Classification of obstruction for separation of semialgebraic sets in dimension 3

Sin resume

Reparametrizing Swung Surfaces over the Reals

Let K⊆R be a computable subfield of the real numbers (for instance, Q). We present an algorithm to decide whether a given parametrization of a rational swung surface, with coefficients in K(i), can be reparametrized over a real (i.e., embedded in R) finite field extension of K. Swung surfaces include, in particular, surfaces of revolution

Ultraquadrics associated to affine and projective automorphisms

The concept of ultraquadric has been introduced by the authors as a tool to algorithmically solve the problem of simplifying the coefficients of a given rational parametrization in K(a) (t1, ..., tn) of an algebraic variety of arbitrary dimension over a field extension K(a). In this context, previous work in the one-dimensional case has shown the importance of mastering the geometry of 1-dimensional ultraquadrics (hypercircles). In this paper we study, for the first time, the properties of some higher dimensional ultraquadrics, namely, those associated to automorphisms in the field K(a) (t1, ..., tn), defined by linear rational (with common denominator) or by polynomial (with inverse also polynomial) coordinates. We conclude, among many other observations, that ultraquadrics related to polynomial automorphisms can be characterized as varieties K-isomorphic to linear varieties, while ultraquadrics arising from projective automorphisms are isomorphic to the Segre embedding of a blowup of the projective space along an ideal and, in some general case, linearly isomorphic to a toric variety. We conclude with some further details about the real-complex, 2-dimensional case, showing, for instance, that this family of ultraquadrics can be presented as a collection of ruled surfaces described by pairs of hypercircles. © 2014, Springer-Verlag Berlin Heidelberg

Proof of the Hyperplane Zeros Conjecture of Lagarias and Wang

We prove that a real analytic subset of a torus group that is contained in its image under an expanding endomorphism is a finite union of translates of closed subgroups. This confirms the hyperplane zeros conjecture of Lagarias and Wang for real analytic varieties. Our proof uses real analytic geometry, topological dynamics and Fourier analysis.Comment: 25 page

Uniform bounds on complexity and transfer of global properties of Nash functions

We show that the complexity of semialgebraic sets and mappings can be used to parametrize Nash sets and mappings by Nash families. From this we deduce uniform bounds on the complexity of Nash functions that lead to first-order descriptions of many properties of Nash functions and a good behaviour under real closed field extension (e.g. primary decomposition). As a distinguished application, we derive the solution of the extension and global equations problems over arbitrary real closed fields, in particular over the field of real algebraic numbers. This last fact and a technique of change of base are used to prove that the Artin-Mazur description holds for abstract Nash functions on the real spectrum of any commutative ring, and solve extension and global equations in that abstract setting. To complete the view, we prove the idempotency of the real spectrum and an abstract version of the separation problem. We also discuss the conditions for the rings of abstract Nash functions to be noetherian

Assessment of Cannabidiol and ∆9-Tetrahydrocannabiol in Mouse Models of Medulloblastoma and Ependymoma

Children with medulloblastoma and ependymoma are treated with a multidisciplinary approach that incorporates surgery, radiotherapy, and chemotherapy; however, overall survival rates for patients with high-risk disease remain unsatisfactory. Data indicate that plant-derived cannabinoids are effective against adult glioblastoma; however, preclinical evidence supporting their use in pediatric brain cancers is lacking. Here we investigated the potential role for ∆9- tetrahydrocannabinol (THC) and cannabidiol (CBD) in medulloblastoma and ependymoma. Dose- dependent cytotoxicity of medulloblastoma and ependymoma cells was induced by THC and CBD in vitro, and a synergistic reduction in viability was observed when both drugs were combined. Mechanistically, cannabinoids induced cell cycle arrest, in part by the production of reactive oxygen species, autophagy, and apoptosis; however, this did not translate to increased survival in orthotopic transplant models despite being well tolerated. We also tested the combination of cannabinoids with the medulloblastoma drug cyclophosphamide, and despite some in vitro synergism, no survival advantage was observed in vivo. Consequently, clinical benefit from the use of cannabinoids in the treatment of high-grade medulloblastoma and ependymoma is expected to be limited. This study emphasizes the importance of preclinical models in validating therapeutic agent efficacy prior to clinical trials, ensuring that enrolled patients are afforded the most promising therapies available

Complexity of global semianalytic sets in a real analytic manifold of dimension 2

Let X subset of R-n be a real analytic manifold of dimension 2. We study the stability index of X, s(X), that is the smallest integer s such that any basic open subset of X can be written using s global analytic functions. We show that s(X) = 2 as it happens in the semialgebraic case. Also, we prove that the Hormander-Lojasiewicz inequality and the Finiteness Theorem hold true in this context. Finally, we compute the stability index for basic closed subsets, S, and the invariants t and (t) over bar for the number of unions of open (resp. closed) basic sets required to describe any open (resp. closed) global semianalytic set