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We present algebraic techniques to analyze state space models in the areas of structural identifiability, observability, and indistinguishability. While the emphasis is on surveying existing algebraic tools for studying ODE systems, we also…
Quivers, gauge theories and singular geometries are of great interest in both mathematics and physics. In this note, we collect a few open questions which have arisen in various recent works at the intersection between gauge theories,…
Symmetry plays a central role in accelerating symbolic computation involving polynomials. This chapter surveys recent developments and foundational methods that leverage the inherent symmetries of polynomial systems to reduce complexity,…
Geometric algebra is an optimal frame work for calculating with vectors. The geometric algebra of a space includes elements that represent all the its subspaces (lines, planes, volumes, ...). Conformal geometric algebra expands this…
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…
We study the relation between algebraic structures and Graph Theory. We have defined five different weighted digraphs associated to a finite dimensional algebra over a field in order to tackle important properties of the associated…
We discuss issues of problem formulation for algorithms in real algebraic geometry, focussing on quantifier elimination by cylindrical algebraic decomposition. We recall how the variable ordering used can have a profound effect on both…
Since its introduction by Gauss, Matrix Algebra has facilitated understanding of scientific problems, hiding distracting details and finding more elegant and efficient ways of computational solving. Today's largest problems, which often…
This article surveys the development of the theory of algebraic geometry codes since their discovery in the late 70's. We summarize the major results on various problems such as: asymptotic parameters, improved estimates on the minimum…
In a previous paper, the general approach for treatment of algebraic equations of different order in gravity theory was exposed, based on the important distinction between covariant and contravariant metric tensor components. In the present…
Achieving disentangled representations of information is one of the key goals of deep network based machine learning system. Recently there are more discussions on this issue. In this paper, by comparing the geometric structure of…
A method to the explict solutions of general systems of algebraic equations is presented via the metric form of affiliated K\"ahler manifolds. The solutions to these systems arise from sets of geodesic second order non-linear differential…
With distributed computing and mobile applications, synchronizing diverging replicas of data structures is a more and more common problem. We use algebraic methods to reason about filesystem operations, and introduce a simplified definition…
Gauge symmetries indicate redundancies in the description of the relevant degrees of freedom of a given field theory and restrict the nature of observable quantities. One of the problems faced by emergent theories of relativistic fields is…
Absolute parallelism (AP) geometry is frequently used for physical applications. Although it is wider than Riemannian geometry, it has two main defects. The first is that its path equation does not represent physical trajectories of any…
The theory uses methods and language of linear algebra to study nonlinear spaces. These techniques can be used particularly to describe analytic geometry of non-linear elliptic, hyperbolic, De Sitter and Anti de Sitter spaces. The main…
In these notes we present preliminary results on quantum-like algorithms where tensor product is replaced by geometric product. Such algorithms possess the essential properties typical of quantum computation (entanglement, parallelism) but…
This paper is to give an overview of AMG methods for solving large scale systems of equations such as those from the discretization of partial differential equations. AMG is often understood as the acronym of "Algebraic Multi-Grid", but it…
A new method for solving systems of linear algebraic equations of a special type arising in solving problems of image reconstruction has been proposed. This method, due to a certain symmetry of the matrix and the choice of the voxel…
The development of computational techniques in the last decade has made possible to attack some classical problems of algebraic geometry. In this survey, we briefly describe some open problems related to algebraic curves which can be…