arXiv subjects

Guillaume Moroz

Publications and source records attributed to Guillaume Moroz.

21 records · Page 2Linked to original sources

arXiv2011

Uniqueness domains and non singular assembly mode changing trajectories

Parallel robots admit generally several solutions to the direct kinematics problem. The aspects are associated with the maximal singularity free domains without any singular configurations. Inside these regions, some trajectories are possible between two solutions of the direct kinematic problem without meeting any type of singularity: non-singular assembly mode trajectories. An established condition for such trajectories is to have cusp points inside the joint space that must be encircled. This paper presents an approach based on the notion of uniqueness domains to explain this behaviour.

cs.RO↗

arXiv2010

Stability and Bifurcation Analysis of Coupled Fitzhugh-Nagumo Oscillators

Neurons are the central biological objects in understanding how the brain works. The famous Hodgkin-Huxley model, which describes how action potentials of a neuron are initiated and propagated, consists of four coupled nonlinear differential equations. Because these equations are difficult to deal with, there also exist several simplified models, of which many exhibit polynomial-like non-linearity. Examples of such models are the Fitzhugh-Nagumo (FHN) model, the Hindmarsh-Rose (HR) model, the Morris-Lecar (ML) model and the Izhikevich model. In this work, we first prescribe the biologically relevant parameter ranges for the FHN model and subsequently study the dynamical behaviour of coupled neurons on small networks of two or three nodes. To do this, we use a computational real algebraic geometry method called the Discriminant Variety (DV) method to perform the stability and bifurcation analysis of these small networks. A time series analysis of the FHN model can be found elsewhere in related work[15].

q-bio.NC↗

arXiv2006

Complexity of Resolution of Parametric Systems of Polynomial Equations and Inequations

Consider a system of n polynomial equations and r polynomial inequations in n indeterminates of degree bounded by d with coefficients in a polynomial ring of s parameters with rational coefficients of bit-size at most $σ$. From the real viewpoint, solving such a system often means describing some semi-algebraic sets in the parameter space over which the number of real solutions of the considered parametric system is constant. Following the works of Lazard and Rouillier, this can be done by the computation of a discriminant variety. In this report we focus on the case where for a generic specialization of the parameters the system of equations generates a radical zero-dimensional ideal, which is usual in the applications. In this case, we provide a deterministic method computing the minimal discriminant variety reducing the problem to a problem of elimination. Moreover, we prove that the degree of the computed minimal discriminant variety is bounded by $D:=(n+r)d^{(n+1)}$ and that the complexity of our method is $σ^{\mathcal{O}(1)} D^{\mathcal{O}(n+s)}$ bit-operations on a deterministic Turing machine.

cs.SC↗