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A new projection operator based on cylindrical algebraic decomposition (CAD) is proposed. The new operator computes the intersection of projection factor sets produced by different CAD projection orders. In other words, it computes the gcd…
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…
One of basic difficulties of machine learning is handling unknown rotations of objects, for example in image recognition. A related problem is evaluation of similarity of shapes, for example of two chemical molecules, for which direct…
We propose a new geometric method of IR factorization in sector decomposition. The problem is converted into a set of problems in convex geometry. The latter problems are solved using algorithms in combinatorial geometry. This method…
We present a new methodology for utilising machine learning technology in symbolic computation research. We explain how a well known human-designed heuristic to make the choice of variable ordering in cylindrical algebraic decomposition may…
Traditional TCAD simulation has succeeded in predicting and optimizing the device performance; however, it still faces a massive challenge - a high computational cost. There have been many attempts to replace TCAD with deep learning, but it…
While invaluable for many computer vision applications, decomposing a natural image into intrinsic reflectance and shading layers represents a challenging, underdetermined inverse problem. As opposed to strict reliance on conventional…
Constructing complex computation from simpler building blocks is a defining problem of computer science. In algebraic automata theory, we represent computing devices as semigroups. Accordingly, we use mathematical tools like products and…
A novel high-order numerical scheme is proposed to compute the covariant derivative, particularly for divergence and curl, on any curved surface. The proposed scheme does not require the construction of a curved axis or metric tensor, which…
The purpose of this paper is to explore the question "to what extent could we produce formal, machine-verifiable, proofs in real algebraic geometry?" The question has been asked before but as yet the leading algorithms for answering such…
Current quantum computers can only solve optimization problems of a very limited size. For larger problems, decomposition methods are required in which the original problem is broken down into several smaller sub-problems. These are then…
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…
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…
Validation is a major challenge in differentiable programming. The state of the art is based on algorithmic differentiation. Consistency of first-order tangent and adjoint programs is defined by a well-known first-order differential…
We survey both old and new developments in the theory of algorithms in real algebraic geometry -- starting from effective quantifier elimination in the first order theory of reals due to Tarski and Seidenberg, to more recent algorithms for…
We show that topological phases include disordered materials if the underlying invariant is interpreted as originating from coarse geometry. This coarse geometric framework, grounded in physical principles, offers a natural setting for the…
A fast algorithm (linear in the degrees of freedom) for the solution of linear variable-coefficient rational-order fractional integral and differential equations is described. The approach is related to the ultraspherical method for…
Our topic is the use of machine learning to improve software by making choices which do not compromise the correctness of the output, but do affect the time taken to produce such output. We are particularly concerned with computer algebra…
In this paper, a new triangular decomposition algorithm is proposed for ordinary differential polynomial systems, which has triple exponential computational complexity. The key idea is to eliminate one algebraic variable from a set of…
The concept of comprehensive triangular decomposition (CTD) was first introduced by Chen et al. in their CASC'2007 paper and could be viewed as an analogue of comprehensive Grobner systems for parametric polynomial systems. The first…