Related papers: Dynamics on Trees of Spheres
This note is devoted to the definition of moduli spaces of rational tropical curves with n marked points. We show that this space has a structure of a smooth tropical variety of dimension n-3. We define the Deligne-Mumford compactification…
This article introduces moduli spaces of coloured graphs on which Feynman amplitudes can be viewed as 'discrete' volume densities. The basic idea behind this construction is that these moduli spaces decompose into disjoint unions of open…
We introduce and investigate the concept of Stratified Algebra, a new algebraic framework equipped with a layer-based structure on a vector space. We formalize a set of axioms governing intra-layer and inter-layer interactions, study their…
We give four applications of Zalcman's lemma to the dynamics of rational maps on the Riemann sphere: a parameter analogue of a proof of the density of repelling cycles in the Julia sets;similarity between the Mandelbrot set and the Julia…
In our previous work, we introduced the random $k$-cut number for rooted graphs. In this paper, we show that the distribution of the $k$-cut number in complete binary trees of size $n$, after rescaling, is asymptotically a periodic function…
We give a geometric interpretation of color-kinematics duality between tree-level scattering amplitudes of gauge and gravity theories. Using their representation as intersection numbers we show how to obtain Bern-Carrasco-Johansson…
This survey paper begins with the description of the duality between arc systems and ribbon graphs embedded in a punctured surface. Then we explain how to cellularize the moduli space of curves in two different ways: using Jenkins-Strebel…
We analyze a real one-parameter family of quasiconformal deformations of a hyperbolic rational map known as {\em spinning}. We show that under fairly general hypotheses, the limit of spinning either exists and is unique, or else converges…
We study discrete Laplacians on two-dimensional lattices under modular iterations, focusing on the emergence of nontrivial large-scale patterns. While purely binary or constant modular sequences quickly collapse into strict periodicity, the…
Regular tree grammars and regular path expressions constitute core constructs widely used in programming languages and type systems. Nevertheless, there has been little research so far on reasoning frameworks for path expressions where node…
Many systems of interest in science and engineering are made up of interacting subsystems. These subsystems, in turn, could be made up of collections of smaller interacting subsystems and so on. In a series of papers David Spivak with…
We consider the multifractal formalism for the dynamics of semigroups of rational maps on the Riemann sphere and random complex dynamical systems. We elaborate a multifractal analysis of level sets given by quotients of Birkhoff sums with…
Motivated by problems in the neural networks setting, we study moduli spaces of double framed quiver representations and give both a linear algebra description and a representation theoretic description of these moduli spaces. We define a…
The spacetime discreteness of causal set theory has enabled the formulation of novel spacetime dynamics. In these so-called "growth" dynamics, a causal set spacetime is generated probabilistically by means of a random walk on certain tree…
In this paper we investigate undirected discrete graphical tree models when all the variables in the system are binary, where leaves represent the observable variables and where all the inner nodes are unobserved. A novel approach based on…
Many complex systems have natural representations as multi-layer networks. While these formulations retain more information than standard single-layer network models, there is not yet a fully developed theory for computing network metrics…
For a finite dimensional vector space equipped with a $\mathbb C$-algebra structure, one can define rational maps using the algebraic structure. In this paper, we describe the growth of the degree sequences for this type of rational maps.
Our aim is to give some insights about how to approach the formal description of situations where one has to conciliate several contradictory statements, rules, laws or ideas. We show that such a conciliation structure can be naturally…
We proved the so called complex bounds for multimodal, infinitely renormalizable analytic maps with bounded combinatorics: deep renormalizations have polynomial-like extensions with definite modulus. The complex bounds is the first step to…
We present a new technique in order to quantify the dynamics of spatially extended systems. Using a test on the existence of unstable periodic orbits, we identify intermediate spatial scales, wherein the dynamics is characterized by maximum…