Nathaniel Dean asks the following: Is it possible to find four nonconcyclic points on the parabola y = x2 such that each of the six distances between pairs of points is rational? We demonstrate that there is a correspondence between all rational points satisfying this condition and orbits under a particular group action of rational points on a fiber product of (three copies of) an elliptic surf...

A new rational Krylov method for the efficient solution of nonlinear eigenvalue problems, A(λ)x = 0, is proposed. This iterative method, called fully rational Krylov method for nonlinear eigenvalue problems (abbreviated as NLEIGS), is based on linear rational interpolation and generalizes the Newton rational Krylov method proposed in [R. Van Beeumen, K. Meerbergen, and W. Michiels, SIAM J. Sci....

Abstra t. We introdu e a new omputationally e ient algorithm for onstru ting near optimal rational approximations of large (onedimensional) data sets. In ontrast to wavelet-type approximations, these new approximations are e e tively shift invariant. We note that the omplexity of urrent algorithms for omputing near optimal rational approximations prevents their use for large data sets. In order...

The multiplicative theory of a set of numbers is the theory of the structure of that set with (solely) the multiplication operation (that set is taken to be multiplicative, i.e., closed under multiplication). In this paper we study the multiplicative theories of the complex, real and (positive) rational numbers. These theories are known to be decidable; here we present some explicit axiomatizat...

Simple geometric objects and transformations appear in representations and algorithms of geometric facilities in computer applications such as modelling, robotics, or graphics. Usually, these applications only support objects and transformations fully describable by rational parameters, and a computer display of points of the objects at least implicitly requires points with rational coordinates...

In this paper we exploit Answer Set Programming (ASP) for reasoning in a rational extension SROEL(⊓, ×)R T of the low complexity description logic SROEL(⊓,×), which underlies the OWL EL ontology language. In the extended language, a typicality operator T is allowed to define concepts T(C) (typical C’s) under a rational semantics. It has been proven that instance checking under rational entailme...

We study pairs (f,Γ) consisting of a non-Archimedean rational function f and a finite set of vertices Γ in the Berkovich projective line, under a certain stability hypothesis. We prove that stability can always be attained by enlarging the vertex set Γ. As a byproduct, we deduce that meromorphic maps preserving the fibers of a rationally-fibered complex surface are algebraically stable after a ...

Following Meyer’s argument [Phys. Rev. Lett. 83, 3751 (1999)] the set of all directions in space is replaced by the dense subset of rational directions. The result conflicts with Euclidean geometry. Meyer’s claim [1] that “finite precision measurement nullifies the KochenSpecker theorem” (that is, makes it irrelevant to physics) and some of its generalizations [2] have caused considerable contr...