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Technology has helped to innovate in the teaching-learning process. Today's students are more demanding actors when it comes to the environment, they have at their disposal to learn, experiment and develop critical thinking. The area of…

In this paper we survey the theory of holomorphic approximation, from the classical 19th century results of Runge and Weierstrass, continuing with the 20th century work of Oka and Weil, Mergelyan, Vitushkin and others, to the most recent…

Many well-known graph drawing techniques, including force directed drawings, spectral graph layouts, multidimensional scaling, and circle packings, have algebraic formulations. However, practical methods for producing such drawings…

Metric spaces are a fundamental component of mathematics and have a paramount importance as a framework for measuring distance. They can be found in many different branches of mathematics, such as analysis and topology. This paper offers an…

We introduce a novel formulation for geometry on discrete points. It is based on a universal differential calculus, which gives a geometric description of a discrete set by the algebra of functions. We expand this mathematical framework so…

Graphs are a central object of study in various scientific fields, such as discrete mathematics, theoretical computer science and network science. These graphs are typically studied using combinatorial, algebraic or probabilistic methods,…

In this paper, by taking into account the beginning of the hypersurface theory in Euclidean space $E^4$, a practical method for the matrix of the Weingarten map (or the shape operator) of an oriented hypersurface $M^3$ in $E^4$ is obtained.…

We review remarkable results in several mathematical scenarios, including graph theory, division algebras, cross product formalism and matroid theory. Specifically, we mention the following subjects: (1) the Euler relation in graph theory,…

We survey some aspects of the theory of elliptic surfaces and give some results aimed at determining the Picard number of such a surface. For the surfaces considered, this will be equivalent to determining the Mordell-Weil rank of an…

The good mesh quality of an evolving discretized surface or domain is often compromised during time evolution. In recent years this phenomena have been overcome in a couple of ways, one of them uses arbitrary Lagrangian Eulerian maps.…

Gaussian graphical models are used throughout the natural sciences, social sciences, and economics to model the statistical relationships between variables of interest in the form of a graph. We here provide a pedagogic introduction to…

This essay considers the special character of mathematical reasoning, and draws on observations from interactive theorem proving and the history of mathematics to clarify the nature of formal and informal mathematical language. It proposes…

Topological transforms have been very useful in statistical analysis of shapes or surfaces without restrictions that the shapes are diffeomorphic and requiring the estimation of correspondence maps. In this paper we introduce two…

We provide an historical overview of how advances in technology influenced high school and university mathematical competitions in the United States and at the International Mathematical Olympiad. While students are not allowed the usage of…

The evolution of infrastructure networks such as roads and streets are of utmost importance to understand the evolution of urban systems. However, datasets describing these spatial objects are rare and sparse. The database presented here…

We prove a conjecture due to Sturmfels and Uhler concerning the degree of the projective variety associated to the Gaussian graphical model of the cycle. We involve new methods based on the intersection theory in the space of complete…

In this pages I give an overview of the relationship between Model Theory, Arithmetic and Algebraic Geometry. The topics will be the basic ones in the area, so this is just an invitation, in the presentation of topics I mainly follow the…

Pythagoras' theorem, the area of a triangle as one half the base times the height, and Heron's formula are amongst the most important and useful results of ancient Greek geometry. Here we look at all three in a new and improved light, using…

This essay discusses the development of key geometric ideas in the 19th century which led to the formulation of the concept of an abstract manifold (which was not necessarily tied to an ambient Euclidean space) by Hermann Weyl in 1913. This…

The enduring legacy of Euclidean geometry underpins classical machine learning, which, for decades, has been primarily developed for data lying in Euclidean space. Yet, modern machine learning increasingly encounters richly structured data…