Related papers: Constructing an expanding metric for dynamical sys…

Embedding into hyperbolic space is emerging as an effective representation technique for datasets that exhibit hierarchical structure. This development motivates the need for algorithms that are able to effectively extract knowledge and…

We prove that unicritical polynomials $f(z)=z^d+c$ which are semihyperbolic, i.e., for which the critical point $0$ is a non-recurrent point in the Julia set, are uniformly expanding on the Julia set with respect to the metric $\rho(z)…

These notes develop aspects of perturbation theory of matrices related to so-called diagonalisation schemes. Primary focus is on constructive tools to derive asymptotic expansions for small/large parameters of eigenvalues and…

This paper investigates the notion of learning user and item representations in non-Euclidean space. Specifically, we study the connection between metric learning in hyperbolic space and collaborative filtering by exploring Mobius…

We find the explicit local models of isolated singularities of conformal hyperbolic metrics by Complex Analysis, which is interesting in its own and could potentially be extended to high-dimensional case.

This study aims at finding a method for constructing molecular dynamics like models using the formalism of cellular automata for fast simulation of fluid dynamic systems (including compressible phenomena). In as much as the results…

Let $\Sigma$ be a compact, orientable surface of negative Euler characteristic, and let $h$ be a complete hyperbolic metric on $\Sigma$. A geodesic curve $\gamma$ in $\Sigma$ is filling, if it cuts the surface into topological disks and…

A problem based on the Extended Euclidean Algorithm applied to a class of polynomials with many factors is presented and believed to be hard. If so, it is a one-way function well suited for applications in digital signicatures.

In this paper we introduce the notion of parabolic-like mapping, which is an object similar to a polynomial-like mapping, but with a parabolic external class, i.e. an external map with a parabolic fixed point. We prove a straightening…

Despite broad interest in self-organizing systems, there are few quantitative, experimentally-applicable criteria for self-organization. The existing criteria all give counter-intuitive results for important cases. In this Letter, we…

We estimate the upper box and Hausdorff dimensions of the Julia set of an expanding semigroup generated by finitely many rational functions, using the thermodynamic formalism in ergodic theory. Furthermore, we show Bowen's formula, and the…

Discrete dynamical systems defined by the iteration of a polynomial map of the unit simplex to itself appear in the context of population genetic systems evolving under mutation, recombination and weak selection. Although exceptional…

Let $K$ be a finite extension of the field $\mathbb{Q}_p$ of $p$-adic numbers. A rational map $\phi\in K(z)$ of degree at least $2$ is subhyperbolic if each critical point in the $\mathbb{C}_p$-Julia set of $\phi$ is eventually periodic. We…

Hyperbolic space is a natural setting for mining and visualizing data with hierarchical structure. In order to compute a hyperbolic embedding from comparison or similarity information, one has to solve a hyperbolic distance geometry…

We produce algorithms to detect whether a complex affine variety computed and presented numerically by the machinery of numerical algebraic geometry corresponds to an associated component of a polynomial ideal.

We describe an algorithm that, given a k-tuple of permutations representing the monodromy of a rational map, constructs an arbitrarily precise floating-point complex approximation of that map. We then explain how it has been used to study a…

$ $Deriving a robot's equation of motion typically requires placing multiple coordinate frames, commonly using the Denavit-Hartenberg convention to express the kinematic and dynamic relationships between segments. This paper presents an…

The goal of this paper is to exhibit and analyze an algorithm that takes a given closed orientable hyperbolic surface and outputs an explicit Dirichlet domain. The input is a fundamental polygon with side pairings. While grounded in…

We study large uniform random maps with one face whose genus grows linearly with the number of edges. They can be seen as a model of discrete hyperbolic geometry. In the past, several of these hyperbolic geometric features have been…

This paper describes a metric for measuring the success of a complex system composed of agents performing autonomous behaviours. Because of the difficulty in evaluating such systems, this metric will help to give an initial indication as to…