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Romain Bondil - One of the best experts on this subject based on the ideXlab platform.

  • Fine polar invariants of Minimal singularities of surface
    arXiv: Algebraic Geometry, 2004
    Co-Authors: Romain Bondil
    Abstract:

    We consider the polar curves $\PSO$ arising from generic projections of a germ $(S,0)$ of complex surface Singularity onto $\C^2$. Taking $(S,0)$ to be a Minimal Singularity of normal surface (i.e. a rational Singularity with reduced tangent cone), we give the $\delta$-invariant of these polar curves, as well as the equiSingularity-type of their generic plane projections, which are also the discriminants of generic projections of $(S,0)$. These two (equiSingularity)-data for $\PSO$ are described in term, on the one side of the geometry of the tangent cone of $(S,0)$ and on the other side of the limit-trees introduced by T. de Jong and D. van Straten for the deformation theory of these Minimal singularities. These trees give a combinatorial device for the description of the polar curve which makes it much clearer than in our previous Note on the subject. This previous work mainly relied on a result of M. Spivakovsky. Here we give a geometrical proof via deformations (on the tangent cone, and what we call Scott deformations) and blow-ups, although we need Spivakovsky's result at some point, extracting some other consequences of it along the way.

  • discriminant of a generic projection of a Minimal normal surface Singularity
    Comptes Rendus Mathematique, 2003
    Co-Authors: Romain Bondil
    Abstract:

    Let (S, 0) be a rational complex surface Singularity with reduced fun- damental cycle, also known as a Minimal Singularity. Using a fundamental result by M. Spivakovsky, we explain how to get a Minimal resolution of the discriminant curve for a generic projection of (S, 0) onto (C 2 , 0) from the resolution of (S, 0). The material in this Note is organized as follows : § 1 recalls the definitions of polar curves, discriminants and a remarkable property of transversality due to Briancon-Henry and Teissier (thm. 1.2). For Minimal surface singularities, a theorem due to M. Spivakovsky describes the behavior of the generic polar curve (cf. § 2). We use this theorem in § 3 to prove two lemmas relating on the one side the resolution of the generic polar curve to the resolution of a Minimal surface Singularity, and on the other side, the polar curve and the discriminant. Gathering these results, we give our main theorem in § 4, which provides us with a combinatorial way to describe the discriminant. 1 Polar curves and discriminants Let (S,0) be a normal complex surface Singularity (S,0), embedded in (C N ,0): for any (N − 2)-dimensional vector subspace D of C N , we consider a linear projection C N → C 2 with kernel D and denote by pD : (S,0) → (C 2 ,0), the restriction of this projection to (S,0). Restricting ourselves to the D such that pD is finite, and considering a small representative S of the germ (S,0), we define, as in (11) (2.2.2), the polar curve C1(D) of the germ (S,0) relative to the direction D, as the closure in S of the critical locus of the restriction of pD to S \ {0}. As explained in loc. cit., it makes sense to say that for an open dense subset of the Grassmann manifold G(N − 2,N) of (N − 2)-planes in C N , the space curve C1(D) are equisingular

  • Discriminant of a generic projection of a Minimal normal surface Singularity
    2003
    Co-Authors: Romain Bondil
    Abstract:

    Let $(S,0)$ be a rational complex surface Singularity with reduced fundamental cycle, also known as a {\em Minimal} Singularity. Using a fundamental result by M. Spivakovsky, we explain how to get a Minimal resolution of the discriminant curve for a generic projection of $(S,0)$ onto $(\C^2,0)$ directly from the resolution graph of $(S,0)$.

Lauga Eric - One of the best experts on this subject based on the ideXlab platform.

  • Direct vs indirect hydrodynamic interactions during bundle formation of bacterial flagella
    2020
    Co-Authors: Chamolly Alexander, Lauga Eric
    Abstract:

    Most motile bacteria swim in viscous fluids by rotating multiple helical flagellar filaments. These semi-rigid filaments repeatedly join ('bundle') and separate ('unbundle'), resulting in a two-gait random walk-like motion of the cell. In this process, hydrodynamic interactions between the filaments are known to play an important role and can be categorised into two distinct types: direct interactions mediated through flows that are generated through the actuation of the filaments themselves, and indirect interactions mediated through the motion of the cell body (i.e. flows induced in the swimming frame that result from propulsion). To understand the relative importance of these two types of interactions, we study a Minimal Singularity model of flagellar bundling. Using hydrodynamic images, we solve for the flow analytically and compute both direct and indirect interactions exactly as a function of the length of the flagellar filaments and their angular separation. We show (i) that the generation of thrust by flagella alone is sufficient to drive the system towards a bundled state through both types of interaction; (ii) that indirect advection dominates for long filaments and at wide separation, i.e. primarily during the early stages of the bundling process; and (iii) that, in contrast, direct interactions dominate when flagellar filaments are in each other's wake, which we characterise mathematically. We further introduce a numerical elastohydrodynamic model that allows us to compute the dynamics of the helical axes of each flagellar filament while analysing direct and indirect interactions separately. With this we show (iv) that the shift in balance between direct and indirect interactions is non-monotonic during the bundling process, with a peak in direct dominance, and that different sections of the flagella are affected by these changes to different extents.Comment: 34 pages, 9 figure

Eric Lauga - One of the best experts on this subject based on the ideXlab platform.

