J. O. González-Cervantes

arXiv2026

Some global operators and the material derivative

The theory of the operator $$G(x) = |\underline{x}|^2 \frac{\partial }{\partial x_0} + \underline{x} \sum_{j=1}^n x_j \frac{\partial }{\partial x_j} $$ is deeply associated with the slice monogenic function theory and has grown in recent years. In particular, for $n=3$ the quaternionic version of $G$ has been recently used to study the quaternionic slice regular function theory. This work extends the study of the $G$ operator in two senses: a) Clifford's analysis structure. The function theory induced by the operator \begin{align*}\mathcal H_a (x) = {\underline a} ( {x}) \frac{\partial }{\partial x_0} - \sum_{i=1}^n \left( \sum_{j=1}^n a_j ( {x}) \frac{\partial (a^{-1})_i}{\partial y_j}\circ a ( {x}) \right) \frac{\partial}{\partial x_i}, \end{align*} where $a$ is a function with certain properties with domain in $\mathbb R^{n+1}$ is presented extending the already known results of the $G$. Also some properties of the material derivative are presented as consequences of function theory induced by $\mathcal H_a$. b) Structure of quaternionic analysis. In particular, the case $n=3$ is approached from the point of view of quaternionic analysis.

math.CV↗

arXiv2017

An Approach with Toric Varieties for Singular Learning Machines

The Computational Algebraic Geometry applied in Algebraic Statistics; are beginning to exploring new branches and applications; in artificial intelligence and others areas. Currently, the development of the mathematics is very extensive and it is difficult to see the immediate application of few theorems in different areas, such as is the case of the Theorem 3.9 given in [10] and proved in part of here. Also this work has the intention to show the Hilbert basis as a powerful tool in data science; and for that reason we compile important results proved in works by, S. Watanabe [27], D. Cox, J. Little and H. Schenck [8], B. Sturmfels [16] and G. Ewald [10]. In this work we study, first, the fundamental concepts in Toric Algebraic Geometry. The principal contribution of this work is the application of Hilbert basis (as one realization of Theorem 3.9) for the resolution of singularities with toric varieties, and a background in Lattice Polytope. In the second part we apply this theorem to problems in statistical learning, principally in a recent area as is the Singular Learning Theory. We define the singular machines and the problem of Singular Learning through the computing of learning curves on these statistical machines. We review and compile results on the work of S. Watanabe in Singular Learning Theory, ref.; [17], [20], [21], also revising the important result in [26], about almost the machines are singular, we formalize this theory withtoric resolution morphism in a theorem proved here (Theorem 5.4), characterizing these Learning Machines as toric varieties, and we reproduce results previously published in Singular Statistical Learning seen in [19], [20], [23].

math.AG↗