On the complements of 3-dimensional convex polyhedra as polynomial images of R3{\mathbb R}^3

We prove that the complement ${\mathcal S}:={\mathbb R}^3\setminus{\mathcal K}$ of a 3-dimensional convex polyhedron ${\mathcal K}\subset{\mathbb R}^3$ and its closure $\overline{{\mathcal S}}$ are polynomial images of ${\mathbb R}^3$. The former techniques cannot be extended in general to represent such semialgebraic sets ${\mathcal S}$ and $\overline{{\mathcal S}}$ as polynomial images of ${\mathbb R}^n$ if $n\geq4$.Comment: 12 pages, 1 figur

Demostraciones geométricas automáticas en GeoGebra: casos prácticos

En este artículo presentamos algunos ejemplos del uso de los comandos Comprueba[] y CompruebaDetalles[] que GeoGebra incorpora en sus nuevas versiones, analizando su eficacia a la hora de resolver problemas geométricos de diversa dificultad. También comprobamos la importancia del modo en que se realizan las construcciones geométricas a la hora de aplicar dichos comandos

Demostraciones geométricas automáticas en GeoGebra

En este artículo describimos algunos fundamentos sobre los que se basa la capacidad que tienen muchos programas actuales de software matemático para demostrar enunciados geométricos, y explicamos cómo hacer uso de este recurso en las últimas versiones de GeoGebra

Morphocultural and molecular characterization of Colletotrichum gloeosporioides isolates pathogenic to papaya

Vinte e nove culturas monospóricas de Colletotrichum, isoladas de frutos e pecíolos de mamoeiro (Carica papaya), foram caracterizadas quanto à morfologia dos conídios e apressórios, coloração e crescimento das colônias, sensibilidade ao benomyl, presença de setas e do teleomorfo, PCR com primers taxon-específicos e análise de PCR-RFLP da região ITS. Os 29 isolados foram identificados como C. gloeosporioides com base na morfologia dos conídios e apressórios, tendo a maioria dos isolados conídios cilíndricos e/ou obclavados e apressórios lobados ou fracamente lobados, em contraste com C. acutatum, isolado de morango (Fragaria x ananassa), que apresentou conídios fusiformes e apressórios circulares e lisos. Presença de setas e do teleomorfo, cor de colônia, sensibilidade ao benomyl e velocidade de crescimento variaram conforme o isolado e sofreram influência do meio de cultura usado. Todos os isolados de mamão e quatro de outras hospedeiras, manga (Mangifera indica), morango e maçã (Malus domestica), foram patogênicos a frutos de mamão cv. Sunrise Solo, mas com variabilidade em agressividade. PCR com o primer específico para C. gloeosporioides, CgInt, confirmou a identidade de apenas quatro isolados de mamão e dois isolados apresentaram reação positiva com o primer CaInt2, específico para C. acutatum. A maioria dos isolados de mamão (23) não reagiu com nenhum dos primers. Por outro lado, a análise de restrição da região ITS do rDNA, com RsaI, gerou perfis distintos entre C. gloeosporioides e C. acutatum e mostrou uniformidade entre os isolados de mamão. _________________________________________________________________________________ ABSTRACTTwenty-nine monoconidial cultures of Colletotrichum isolated from papaya (Carica papaya) petioles and fruits were characterized by conidial and appressoria morphology, colony color, growth rate, sensitivity to benomyl, presence of setae, presence of the teleomorph, PCR with taxon-specific primers and analysis of PCR-RFLP of the ITS region. The 29 isolates from papaya were identified as C. gloeosporioides, based mainly on conidial and appressoria morphology, with most isolates producing cylindrical and/or obclavate conidia and entirely or weakly lobed appressoria, in contrast with the strawberry (Fragaria x ananassa) isolate of C. acutatum, which produced fusiform conidia and circular appressoria with entire edges. Presence of setae, teleomorphic stage, colony color, sensitivity to benomyl and growth rate were variable among isolates and influenced by the culture medium. All papaya isolates and four isolates (C. gloeosporioides and C. acutatum) from other hosts, mango (Mangifera indica), strawberry and apple (Malus domestica), were pathogenic to papaya fruits cv. Sunrise Solo, producing similar symptoms, but with variability in aggressiveness. PCR with C. gloeosporioidesspecific primer, CgInt, confirmed the identity of four papaya isolates. Two other isolates reacted with C. acutatum-specific primer, CaInt2. The majority of papaya isolates (23), however, did not react with any of the primers tested. In contrast, RFLP analysis of the amplified ITS region with RsaI, generated distinct patterns that could differentiate between the two species, C. gloeosporioides and C. acutatum, and showed uniformity among papaya isolates

On polynomial images of a closed ball

In this work we approach the problem of determining which (compact) semialgebraic subsets of ${\mathbb R}^n$ are images under polynomial maps $f:{\mathbb R}^m\to{\mathbb R}^n$ of the closed unit ball $\overline{{\mathcal B}}_m$ centered at the origin of some Euclidean space ${\mathbb R}^m$ and that of estimating (when possible) which is the smallest $m$ with this property. Contrary to what happens with the images of ${\mathbb R}^m$ under polynomial maps, it is quite straightforward to provide basic examples of semialgebraic sets that are polynomial images of the closed unit ball. For instance, simplices, cylinders, hypercubes, elliptic, parabolic or hyperbolic segments (of dimension $n$) are polynomial images of the closed unit ball in ${\mathbb R}^n$. The previous examples (and other basic ones proposed in the article) provide a large family of `$n$-bricks' and we find necessary and sufficient conditions to guarantee that a finite union of `$n$-bricks' is again a polynomial image of the closed unit ball either of dimension $n$ or $n+1$. In this direction, we prove: {\em A finite union ${\mathcal S}$ of $n$-dimensional convex polyhedra is the image of the $n$-dimensional closed unit ball $\overline{{\mathcal B}}_n$ if and only if ${\mathcal S}$ is connected by analytic paths}. The previous result can be generalized using the `$n$-bricks' mentioned before and we show: {\em If ${\mathcal S}_1,\ldots,{\mathcal S}_\ell\subset{\mathbb R}^n$ are `$n$-bricks', the union ${\mathcal S}:=\bigcup_{i=1}^\ell{\mathcal S}_i$ is the image of the closed unit ball $\overline{{\mathcal B}}_{n+1}$ of ${\mathbb R}^{n+1}$ under a polynomial map $f:{\mathbb R}^{n+1}\to{\mathbb R}^n$ if and only if ${\mathcal S}$ is connected by analytic paths}.Comment: 41 pages, 18 page