On the Set of Locally Convex Topologies Compatible with a Given Topology on a Vector Space: Cardinality Aspects

For a topological vector space (X, τ ), we consider the family LCT (X, τ ) of all locally convex topologies defined on X, which give rise to the same continuous linear functionals as the original topology τ . We prove that for an infinite-dimensional reflexive Banach space (X, τ ), the cardinality of LCT (X, τ ) is at least c.

Mackey topology on locally convex spaces and on locally quasi-convex groups. Similarities and historical remarks

A counterpart of the Mackey–Arens Theorem for the class of locally quasi-convex topological Abelian groups (LQC-groups) was initiated in Chasco et al. (Stud Math 132(3):257–284, 1999). Several authors have been interested in the problems posed there and have done clarifying contributions, although the main question of that source remains open. Some differences between the Mackey Theory for locally convex spaces and for locally quasi-convex groups, stem from the following fact: The supremum of all compatible locally quasi-convex topologies for a topological abelian group G may not coincide with the topology of uniform convergence on the weak quasi-convex compact subsets of the dual groupG∧. Thus, a substantial part of the classical Mackey–Arens Theorem cannot be generalized to LQC-groups. Furthermore, the mentioned fact gives rise to a grading in the property of “being a Mackey group”, as defined and thoroughly studied in Díaz Nieto and Martín-Peinador (Proceedings in Mathematics and Statistics 80:119–144, 2014). At present it is not known—and this is the main open question—if the supremum of all the compatible locally quasi-convex topologies on a topological group is in fact a compatible topology. In the present paper we do a sort of historical review on the Mackey Theory, and we compare it in the two settings of locally convex spaces and of locally quasi-convex groups. We point out some general questions which are still open, under the name of Problems.

Bathymetric Terrain Model of the Puerto Rico Trench and the Northeastern Caribbean Region for Marine Geological Investigations

Multibeam bathymetric data collected in the Puerto Rico Trench and northeastern Caribbean region are compiled into a seamless bathymetric terrain model for broad-scale geological investigations of the trench system. These data, collected during eight separate surveys between 2002 and 2013 and covering almost 180,000 square kilometers, are published here in large-format map sheet and digital spatial data. This report describes the common multibeam data collection and processing methods used to produce the bathymetric terrain model and corresponding data-source polygon. Details documenting the complete provenance of the data are provided in the metadata in the Data Catalog section.

Optimized square Fresnel zone plates for microoptics applications

Polygonal Fresnel zone plates with a low number of sides have deserved attention in micro and nanoptics, because they can be straightforwardly integrated in photonic devices, and, at the same time, they represent a balance between the high-focusing performance of a circular zone plate and the easiness of fabrication at micro and nano-scales of polygons. Among them, the most representative family are Square Fresnel Zone Plates (SFZP). In this work, we propose two different customized designs of SFZP for optical wavelengths. Both designs are based on the optimization of a SFZP to perform as close as possible as a usual Fresnel Zone Plate. In the first case, the criterion followed to compute it is the minimization of the difference between the area covered by the angular sector of the zone of the corresponding circular plate and the one covered by the polygon traced on the former. Such a requirement leads to a customized polygon-like Fresnel zone. The simplest one is a square zone with a pattern of phases repeating each five zones. On the other hand, an alternative SFZP can be designed guided by the same criterion but with a new restriction. In this case, the distance between the borders of different zones remains unaltered. A comparison between the two lenses is carried out. The irradiance at focus is computed for both and suitable merit figures are defined to account for the difference between them.

Optimal phase distributions for polygonal Fresnel lenses

Polygonal Fresnel zone plates can be configured in a variety of forms depending on the number of sides of the polygon and the number of phase steps used. This contribution deals with some specific polygonal designs that tessellate the plane: triangles, squares, and hexagons. The phase distribution is chosen as a continuous one to form a polygonal kinoform. The selected designs have been simulated and its behaviour compared. Although their performance is worse than the circular Fresnel plate, they may present some other advantages as the tessellation capability, and the possibility to fabricate them as extruded profiles.