Challenges and opportunities in machine learning for geometry

Over the past few decades, the mathematical community has accumulated a significant amount of pure mathematical data, which has been analyzed through supervised, semi-supervised, and unsupervised machine learning techniques with remarkable results, e.g., artificial neural networks, support vector machines, and principal component analysis. Therefore, we consider as disruptive the use of machine learning algorithms to study mathematical structures, enabling the formulation of conjectures via numerical algorithms. In this paper, we review the latest applications of machine learning in the field of geometry. Artificial intelligence can help in mathematical problem solving, and we predict a blossoming of machine learning applications during the next years in the field of geometry. As a contribution, we propose a new method for extracting geometric information from the point cloud and reconstruct a 2D or a 3D model, based on the novel concept of generalized asymptotes.Agencia Estatal de Investigació

Design and implementation of symbolic algorithms for the computation of generalized asymptotes

In this paper we present two algorithms for computing the g-asymptotes or generalized asymptotes, of a plane algebraic curve, C , implicitly or parametrically defined. The asymptotes of a curve C reflect the status of C at points with sufficiently large coordinates. It is well known that an asymptote of a curve C is a line such that the distance between C and the line approaches zero as they tend to infinity. However, a curve C may have more general curves than lines describing the status of C at infinity. These curves are known as g-asymptotes or generalized asymptotes. The pseudocodes of these algorithms are presented, as well as the corresponding implementations. For this purpose, we use the algebra software Maple. A comparative analysis of the algorithms is carried out, based on some properties of the input curves and their results to analyze the efficiency of the algorithms and to establish comparative criteria. The results presented in this paper are a starting point to generalize this study to surfaces or to curves defined by a non-rational parametrization, as well as to improve the efficiency of the algorithms. Additionally, the methods developed can provide a new and different approach in prediction (regression) or classification algorithms in the machine learning field.Agencia Estatal de Investigació

Asymptotic behavior of parametric algebraic surfaces

Starting from the concept of infinite branches and approximation surfaces, we present a method to compute infinite branches and surfaces having the same asymptotic behavior as an input parametric surface. The results obtained in this work represent a breakthrough for the study of surfaces and their applications.Agencia Estatal de Investigació

Computing branches and asymptotes of meromorphic functions

In this paper, we first summarize the existing algorithms for computing all the generalized asymptotes of a plane algebraic curve implicitly or parametrically defined. From these previous results, we derive a method that allows to easily compute the whole branch and all the generalized asymptotes of a ?special? curve defined in n-dimensional space by a parametrization that is not necessarily rational. So, some new concepts and methods are established for this type of curves. The approach is based on the notion of perfect curves introduced from the concepts and results presented in previous papers.Agencia Estatal de Investigació

An effective algorithm for computing the asymptotes of an implicit curve

In this paper, we first summarize the algorithm presented in Blasco and Pérez-Díaz (2014) for computing the generalized asymptotes of algebraic curves implicitly defined. This algorithm is based on the computation of Puiseux series. The main and very important contribution of this paper is a new and efficient method that allows to easily compute all the generalized asymptotes of an algebraic plane curve implicitly defined by just solving a triangular system of equations. The method can be easily generalized to the case of algebraic curves implicitly defined in the n-dimensional space.Agencia Estatal de Investigació

Detecting and parametrizing polynomial surfaces without base points

Given an algebraic surface implicitly defined by an irreducible polynomial, we present a method that decides whether or not this surface can be parametrized by a polynomial parametrization without base points and, in the affirmative case, we show how to compute this parametrization.Agencia Estatal de Investigació

Determining the asymptotic family of an implicit curve

In this paper we deal with the following problem: given an algebraic plane curve C, implicitly defined, we determine its “asymptotic family”, that is, the set of algebraic curves that have the same asymptotic behavior as C.Agencia Estatal de Investigació

Some New Symbolic Algorithms for the Computation of Generalized Asymptotes

We present symbolic algorithms for computing the g-asymptotes, or generalized asymptotes, of a plane algebraic curve, C, implicitly or parametrically defined. The g-asymptotes generalize the classical concept of asymptotes of a plane algebraic curve. Both notions have been previously studied for analyzing the geometry and topology of a curve at infinity points, as well as to detect the symmetries that can occur in coordinates far from the origin. Thus, based on this research, and in order to solve practical problems in the fields of science and engineering, we present the pseudocodes and implementations of algorithms based on the Puiseux series expansion to construct the g-asymptotes of a plane algebraic curve, implicitly or parametrically defined. Additionally, we propose some new symbolic methods and their corresponding implementations which improve the efficiency of the preceding. These new methods are based on the computation of limits and derivatives; they show higher computational performance, demanding fewer hardware resources and system requirements, as well as reducing computer overload. Finally, as a novelty in this research area, a comparative analysis for all the algorithms is carried out, considering the properties of the input curves and their outcomes, to analyze their efficiency and to establish comparative criteria between them.Agencia Estatal de Investigació

Transdiagnostic neurocognitive deficits in patients with type 2 diabetes mellitus, major depressive disorder, bipolar disorder, and schizophrenia: a 1-year follow-up study

Background Neurocognition impairments are critical factors in patients with major depressive disorder (MDD), bipolar disorder (BD), and schizophrenia (SZ), and also in those with somatic diseases such as type 2 diabetes mellitus (T2DM). Intriguingly, these severe mental illnesses are associated with an increased co-occurrence of diabetes (direct comorbidity). This study sought to investigate the neurocognition and social functioning across T2DM, MDD, BD, and SZ using a transdiagnostic and longitudinal approach. Methods A total of 165 participants, including 30 with SZ, 42 with BD, 35 with MDD, 30 with T2DM, and 28 healthy controls (HC), were assessed twice at a 1-year interval using a comprehensive, integrated test battery on neuropsychological and social functioning. Results Common neurocognitive impairments in somatic and psychiatric disorders were identified, including deficits in short-term memory and cognitive reserve (p < 0.01, η²p=0.08–0.31). Social functioning impairments were observed in almost all the disorders (p < 0.0001; η²p=0.29–0.49). Transdiagnostic deficits remained stable across the 1-year follow-up (p < 0.001; η²p=0.13–0.43) and could accurately differentiate individuals with somatic and psychiatric disorders (χ² = 48.0, p < 0.0001). Limitations The initial sample size was small, and high experimental mortality was observed after follow-up for one year. Conclusions This longitudinal study provides evidence of some possible overlap in neurocognition deficits across somatic and psychiatric diagnostic categories, such as T2DM, MDD, BD, and SZ, which have high comorbidity. This overlap may be a result of shared genetic and environmental etiological factors. The findings open promising avenues for research on transdiagnostic phenotypes of neurocognition in these disorders, in addition to their biological bases