### [hal-01452778] On measures in sub-Riemannian geometry

16 février 2017

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Une partie des prépublications de l’Institut de Mathématiques de Bourgogne est disponible sur la plateforme HAL :

16 février 2017

In \citegjha we give a detailed analysis of spherical Hausdorff measures on sub-Riemannian manifolds in a general framework, that is, without the assumption of equiregularity. The present paper is devised as a complement of this analysis, with both new results and open questions. The first aim (...)

16 février 2017

This paper deals with the non-uniqueness of the solutions of an analysis-Lasso regularization. Most of previous works in this area is concerned with the case where the solution set is a singleton, or to derive guarantees to enforce uniqueness. Our main contribution consists in providing a (...)

16 février 2017

We show that, up to topological conjugation, the equivalence class of a Morse-Smale diffeomorphism without heteroclinic curves on 3-manifold is completely defined by an em- bedding of two-dimensional stable and unstable heteroclinic laminations to a characteristic (...)

15 février 2017

In \citegjha we give a detailed analysis of spherical Hausdorff measures on sub-Riemannian manifolds in a general framework, that is, without the assumption of equiregularity. The present paper is devised as a complement of this analysis, with both new results and open questions. The first aim (...)

6 février 2017

In \citegjha we give a detailed analysis of spherical Hausdorff measures on sub-Riemannian manifolds in a general framework, that is, without the assumption of equiregularity. The present paper is devised as a complement of this analysis, with both new results and open questions. The first aim (...)

30 janvier 2017

In this article, we provide a comparative geometric and numerical analysis of optimal strokes for two different rigid links swimmer models at low Reynolds number: the Copepod swimmer (a symmetric swimmer recently introduced by [31]) and the long-standing three-link Purcell swimmer [29]. The (...)

23 janvier 2017

In [17], based on the observation of copepods, Takagi proposed a model to analyze the swimming of microorganisms using sinusoidal paddling or sequential paddling followed by a recovery stroke in unison, and they are compared with the concept of efficiency. Our aim is to provide an (...)

7 janvier 2017

A smooth complex variety satisfies the Generalized Jacobian Conjecture if all its etale endomorphisms are proper. We study the equivariant version of the conjecture for ℚ-acyclic surfaces of negative Kodaira dimension and infinite algebraic groups. We show that it holds for groups other than ℂ∗, (...)

8 décembre 2016

We focus on the maximum regularization parameter for anisotropic total-variation denoising. It corresponds to the minimum value of the regularization parameter above which the solution remains constant. While this value is well know for the Lasso, such a critical value has not been investigated (...)

8 décembre 2016

In this paper, we propose a new framework to remove parts of the systematic errors affecting popular restoration algorithms, with a special focus for image processing tasks. Generalizing ideas that emerged for l1 regularization, we develop an approach re-fitting the results of standard methods (...)

1er décembre 2016

We give a complete and unified description – under some stability assumptions – of the functional scaling limits associated with some persistent random walks for which the recurrent or transient type is studied in [1]. As a result, we highlight a phase transition phenomenon with respect to the (...)

1er décembre 2016

Minimum time control of slow-fast systems is considered. In the case of only one fast angle, averaging techniques are available for such systems. The approach introduced in Dargent (2014) and Bombrun et al. (2013) is recalled, then extended to time dependent systems by means of a suitable (...)

30 novembre 2016

The goal of this paper is to show how non-parametric statistics can be used to solve chance-constrained optimization and optimal control problems by reformulating them into deterministic ones, focusing on the details of the algorithmic approach. We use the Kernel Density Estimation method to (...)

28 novembre 2016

Let $(W,S)$ be a Coxeter system, let $G$ be a group of symmetries of $(W,S)$ and let $f : W \to \GL (V)$ be the linear representation associated with a root basis $(V, \langle .,. \rangle, \Pi)$. We assume that $G \subset \GL (V)$, and that $G$ leaves invariant $\Pi$ and $\langle .,. \rangle$. (...)

22 novembre 2016

For general dimension $d$ we prove the equidistribution of energy at the micro-scale in $\mathbb R^d$, for the optimal point configurations appearing in Coulomb gases at zero temperature. At the microscopic scale, i.e. after blow-up at the scale corresponding to the interparticle distance, in (...)

10 novembre 2016

We classify closed curves isomorphic to the affine line in the complement of a smooth rational projective plane conic Q. Over a field of characteristic zero, we show that up to the action of the subgroup of the Cremona group of the plane consisting of birational endomorphisms restricting to (...)

