On the Doubling Period Reversible Cusps in R3

We deal with normal forms of germs at the origin of reversible diffeomorphisms of the space whose lineae part is unipotent with two negative eigenvalues. The computing of these normal forms involves effective algebraic geometry algorithms. We also study generic one-paramater deformations

Determinantal sets, singularities and application to optimal control in medical imagery

Control theory has recently been involved in the field of nuclear magnetic resonance imagery. The goal is to control the magnetic field optimally in order to improve the contrast between two biological matters on the pictures. Geometric optimal control leads us here to analyze mero-morphic vector fields depending upon physical parameters , and having their singularities defined by a deter-minantal variety. The involved matrix has polynomial entries with respect to both the state variables and the parameters. Taking into account the physical constraints of the problem, one needs to classify, with respect to the parameters, the number of real singularities lying in some prescribed semi-algebraic set. We develop a dedicated algorithm for real root classification of the singularities of the rank defects of a polynomial matrix, cut with a given semi-algebraic set. The algorithm works under some genericity assumptions which are easy to check. These assumptions are not so restrictive and are satisfied in the aforementioned application. As more general strategies for real root classification do, our algorithm needs to compute the critical loci of some maps, intersections with the boundary of the semi-algebraic domain, etc. In order to compute these objects, the determinantal structure is exploited through a stratifi-cation by the rank of the polynomial matrix. This speeds up the computations by a factor 100. Furthermore, our implementation is able to solve the application in medical imagery, which was out of reach of more general algorithms for real root classification. For instance, computational results show that the contrast problem where one of the matters is water is partitioned into three distinct classes

Du quadrant vetustior à l’horologium viatorum d’Hermann de Reichenau : étude du manuscrit Vaticano, BAV Ott. lat. 1631, f. 16-17v

Le fragment d’une correspondance entre deux clercs latins conservé dans Vaticano, BAV Ott. lat. 1631, f. 16-17v, et le schéma de quadrant reproduit dans London, BL Royal 15 B IX, f. 60v, nous prouvent que, vers le milieu du XIe siècle, un instrument nouveau, un quadrant horaire, a été importé du monde arabe, sans doute en même temps que le cadran solaire cylindrique décrit dans le De horologio viatorum. Il s’agit d’un quart de cercle qui porte un diagramme horaire construit sur une échelle zodiacale. Il se situe, en termes de chronologie relative, entre le quadrant vetustissimus et le quadrant vetus. Un faisceau d’indices probants permet de supposer, d’une part, que l’auteur de la lettre, B, Berengarius, et son destinataire, W, Werinherus, appartiennent au cercle d’Hermann de Reichenau ; d’autre part, que la section de texte conservée dans Vaticano, BAV Ott. lat. 1631, f. 16-17v, appartient à une lettre qui devait comporter, dans son état primitif, la description du quadrant vetustior transmis par Vaticano, BAV Ott. lat. 1631, f. 16-17v, la figure reproduite dans London, BL Royal 15 B IX, f. 60v, le De horologio viatorum. Il est désormais impossible d’attribuer à Hermann de Reichenau la paternité du De horologio viatorum. La lettre de B à W est la source des chapitres De utilitatibus astrolabii II, 6 / Geometria Incerti Auctoris III, 6, ce qui donne un terminus post quem pour les collections De utilitatibus astrolabii II et Geometria Incerti Auctoris III, classe D, produites dans l’entourage de Berengarius.The fragment of a correspondence between two Latin scholars, kept in Vaticano, BAV Ott. lat. 1631, f. 16-17v, and the diagram reproducing a horary quadrant in London, BL Royal 15 B IX, f. 60v, prove that, about 1050, an Arabic instrument, unknown untilthen to the Latin scholars, was introduced into the Occident, probably at the same timeas the cylindrical sundial described in the De horologio viatorum. This quadrant, on which is drawn a horary diagram founded on a zodiacal scale, was imported after thevetustissimus quadransand before thevetus quadrans. We can suggest, in all probability, that the author of the letter, B, Berengarius, and his correspondent, W, Werinherus,were on familiar terms with Hermann the Lamer. Moreover, the fragment preserved in Vaticano, BAV Ott. lat. 1631, f. 16-17v, is an extract from a letter which, in the beginning, may have included the account about the horary quadrant, the diagram transcribed in London, BL Royal 15 B IX, f. 60v, and the De horologio viatorum. So Hermann the Lamer can’t be the author of the De horologio viatorum. Moreover, the letter of B is the source of the chapters De utilitatibus II, 6 and Geometria Incerti Auctoris III, 6. So the collections De utilitatibus II and Geometria Incerti Auctoris III (class D) came after the correspondence between B and W

Kummer's quartic surface associated to the Clebsch top (Symmetry and Singularity of Geometric Structures and Differential Equations)

This note deals with the Kummer surface associated to the Clebsch top with Weber's condition. The Clebsch top is an integrable system describing the rotational motion of a rigid body in an ideal fluid, under special conditions. When restricted to a specific symplectic leaf, given through the so-called Weber condition, there is an associated Kummer surface given as a quartic algebraic surface in CP3 with 16 double points. The explicit conditions of these singular points are given by theoretical computations and are verified by numerical computation through Grabner bases

Optimal Control of an Ensemble of Bloch Equations with Applications in MRI

International audienceThe optimal control of an ensemble of Bloch equations describing the evolution of an ensemble of spins is the mathematical model used in Nuclear Resonance Imaging and the associated costs lead to consider Mayer optimal control problems. The Maximum Principle allows to parameterize the optimal control and the dynamics is analyzed in the framework of geometric optimal control. This lead to numerical implementations or suboptimal controls using averaging principle

Integration of a Dirac comb and the Bernoulli polynomials

International audienceFor any positive integer $n$, we consider the ordinary differential equations of the form $y^{(n)} = 1 - Ш + F$ where $Ш$ denotes the Dirac comb distribution and $F$ is a piecewise-$\mathcal{C}^\infty$ periodic function with null average integral. We prove the existence and uniqueness of periodic solutions of maximal regularity. Above all, these solutions are given by means of finite explicit formulae involving a minimal number of Bernoulli polynomials. We generalize this approach to a larger class of differential equations for which the computation of periodic solutions is also sharp, finite and effective

Periodic Solutions of a Class of Non-autonomous Discontinuous Second-Order Differential Equations

International audienceWe consider the second-order discontinuous differential equation y('') + eta sgn(y) = & x1d703;& xdf03;y + alpha sin(beta t) where the parameters eta, & x1d703;& xdf03;, alpha, and beta are real. The main goal is to discuss the existence of periodic solutions. Under explicit conditions, the number of such solutions is given. Furthermore, for each of these periodic solutions, an explicit formula is provided

Algebraic geometric classification of the singular flow in the contrast imaging problem in nuclear magnetic resonance

International audienceThe analysis of the contrast problem in NMR medical imaging is essentially reduced to the analysis of the so-called singular trajectories of the system modeling the problem: a coupling of two spin 1/2 control systems. They are solutions of a constraint Hamiltonian vector field and restricting the dynamics to the zero level set of the Hamiltonian they define a vector field on B1 x B2, where B1 and B2 are the Bloch balls of the two spin particles. In this article we classify the behaviors of the solutions in relation with the relaxation parameters using the concept of feedback classification. The optimality status is analyzed using the feedback invariant concept of conjugate points