Relative Morsification Theory

In this paper we develope a Morsification Theory for holomorphic functions defining a singularity of finite codimension with respect to an ideal, which recovers most previously known Morsification results for non-isolated singulatities and generalize them to a much wider context. We also show that deforming functions of finite codimension with respect to an ideal within the same ideal respects the Milnor fibration. Furthermore we present some applications of the theory: we introduce new numerical invariants for non-isolated singularities, which explain various aspects of the deformation of functions within an ideal; we define generalizations of the bifurcation variety in the versal unfolding of isolated singularities; applications of the theory to the topological study of the Milnor fibration of non-isolated singularities are presented. Using intersection theory in a generalized jet-space we show how to interprete the newly defined invariants as certain intersection multiplicities; finally, we characterize which invariants can be interpreted as intersection multiplicities in the above mentioned generalized jet space.Comment: 56 pages, some typos correcte

The L\^e numbers of the square of a function and their applications

L\^e numbers were introduced by Massey with the purpose of numerically controlling the topological properties of families of non-isolated hypersurface singularities and describing the topology associated with a function with non-isolated singularities. They are a generalization of the Milnor number for isolated hypersurface singularities. In this note the authors investigate the composite of an arbitrary square-free f and $z^2$. They get a formula for the L\^e numbers of the composite, and consider two applications of these numbers. The first application is concerned with the extent to which the L\^e numbers are invariant in a family of functions which satisfy some equisingularity condition, the second is a quick proof of a new formula for the Euler obstruction of a hypersurface singularity. Several examples are computed using this formula including any X defined by a function which only has transverse D(q,p) singularities off the origin.Comment: 14 page

Nash problem for surface singularities is a topological problem

AbstractWe address Nash problem for surface singularities using wedges. We give a refinement of the characterisation in A. Reguera-López (2006) [32] of the image of the Nash map in terms of wedges. Our improvement consists in a characterisation of the bijectivity of the Nash mapping using wedges defined over the base field, which are convergent if the base field is C, and whose generic arc has transverse lifting to the exceptional divisor. This improves the results of M. Lejeune-Jalabert and A. Reguera (2008) [16] for the surface case. In the way to do this we find a reformulation of Nash problem in terms of branched covers of normal surface singularities. As a corollary of this reformulation we prove that the image of the Nash mapping is characterised by the combinatorics of a resolution of the singularity, or, what is the same, by the topology of the abstract link of the singularity in the complex analytic case. Using these results we prove several reductions of the Nash problem, the most notable being that, if Nash problem is true for singularities having rational homology sphere links, then it is true in general

Pío Wandosell: un empresario industrial, minero-fundidor, contratista, terrateniente, banquero, comerciante y político de la segunda mitad del Siglo XIX en la provincia de Murcia

El protagonista de este trabajo, Pío Wandosell Gil, quien nació en Alhama de Almería en 1847, fue uno de los más importantes empresarios industriales, relacionados con la minería y la metalurgia del plomo, contratista de obra pública, y político republicano de la segunda mitad del siglo XIX y principios del XX en la provincial de Murcia. El fue una de las figuras más relevantes de esa época, desde el punto de vista político, económico y social, pero por diversas razones su vida había permanecido casi desconocida hasta la fecha.The protagonist of this essay, Pío Wandosell Gil, who was born in Alhama de Almería in 1847, was one of the most important industrial businessman, related with the mining industry and the lead metallurgy, public building contractor, and republican politician of the second half of the nineteenth century and the beguining of the twenty century in the Murcia province. He was one of the most relevant figures at that time, from the economical, social and political point of view, but for many reasons his life had remained almost hidden to date.Arte y Humanidade

Un ejemplo de industrialización en Murcia en el Siglo XIX: Alejandro Delgado y cía, sociedad en comandita

La economía murciana de la segunda mitad del siglo XIX tuvo un gran desarrollo, impulsado por el crecimiento de la demanda internacional, y caracterizado por la modernización tecnológica de las sociedades mercantiles creadas para aprovechar oportunidades empresariales, y verdaderas protagonistas de ese desarrollo. Uno de los sectores con mayor crecimiento fue el agroalimentario y sus empresas auxiliares, sobre todo, las fábricas de embalajes de madera para la exportación, y aunque la mayoría de las grandes firmas conserveras fabricaban sus propios envases, hubo empresas independientes, como la sociedad comanditaria Alejandro Delgado y Cia., que tuvo un constante desarrollo desde su nacimiento en 1887.The regional economy of Murcia in the second half of the 19th century had a great development, driven by the growth of international demand, and characterized by the technological modernization of the companies created to take advantage of business opportunities, and real protagonists of this development. One of the fastest expanding economic sectors was the agribusiness, and its subsidiary companies, particularly factories of wooden packing for export. Although most of the biggest canning companies had their own manufacturing packing workshops, there were independent companies such as the society Alejandro Delgado y Cia., which had a constant development since its founding in 1887.Arte y Humanidade

Participación del cartagenero José Ramón de Leyva en los actos revolucionarios que se iniciaron en el Virreinato de la Nueva Granada el 20 de Julio de 1810

El protagonista de este trabajo, el cartagenero José de Leyva de Salas, fue un hombre de dos mundos: militar de infantería e intelectual, ilustrado y erudito, que después de servir veintiséis años en el ejército borbónico del siglo XVIII desembarcó en el Virreinato de la Nueva Granada como Secretario del Virrey. Sus diecinueve años ejerciendo el cargo de Secretario del Virrey lo llevaron a enraizar de una forma destacable con la sociedad de su tiempo en la colonia, y a participar de una forma destacada en los sucesos políticos de la primera República de Colombia entre 1810 y 1816The protagonist of this essay, Jose Ramón de Leyva, who was born in Cartagena, was a man who belonged to two different worlds, an infantryman and an intellectual, enlightened, an erudite person, who was appointed to the Viceroyalty of New Granada, as the Viceroy’s Secretary, after twenty six years serving in the 18th century Bourbon army. His nineteen years performing as the Viceroy’s Secretary led him to take root in a very remarkable way with his contemporary society at the Colony, and to occupy a choice position in the political happenings of the first Colombian Republic, between 1810 and 1816.Arte y Humanidade

La construcción del dique seco del arsenal de Cartagena a finales del XIX: una necesidad militar, económica y social

La construcción de un dique seco de carenas en la ciudad de Cartagena entre 1896 y 1904 se convirtió en un acontecimiento de capital importancia militar, económica y social para la provincia de Murcia en los últimos años del siglo XIX y primeros del XX. La obra fue adjudicada por concurso al empresario fundidor unionense Pio Wandosell Gil. No fue una construcción fácil porque hubo muchos conflictos con la autoridad militar del arsenal e imprevistos de todo tipo. Las obras se prolongaron durante siete años, tres más de los previstos, porque ocurrieron gran cantidad de incidencias que retrasaron la entrega definitiva.The construction of a military dry dock in Cartagena between 1896 and 1904 became a major military, economic and social event for the province of Murcia in the last 19 th and early 20 th century. The project was awarded through a tender process to Pío Wandosell Gil, a foundry entre preneur from La Unión. It was not an easy construction due to the many conflicts with the military authority in the arsenal and various unforesseen events. The construction lasted for seven years, three more longer than predicted, because of the aforementioned incidents and the subsequent delay in the final delivery.Arte y Humanidade