3 novembre 2016

In this paper we consider the degeneracies of the third type. More exact, the perturbations of the Darboux integrable foliation with a triple point, i.e. the case where three of the curves P i = 0 meet at one point, are considered. Assuming that this is the only non-genericity, we prove that (...)

28 octobre 2016

Given a solution to a recursive distributional equation, a natural (and non-trivial) question is whether the corresponding recursive tree process is endogenous. That is, whether the random environment almost surely defines the tree process. We propose a new method of proving endogeny, which (...)

28 octobre 2016

In this article, the model of swimming at low Reynolds number introduced by D. Takagi (2015) to analyze the displacement of an abundant variety of zooplankton is used as a testbed to analyze the motion of symmetric microswimmers in the framework of optimal control theory assuming that the (...)

27 octobre 2016

In this paper we consider the degeneracies of the third type. More exact, the perturbations of the Darboux integrable foliation with a triple point, i.e. the case where three of the curves P i = 0 meet at one point, are considered. Assuming that this is the only non-genericity, we prove that (...)

25 octobre 2016

In this article, the model of swimming at low Reynolds number introduced by D. Takagi (2015) to analyze the displacement of an abundant variety of zooplankton is used as a testbed to analyze the motion of symmetric microswimmers in the framework of optimal control theory assuming that the (...)

25 octobre 2016

In this article, the model of swimming at low Reynolds number introduced by D. Takagi (2015) to analyze the displacement of an abundant variety of zooplankton is used as a testbed to analyze the motion of symmetric microswimmers in the framework of optimal control theory assuming that the (...)

13 octobre 2016

The Initial-Boundary Value Problem for the heat equation is solved by using a new algorithm based on a random walk on heat balls. Even if it represents a sophisticated generalization of the Walk on Spheres (WOS) algorithm introduced to solve the Dirich-let problem for Laplace's equation, its (...)

5 octobre 2016

We construct algebraic families of exotic affine 3-spheres, that is, smooth affine threefolds diffeomorphic to a non-degenerate smooth complex affine quadric of dimension 3 but non algebraically isomorphic to it. We show in particular that for every smooth topologically contractible affine (...)

15 septembre 2016

In this paper, we propose a new framework to remove parts of the systematic errors affecting popular restoration algorithms, with a special focus for image processing tasks. Generalizing ideas that emerged for l1 regularization, we develop an approach re-fitting the results of standard methods (...)

13 septembre 2016

For general dimension $d$ we prove the equidistribution of energy at the micro-scale in $\mathbb R^d$, for the optimal point configurations appearing in Coulomb gases at zero temperature. At the microscopic scale, i.e. after blow-up at the scale corresponding to the interparticle distance, in (...)

2 septembre 2016

In control theory, the term chattering is used to refer to strong oscillations of controls, such as an infinite number of switchings over a compact interval of times. In this paper we focus on three typical occurences of chattering: the Fuller phenomenon, referring to situations where an (...)

30 août 2016

Qui est donc Alexandre Véronnet ? Nous allons voir qu'Alexandre Véronnet est un scientifique tout à fait étonnant ayant eu un parcours particulièrement original. Prêtre, il quitte l’Eglise à la fin de la Première Guerre mondiale. Il soutient une thèse en Astronomie en 1912. Le premier conflit mondial, (...)

3 août 2016

In this article, we make a comparative geometric and numeric analysis of the optimal strokes at low Reynolds number using two specific rigid links swimmers: the Copepod swimmer, a symmetric swimmer introduced recently 30 and the historical three-link Pur-cell swimmer 27 where the cost to (...)

22 juillet 2016

This paper extends recent results by the first author and T. Pock (ICG, TU Graz, Austria) on the acceleration of alternating minimization techniques for quadratic plus nons-mooth objectives depending on two variables. We discuss here the strongly convex situation, and how " fast " (...)

8 juillet 2016

In this paper we conjecture that the Links-Gould invariant of links, that we know is a generalization of the Alexander-Conway polynomial, shares some of its classical features. In particular it seems to give a lower bound for the genus of links and to provide a criterion for fiberedness of (...)

4 juillet 2016

We show that a del Pezzo fibration π : V → W of degre d contains a vertical open cylinder, that is, an open subset whose intersection with the generic fiber of π is isomorphic to Z × A 1 K for some quasi-projective variety Z defined over the function field K of W , if and only if d ≥ 5 and π : V → (...)

30 juin 2016

The Peregrine breather is widely discussed as a model for rogue waves in deep water. We present here a detailed numerical study of perturbations of the Peregrine breather as a solution to the nonlinear Schr\"odinger (NLS) equations. We first address the modulational instability of the (...)

