Related papers: Riemann surfaces for integer counting processes
We detail the theory of Discrete Riemann Surfaces. It takes place on a cellular decomposition of a surface, together with its Poincar\'e dual, equipped with a discrete conformal structure. A lot of theorems of the continuous theory follow…
A procedure for counting the number of eigenvalues of a matrix in a region surrounded by a closed curve is presented. It is based on the application of the residual theorem. The quadrature is performed by evaluating the principal argument…
Processes having the same bridges are said to belong to the same reciprocal class. In this article we analyze reciprocal classes of Markov counting processes by identifying their reciprocal invariants and we characterize them as the set of…
We show that for a large class of marked point processes there exists a random measure m with the predictable representation property such that iterated integrals with respect to m span the space of square integrable random variables.
This work uses a simple quantum computer model to discuss the randomness of bit strings originated from integer sequences. The considered quantum computer model has three elements: a processing unit responsible for a mathematical operation,…
Exploiting the variational interpretation of kernel interpolation we exhibit a direct connection between interpolation and regression, where interpolation appears as a limiting case of regression. By applying this framework to point clouds…
We present an algorithm for computing equations of canonically embedded Riemann surfaces with automorphisms. A variant of this algorithm with many heuristic improvements is used to produce equations of Riemann surfaces $X$ with large…
We extend a classical approximation result of harmonic functions in planar domains due to Bernstein and Walsch to the setting of harmonic functions in Riemann surfaces. This result gives an exact characterization of the rate at which a…
The goal of this work is to formally abstract a Markov process evolving in discrete time over a general state space as a finite-state Markov chain, with the objective of precisely approximating its state probability distribution in time,…
For any positive integer $r$, we construct a smooth complex projective rational surface which has at least $r$ real forms not isomorphic over $\mathbb{R}$.
We equip many non compact non simply connected surfaces with smooth Riemannian metrics whose isoperimetric profile is smooth, a highly non generic property. The computation of the profile is based on a calibration argument, a rearrangement…
This paper develops a theory of clustering and coding which combines a geometric model with a probabilistic model in a principled way. The geometric model is a Riemannian manifold with a Riemannian metric, ${g}_{ij}({\bf x})$, which we…
Edge-contraction operations form an effective tool in various graph enumeration problems, such as counting Grothendieck's dessins d'enfants and simple and double Hurwitz numbers. These counting problems can be solved by a mechanism known as…
We study qualitative aspects of the Welschinger-like $\mathbb Z$-valued count of real rational curves on primitively polarized real $K3$ surfaces. In particular, we prove that with respect to the degree of the polarization, at logarithmic…
Translation surfaces can be defined in an elementary way via polygons, and arise naturally in in the study of various basic dynamical systems. They can also be defined as Abelian differentials on Riemann surfaces, and have moduli spaces…
We count certain abelian surfaces with potential quaternionic multiplication defined over a number field $K$ by counting points of bounded height on some genus zero Shimura curves.
Tropical geometry is a piecewise linear "shadow" of algebraic geometry. It allows for the computation of several cohomological invariants of an algebraic variety. In particular, its application to enumerative algebraic geometry led to…
Kernel embeddings have emerged as a powerful tool for representing probability measures in a variety of statistical inference problems. By mapping probability measures into a reproducing kernel Hilbert space (RKHS), kernel embeddings enable…
Gaussian processes are widely employed as versatile modelling and predictive tools in spatial statistics, functional data analysis, computer modelling and diverse applications of machine learning. They have been widely studied over…
The paper presents an analog of the old result by the author and V. Voevodsky, according to which a Riemann surface admits a conformal structure, defined by an equilateral triangulation, if and only if the corresponding algebraic curve can…