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Recent results by Harrow et. al. and by Ta-Shma, suggest that quantum computers may have an exponential advantage in solving a wealth of linear algebraic problems, over classical algorithms. Building on the quantum intuition of these…
The purpose of a program analysis is to compute an abstract meaning for a program which approximates its dynamic behaviour. A compositional program analysis accomplishes this task with a divide-and-conquer strategy: the meaning of a program…
Many combinatorial problems can be formulated as a polynomial optimization problem that can be solved by state-of-the-art methods in real algebraic geometry. In this paper we explain many important methods from real algebraic geometry, we…
In computer algebra there are different ways of approaching the mathematical concept of functions, one of which is by defining them as solutions of differential equations. We compare different such approaches and discuss the occurring…
As soon as a new technology emerges, the education community explores its affordances and the possibilities to apply it in education. In this paper, we analyze sessions with ChatGPT around topics in basic Linear Algebra. We reflect the…
We present axiomatisations for a number of partial function signatures that include domain restriction, modelled as a right normal band operation. Other operations considered are override and update, difference, minus, intersection,…
We present a form of algebraic reasoning for computational objects which are expressed as graphs. Edges describe the flow of data between primitive operations which are represented by vertices. These graphs have an interface made of…
Linear algebraic expressions are the essence of many computationally intensive problems, including scientific simulations and machine learning applications. However, translating high-level formulations of these expressions to efficient…
This short expository paper outlines applications of computer algebra to the implication problem of conditional independence for Gaussian random variables. We touch on certificates for validity and invalidity of inference rules from 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…
Stochastic models are highly relevant tools in science, engineering, and society. Recent work suggests emerging quantum computing technologies can substantially decrease the memory requirements for simulating stochastic models. Here we show…
If we define classical foundational concepts constructively, and introduce non-algorithmic effective methods into classical mathematics, then we can bridge the chasm between truth and provability, and define computational methods that are…
This research started with an algebra for reasoning about rely/guarantee concurrency for a shared memory model. The approach taken led to a more abstract algebra of atomic steps, in which atomic steps synchronise (rather than interleave)…
A tutorial of the Mathematica package CGAlgebra, for conformal geometric algebra calculations is presented. Using rule-based programming, the 5-dimensional conformal geometric algebra is implemented and defined functions simplify the…
Abstraction is a fundamental tool for reasoning about complex systems. Program abstraction has been utilized to great effect for analyzing deterministic programs. At the heart of program abstraction is the relationship between a concrete…
Symbolic integration is an important module of a typical Computer Algebra System. As for now, Mathematica, Matlab, Maple and Sage are all mainstream CAS. They share the same framework for symbolic integration at some points. In this book…
We observe that the vocabulary used to construct the "answer" to problems in computer algebra can have a dramatic effect on the computational complexity of solving that problem. We recall a formalization of this observation and explain the…
These lecture notes focus on some numerical linear algebra algorithms in scientific computing. We assume that students are familiar with elementary linear algebra concepts such as vector spaces, systems of equations, matrices, norms,…
Real algebraic geometry adapts the methods and ideas from (complex) algebraic geometry to study the real solutions to systems of polynomial equations and polynomial inequalities. As it is the real solutions to such systems modeling…
Mathematical methods provide useful framework for the analysis and design of complex systems. In newer contexts such as biology, however, there is a need to both adapt existing methods as well as to develop new ones. Using a combination of…