Infinitesimal Hartman-Grobman Theorem in Dimension Three

In this paper we give the main ideas to show that a real analytic vector field in R3 with a singular point at the origin is locally topologically equivalent to its principal part defined through Newton polyhedra under non-degeneracy conditions.This research was partially supported by the Spanish Government (MTM2010-15471 (subprogram MTM))

Stratification of three-dimensional real flows II: A generalization of Poincar\'e's planar sectorial decomposition

Let $\xi$ be an analytic vector field in $\mathbb{R}^3$ with an isolated singularity at the origin and having only hyperbolic singular points after a reduction of singularities $\pi:M\to\mathbb{R}^3$. Assuming certain conditions to be specified throughout the work at hand, we establish a theorem of stratification of the dynamics of $\xi$ that generalizes to dimension three the classical one, coming from Poincar\'{e}, about the decomposition of the dynamics of an analytic planar vector field into {\em parabolic}, {\em elliptic} or {\em hyperbolic} invariant sectors

Escuelas doctorales intercontinentales: una herramienta para la internacionalización universitaria

En la mayoría de las universidades españolas tenemos mucha fuerza y hay otros lugares del mundo en los que están deseando recibir lo que nosotros podemos ofrecer. Por esta razón el marco internacional es imprescindible. Si, por ejemplo, quisiéramos hacer un grado internacional en alguna materia, no sería suficiente con aportar los profesores; sería necesario pensar en el calendario, a quién va dirigido, en qué horario se van a dar las clases, etc. Para que eso funcione con unos criterios firmes es necesario contar con una estructura definida que coordine el proceso y fije unos parámetros comunes. Lo ideal sería la existencia de un centro de formación internacional y de posgrado que se ocupe de intentar avanzar en la consolidación de este tipo de actividades para que al final exista una especie de oferta internacional de las universidades que sea real. Presentaremos el caso de la Escuela Doctoral Intercontinental PUCP-UVa, en la que la UA intervino en el año 2013, como ejemplo exitoso de internacionalización de la universidad aprovechando los últimos recursos que ofrece la tecnología

A new invariant for cyclic orbit flag codes

In the network coding framework, given a prime power $q$ and the vector space $\mathbb{F}_q^n$, a constant type flag code is a set of nested sequences of $\mathbb{F}_q$-subspaces (flags) with the same increasing sequence of dimensions (the type of the flag). If a flag code arises as the orbit under the action of a cyclic subgroup of the general linear group over a flag, we say that it is a cyclic orbit flag code. Among the parameters of such a family of codes, we have its best friend, that is the largest field over which all the subspaces in the generating flag are vector spaces. This object permits to compute the cardinality of the code and estimate its minimum distance. However, as it occurs with other absolute parameters of a flag code, the information given by the best friend is not complete in many cases due to the fact that it can be obtained in different ways. In this work, we present a new invariant, the best friend vector, that captures the specific way the best friend can be unfolded. Furthermore, throughout the paper we analyze the strong underlying interaction between this invariant and other parameters such as the cardinality, the flag distance, or the type vector, and how it conditions them. Finally, we investigate the realizability of a prescribed best friend vector in a vector space

Consistent Flag Codes

In this paper we study flag codes on Fnq, being Fq the finite field with q elements. Special attention is given to the connection between the parameters and properties of a flag code and the ones of a family of constant dimension codes naturally associated to it (the projected codes). More precisely, we focus on consistent flag codes, that is, flag codes whose distance and size are completely determined by their projected codes. We explore some aspects of this family of codes and present examples of them by generalizing the concepts of equidistant and sunflower subspace code to the flag codes setting. Finally, we present a decoding algorithm for consistent flag codes that fully exploits the consistency condition.The authors receive financial support from Ministerio de Ciencia e Innovación (PID2019-108668GB-I00). The second author is supported by Generalitat Valenciana and Fondo Social Europeo (ACIF/2018/196)

On Generalized Galois Cyclic Orbit Flag Codes

Flag codes that are orbits of a cyclic subgroup of the general linear group acting on flags of a vector space over a finite field, are called cyclic orbit flag codes. In this paper, we present a new contribution to the study of such codes, by focusing this time on the generating flag. More precisely, we examine those ones whose generating flag has at least one subfield among its subspaces. In this situation, two important families arise: the already known Galois flag codes, in case we have just fields, or the generalized Galois flag codes in other case. We investigate the parameters and properties of the latter ones and explore the relationship with their underlying Galois flag code.This research was funded by Ministerio de Ciencia e Innovación (grant number PID2019-108668GB-I00) and Generalitat Valenciana y Fondo Social Europeo (Grant number ACIF/2018/196)

Flag Codes: Distance Vectors and Cardinality Bounds

Given Fq the finite field with q elements and an integer n > 2, a flag is a sequence of nested subspaces of Fnq and a flag code is a nonempty set of flags. In this context, the distance between flags is the sum of the corresponding subspace distances. Hence, a given flag distance value might be obtained by many different combinations. To capture such a variability, in the paper at hand, we introduce the notion of distance vector as an algebraic object intrinsically associated to a flag code that encloses much more information than the distance parameter itself. Our study of the flag distance by using this new tool allows us to provide a fine description of the structure of flag codes as well as to derive bounds for their maximum possible size once the minimum distance and dimensions are fixed.The authors received financial support of Ministerio de Ciencia e Innovación (PID2019-108668GB-I00). The second author is supported by Generalitat Valenciana and Fondo Social Europeo (ACIF/2018/196)

Flag Codes: Distance Vectors and Cardinality Bounds

Given Fq the finite field with q elements and an integer n > 2, a flag is a sequence of nested subspaces of Fnq and a flag code is a nonempty set of flags. In this context, the distance between flags is the sum of the corresponding subspace distances. Hence, a given flag distance value might be obtained by many different combinations. To capture such a variability, in the paper at hand, we introduce the notion of distance vector as an algebraic object intrinsically associated to a flag code that encloses much more information than the distance parameter itself. Our study of the flag distance by using this new tool allows us to provide a fine description of the structure of flag codes as well as to derive bounds for their maximum possible size once the minimum distance and dimensions are fixed.The authors received financial support of Ministerio de Ciencia e Innovación (PID2019-108668GB-I00). The second author is supported by Generalitat Valenciana and Fondo Social Europeo (ACIF/2018/196)