An elementary approach to the homological properties of constant-rank operators

We give a simple and constructive extension of Rai\c{t}\u{a}'s result that every constant-rank operator possesses an exact potential and an exact annihilator. Our construction is completely self-contained and provides an improvement on the order of the operators constructed by Rai\c{t}\u{a}, as well as the order of the explicit annihilators for elliptic operators due to Van Schaftingen. We also give an abstract construction of an optimal annihilator for constant-rank operators, which extends the optimal construction of Van Schaftingen for elliptic operators. Lastly, we establish a generalized Poincar\'e lemma for constant-rank operators and homogeneous spaces on $\mathbb{R}^d$, and we prove that the existence of potentials on spaces of periodic maps requires a strictly weaker condition than the constant-rank property.Comment: v3 22 pages, we added an an observation about the homology associated with operators acting on periodic maps, comments still welcome

The Irregular Ribbed Vault of the Sacristy of the Cathedral of Saint-Jean Baptiste in Perpignan

The vault of the sacristy of the Cathedral of Saint-Jean Baptiste in Perpignan (France), constructed by the Majorcan architect Guillem Sagrera between 1433 and 1447, is an outstanding, yet strikingly unknown, example of rib vaulting. This paper analyzes the overall construction of the form of the vault, characterized by its highly irregular perimeter, with particular attention to an isolated decorated corbel which solves the problem of the wall support of a group of six ribs and is in stark contrast with the rest of the supports, which are completely unadorned. Given the extreme rigour of Sagrera in all his works (and this one in particular), this apparent “capriccio” must be justified not only by decorative or formal requirements, but also by the constructive logic of Gothic vaulting syste

Construcción, dibujo y geometría en la transición entre Gótico y Renacimiento

El artículo examina algunos problemas centrales del dibujo de arquitectura en la época bajomedieval y renacentista, teniendo en cuenta tanto los estudios clásicos en este campo como las aportaciones más recientes; intenta ofrecer una visión panorámica pero integrada de varias de estas cuestiones esenciales. En particular, se tratan las conexiones entre la geometría culta y las prácticas canteriles de trazado, la relación entre trazas en papel y monteas sobre soportes rígidos, la aparición de la proyección ortogonal, las diferencias entre las elevaciones medievales y nuestros alzados, los nuevos resultados geométricos que debe suministrar la traza de origen medieval para facilitar la ejecución de los diseños renacentistas, el empleo de algunos procedimientos de trazado de origen renaciente para materializar bóvedas de crucería, invirtiendo el orden cronológico que cabría esperar, y la peculiar síntesis entre ambos mundos que ofrecen las bóvedas por cruceros, es decir, bóvedas de casetones de aspecto clásico materializadas por nervios y plementería

Construcción, dibujo y geometría en la transición entre Gótico y Renacimiento

[ES] El artículo examina algunos problemas centrales del dibujo de arquitectura en la época bajomedieval y renacentista, teniendo en cuenta tanto los estudios clásicos en este campo como las aportaciones más recientes; intenta ofrecer una visión panorámica pero integrada de varias de estas cuestiones esenciales. En particular, se tratan las conexiones entre la geometría culta y las prácticas canteriles de trazado, la relación entre trazas en papel y monteas sobre soportes rígidos, la aparición de la proyección ortogonal, las diferencias entre las elevaciones medievales y nuestros alzados, los nuevos resultados geométricos que debe suministrar la traza de origen medieval para facilitar la ejecución de los diseños renacentistas, el empleo de algunos procedimientos de trazado de origen renaciente para materializar bóvedas de crucería, invirtiendo el orden cronológico que cabría esperar, y la peculiar síntesis entre ambos mundos que ofrecen las bóvedas por cruceros, es decir, bóvedas de casetones de aspecto clásico materializadas por nervios y plementería. [EN] This paper deals with some central problems in Late Mediaeval and Renaissance architectural drawing, taking into account some classical studies in this field, as well as many recent contributions. It endeavors to present a panoramic, yet integrated vision of a number of these crucial issues. In particular, it deals with the connections between learned geometry and stonecutters’ tracing methods, the relationship between drawings in paper and tracings on walls and floors, the emergence of orthogonal projection, the different approaches of Late Mediaeval elevations and their modern counterparts, the new geometrical results offered by a tracing system with Mediaeval roots in order to face the requirements of Renaissance construction, the use of Renaissance tracing methods in order to innovate and improve rib vault construction, reversing the chronological succession that could be expected, and the idiosyncratic synthesis offered by ‘por cruceros’ vaults, that is, classical coffered vaults solved by a network of ribs supporting severie

Multiscale assessment of woody species recruitment in Mediterranean shrublands: facilitation and beyond

Forest recovery in Mediterranean environments is influenced by factors such as aridity, herbivory and facilitation by shrubs, as well as by seed limitation in the case of highly fragmented forests. How these various factors interact can determine the direction of secondary succession, yet these interactions are poorly understood. We assessed the relative importance of several factors in forest species recruitment in Retama sphaerocarpa (L.) Boiss (Retama) shrublands at different spatial scales.Ministerio de Economía y CompetitividadComunidad de Madri

Functionalism and Caprice in Stonecutting. The Case of the Nativity Chapel in Burgos Cathedral

Starting from the inaugural text of Philibert de L'Orme, stereotomic treatises and manuscripts are subject to the opposing forces of reason and fancy. The Nativity Chapel in Burgos Cathedral provides an outstanding case study on this subject. It was built in 1571-1582 by Martín de Bérriz and Martín de la Haya, using an oval vault resting on trumpet squinches to span a rectangular bay. Bed joints and rib axes are not planar curves, as usual in oval vaults. This warping is not capricious; we shall argue that it is the outcome of a systematic tracing method. As a result of this process, the slope of the bed joints increases slightly in the first courses, but stays fairly constant after the third course; this solution prevents the upper courses from slipping. Thus, in the Nativity Chapel of Burgos Cathedral, the constraints of masonry construction fostered a singular solution verging on capriccio. It is also worthwhile to remark that the warping of the joints is not easily appreciable to the eye and that the tracing process does not seem to start from a previous conception of the resulting form. All this suggests that we should be quite careful when talking about the whimsical character of Late Gothic and Early Renaissance; in some occasions, apparent caprice is the offspring of practical thinking