An algorithm for the determination of graphs associated to fold maps between closed surfaces

The aim of this paper is to introduce a computational tool that checks theoretical conditions in order to determine whether a weighted graph, as a topological invariant of stable maps, can be associated to stable maps without cusps (ie, fold maps) from closed surfaces to the projective plan.     

On Subordination Chains with Normalization Given by a Time-dependent Linear Operator

In this paper we are concerned with solutions, in particular with univalent solutions, of the Loewner differential equation associated with non-normalized subordination chains on the Euclidean unit ball B ⁿ in \mathbbCⁿ{\mathbb{C}^n}. The main result is a generalization to higher dimensions of a well known result due to Becker. Various particular cases of this result have been recently obtained for subordination chains with normalization Df(0,t)=e^tI_n{Df(0,t)=e^tI_n} or Df(0, t) = e ^tA , t ≥ 0, where A ? L(\mathbbCⁿ,\mathbbCⁿ){A\in L(\mathbb{C}^n,\mathbb{C}^n)}. We also determine the form of the standard solutions to the Loewner differential equation associated with generalized spirallike mappings. In the last section we obtain the form of the solution in the presence of coefficient bounds.     

A general criterion to determine the number of change-points

A general criterion is proposed to determine the number K of the change-points in a parametric nonlinear multi-response model. Schwarz criterion is a particular case. The change-points depend on regressor values and not on instant of measure. We prove that the proposed estimator for K is consistent. Simulation results, using Monte Carlo technique, for nonlinear models which have numerous applications, support the relevance of the theory.     

Deformation Techniques for Sparse Systems

We exhibit a probabilistic symbolic algorithm for solving zero-dimensional sparse systems. Our algorithm combines a symbolic homotopy procedure, based on a flat deformation of a certain morphism of affine varieties, with the polyhedral deformation of Huber and Sturmfels. The complexity of our algorithm is cubic in the size of the combinatorial structure of the input system. This size is mainly represented by the cardinality and mixed volume of Newton polytopes of the input polynomials and an arithmetic analogue of the mixed volume associated to the deformations under consideration. Research was partially supported by the following grants: UBACyT X112 (2004–2007), UBACyT X847 (2006–2009), PIP CONICET 2461, PIP CONICET 5852/05, ANPCyT PICT 2005 17-33018, UNGS 30/3005, MTM2004-01167 (2004–2007), MTM2007-62799 and CIC 2007–2008.     

Quasiconformal extension of biholomorphic mappings in several complex variables

Letf(z, t) be a subordination chain fort ∈ [0, α], α>0, on the Euclidean unit ballB inC ⁿ. Assume thatf(z) =f(z, 0) is quasiconformal. In this paper, we give a sufficient condition forf to be extendible to a quasiconformal homeomorphism on a neighbourhood of . We also show that, under this condition,f can be extended to a quasiconformal homeomorphism of onto itself and give some applications. Partially supported by Grant-in-Aid for Scientific Research (C) no. 14540195 from Japan Society for the Promotion of Science, 2004.     

The Computational Complexity of the Chow Form

We present a bounded probability algorithm for the computation of the Chowforms of the equidimensional components of an algebraic variety. In particular, this gives an alternative procedure for the effective equidimensional decomposition of the variety, since each equidimensional component is characterized by its Chow form. The expected complexity of the algorithm is polynomial in the size and the geometric degree of the input equation system defining the variety. Hence it improves (or meets in some special cases) the complexity of all previous algorithms for computing Chow forms. In addition to this, we clarify the probability and uniformity aspects, which constitutes a further contribution of the paper. The algorithm is based on elimination theory techniques, in line with the geometric resolution algorithm due to M. Giusti, J. Heintz, L. M. Pardo, and their collaborators. In fact, ours can be considered as an extension of their algorithm for zero-dimensional systems to the case of positive-dimensional varieties. The key element for dealing with positive-dimensional varieties is a new Poisson-type product formula. This formula allows us to compute the Chow form of an equidimensional variety from a suitable zero-dimensional fiber. As an application, we obtain an algorithm to compute a subclass of sparse resultants, whose complexity is polynomial in the dimension and the volume of the input set of exponents. As another application, we derive an algorithm for the computation of the (unique) solution of a generic overdetermined polynomial equation system.     

Early error detection: an action-research experience teaching vector calculus

This paper describes an action-research experience carried out with second year students at the School of Engineering of the National University of Entre Ríos, Argentina. Vector calculus students played an active role in their own learning process. They were required to present weekly reports, in both oral and written forms, on the topics studied, instead of merely sitting and watching as the teacher solved problems on the blackboard. The students were also asked to perform computer assignments, and their learning process was continuously monitored. Among many benefits, this methodology has allowed students and teachers to identify errors and misconceptions that might have gone unnoticed under a more passive approach.     

Approximation of a population dynamics model by parabolic regularization

The basic linear model for describing an age structured population spreading in a limited habitat is considered with the purpose of investigating an approximation procedure based on parabolic regularization. In fact, a viscosity model is introduced by considering an appropriate approximating regularized parabolic problem and it is proved that the sequence of the approximating solutions tends to the solution to the original problem. The advantage of this approach is that it leads to the numerical solution of a parabolic problem that has more stable solutions than the hyperbolic‐parabolic original problem and avoids the restrictions (compatibility conditions) needed to treat the latter. Moreover, for the solution of the approximating problem, it is possible to take advantage of established software packages dedicated to parabolic problems. Some examples of the approach are provided using COMSOL Multiphysics. Copyright © 2012 John Wiley & Sons, Ltd.