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Coalgebra is a currently quite active field, which aims to look at generic state-based systems (most prominently automata) from a very abstract point of view, mainly using tools from category theory. One of its achievements is to give a…
Cylindrical algebraic decomposition (CAD) is an important tool for the study of real algebraic geometry with many applications both within mathematics and elsewhere. It is known to have doubly exponential complexity in the number of…
A systematic method is presented for the construction and classification of algebras of gauge transformations for arbitrary high rank tensor gauge fields. For every tensor gauge field of a given rank, the gauge transformation will be…
The field of quantum computing is at an exciting time where we are constructing novel hardware, evaluating algorithms, and finding out what works best. As qubit technology grows and matures, we need to be ready to design and program larger…
Quaternion symmetry is ubiquitous in the physical sciences. As such, much work has been afforded over the years to the development of efficient schemes to exploit this symmetry using real and complex linear algebra. Recent years have also…
Background: Many biological systems are modeled qualitatively with discrete models, such as probabilistic Boolean networks, logical models, Petri nets, and agent-based models, with the goal to gain a better understanding of the system. The…
We consider the question: what is the abstraction that should be implemented by the computational engine of a machine learning system? Current machine learning systems typically push whole tensors through a series of compute kernels such as…
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…
Solving systems of polynomial equations, particularly those with finitely many solutions, is a crucial challenge across many scientific fields. Traditional methods like Gr\"obner and Border bases are fundamental but suffer from high…
In this article we present a method to implement orthogonal polynomials and many other special functions in Computer Algebra systems enabling the user to work with those functions appropriately, and in particular to verify different types…
Dynamical systems are capable of performing computation in a reservoir computing paradigm. This paper presents a general representation of these systems as an artificial neural network (ANN). Initially, we implement the simplest dynamical…
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…
Cellular automata are discrete and computational models thatcan be shown as general models of complexity. They are used in varied applications to derive the generalized behavior of the presented model. In this paper we have took one such…
The advent of memristive devices offers a promising avenue for efficient and scalable analog computing, particularly for linear algebra operations essential in various scientific and engineering applications. This paper investigates the…
Quantum principal component analysis (QPCA) ignited a new development toward quantum machine learning algorithms. Initially showcasing as an active way for analyzing a quantum system using the quantum state itself, QPCA also found potential…
Qubit models and methods for improving the performance of software and hardware for analyzing digital devices through increasing the dimension of the data structures and memory are proposed. The basic concepts, terminology and definitions…
The power of Clifford or, geometric, algebra lies in its ability to represent geometric operations in a concise and elegant manner. Clifford algebras provide the natural generalizations of complex, dual numbers and quaternions into…
In the world of linear algebra computation, a well-established standard exists called BLAS(Basic Linear Algebra Subprograms). This standard has been crucial for the development of software using linear algebra operations. Its benefits…
We give illustrative examples of how the computer algebra system OSCAR can support research in commutative algebra and algebraic geometry. We start with a thorough introduction to Groebner basis techniques, with particular emphasis on the…
Existing computer algebra packages do not fully support quantum mechanics calculations in Dirac's notation. I present the foundation for building such support: a mathematical system for the symbolic manipulation of expressions used in the…