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

Inside the Mathematics class. Sociological perspectives on participation, inclusion, and enhancement

Resenya de: Gellert, U., Knipping, C., & Straehler-Pohl, H. (2018). Inside the Mathematics Class. Sociological Perspectives on Participation, Inclusion, and Enhancement. Cham: Springer

El despido plural en las pequeñas empresas

[Resumen] Este trabajo trata de los umbrales que trazan las fronteras entre las pequeñas y las medianas empresas, según el Derecho español y europeo sobre despidos colectivos.[Abstract] This work deals with the thresholds tracing the borders between small and middle-sized enterprises, under the Spanish and European Law about collective dismissals

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

Caracterización cohomológica de fibrados universales de la Grassmaniana de rectas

Tesis inédita de la Universidad Complutense de Madrid, Facultad de Ciencias Matemáticas, Departamento de Álgebra, leída el 22-07-2015La tesis Caracterización Cohomológica de los Fibrados Universales de la Grasmanniana de Rectas se dedica a dar una caracterización mediante anulaciones de cohomología de los productos simétricos de uno de los fibrados de la Grassmanniana de rectas, en particular, del fibrado cociente que tiene rango 2. Probablemente uno de los resultados más importantes sobre fibrados vectoriales sobre un espacio proyectivo es el criterio de Horrocks que dice que un fibrado vectorial sobre el espacio proyectivo de dimension n escinde si y sólo si no tiene cohomología intermedia. Este teorema se ha generalizado a otras variedades projectivas. Por ejemplo G. Ottaviani caracterizó dichos fibrados vectoriales sobre las Grassmannianas de k planos y sobre cuádricas lisas. Una mejora del criterio de Ottaviani, que será el punto de inicio de esta tesis, lo dieron E. Arrondo y F. Malaspina para el caso de las Grassmannianas de rectas. La idea que hay detrás de la prueba de este resultado es bastante simple. Si un fibrado vectorial tiene como sumando directo un fibrado de linea, ésto es equivalente a tener las aplicaciones del fibrado lineal al fibrado que queremos caracterizar y viceversa, cuya composición es no nula. La parte difícil es relacionar la composición de dichas aplicaciones con un par perfecto dado por la dualidad de Serre. Uno podría estar interesado en caracterizar cohomológicamente otros fibrados vectoriales, y no sólo los fibrados lineales. Un resultado del tipo que estamos buscando está hecho por E. Arrondo y B. Graña en el que caracterizan la suma directa de twists de fibrados lineales y del fibrado cociente en la Grassmanniana de rectas en el espacio proyectivo de dimensión 4. Tomaron como punto de partida el criterio dado por G. Ottaviani y que da una caracterización de sumas directas de fibrados lineales. Ya que el fibrado cociente no escinde, tiene que haber alguna hipótesis de ese teorema que no la satisfaga. Hay exactamente una de ellas, y uno es capaz de probar que el resto de hipótesis nos sirven para caracterizar sumas directas de twists de fibrados lineales y de fibrados cocientes. Nuestro objetivo es dar una caracterización para sumas directas de twists de los productos simétricos del fibrado cociente en la Grassmanniana de rectas en el espacio proyectivo de dimensión n. Daremos dicha clasificación para productos simétricos de orden no mayor de n-2. Como ideas principales, usaremas las que ya hemos señalado. Empezaremos por el criterio de escisión dado por E. Arrondo y F. Malaspina. Contrairamente a lo que ocurre en la prueba de E. Arrondo y B. Graña, quitando la única hipótesis del criterio de escisión no será suficiente para caracterizar las sumas directas de twists de fibrados lineales y fibrados cocientes, así que necesitaremos añadir más hipótesis para dicha caracterización. La idea de encontrar un sumando directo del fibrado cociente estará relacionada con la dualidad de Serre. Con todas estas ideas en la cabeza, uno puede ir quitando paso a paso condiciones y añadiendo algunas más, hasta llegar a la clasificación que queríamos. Por otro lado, las técnicas de categorías derivadas es también un modo natural para llegar al tipo de caracterizaciones que acabamos de mencionar. Por lo tanto también estudiamos esos métodos, en particular el teorema de Beilinson, para el caso proyectivo y para Grassmannianas, mediante el cual pudimos realizar un critetio de escisión para la Grassmanniana de rectas. Seguimos los pasos dados por V. Ancona y G. Ottaviani en el caso del espacio proyectivo. Comparamos el criterio que obtenemos con el obtenido por G. Ottaviani, mostrando que es más fuerte, y con el realizado por E. Arrondo and F. Malaspina. En este segundo caso podemos ver fácilmente que esta nueva caracterización hecha usando el teorema de Beilinson tiene muchas más condiciones que la que nosotros hemos tomado como nuestro punto de partida.Depto. de Álgebra, Geometría y TopologíaFac. de Ciencias MatemáticasTRUEunpu

