Best rank-k approximations for tensors: generalizing Eckart-Young

Joint work with Jan Draisma and Giorgio Ottaviani. Given a tensor f in a Euclidean tensor space, we are interested in the critical points of the distance function from f to the set of tensors of rank at most k, which we call the critical rank-at-most-k tensors for f. When f is a matrix, the critical rank-one matrices for f correspond to the singular pairs of f . The critical rank-one tensors for f lie in a linear subspace H_f, the critical space of f. Our main result is that, for any k, the critical rank-at-most-k tensors for a sufficiently general f also lie in the critical space H_f. This is the part of Eckart-Young Theorem that generalizes from matrices to tensors. Moreover, we show that when the tensor format satisfies the triangle inequalities, the critical space H_f is spanned by the complex critical rank-one tensors. Since f itself belongs to H_f, we deduce that also f itself is a linear combination of its critical rank-one tensors. For simplicity, we will focus on binary forms during the talk.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

Best rank-kk approximations for tensors: generalizing Eckart-Young

Given a tensor $f$ in a Euclidean tensor space, we are interested in the critical points of the distance function from $f$ to the set of tensors of rank at most $k$, which we call the critical rank-at-most-$k$ tensors for $f$. When $f$ is a matrix, the critical rank-one matrices for $f$ correspond to the singular pairs of $f$. The critical rank-one tensors for $f$ lie in a linear subspace $H_f$, the critical space of $f$. Our main result is that, for any $k$, the critical rank-at-most-$k$ tensors for a sufficiently general $f$ also lie in the critical space $H_f$. This is the part of Eckart-Young Theorem that generalizes from matrices to tensors. Moreover, we show that when the tensor format satisfies the triangle inequalities, the critical space $H_f$ is spanned by the complex critical rank-one tensors. Since $f$ itself belongs to $H_f$, we deduce that also $f$ itself is a linear combination of its critical rank-one tensors

Benzú Rockshelter and Sima de las Palomas de Teba. A preliminary and comparative research of the lithic technology

Presentamos un avance de estudio comparativo entre dos yacimientos con tecnología de Modo 3, para intentar dar un paso más en la comprensión del Estrecho de Gibraltar como posible región histórica que una el sur de la Península Ibérica con el norte del Magreb.We have the intention to offer a preliminary and comparative research between two lithic technology Mode 3 sites. With this paper, we will try to understand the Strait of Gibraltar as an historic area that connects the south of Iberian Peninsula with the north of Magreb

Modos de vida y de producción de las sociedades cazadoras-recolectoras pescadoras y mariscadoras en el Abrigo de Benzú hace 170.000 años.

Póster presentado en el Congreso internacional "Arqueoworld", organizado en Valencina de la Concepción (Sevilla)

Europeas y Españolas contra Napoleón. Un estudio comparado

L'article posa l'accent en la necessitat de no estar exclusivament circumscrits a les disposicions pel que fa al binomi Dona/Guerra, el que ens podria portar a atorgar un significat de singularitat erroni als casos analitzats, erigint-se, en el seu aïllament, en prototips genuïns de l'expressió patriòtica nacional. Per això aquest treball tracta de posar en connexió els estudis nacionals amb els d'altres països amb la intenció de posa de relleu les diferències o les concomitàncies existents entre les vivències de les europees, incloses les espanyoles, davant les guerres napoleòniques.El artículo hace hincapié en la necesidad de no permanecer exclusivamente circunscritos a los modelos nacionales en lo que se refiere al binomio Mujer/Guerra, lo que nos podría llevar a otorgar un significado de singularidad erróneo a los casos analizados, erigiéndolos, en su aislamiento, en prototipos genuinos de la expresión patriótica nacional. Por ello este trabajo trata de poner en conexión los estudios nacionales con los de otros paises con la intención de pone de relieve las diferencias o las concomitancias existentes entre las vivencias de las europeas, incluidas las españolas, ante las guerras napoleónicas.The article emphasizes the need not remain exclusively confined to national patterns in regard to the binomial Women/War, which could lead us to give a wrong meaning of uniqueness to the cases studied, created, in their isolation, Prototype genuine national patriotic expression. Hence this paper tries to national studies in connection with other countries with the intention of highlighting the differences or similarities between the experiences of Europe, including the Spanish ones, before the Napoleonic wars

Mujeres y ciudadanía. Del antiguo régimen a la revolución liberal

El artículo regoge la ponencia presentada por Gloria Espigado en las Jornadas Mujer y ciudadanía: del Antiguo Régimen a la revolución liberal. El artículo trata la relación de las españolas con el primer liberalismo, concretamente acerca de las Cortes de 1812.L'article recull la ponència presentada per Gloria Espigado a les jornades Mujer y ciudadanía: del Antiguo Régimen a la Revolución liberal. L'article tracta la relació de les espanyoles amb el primer liberalisme, concretament entorn les Corts de 1812.The article present Glory Espigado paper presented in debate: Woman and citizenship: of the Old State to the liberal revolution. The article treats the relation of the Spaniards with the first liberalism, concretely about the Courts of 1812

Celebrative Talk. Enrique's Homemade Algebraic Geometry.

During the talk, Enrique Arrondo's life and mathematical research were highlighted, intertwining his career with a touch of history and significant photos that captured important moments in Enrique's life. Enrique's fascination with mathematics was ignited at an early age, propelling him to become a prodigious thinker. Scholarships and fellowships paved the way for him in his academic pursuits, both locally and internationally. Enrique's contributions to the field of mathematics were remarkable. His interest in algebraic geometry led to numerous publications in prestigious journals, cementing his reputation as an exceptional mind in his field by continuing to advance the resolution of Hartshorne's conjecture. A summary was made of each of the topics he covered, including congruences, vector fibrations without intermediate cohomology, projections on Grassmannian varieties and his advances related with low codimension bundles. In addition, the new lines of research that were created from the results of Enrique's research were presented in the talk.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech