arXiv subjects

Jérôme Michaud

Publications and source records attributed to Jérôme Michaud.

11 recordsLinked to original sources

arXiv2026

Cultural, Structural, and Mediated Routes to Polarization in a Nonlinear Mean-Field Model

Collective polarization can arise through mechanisms acting at different stages of social communication, but similar polarized outcomes need not imply dynamically equivalent processes. We introduce a nonlinear two-group mean-field model that distinguishes three routes to polarization: individual content filtering, assortativity, and receiver-side message tailoring. Sender-side reformulation is included as a complementary mediation process that transforms source signals before they are mixed. These message transformations are motivated by AI-mediated cultural transmission, in which intermediaries can reformulate messages or adapt them to receivers. Using common and difference coordinates, we derive general conditions for instabilities toward consensus and balanced polarization and analyze the resulting stationary branches in the full two-dimensional phase plane. Individual filtering can generate polarization through state-dependent updating, assortativity can preserve group differences in the social field, and receiver tailoring can create a polarizing feedback even under complete mixing. By contrast, sender-side reformulation affects the polarization mode only when assortativity preserves differentiated source signals. Although these mechanisms can produce identical local instability conditions or balanced-branch amplitudes, they remain distinguishable through transverse stability, asymmetric equilibria, multistability, and basin geometry. In particular, balanced polarization may emerge as a saddle before becoming a stable attractor, while asymmetric saddles organize the boundary between consensus and polarization basins. These results show that collective polarization depends not only on feedback strength, but also on where feedback enters the communication process.

physics.soc-ph↗

arXiv2026

Largest-dihedral-angle bisection algorithm does not preserve mesh regularity for tetrahedral partitions

We present a fixed nondegenerate tetrahedron with a nested largest-dihedral-anglebisection branch that collapses onto a unit edge. The selected dihedral angles to bisect are uniquely largest ones at every bisection step. Some face angles of the branch elements tend to zero, the corresponding diameter-to-inradius ratio diverges, and the diameters of the elements produced do not tend to zero, although every dihedral angle stays uniformly away from both 0 and π and all face and dihedral angles satisfy a uniform maximum-angle bound. The proof is based on an exact invariant-range argument. This example shows that the largest-dihedral-angle bisection algorithm can produce degenerating tetrahedral partitions.

math.NA↗

arXiv2026

On Triangulations Generated by the Largest-Angle $n$-Section Algorithm

We define a mesh refinement algorithm based on the rule of dividing the largest angles of triangular elements of planar partitions in focus into $n$ equal parts, and analyse the (geometric) properties of triangulations generated by this technique. This largest-angle $n$-section rule is compared with the classical longest-edge $n$-section rule, where it is the longest edges which are split into $n$ equal parts. The longest-edge bisection and trisection are known to produce nondegenerate triangulations (possibly with hanging nodes), but the longest-edge $n$-sections with $n\geq 4$ always produce (infinite) sequences of triangles with minimum angles tending to zero (moreover, their relevant maximum angles tend to $π$), thus breaking the minimum and maximum angle conditions. We show that this degeneration effect is not a consequence of $n$-section itself. For every $n\geq 2$, the largest-angle $n$-sections produce partitions satisfying the minimum angle condition (and, therefore, the maximum angle condition). More precisely, if the initial triangle has its smallest angle $γ_0>0$, then all descendant triangles have angles bounded below by $m_n=\min\left\{γ_0,\fracπ{3n}\right\},$ and, correspondingly, bounded above by $π-2m_n<π$. We also show that the recursive largest-angle $n$-section algorithm always produces a family of triangular partitions, i.e. the maximum diameter of level-$k$ descendants tends to zero as $k \to \infty$.

cs.CG↗

arXiv2026

Dynamics of the Longest-Edge Altitude Bisection Algorithm

We study a longest-edge based refinement scheme for triangulations, termed the longest-edge altitude bisection (LEAB), in which each triangle is subdivided by dropping the altitude from the vertex opposite to its longest edge. Using the normalized shape space of triangles introduced by Perdomo and Plaza in: Properties of triangulations obtained by the longest-edge bisection. \emph{Cent. Eur. J. Math.}, 12(12) (2014), 1796-1810, we show that LEAB admits a simple geometric description: the normalized left and right children of a triangle in focus are obtained by intersecting the geodesic of right triangles with rays issued from the endpoints of the longest edge and explicit formulas for the mappings are derived. This characterization implies an interesting observation that the associated refinement dynamics collapse the entire shape space onto the right-triangle geodesic in a single step and that every point on this geodesic is fixed. Two-sided bounds for the contraction of the mesh size (discretization parameter) are derived. Also, applications and limitations of the method are briefly discussed.

math.NA↗

arXiv2025

On the Orbits of Similarity Classes of Tetrahedra Generated by the Longest-Edge Bisection Algorithm

In this work, we study the dynamics of similarity classes of tetrahedra generated by the longest-edge bisection (LEB) algorithm. Building on the normalization strategy introduced by Perdomo and Plaza for triangles, we construct a canonical representation of tetrahedra in a normalized space embedded in the product of the hyperbolic half-plane and the hyperbolic half-space model. This representation allows us to define the left and right refinement maps, $Φ_L$ and $Φ_R$, acting on the space of normalized tetrahedral shapes, and to study their iterative orbits as discrete dynamical systems. Using these maps, we show that the orbit of the space-filling Sommerville tetrahedron contains only 4 similarity classes, 3 of which form an attractive cycle corresponding to the orbit of the path tetrahedron. We also show that small perturbations of elements in those orbits still lead to finite orbits. In addition, we study small perturbations of the regular tetrahedron and show that their orbits are also finite. Extensive numerical exploration of orbits for the other types of tetrahedra suggests that the LEB algorithm does not produce degenerating tetrahedra. Our framework provides a geometric and dynamical foundation for analyzing the shape evolution of tetrahedral meshes and offers a possible route toward an analytic proof of the non-degeneracy property for the tetrahedral partitions generated by the LEB refinements. This property is highly desired in e.g. the finite element methods (FEMs).