The hyperdeterminant vanishes on all but two Schur functors

We recall the notion of hyperdeterminant of a multidimensional matrix (tensor). We prove that if we restrict the hyperdeterminant to a skew-symmetric tensor p V ⊆ V ⊗p with p ≥ 3 then it vanishes. The hyperdeterminant also vanishes when we restrict it to the space Γλ ⊗ SλV ⊂ V ⊗p where λ is a Young diagram with p boxes and λ2 ≥ 2 or λ3 ≥ 1

Audiovisual Translation in Teaching Foreign Languages: Contributions of Dubbing to Develop Fluency and Pronunciation in Spontaneous Conversations

The development of new technologies has led to a greater focus on the active use of audiovisual translation in teaching foreign languages in recent years. This paper constitutes a descriptive pilot research, which primarily looks at the interface between the activity of active dubbing and the oral expression of English students of Spanish. After triangulating the data, it has been established that students have improved their speed by an average of 22 words per minute in only six weeks, their confidence and comfort has considerably increased when expressing orally and, concurrently, students have developed other learning areas indirectly.El desarrollo de las nuevas tecnologías ha supuesto una mayor atención en el uso activo de la traducción audiovisual dentro de la enseñanza de lenguas extranjeras en los últimos años. Este estudio constituye una investigación piloto descriptiva que analiza primordialmente la relación entre la actividad del doblaje activo y la expresión oral de estudiantes ingleses de español. Tras la triangulación de datos, se ha constatado que los alumnos han mejorado en una media de 22 palabras por minuto en solo seis semanas, ha aumentado considerablemente su confianza y comodidad al expresarse oralmente y, a su vez, han desarrollado otras áreas de aprendizaje indirectamente

The hyperdeterminant vanishes on all but two Schur functors

We recall the notion of hyperdeterminant of a multidimensional matrix (tensor). We prove that if we restrict the hyperdeterminant to a skew-symmetric tensor with then it vanishes. The hyperdeterminant also vanishes when we restrict it to the space where λ is a Young diagram with p boxes and or

Intralingual dubbing as a tool for developing speaking skills

Communicating verbally with others is one of the main features of human behaviour, but the time employed in class to practise this skill is often insufficient. In an attempt to overcome the need to practise oral conversations and help students feel less anxious in foreign language (FL) contexts, new techniques traditionally employed in audiovisual translation (AVT) has proved to have a positive impact on FL learning. This paper examines the relationship between intralingual dubbing (students' replacement of the original voices of actors in 1 minute long clips) and FL oral expression. The main aim is to provide objective evidence that the use of intralingual dubbing can enhance speed, intonation and pronunciation when speaking spontaneously in Spanish as a FL. A total of 47 participants aged 16-18 with a B1 level of Spanish dubbed videos for 12 weeks. Data is triangulated both qualitatively and quantitatively. Results confirm the main hypothesis and serve as evidence to support theoretical aspects of the inclusion of active AVT techniques in FL speaking classes

An Exploratory analysis of the representations of functions in the university entrance exam in Spain and Iran

The University Entrance Exam (UEE) has an important impact on how math instructors teach mathematics to prepare students for this exam as well as how their students learn mathematics. Since the UEE is like a bridge between high school and university, its importance becomes twofold. However, research regarding this issue is very limited in mathematics education. The purpose of this research is to conduct an exploratory analysis of mathematics questions from the UEE in Spain and Iran from the point of view of representations of functions (i.e., algebraic, graphical, numerical and explanations with words). For this, the mathematics questions related to derivatives and integrals, which are two of the most important concepts in mathematics, from the last five years of the UEE in these two countries were considered to analyze them from the point of view of different representations. The results showed that most of the derivative and integral questions in the UEE in both countries were related to the algebraic representation. However, a very small percentage of the derivative and integral questions were related to the graphical and numerical representations and the relationships between the representations. At the end, suggestions to the curriculum planners and designers of mathematics questions of the UEEs are given