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Many statistical models are algebraic in that they are defined by polynomial constraints or by parameterizations that are polynomial or rational maps. This opens the door for tools from computational algebraic geometry. These tools can be…
The classical approach to solvability of a mathematical problem is to define a method which includes certain rules of operation or algorithms. Then using the defined method, one can show that some problems are solvable or not solvable or…
Estimating free energy differences quantifies thermodynamic preferences in molecular interactions, which is central to chemistry and drug discovery. Despite fruitful progress, existing methods still face key limitations: classical…
Cylindrical Algebraic Decomposition (CAD) algorithms typically produce a decomposition adapted to a finite family of semi-algebraic sets $\mathcal{F}$ (i.e. every member of $\mathcal{F}$ is a union of cells). Different algorithms may…
Addressing irregular cutting and packing (C&P) optimization problems poses two distinct challenges: the geometric challenge of determining whether or not an item can be placed feasibly at a certain position, and the optimization challenge…
Geometric programming (GP) is a well-known optimization tool for dealing with a wide range of nonlinear optimization and engineering problems. In general, it is assumed that the parameters of a GP problem are deterministic and accurate.…
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…
Canonical Polyadic Decomposition (CPD) of a third-order tensor is a minimal decomposition into a sum of rank-$1$ tensors. We find new mild deterministic conditions for the uniqueness of individual rank-$1$ tensors in CPD and present an…
Stable computational algorithms for the approximate solution of the Cauchy problem for nonstationary problems are based on implicit time approximations. Computational costs for boundary value problems for systems of coupled multidimensional…
The cylindrical algebraic covering method was originally proposed to decide the satisfiability of a set of non-linear real arithmetic constraints. We reformulate and extend the cylindrical algebraic covering method to allow for checking the…
Computing the topology of an algebraic plane curve $\mathcal{C}$ means to compute a combinatorial graph that is isotopic to $\mathcal{C}$ and thus represents its topology in $\mathbb{R}^2$. We prove that, for a polynomial of degree $n$ with…
The computation of triangular decompositions are based on two fundamental operations: polynomial GCDs modulo regular chains and regularity test modulo saturated ideals. We propose new algorithms for these core operations relying on modular…
There are many numerical methods for solving partial different equations (PDEs) on manifolds such as classical implicit, finite difference, finite element, and isogeometric analysis methods which aim at improving the interoperability…
We present a novel certified and complete algorithm to compute arrangements of real planar algebraic curves. It provides a geometric-topological analysis of the decomposition of the plane induced by a finite number of algebraic curves in…
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…
Logic-based Benders decomposition (LBBD) is a substantial generalization of classical Benders decomposition that, in principle, allows the subproblem to be any optimization problem rather than specifically a linear or nonlinear programming…
The purpose of this note is to survey a methodology to solve systems of polynomial equations and inequalities. The techniques we discuss use the algebra of multivariate polynomials with coefficients over a field to create large-scale linear…
The evaluation of robustness and reliability of realistic structures in the presence of uncertainty involves costly numerical simulations with a very high number of evaluations. This motivates model order reduction techniques like the…
Simplifying the geometry of a CAD model using defeaturing techniques enables more efficient discretisation and subsequent simulation for engineering analysis problems. Understanding the effect this simplification has on the solution helps…
This paper shows how to build a formal analytical solution for a differential equation of arbitrary order and with variable coefficients. It proofs that the most known approximated solutions for such a problem can be derived from the…