20 juin 2016

We construct solutions to the Kadomtsev-Petviashvili equation (KPI) in terms of Fredholm determinants. We deduce solutions written as a quotient of wronskians of order 2N. These solutions called solutions of order N depend on 2N − 1 parameters. When one of these parameters tends to zero, we (...)

20 juin 2016

We show that the nonlinear stage of modulational instability induced by parametric driving in the defocusing nonlinear Schrödinger equation can be accurately described by combining mode truncation and averaging methods, valid in the strong driving regime. The resulting integrable oscillator (...)

20 juin 2016

In this paper, we propose a new framework to remove parts of the systematic errors affecting popular restoration algorithms, with a special focus for image processing tasks. Extending ideas that emerged for l1 regularization, we develop an approach that can help re-fitting the results of (...)

10 juin 2016

In this article, we make a comparative geometric and numeric analysis of the optimal strokes at low Reynolds number using two specific rigid links swimmers: the Copepod swimmer, a symmetric swimmer introduced recently 28 and the historical three-link Pur-cell swimmer 25 where the cost to (...)

9 juin 2016

Oleg Viro studied in arXiv:math/0204290 two interpretations of the (multivariable) Alexander polynomial as a quantum link invariant: either by considering the quasi triangular Hopf algebra associated to $U_q sl(2)$ at fourth roots of unity, or by considering the super Hopf algebra $U_q (...)

9 mai 2016

We consider the orbital stability of the relative equilibria of Hamiltonian dynamical systems on Banach spaces, in the presence of a multi-dimensional invariance group for the dynamics. We present a generalization of the Vakhitov-Kolokolov slope condition to this higher dimensional setting, and (...)

8 avril 2016

We present a new approach to calculating the higher Frobenius-Schur indicators for the simple modules over the Drinfeld double of a finite group. In contrast to the formula by Kashina-Sommerhäuser-Zhu that involves a sum over all group elements satisfying a certain condition, our formula (...)

7 avril 2016

The twelfth Peregrine breather (P12 breather) solution to the focusing one dimensional nonlinear Schrödinger equation (NLS) with its twenty two real parameters deformations solutions to the NLS equation are explicitly constructed here. New families of quasi-rational solutions of the NLS equation (...)

7 avril 2016

— The objective of this article is to make a geometric and numerical analysis about the optimal displacements of a larval copepod swimming at low Reynolds number. A simplified model of locomotion is analyzed in the framework of Sub-Riemannian geometry. In particular, the role of both normal and (...)

25 mars 2016

This series of notes have their origin into accelerated course in optimal control given to multidisciplinary Phd students in Mathematics, Physics or Control Engineering and the purpose is to present the seminal results of this theory in view of applications to concrete problems. For that we (...)

14 mars 2016

The 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 (...)

11 mars 2016

The maximum principle combined with numerical methods is a powerful tool to compute solutions for optimal control problems. This approach turns out to be extremely useful in applications, including solving problems which require establishing periodic trajectories for Hamiltonian systems, (...)

11 mars 2016

The objective of this article is to make a geometric and numerical analysis about the optimal displacements of a larval copepod swimming at low Reynolds number. A simplified model of locomotion is analyzed in the framework of Sub-Riemannian geometry. In particular, the role of both normal and (...)

4 mars 2016

We show a simple and easily implementable solution to the word problem for virtual braid groups.

1er mars 2016

Given a symmetric non degenerated bilinear form b on a vector space V , G. Pinczon and R. Ushirobira defined a bracket , on the space of multilinear skewsymmetric forms on V. With this bracket, the quadratic Lie algebra structure equation on (V, b) becomes simply Ω, Ω = 0. We characterize (...)

23 février 2016

In the note added in proof of the seminal paper [Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. of Math. 92 (1970), 102-163], Ebin and Marsden introduced the so-called correct Laplacian for the Navier-Stokes equation on a compact Riemannian manifold. In the spirit of (...)

23 février 2016

In this article, one gives a classification of the solutions to the one dimensional nonlinear focusing Schrödinger equation (NLS) by considering the modulus of the solutions in the (x, t) plan in the cases of orders 3 and 4. For this, we use a representation of solutions to NLS equation as a (...)

13 février 2016

Bifurcating autoregressive processes, which can be seen as an adaptation of au-toregressive processes for a binary tree structure, have been extensively studied during the last decade in a parametric context. In this work we do not specify any a priori form for the two autoregressive functions (...)

8 février 2016

Ribbon 2-knotted objects are locally flat embeddings of surfaces in 4-space which bound immersed 3-manifolds with only ribbon singularities. They appear as topological realizations of welded knotted objects, which is a natural quotient of virtual knot theory. In this paper, we consider ribbon (...)