math.NA↗

arXiv2024

Opening the black box of language acquisition

Recent advances in large language models using deep learning techniques have renewed interest on how languages can be learned from data. However, it is unclear whether or how these models represent grammatical information from the learned languages. In addition, the models must be pre-trained on large corpora before they can be used. In this work, we propose an alternative, more transparent and cognitively plausible architecture for learning language. Instead of using deep learning, our approach uses a minimal cognitive architecture based on sequence memory and chunking. The learning mechanism is based on the principles of reinforcement learning. We test our architecture on a number of natural-like toy languages. Results show that the model can learn these artificial languages from scratch and extract grammatical information that supports learning. Our study demonstrates the power of this simple architecture and stresses the importance of sequence memory as a key component of the language learning process. Since other animals do not seem to have a faithful sequence memory, this may explain why only humans have developed complex languages.

cs.CL↗

arXiv2018

Social Influence with Recurrent Mobility with multiple options

In this paper, we discuss the possible generalizations of the Social Influence with Recurrent Mobility (SIRM) model developed in Phys. Rev. Lett. 112, 158701 (2014). Although the SIRM model worked approximately satisfying when US election was modelled, it has its limits: it has been developed only for two-party systems and can lead to unphysical behaviour when one of the parties has extreme vote share close to 0 or 1. We propose here generalizations to the SIRM model by its extension for multi-party systems that are mathematically well-posed in case of extreme vote shares, too, by handling the noise term in a different way. In addition, we show that our method opens new applications for the study of elections by using a new calibration procedure, and makes possible to analyse the influence of the "free will" (creating a new party) and other local effects for different commuting network topologies.

physics.soc-ph↗

arXiv2017

Continuous time limits of the Utterance Selection Model

In this paper, we derive new continuous time limits of the Utterance Selection Model (USM) for language change (Baxter et al., Phys. Rev. E {\bf 73}, 046118, 2006). This is motivated by the fact that the Fokker-Planck continuous time limit derived in the original version of the USM is only valid for a small range of parameters. We investigate the consequences of relaxing these constraints on parameters. Using the normal approximation of the multinomial approximation, we derive a new continuous time limit of the USM in the form of a weak-noise stochastic differential equation. We argue that this weak noise, not captured by the Kramers-Moyal expansion, can not be neglected. We then propose a coarse-graining procedure, which takes the form of a stochastic version of the \emph{heterogeneous mean field} approximation. This approximation groups the behaviour of nodes of same degree, reducing the complexity of the problem. With the help of this approximation, we study in detail two simple families of networks: the regular networks and the star-shaped networks. The analysis reveals and quantifies a finite size effect of the dynamics. If we increase the size of the network by keeping all the other parameters constant, we transition from a state where conventions emerge to a state when no convention emerges. Furthermore, we show that the degree of a node acts as a time scale. For heterogeneous networks such as star-shaped networks, the time scale difference can become very large leading to a noisier behaviour of highly connected nodes.

physics.soc-ph↗

arXiv2016

The IDSA and the homogeneous sphere: Issues and possible improvements

In this paper, we are concerned with the study of the Isotropic Diffusion Source Approximation (IDSA) (Baxter et al., Phys. Rev. E 73, 046118, 2006) of radiative transfer. After having recalled well-known limits of the radiative transfer equation, we present the IDSA and adapt it to the case of the homogeneous sphere. We then show that for this example the IDSA suffers from severe numerical difficulties. We argue that these difficulties originate in the min-max switch coupling mechanism used in the IDSA. To overcome this problem we reformulate the IDSA to avoid the problematic coupling. This allows us to access the modeling error of the IDSA for the homogeneous sphere test case. The IDSA is shown to overestimate the streaming component, hence we propose a new version of the IDSA which is numerically shown to be more accurate than the old one. Analytical results and numerical tests are provided to support the accuracy of the new proposed approximation.

math.NA↗

arXiv2012

A Mathematical Description of the IDSA for Supernova Neutrino transport, its discretization and a comparison with a finite volume scheme for Boltzmann's Equation

In this paper we give an introduction to the Boltzmann equation for neutrino transport used in core collapse supernova models as well as a detailed mathematical description of the \emph{Isotropic Diffusion Source Approximation} (IDSA). Furthermore, we present a numerical treatment of a reduced Boltzmann model problem based on time splitting and finite volumes and revise the discretization of the IDSA for this problem. Discretization error studies carried out on the reduced Boltzmann model problem and on the IDSA show that the errors are of order one in both cases. By a numerical example, a detailed comparison of the reduced model and the IDSA is carried out and interpreted. For this example the IDSA modeling error with respect to the reduced Boltzmann model is numerically determined and localized.

astro-ph.SR↗

arXiv2011

The Kolmogorov-Smirnov test for the CMB

We investigate the statistics of the cosmic microwave background using the Kolmogorov-Smirnov test. We show that, when we correctly de-correlate the data, the partition function of the Kolmogorov stochasticity parameter is compatible with the Kolmogorov distribution and, contrary to previous claims, the CMB data are compatible with Gaussian fluctuations with the correlation function given by standard Lambda-CDM. We then use the Kolmogorov-Smirnov test to derive upper bounds on residual point source power in the CMB, and indicate the promise of this statistics for further datasets, especially Planck, to search for deviations from Gaussianity and for detecting point sources and Galactic foregrounds.

astro-ph.CO↗