  • Direct vs indirect hydrodynamic interactions during bundle formation of bacterial flagella.
    arXiv: Fluid Dynamics, 2020
    Co-Authors: Alexander Chamolly, Eric Lauga
    Abstract:

    Most motile bacteria swim in viscous fluids by rotating multiple helical flagellar filaments. These semi-rigid filaments repeatedly join ('bundle') and separate ('unbundle'), resulting in a two-gait random walk-like motion of the cell. In this process, hydrodynamic interactions between the filaments are known to play an important role and can be categorised into two distinct types: direct interactions mediated through flows that are generated through the actuation of the filaments themselves, and indirect interactions mediated through the motion of the cell body (i.e. flows induced in the swimming frame that result from propulsion). To understand the relative importance of these two types of interactions, we study a Minimal Singularity model of flagellar bundling. Using hydrodynamic images, we solve for the flow analytically and compute both direct and indirect interactions exactly as a function of the length of the flagellar filaments and their angular separation. We show (i) that the generation of thrust by flagella alone is sufficient to drive the system towards a bundled state through both types of interaction; (ii) that indirect advection dominates for long filaments and at wide separation, i.e. primarily during the early stages of the bundling process; and (iii) that, in contrast, direct interactions dominate when flagellar filaments are in each other's wake, which we characterise mathematically. We further introduce a numerical elastohydrodynamic model that allows us to compute the dynamics of the helical axes of each flagellar filament while analysing direct and indirect interactions separately. With this we show (iv) that the shift in balance between direct and indirect interactions is non-monotonic during the bundling process, with a peak in direct dominance, and that different sections of the flagella are affected by these changes to different extents.

Eric Sommers - One of the best experts on this subject based on the ideXlab platform.

  • Generic singularities of nilpotent orbit closures
    Advances in Mathematics, 2017
    Co-Authors: Daniel Juteau, Paul Levy, Eric Sommers
    Abstract:

    Abstract According to a theorem of Brieskorn and Slodowy, the intersection of the nilpotent cone of a simple Lie algebra with a transverse slice to the subregular nilpotent orbit is a simple surface Singularity. At the opposite extremity of the poset of nilpotent orbits, the closure of the Minimal nilpotent orbit is also an isolated symplectic Singularity, called a Minimal Singularity. For classical Lie algebras, Kraft and Procesi showed that these two types of singularities suffice to describe all generic singularities of nilpotent orbit closures: specifically, any such Singularity is either a simple surface Singularity, a Minimal Singularity, or a union of two simple surface singularities of type . In the present paper, we complete the picture by determining the generic singularities of all nilpotent orbit closures in exceptional Lie algebras (up to normalization in a few cases). We summarize the results in some graphs at the end of the paper. In most cases, we also obtain simple surface singularities or Minimal singularities, though often with more complicated branching than occurs in the classical types. There are, however, six singularities that do not occur in the classical types. Three of these are unibranch non-normal singularities: an -variety whose normalization is , an -variety whose normalization is , and a two-dimensional variety whose normalization is the simple surface Singularity . In addition, there are three 4-dimensional isolated singularities each appearing once. We also study an intrinsic symmetry action on the singularities, extending Slodowy's work for the Singularity of the nilpotent cone at a point in the subregular orbit.

Jun Morimoto - One of the best experts on this subject based on the ideXlab platform.

  • ICRA - Orientation in Cartesian space dynamic movement primitives
    2014 IEEE International Conference on Robotics and Automation (ICRA), 2014
    Co-Authors: Ales Ude, Bojan Nemec, Tadej Petrič, Jun Morimoto
    Abstract:

    Dynamic movement primitives (DMPs) were pro- posed as an efficient way for learning and control of complex robot behaviors. They can be used to represent point-to-point and periodic movements and can be applied in Cartesian or in joint space. One problem that arises when DMPs are used to define control policies in Cartesian space is that there exists no Minimal, Singularity-free representation of orientation. In this paper we show how dynamic movement primitives can be defined for non Minimal, Singularity free representations of orientation, such as rotation matrices and quaternions. All of the advantages of DMPs, including ease of learning, the ability to include coupling terms, and scale and temporal invariance, can be adopted in our formulation. We have also proposed a new phase stopping mechanism to ensure full movement reproduction in case of perturbations. I. INTRODUCTION