Moreover, when it is applied to real-data-based instances, the obtained solutions considerably reduce transportation costs over the planning horizon.     

Λ-buildings and base change functors

We prove an analog of the base change functor of Λ-trees in the setting of generalized affine buildings. The proof is mainly based on local and global combinatorics of the associated spherical buildings. As an application we obtain that the class of generalized affine buildings is closed under taking ultracones and asymptotic cones. Other applications involve a complex of groups decompositions and fixed point theorems for certain classes of generalized affine buildings.     

A Note on the Subcritical Two Dimensional Keller-Segel System

The existence of solution for the 2D-Keller-Segel system in the subcritical case, i.e. when the initial mass is less than 8π, is reproved. Instead of using the entropy in the free energy and free energy dissipation, which was used in the proofs (Blanchet et al. in SIAM J. Numer. Anal. 46:691–721, 2008; Electron. J. Differ. Equ. Conf. 44:32, 2006 (electronic)), the potential energy term is fully utilized by adapting Delort’s theory on 2D incompressible Euler equation (Delort in J. Am. Math. Soc. 4:553–386, 1991).     

Separable geodesic action slicing in stationary spacetimes

A simple observation about the action for geodesics in a stationary spacetime with separable geodesic equations leads to a natural class of slicings of that spacetime whose orthogonal geodesic trajectories represent the world lines of freely falling fiducial observers. The time coordinate function can then be taken to be the observer proper time, leading to a unit lapse function, although the time coordinate lines still follow Killing trajectories to retain the explicitly stationary nature of the coordinate grid. This explains some of the properties of the original Painlevé-Gullstrand coordinates on the Schwarzschild spacetime and their generalization to the Kerr-Newman family of spacetimes, reproducible also locally for the Gödel spacetime. For the static spherically symmetric case the slicing can be chosen to be intrinsically flat with spherically symmetric geodesic observers, leaving all the gravitational field information in the shift vector field.     

Discrete Laplace cycles of period four

We study discrete conjugate nets whose Laplace sequence is of period four. Corresponding points of opposite nets in this cyclic sequence have equal osculating planes in different net directions, that is, they correspond in an asymptotic transformation. We show that this implies that the connecting lines of corresponding points form a discrete W-congruence. We derive some properties of discrete Laplace cycles of period four and describe two explicit methods for their construction.