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We study star product algebras of analytic functions for which the power series defining the products converge absolutely. Such algebras arise naturally in deformation quantization theory and in noncommutative quantum field theory. We…
We represent vector bundles over a regular algebraic curve as pairs of lattices over the maximal orders of its function field and we give polynomial time algorithms for several tasks: computing determinants of vector bundles, kernels and…
A real vector space combined with an inverse for vectors is sufficient to define a vector continued fraction whose parameters consist of vector shifts and changes of scale. The choice of sign for different components of the vector inverse…
Different (not only by sign) affine connections are introduced for contravariant and covariant tensor fields over a differentiable manifold by means of a non-canonical contraction operator, defining the notion dual bases and commuting with…
We introduce the concept of paravectors to describe the geometry of points in a three dimensional space. After defining a suitable product of paravectors, we introduce the concepts of biparavectors and triparavectors to describe line…
Vectors fields defined on surfaces constitute relevant and useful representations but are rarely used. One reason might be that comparing vector fields across two surfaces of the same genus is not trivial: it requires to transport the…
A class of associative (super) algebras is presented, which naturally generalize both the symmetric algebra $Sym(V)$ and the wedge algebra $\wedge (V)$, where $V$ is a vector-space. These algebras are in a bijection with those subsets of…
Universal algebraic geometry is generalised from solutions of equations in a single algebra to the study of $\varphi$- or $K$-spectra, akin to the prime spectrum of a ring. We explore their basic properties and constructions, give a…
In this paper, some real-world motivated examples are provided illustrating the power of linear algebra tools as the product of matrices, determinants, eigenvalues and eigenvectors. In this sense, some practical applications related to…
This article is an expository account of the theory of twisted commutative algebras, which simply put, can be thought of as a theory for handling commutative algebras with large groups of linear symmetries. Examples include the coordinate…
We construct examples of non-isomorphic algebraic vector bundles on the punctured affine space with isomorphic pullbacks to the smooth quadric.
In this paper we introduce geometric tools to study the families of rational vector fields of a given degree over $\mathbb C\mathbb P^1$. To a generic vector field of such a parametric family we associate several geometric objects: a…
This article presents a natural extension of the tensor algebra. In addition to "left multiplications" by vectors, we can consider "derivations" by covectors as basic operators on this extended algebra. These two types of operators satisfy…
Metric algebras are metric variants of $\Sigma$-algebras. They are first introduced in the field of universal algebra to deal with algebras equipped with metric structures such as normed vector spaces. Recently a similar notion of…
There are two problems Analytical Geometry with facing anyone who studies this discipline: define the nature of the locus represented by the general equation 2do degree in two or three variables: That curve represents the plane? What…
We introduce many new generalizations of Poisson algebras which can be constructed inside the associative algebra of linear transformations over a vector space.
We introduce in the framework of the linear approximation of General relativity a natural distinction between General gauge transformations generated by any vector field and those Special ones for which this vector field is a gradient. This…
We present an introduction to the theory of algebraic geometry codes. Starting from evaluation codes and codes from order and weight functions, special attention is given to one-point codes and, in particular, to the family of Castle codes.
The topics of Convexity and Concavity and Envelopes are central in Complex Analysis and extensively investigated. The aim of this paper is to find a possible counterpart in Algebraic Geometry. The article presents preliminary results on…
An analogue of the total variation prior for the normal vector field along the boundary of piecewise flat shapes in 3D is introduced. A major class of examples are triangulated surfaces as they occur for instance in finite element…