Optimal configurations of lines and a statistical application

Motivated by the construction of confidence intervals in statistics, we study optimal configurations of $2^d-1$ lines in real projective space $RP^{d-1}$. For small $d$, we determine line sets that numerically minimize a wide variety of potential functions among all configurations of $2^d-1$ lines through the origin. Numerical experiments verify that our findings enable to assess efficiently the tightness of a bound arising from the statistical literature.Comment: 13 page

Reduced-order modeling of transonic flows around an airfoil submitted to small deformations

A reduced-order model (ROM) is developed for the prediction of unsteady transonic flows past an airfoil submitted to small deformations, at moderate Reynolds number. Considering a suitable state formulation as well as a consistent inner product, the Galerkin projection of the compressible flow Navier–Stokes equations, the high-fidelity (HF) model, onto a low-dimensional basis determined by Proper Orthogonal Decomposition (POD), leads to a polynomial quadratic ODE system relevant to the prediction of main flow features. A fictitious domain deformation technique is yielded by the Hadamard formulation of HF model and validated at HF level. This approach captures airfoil profile deformation by a modification of the boundary conditions whereas the spatial domain remains unchanged. A mixed POD gathering information from snapshot series associated with several airfoil profiles can be defined. The temporal coefficients in POD expansion are shape-dependent while spatial POD modes are not. In the ROM, airfoil deformation is introduced by a steady forcing term. ROM reliability towards airfoil deformation is demonstrated for the prediction of HF-resolved as well as unknown intermediate configurations

Optimal Alignments for Designing Urban Transport Systems: Application to Seville

The achievement of some of the Sustainable Development Goals (SDGs) from the recent 2030 Agenda for Sustainable Development has drawn the attention of many countries towards urban transport networks. Mathematical modeling constitutes an analytical tool for the formal description of a transportation system whereby it facilitates the introduction of variables and the definition of objectives to be optimized. One of the stages of the methodology followed in the design of urban transit systems starts with the determination of corridors to optimize the population covered by the system whilst taking into account the mobility patterns of potential users and the time saved when the public network is used instead of private means of transport. Since the capture of users occurs at stations, it seems reasonable to consider an extensive and homogeneous set of candidate sites evaluated according to the parameters considered (such as pedestrian population captured and destination preferences) and to select subsets of stations so that alignments can take place. The application of optimization procedures that decide the sequence of nodes composing the alignment can produce zigzagging corridors, which are less appropriate for the design of a single line. The main aim of this work is to include a new criterion to avoid the zigzag effect when the alignment is about to be determined. For this purpose, a curvature concept for polygonal lines is introduced, and its performance is analyzed when criteria of maximizing coverage and minimizing curvature are combined in the same design algorithm. The results show the application of the mathematical model presented for a real case in the city of Seville in Spain.Ministerio de Economía y Competitividad MTM2015-67706-