25 janvier 2016

Ribbon 2-knotted objects are locally flat embeddings of surfaces in 4-space which bound immersed 3-manifolds with only ribbon singularities. They appear as topological realizations of welded knotted objects, which is a natural quotient of virtual knot theory. In this paper, we consider ribbon (...)

13 janvier 2016

In a first part, we prove Bernstein-type deviation inequalities for bifurcating Markov chains (BMC) under a geometric ergodicity assumption, completing former results of Guyon and Bitseki Penda, Djellout and Guillin. These preliminary results are the key ingredient to implement nonparametric (...)

7 janvier 2016

In a first part, we prove Bernstein-type deviation inequalities for bifurcating Markov chains (BMC) under a geometric ergodicity assumption, completing former results of Guyon and Bitseki Penda, Djellout and Guillin. These preliminary results are the key ingredient to implement nonparametric (...)

22 décembre 2015

Qui est donc Alexandre Véronnet ? C'est à cette question que nous allons répondre. Nous allons voir que Alexandre Véronnet est un scientifique tout à fait étonnant ayant eu un parcours particulièrement original.

15 décembre 2015

The Peregrine breather of order eleven (P11 breather) solution to the focusing one dimensional nonlinear Schrödinger equation (NLS) is explicitly constructed here. Deformations of the Peregrine breather of order 11 with 20 real parameters solutions to the NLS equation are also given : when all (...)

15 décembre 2015

Based on a representation in terms of determinants of order 2N , an attempt to classification of quasi rational solutions to the one dimensional focusing nonlinear Schrödinger equation (NLS) is given and several conjectures about the structure of the solutions are also formulated. These (...)

1er décembre 2015

This paper gives a connection between well chosen reductions of the Links-Gould invariants of oriented links and powers of the Alexander-Conway polynomial. We prove these formulas by showing the representations of the braid groups we derive the specialized Links-Gould polynomials from can be (...)

30 novembre 2015

Based on a representation in terms of determinants of order 2N , an attempt to classification of quasi rational solutions to the one dimensional focusing nonlinear Schrödinger equation (NLS) is given and several conjectures about the structure of the solutions are also formulated. These (...)

6 novembre 2015

The Peregrine breather of order eleven (P11 breather) solution to the focusing one dimensional nonlinear Schrödinger equation (NLS) is explicitly constructed here. Deformations of the Peregrine breather of order 11 with 20 real parameters solutions to the NLS equation are also given : when all (...)

21 octobre 2015

We consider the quotient of welded knotted objects under several equivalence relations, generated respectively by self-crossing changes, ∆ moves, self-virtualizations and forbidden moves. We prove that for welded objects up to forbidden moves or classical objects up to ∆ moves, the notions of (...)

20 octobre 2015

We consider the quotient of welded knotted objects under several equivalence relations, generated respectively by self-crossing changes, ∆ moves, self-virtualizations and forbidden moves. We prove that for welded objects up to forbidden moves or classical objects up to ∆ moves, the notions of (...)

6 octobre 2015

The maximum principle combined with numerical methods is a powerful tool to compute solutions for optimal control problems. This approach turns out to be extremely useful in applications, including solving problems which require establishing periodic trajectories for Hamiltonian systems, (...)

25 septembre 2015

We describe a method to construct hypersurfaces of the complex affine $n$-space with isomorphic $\mathbbC^*$-cylinders. Among these hypersurfaces, we find new explicit counterexamples to the Laurent Cancellation Problem, i.e. hypersurfaces that are non isomorphic, although their (...)

22 septembre 2015

Estimating high level quantiles of aggregated variables (mainly sums or weighted sums) is crucial in risk management for many application fields such as finance, insurance, environment... This question has been widely treated but new efficient methods are always welcome; especially if they (...)

13 septembre 2015

We consider a walker that at each step keeps the same direction with a probability that depends on the time already spent in the direction the walker is currently moving. In this paper, we study some asymptotic properties of this persistent random walk and give the conditions of recurrence or (...)

11 septembre 2015

The purpose of this work is to establish a counterpart for the Navier-Stokes equation of Brenier's generalized flows for the Euler equation. We introduce a class of semi-martingales on a compact Riemannian manifold which will be generalized solutions of the Navier-Stokes equation. We prove that (...)

4 septembre 2015

The purpose of this note is to establish a probabilistic representation formula for Navier–Stokes equations on a compact Riemannian manifold. To this end, we first give a geometric interpretation of Constantin and Iyer's representation formula for the Navier–Stokes equation, then extend it to a (...)

