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Nicolas Doyon

Publications and source records attributed to Nicolas Doyon.

10 recordsLinked to original sources
arXiv2026

On the convergence of doubly stochastic Markov chains

We characterize the asymptotic behavior of time-homogeneous doubly stochastic Markov chains. Our investigation revolves around understanding the dynamics of products of doubly stochastic matrices, which in turn allows us to fully characterize three distinct behaviors: cyclicity, convergence towards a special equilibrium matrix, and divergence. Notably, we introduce a novel and comprehensive sufficient condition for the convergence of an infinite product of doubly stochastic matrices.

math.PR↗
arXiv2025

On the minimal length of addition chains

We denote by $\ell(n)$ the minimal length of an addition chain leading to $n$ and we define the counting function $$ F(m,r):=\#\left\{n\in[2^m, 2^{m+1}):\ell(n)\le m+r\right\}, $$ where $m$ is a positive integer and $r\ge 0$ is a real number. We show that for $0< c<\log 2$ and for any $\varepsilon>0$, we have as $m\to \infty$, $$ F\left(m,\frac{cm}{\log m}\right)<\exp\left(cm+\frac{\varepsilon m\log\log m}{\log m}\right) $$ and $$ F\left(m,\frac{cm}{\log m}\right)>\exp\left(cm-\frac{(1+\varepsilon)cm\log\log m}{\log m}\right). $$ This extends a result of Erd\H{o}s which says that for almost all $n$, as $n\to\infty$, $$ \ell(n)=\frac{\log n}{\log 2}+\left(1+o(1)\right)\frac{\log n}{\log \log n}. $$

math.NT↗
arXiv2022

The determining role of covariances in large networks of stochastic neurons

Biological neural networks are notoriously hard to model due to their stochastic behavior and high dimensionality. We tackle this problem by constructing a dynamical model of both the expectations and covariances of the fractions of active and refractory neurons in the network's populations. We do so by describing the evolution of the states of individual neurons with a continuous-time Markov chain, from which we formally derive a low-dimensional dynamical system. This is done by solving a moment closure problem in a way that is compatible with the nonlinearity and boundedness of the activation function. Our dynamical system captures the behavior of the high-dimensional stochastic model even in cases where the mean-field approximation fails to do so. Taking into account the second-order moments modifies the solutions that would be obtained with the mean-field approximation, and can lead to the appearance or disappearance of fixed points and limit cycles. We moreover perform numerical experiments where the mean-field approximation leads to periodically oscillating solutions, while the solutions of the second-order model can be interpreted as an average taken over many realizations of the stochastic model. Altogether, our results highlight the importance of including higher moments when studying stochastic networks and deepen our understanding of correlated neuronal activity.

q-bio.NC↗
arXiv2022

Classes of Holomorphic Multicomplex-Valued Functions Generated by Elliptic-Admissible Involutions

We classify and count the real-algebra involutions of the multicomplex algebra $\mathbb{M}\mathbb{C} (n)$ that map each element of its canonical monomial basis to a signed monomial, using a matrix model over $\mathbb F_2$. An \emph{elliptic-admissible pair} consists of such an involution $\sigma$ and a monomial unit $\mathbf{i}\in\mathbb{I}(n)$ satisfying $\mathbf{i^2}=-1$ and $\sigma(\mathbf{i})=-\mathbf{i}$. The fixed algebra of $\sigma$ is a real form for the complex structure $\mathcal{L}_{\mathbf{i}}$ defined by multiplication by $\mathbf{i}$, and the pair yields a Cauchy--Riemann system. We prove that its solution class depends only on $\mathbf{i}$, not on $\sigma$, and is exactly the class of mappings holomorphic with respect to $\mathcal{L}_{\mathbf{i}}$. Multicomplex holomorphy is recovered as the intersection of the classes associated with the elementary generators. We obtain the analogous characterization for anti-holomorphic classes, describe twisted systems intertwining two such complex structures, and prove that every $\mathbf{i}{}$-holomorphic or $\mathbf{i}$-anti-holomorphic mapping is componentwise harmonic (solutions to Laplace's equation).

