Being motivated by some methods for construction of homothetically indecomposable polytopes, we obtain new methods for construction of families of integrally indecomposable polytopes. As a result, we ...find new infinite families of absolutely irreducible multivariate polynomials over any field F. Moreover, we provide different proofs of some of the main results of Gao 2.
We introduce PSN polytopes whose k-skeleton is combinatorially equivalent to that of a product of r simplices. They simultaneously generalize both neighborly and neighborly
cubical polytopes.
We ...construct PSN polytopes by three different methods, the most versatile of which is an extension of Sanyal & Ziegler’s “projecting deformed products” construction to products of arbitrary simple polytopes. For general r and k, the lowest dimension we achieve is 2k+r+1.
Using topological obstructions similar to those introduced by Sanyal to bound the number of vertices of Minkowski sums, we show that this dimension is minimal if we moreover require the PSN polytope to be obtained as a projection of a polytope combinatorially equivalent to the product of r simplices, when the sum of their dimensions is at least 2k.
The Monotone Upper Bound Problem asks for the maximal number M(d,n) of vertices on a strictly-increasing edge-path on a simple d-polytope with n facets. More specifically, it asks whether the upper ...bound
M(d,n) ≤ Mubt(d,n)
provided by McMullen’s (1970) Upper Bound Theorem is tight, where Mubt(d,n) is the number of vertices of a dual-to-cyclic d-polytope with n facets.
It was recently shown that the upper bound M(d,n) ≤ Mubt(d,n) holds with equality for small dimensions (d ≤ 4: Pfeifle, 2003) and for small corank (n ≤ d + 2: Gärtner et al., 2001). Here we prove that it is not tight in general: In dimension d=6 a polytope with n=9 facets can have Mubt(6,9)=30 vertices, but not more than 27 ≤ M(6,9) ≤ 29 vertices can lie on a strictly-increasing edge-path.
The proof involves classification results about neighborly polytopes, Kalai’s (1988)
concept of abstract objective functions, the Holt-Klee conditions (1998), explicit
enumeration, Welzl’s (2001) extended Gale diagrams, randomized generation of instances,
as well as non-realizability proofs via a version of the Farkas lemma.
In 1988, Kalai 5 extended a construction of Billera and Lee to produce many triangulated(d−1)-spheres. In fact, in view of upper bounds on the number of simplicial d-polytopes by Goodman and Pollack ...2, 3, he derived that for every dimension d ≥ 5, most of these(d − 1)-spheres are not polytopal. However, for d = 4, this reasoning fails. We can now show that, as already conjectured by Kalai, all of his 3-spheres are in fact polytopal.
We also give a shorter proof for Hebble and Lee’s result 4 that the dual graphs of these 4-polytopes are Hamiltonian.
In 1988, Kalai 5 extended a construction of Billera and Lee to produce many triangulated(d−1)-spheres. In fact, in view of upper bounds on the number of simplicial d-polytopes by Goodman and Pollack ...2,3, he derived that for every dimension d ≥ 5, most of these(d−1)-spheres are not polytopal. However, for d=4, this reasoning fails. We can now show that, as already conjectured by Kalai, all of his 3-spheres are in fact polytopal. We also give a shorter proof for Hebble and Lee’s result 4 that the dual graphs of these 4-polytopes are Hamiltonian.
Los diseños cualitativos y sus modelos de análisis en geografía han sido objeto de debate debido a la falta de claridad a la hora de exponer las diferentes opciones metodológicas tomadas por los ...investigadores, y de explicitar las formas de análisis para la producción de resultados desde una perspectiva cualitativa. El presente trabajo describe un dispositivo metodológico cualitativo/espacial utilizado para el estudio de la experiencia espacial de estudiantes de la Región Metropolitana (Chile), así como el modelo de análisis de la información que fue implementado. Se da cuenta de una aproximación topológica de trabajo en la producción de datos (entrevista topológica), así como de un proceso de codificación en base a topos, asegurando la perspectiva espacial de las experiencias de los sujetos. Se presentan dos tipos de resultados obtenidos, el primero de ellos a la reconstrucción de paisajes de interiorización de las experiencias de los sujetos y el segundo a la producción de tecnologías desarrolladas por los estudiantes en los espacios que habitan.
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