4 septembre 2015

In this paper, we will give some remarks on links between the spectral gap of the Ornstein-Uhlenbeck operator on the Riemannian path space with lower and upper bounds of the Ricci curvature on the base manifold; this work was motivated by a recent work of A. Naber on the characterization of the (...)

1er septembre 2015

In [8], we define and partially classify fake real planes, that is, minimal complex surfaces with conjugation whose real locus is diffeomorphic to the euclidean real plane $\mathbbR^2$. Classification results are given up to biregular isomorphisms and up to birational diffeomorphisms. In this (...)

8 août 2015

We study certain pencils of del Pezzo surfaces generated by a smooth del Pezzo surface S of degree less or equal to 3 anti-canonically embedded into a weighted projective space P and an appropriate multiple of a hyperplane H. Our main observation is that every minimal model program relative to (...)

23 juillet 2015

We determine the automorphism groups of Koras-Russell threefolds of the second kind. In particular we show that these groups are semi-direct products of two subgroups, one given by the multiplicative group and the other isomorphic to a polynomial ring in two variables with the addition law. We (...)

22 juillet 2015

We construct families of smooth affine surfaces with pairwise non isomorphic A 1-cylinders but whose A 2-cylinders are all isomorphic. These arise as complements of cuspidal hyperplane sections of smooth projective cubic surfaces.

17 juillet 2015

We consider several classes of knotted objects, namely usual, virtual and welded pure braids and string links, and two equivalence relations on those objects, induced by either self-crossing changes or self-virtualizations. We provide a number of results which point out the differences between (...)

17 juillet 2015

En 1942, le célèbre astronome et physicien belge Georges Lemaître (1894-1966) publie dans le Bulletin de la classe des sciences de l'Académie Royale de Belgique, un article dans lequel il formule une méthode qu'il nomme itération rationnelle. Il redécouvre à la fois le processus du Delta 2 de Aitken et (...)

12 juillet 2015

Lie-Rinehart algebras, also known as Lie algebroids, give rise to Hopf algebroids by a universal enveloping algebra construction, much as the universal enveloping algebra of an ordinary Lie algebra gives a Hopf algebra, of infinite dimension. In finite characteristic, the universal enveloping (...)

11 juillet 2015

We study real rational models of the euclidean affine plane $\mathbbR^2$ up to isomorphisms and up to birational diffeomorphisms. The analogous study in the compact case, that is the classification of real rational models of the real projective plane $\mathbbR\mathbbP^2$ is well known: up to (...)

2 juillet 2015

In this paper we study the Hausdorff volume in a non equiregular sub-Riemannian manifold and we compare it with a smooth volume. We first give the Lebesgue decomposition of the Hausdorff volume. Then we study the regular part, show that it is not commensurable with the smooth volume, and give (...)

30 juin 2015

We develop some methods to compute the Makar-Limanov and Derksen invariants, isomorphism classes and automorphism groups for k-domains B, which are constructed from certain Russell k-domains. We propose tools and techniques to distinguish between k-domains with the same Makar-Limanov and (...)

20 juin 2015

The objective of this note is to made preliminary remarks concerning the optimal locomotion of the Copepod swimmer described by a simple model concerning swimming at low Reynolds number, in the framework of Sub-Riemannian geometry. In particular the role of both normal and abnormal geodesics is (...)

20 juin 2015

This paper gives a connection between well chosen reductions of the Links-Gould invariants of oriented links and powers of the Alexander-Conway polynomial. We prove these formulas by showing the representations of the braid groups we derive the specialized Links-Gould polynomials from can be (...)

20 juin 2015

We present here new solutions of the focusing one dimensional non linear Schrödinger equation which appear as deformations of the Peregrine breather of order 10 with 18 real parameters. With this method, we obtain new families of quasi-rational solutions of the NLS equation, and we obtain (...)

20 juin 2015

Virtual braids are a combinatorial generalization of braids. We present abstract braids as equivalence classes of braid diagrams on a surface, joining two distinguished boundary components. They are identified up to isotopy, compatibility, stability and Reidemeister moves. We show that virtual (...)

18 juin 2015

We show a simple and easily implementable solution to the word problem for virtual braid groups.

30 mai 2015

En 1902, Robert de Montessus de Ballore démontre la convergence d'une fraction continue algébrique associée à une fonction analytique à l'origine et méromorphe dans un domaine contenant l'origine. Aujourd'hui ce théorème est encore cité. Et le nom Montessus de Ballore sert à nommer des généralisations (...)

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