Related papers: Three arithmetic sites
10 years ago or so Bill Helton introduced me to some mathematical problems arising from semidefinite programming. This paper is a partial account of what was and what is happening with one of these problems, including many open questions…
We introduce and study arithmetic polygons. We show that these arithmetic polygons are connected to triples of square pyramidal numbers. For every odd $N\geq3$, we prove that there is at least one arithmetic polygon with $N$ sides. We also…
Bicritical rational functions -- those with precisely two critical points -- include the well-studied families of unicritical polynomials and quadratic rational functions. In this article we lay out general foundations for studying…
An analytical approach to convolution of functions, which appear in perturbative calculations, is discussed. An extended list of integrals is presented.
This paper has two parts. In the first part we construct arithmetic models of Bost-Connes systems for arbitrary number fields, which has been an open problem since the seminal work of Bost and Connes [3]. In particular our construction…
An arithmetical structure on a graph is given by a labeling of the vertices which satisfies certain divisibility properties. In this note, we look at several families of graphs and attempt to give counts on the number of arithmetical…
The increasing data availability and imported analyzing tools from computer science and physical science have sharply changed traditional methodologies of social sciences, leading to a new branch named computational socioeconomics that…
Consider a cohomologically hyperbolic birational self-map defined over the algebraic numbers, for example, a birational self-map in dimension two with the first dynamical degree greater than one, or in dimension three with the first and the…
The study of arboreal Galois representations (that is, Galois groups arising from iteration of polynomial and rational functions) originated with work of Odoni in the 1980s. Beginning in the early 2000s it underwent a period of renewed…
We present all Belyi maps P^1(C) -> P^1(C) having almost simple primitive monodromy groups (not isomorphic to A_n or S_n) containing rigid and rational generating triples of degree between 50 and 250. This also leads to new polynomials…
The arithmetic of the natural numbers can be extended to arithmetic operations on planar binary trees. This gives rise to a non-commutative arithmetic theory. In this exposition, we describe this arithmetree, first defined by Loday, and…
The topological method for the reconstruction of dynamics from time series [K. Mischaikow, M. Mrozek, J. Reiss, A. Szymczak. Construction of Symbolic Dynamics from Experimental Time Series, Physical Review Letters, 82 (1999), 1144-1147] is…
In this paper, rough approximations of Cayley graphs are studied and rough edge Cayley graphs are introduced. Furthermore, a new algebraic definition called pseudo-Cayley graphs containing Cayley graphs is proposed. Rough approximation is…
We introduce a category of dual pairs of finite locally free algebras over a ring. This gives an efficient way to represent finite locally free commutative group schemes. We give a number of algorithms to compute with dual pairs of…
This article is a review of theoretical advances in the research field of algebraic geometry and Bayesian statistics in the last two decades. Many statistical models and learning machines which contain hierarchical structures or latent…
We give an explicit description of the Barr- and Diaconescu covers of the arithmetic site, which are relevant to cohomology. Further, we construct the arithmetic site as the commutative shadow of a non-commutative topological space.
We give a survey of algorithms for computing topological invariants of semi-algebraic sets with special emphasis on the more recent developments in designing algorithms for computing the Betti numbers of semi-algebraic sets. Aside from…
Brooks and Makover introduced an approach to random Riemann surfaces based on associating a dense set of them - Belyi surfaces - with random cubic graphs. In this paper, using Bollobas model for random regular graphs, we examine the…
We prove that arboreal Galois extensions of number fields are never abelian for post-critically finite rational maps and non-preperiodic base points. For polynomials, this establishes a new class of known cases of a conjecture of…
We extract a two-dimensional dynamical system from the theorems of Pappus and Steiner in classical projective geometry. We calculate an explicit formula for this system, and study its elementary geometric properties. Then we use Artin…