math.RA↗
arXiv2022

Beyond Wilson-Cowan dynamics: oscillations and chaos without inhibition

Fifty years ago, Wilson and Cowan developed a mathematical model to describe the activity of neural populations. In this seminal work, they divided the cells in three groups: active, sensitive and refractory, and obtained a dynamical system to describe the evolution of the average firing rates of the populations. In the present work, we investigate the impact of the often neglected refractory state and show that taking it into account can introduce new dynamics. Starting from a continuous-time Markov chain, we perform a rigorous derivation of a mean-field model that includes the refractory fractions of populations as dynamical variables. Then, we perform bifurcation analysis to explain the occurance of periodic solutions in cases where the classical Wilson-Cowan does not predict oscillations. We also show that our mean-field model is able to predict chaotic behavior in the dynamics of networks with as little as two populations.

q-bio.NC↗
arXiv2021

Repetitions of multinomial coefficients and a generalization of Singmaster's conjecture

Given two integers $k\geq 2$ and $a>1$, let $N_k(a)$ stand for the number of multinomial coefficients, with $k$ terms, equal to $a$. We study the behavior of $N_k(a)$ and show that its average and normal orders are equal to $k(k-1)$. We also prove that $N_k(a)=O\left((\log a/\log\log a)^{k-1}\right)$ and make several propositions about extreme results regarding large values of $N_k(a)$.

math.NT↗
arXiv2019

Spectral dimension reduction of complex dynamical networks

Dynamical networks are powerful tools for modeling a broad range of complex systems, including financial markets, brains, and ecosystems. They encode how the basic elements (nodes) of these systems interact altogether (via links) and evolve (nodes' dynamics). Despite substantial progress, little is known about why some subtle changes in the network structure, at the so-called critical points, can provoke drastic shifts in its dynamics. We tackle this challenging problem by introducing a method that reduces any network to a simplified low-dimensional version. It can then be used to describe the collective dynamics of the original system. This dimension reduction method relies on spectral graph theory and, more specifically, on the dominant eigenvalues and eigenvectors of the network adjacency matrix. Contrary to previous approaches, our method is able to predict the multiple activation of modular networks as well as the critical points of random networks with arbitrary degree distributions. Our results are of both fundamental and practical interest, as they offer a novel framework to relate the structure of networks to their dynamics and to study the resilience of complex systems.

physics.soc-ph↗
arXiv2016

TE-TM Electromagnetic modes and states in quantum physics

We propose, as another pedagogical approach, a quantification of the e.m. field better adapted in isolated systems, a quantification of e.m. fields in finite-spacetime which does not rely explicitly on the notion of photon, nor on the general application of gauge. Being based on the development of e.m. field in TE-TM modes and states, it obeys the limit conditions of a finite-spacetime which follows from the solution of an eigenvalue equation and allows to interpret more profoundly some phenomenons, notably: the equivalence of a Pauli principle in stationary TE-TM states, the notion of non-locality in a radiation field, a modified form of the De Broglie analysis and the notion of wave-paquets in finite-spacetime.

quant-ph↗
arXiv2015

$\hbar$ as a Physical Constant of Classical Optics and Electrodynamics

The Planck constant ($\hbar$) plays a pivotal role in quantum physics. Historically, it has been proposed as postulate, part of a genius empirical relationship $E=\hbar ω$ in order to explain the intensity spectrum of the blackbody radiation for which classical electrodynamic theory led to an unacceptable prediction: The ultraviolet catastrophe. While the usefulness of the Planck constant in various fields of physics is undisputed, its derivation (or lack of) remains unsatisfactory from a fundamental point of view. In this paper, the analysis of the blackbody problem is performed with a series expansion of the electromagnetic field in terms of TE, TM modes in a metallic cavity with small losses, that leads to developing the electromagnetic fields in a \textit{complete set of orthonormal functions}. This expansion, based on coupled power theory, maintains both space and time together enabling modeling of the blackbody's evolution toward equilibrium. Reaching equilibrium with a multimodal waveguide analysis brings into consideration the coupling between modes in addition to absorption and emission of radiation. The properties of the modes, such as spectral broadening, losses and lifetime, then progressively become independent of frequency and explains how equilibrium is allowed in good conductor metallic cavities. Based on the free electron relaxation time in gold, a value of $\hbar = 1.02 \times 10^{-34}$ J$\cdot$s for the reduced Planck constant is found and the uncertainty principle is also emerging from this \textit{a priori} classical study. The Planck constant is then obtained no longer as an ad hoc addition but as a natural consequence of the analysis taking boundary conditions into account as into optical resonators. That analysis based on finite-spacetime paradigm, also shine new light on the notion of decoherence in classical optics and electrodynamics.

physics.optics↗