Local tropical linear spaces

In this paper we study general tropical linear spaces locally: For any basis B of the matroid underlying a tropical linear space L, we define the local tropical linear space L_B to be the subcomplex of L consisting of all vectors v that make B a basis of maximal v-weight. The tropical linear space L can then be expressed as the union of all its local tropical linear spaces, which we prove are homeomorphic to Euclidean space. Local tropical linear spaces have a simple description in terms of polyhedral matroid subdivisions, and we prove that they are dual to mixed subdivisions of Minkowski sums of simplices. Using this duality we produce tight upper bounds for their f-vectors. We also study a certain class of tropical linear spaces that we call conical tropical linear spaces, and we give a simple proof that they satisfy Speyer's f-vector conjecture.Comment: 13 pages, 1 figure. Some results are stated in a bit more generality. Minor corrections were also mad

Valuations for matroid polytope subdivisions

We prove that the ranks of the subsets and the activities of the bases of a matroid define valuations for the subdivisions of a matroid polytope into smaller matroid polytopes.Comment: 19 pages. 2 figures; added section 6 + other correction

Positively oriented matroids are realizable

We prove da Silva's 1987 conjecture that any positively oriented matroid is a positroid; that is, it can be realized by a set of vectors in a real vector space. It follows from this result and a result of the third author that the positive matroid Grassmannian (or positive MacPhersonian) is homeomorphic to a closed ball.Comment: 20 pages, 3 figures, references adde

Compressed sensing of data with a known distribution

Compressed sensing is a technique for recovering an unknown sparse signal from a small number of linear measurements. When the measurement matrix is random, the number of measurements required for perfect recovery exhibits a phase transition: there is a threshold on the number of measurements after which the probability of exact recovery quickly goes from very small to very large. In this work we are able to reduce this threshold by incorporating statistical information about the data we wish to recover. Our algorithm works by minimizing a suitably weighted $\ell_1$-norm, where the weights are chosen so that the expected statistical dimension of the corresponding descent cone is minimized. We also provide new discrete-geometry-based Monte Carlo algorithms for computing intrinsic volumes of such descent cones, allowing us to bound the failure probability of our methods.Comment: 22 pages, 7 figures. New colorblind safe figures. Sections 3 and 4 completely rewritten. Minor typos fixe

Tropical ideals

We introduce and study a special class of ideals, called tropical ideals, in the semiring of tropical polynomials, with the goal of developing a useful and solid algebraic foundation for tropical geometry. The class of tropical ideals strictly includes the tropicalizations of classical ideals, and allows us to define subschemes of tropical toric varieties, generalizing Giansiracusa and Giansiracusa [Equations of tropical varieties, Duke Math. J. 165 (2016), 3379–3433]. We investigate some of the basic structure of tropical ideals, and show that they satisfy many desirable properties that mimic the classical setup. In particular, every tropical ideal has an associated variety, which we prove is always a finite polyhedral complex. In addition we show that tropical ideals satisfy the ascending chain condition, even though they are typically not finitely generated, and also the weak Nullstellensatz

Tropical Ideals

International audienceWe introduce and study a special class of ideals over the semiring of tropical polynomials, which we calltropical ideals, with the goal of developing a useful and solid algebraic foundation for tropical geometry. We exploretheir rich combinatorial structure, and prove that they satisfy numerous properties analogous to classical ideals

The implementation of storytelling in the native Language to promote L2 vocabulary in older adults

Este proyecto de aula presenta la implementación de la técnica de narración de cuentos utilizando mitos y leyendas colombianos para enseñar vocabulario en inglés a adultos mayores. Para este proyecto, establecimos algunos objetivos de enseñanza que se dividieron en dos; objetivos generales en los que los adultos mayores reconocen el vocabulario establecidos a través de la producción verbal y no verbal. También los objetivos específicos, por ejemplo, la implementación de la narración de cuentos como un medio para promover el vocabulario en inglés. La metodología utilizada en este proyecto de aula se centró en la andragogía, que es la ciencia de la educación de adultos, en base a esto enseñamos vocabulario en inglés utilizando mitos y leyendas colombianas. Los participantes involucrados fueron siete adultos mayores de una casa de retiro en Pereira, y los tres facilitadores los cuales usaron como instrumentos diarios, observaciones y artefactos de los participantes recolectar evidencia y su posterior análisis. Los resultados indicaron que los participantes pudieron recordar y utilizar el vocabulario presentado durante la sesión de narración de mitos y leyendas. Como conclusión, este proyecto de aula reveló que el uso de mitos y leyendas para enseñar vocabulario en inglés es una técnica favorable la cual permitió a los participantes sentirse cómodos y usar sus conocimientos previos en su idioma nativo con el propósito de aprender vocabulario en inglés

A Grammar Compression Algorithm based on Induced Suffix Sorting

We introduce GCIS, a grammar compression algorithm based on the induced suffix sorting algorithm SAIS, introduced by Nong et al. in 2009. Our solution builds on the factorization performed by SAIS during suffix sorting. We construct a context-free grammar on the input string which can be further reduced into a shorter string by substituting each substring by its correspondent factor. The resulting grammar is encoded by exploring some redundancies, such as common prefixes between suffix rules, which are sorted according to SAIS framework. When compared to well-known compression tools such as Re-Pair and 7-zip, our algorithm is competitive and very effective at handling repetitive string regarding compression ratio, compression and decompression running time