Related papers: Computational Euclid
Projection methods are popular algorithms for iteratively solving feasibility problems in Euclidean or even Hilbert spaces. They employ (selections of) nearest point mappings to generate sequences that are designed to approximate a point in…
When considering geometry, one might think of working with lines and circles on a flat plane as in Euclidean geometry. However, doing geometry in other spaces is possible, as the existence of spherical and hyperbolic geometry demonstrates.…
A coherent mathematical overview of computation and its generalisations is described. This conceptual framework is sufficient to comfortably host a wide range of contemporary thinking on embodied computation and its models.
Mathematical thinking is a fundamental aspect of human cognition. Cognitive scientists have investigated the mechanisms that underlie our ability to thinking geometrically and numerically, to take two prominent examples, and developmental…
In the directed setting, the spaces of directed paths between fixed initial and terminal points are the defining feature for distinguishing different directed spaces. The simplest case is when the space of directed paths is homotopy…
In this era of diverse and heterogeneous computer architectures, the programmability issues, such as productivity and portable efficiency, are crucial to software development and algorithm design. One way to approach the problem is to step…
Quantum computers hold great promise, but it remains a challenge to find efficient quantum circuits that solve interesting computational problems. We show that finding optimal quantum circuits is essentially equivalent to finding the…
Traditional manifold learning algorithms assumed that the embedded manifold is globally or locally isometric to Euclidean space. Under this assumption, they divided manifold into a set of overlapping local patches which are locally…
A non-Euclidean generalization of conditional expectation is introduced and characterized as the minimizer of expected intrinsic squared-distance from a manifold-valued target. The computational tractable formulation expresses the…
Continual learning systems operating in fixed-dimensional spaces face a fundamental geometric barrier: the flat manifold problem. When experience is represented as a linear trajectory in Euclidean space, the geodesic distance between…
A unitary (Euclidean) representation of a quiver is given by assigning to each vertex a unitary (Euclidean) vector space and to each arrow a linear mapping of the corresponding vector spaces. We recall an algorithm for reducing the matrices…
We introduce a generalization of the Euclidean algorithm for rings equipped with an involution, and completely enumerate all isomorphism classes of orders over definite, rational quaternion algebras equipped with an orthogonal involution…
This is the first of a series of papers in which we study deep computations (ultracomputations) and deep iterates, formalizing the ideas of "asymptotic limit" of computations and compositional iterates, respectively. In this first paper of…
Recent algorithmic advances in algebraic automata theory drew attention to semigroupoids (semicategories). These are mathematical descriptions of typed computational processes, but they have not been studied systematically in the context of…
We define the simplest log-euclidean geometry. This geometry exposes a difficulty hidden in Hilbert's list of axioms presented in his "Grundlagen der Geometrie". The list of axioms appears to be incomplete if the foundations of geometry are…
We introduce group surface codes, which are a natural generalization of the $\mathbb{Z}_2$ surface code, and equivalent to quantum double models of finite groups with specific boundary conditions. We show that group surface codes can be…
Cylindrical algebraic decomposition is a classical construction in real algebraic geometry. Although there are many algorithms to compute a cylindrical algebraic decomposition, their practical performance is still very limited. In this…
This article provides a pedagogically oriented introduction to geometric (Clifford) calculus on pseudo-Riemannian manifolds. Unlike usual approaches to the topic, which rely on embedding the geometric algebra either within a tensor algebra…
The aim of this paper is to develop a new axiomatization of planar geometry by reinterpreting the original axioms of Euclid. The basic concept is still that of a line segment but its equivalent notion of betweenness is viewed as a…
Learning faithful graph representations as sets of vertex embeddings has become a fundamental intermediary step in a wide range of machine learning applications. The quality of the embeddings is usually determined by how well the geometry…