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The classical ``computation'' methods in Algebraic Topology most often work by means of highly infinite objects and in fact +are_not+ constructive. Typical examples are shown to describe the nature of the problem. The Rubio-Sergeraert…
In this paper, a set of programs enhancing the Kenzo system is presented. Kenzo is a Common Lisp program designed for computing in Algebraic Topology, in particular it allows the user to calculate homology and homotopy groups of complicated…
Despite the wide variety of input types in machine learning, this diversity is often not fully reflected in their representations or model architectures, leading to inefficiencies throughout a model's lifecycle. This paper introduces an…
To synthesize Maxwell optics systems, the mathematical apparatus of tensor and vector analysis is generally employed. This mathematical apparatus implies executing a great number of simple stereotyped operations, which are adequately…
In this paper we show how certain techniques of image processing, having different scopes, can be joined together under a common "algebraic roof".
Tropical geometry is a piecewise linear "shadow" of algebraic geometry. It allows for the computation of several cohomological invariants of an algebraic variety. In particular, its application to enumerative algebraic geometry led to…
As dynamic and control systems become more complex, relying purely on numerical computations for systems analysis and design might become extremely expensive or totally infeasible. Computer algebra can act as an enabler for analysis and…
This research addresses a new tool for data analysis known as Topological Data Analysis TDA It underlies an area of Mathematics known as Combinatorial Algebra or more recently Algebraic Topology which through making strong use of…
In this paper, three Computational Topology methods (namely effective homology, persistent homology and discrete vector fields) are mixed together to produce algorithms for homological digital image processing. The algorithms have been…
In certain scientific domains, there is a need for tensor operations. To facilitate tensor computations,computer algebra systems are employed. In our research, we have been using Cadabra as the main computer algebra system for several…
We introduce a geometry of interaction model for Mazza's multiport interaction combinators, a graph-theoretic formalism which is able to faithfully capture concurrent computation as embodied by process algebras like the $\pi$-calculus. The…
Over the past two decades, topological data analysis has emerged as a field of applied mathematics with new applications and algorithmic developments appearing rapidly. Two fundamental computations in this field are persistent homology and…
This article deals with OLAP systems based on multidimensional model. The conceptual model we provide, represents data through a constellation (multi-facts) composed of several multi-hierarchy dimensions. In this model, data are displayed…
Compositionality is a key property for dealing with complexity, which has been studied from many points of view in diverse fields. Particularly, the composition of individual computations (or programs) has been widely studied almost since…
In past few decades, tensor algebra also known as multi-linear algebra has been developed and customized as a tool to be used for various engineering applications. In particular, with the help of a special form of tensor contracted product,…
Techniques from higher categories and higher-dimensional rewriting are becoming increasingly important for understanding the finer, computational properties of higher algebraic theories that arise, among other fields, in quantum…
In this paper we present several algorithms related with the computation of the homology of groups, from a geometric perspective (that is to say, carrying out the calculations by means of simplicial sets and using techniques of Algebraic…
We introduce a topology on the space of all isomorphism types represented in a given class of countable models, and use this topology as an aid in classifying the isomorphism types. This mixes ideas from effective descriptive set theory and…
A novel algebra underlying integrable systems is shown to generate and unify a large class of quantum integrable models with given $R$-matrix, through reductions of an ancestor Lax operator and its different realizations. Along with known…
An abstract mathematical framework is presented in this paper as a unification of several deformed or generalized algebra proposed recently in the context of generalized statistical theories intended to treat certain complex thermodynamic…