Pre-Symmetry Sets of 3D shapes

The investigation of 3D euclidean symmetry sets (SS) and medial axis is an important area, due in particular to their various important applications. The pre-symmetry set of a surface M in 3-space (resp. smooth closed curve in 2D) is the set of pairs of points which contribute to the symmetry set, that is, the closure of the set of pairs of distinct points p and q in M, for which there exists a sphere (resp. a circle) tangent to M at p and at q. The aim of this paper is to address problems related to the smoothness and the singularities of the pre-symmetry sets of 3D shapes. We show that the pre-symmetry set of a smooth surface in 3-space has locally the structure of the graph of a function from R^2 to R^2, in many cases of interest.Comment: ACM-class: I.2; I.5; I.4; J.2. Latex, 3 grouped figures. The final version will appear in the proceedings of the First International Workshop on Deep Structure, Singularities and Computer Vision, Maastricht June 200

Reproductive biology of the common torpedo, Torpedo torpedo (Linnaeus, 1758) (Pisces, Torpedinidae) from the coast of Senegal (Eastern Tropical Atlantic)

Among the five species of genus Torpedo recorded from the coast of Senegal, the common torpedo, Torpedo torpedo is that most cornrnonly caught in the area. Adult males and fernales studied were over 300 mm and 310 mm total length (TL), respectively, with the largest male and the largest female recorded being 445 mm and 550 mm TL respectively. Size at birth was between 102 and 125 mm TL (mean 112.12 mm; s.e.m. 5.58). Weight of eggs ranged frorn 6.2 to 8.0 g (rnean: 7.07; s.e.m. 0.5). Gestation lasts longer, from 6 to 8 rnonths, than in the Mediterranean specimens. A calculated chernical balance of development based on rnean dry weights of the fully developed fetuses and ripe oocytes was 1.58 for T. torpedo. This value shows that this torpedinid is not a pure lecithotrophic species, and the role of the rnother during gestation is not negligible. The fecundity (s.1.) of T. torpedo ranges from 5 to 28 in Senegalese specimens as cornpared to 1 to 9 in Mediterranean specirnens. Male ernbryos and fully developed fetuses are more nurnerous than fernales. Among the free-living specimens, nurnber of males and fernales is practically the same. The cornrnon torpedos from the coast of Senegal are larger and the eggs are heavier than Mediterranean specirnens. Key words: Pisces, Torpedinidae, Torpedo torpedo, Reproductive biology, Senegal, Eastern Tropical Atlantic.Among the five species of genus Torpedo recorded from the coast of Senegal, the common torpedo, Torpedo torpedo is that most cornrnonly caught in the area. Adult males and fernales studied were over 300 mm and 310 mm total length (TL), respectively, with the largest male and the largest female recorded being 445 mm and 550 mm TL respectively. Size at birth was between 102 and 125 mm TL (mean 112.12 mm; s.e.m. 5.58). Weight of eggs ranged frorn 6.2 to 8.0 g (rnean: 7.07; s.e.m. 0.5). Gestation lasts longer, from 6 to 8 rnonths, than in the Mediterranean specimens. A calculated chernical balance of development based on rnean dry weights of the fully developed fetuses and ripe oocytes was 1.58 for T. torpedo. This value shows that this torpedinid is not a pure lecithotrophic species, and the role of the rnother during gestation is not negligible. The fecundity (s.1.) of T. torpedo ranges from 5 to 28 in Senegalese specimens as cornpared to 1 to 9 in Mediterranean specirnens. Male ernbryos and fully developed fetuses are more nurnerous than fernales. Among the free-living specimens, nurnber of males and fernales is practically the same. The cornrnon torpedos from the coast of Senegal are larger and the eggs are heavier than Mediterranean specirnens. Key words: Pisces, Torpedinidae, Torpedo torpedo, Reproductive biology, Senegal, Eastern Tropical Atlantic.Among the five species of genus Torpedo recorded from the coast of Senegal, the common torpedo, Torpedo torpedo is that most cornrnonly caught in the area. Adult males and fernales studied were over 300 mm and 310 mm total length (TL), respectively, with the largest male and the largest female recorded being 445 mm and 550 mm TL respectively. Size at birth was between 102 and 125 mm TL (mean 112.12 mm; s.e.m. 5.58). Weight of eggs ranged frorn 6.2 to 8.0 g (rnean: 7.07; s.e.m. 0.5). Gestation lasts longer, from 6 to 8 rnonths, than in the Mediterranean specimens. A calculated chernical balance of development based on rnean dry weights of the fully developed fetuses and ripe oocytes was 1.58 for T. torpedo. This value shows that this torpedinid is not a pure lecithotrophic species, and the role of the rnother during gestation is not negligible. The fecundity (s.1.) of T. torpedo ranges from 5 to 28 in Senegalese specimens as cornpared to 1 to 9 in Mediterranean specirnens. Male ernbryos and fully developed fetuses are more nurnerous than fernales. Among the free-living specimens, nurnber of males and fernales is practically the same. The cornrnon torpedos from the coast of Senegal are larger and the eggs are heavier than Mediterranean specirnens. Key words: Pisces, Torpedinidae, Torpedo torpedo, Reproductive biology, Senegal, Eastern Tropical Atlantic

Numerical Analysis of Three-dimensional Acoustic Cloaks and Carpets

We start by a review of the chronology of mathematical results on the Dirichlet-to-Neumann map which paved the way towards the physics of transformational acoustics. We then rederive the expression for the (anisotropic) density and bulk modulus appearing in the pressure wave equation written in the transformed coordinates. A spherical acoustic cloak consisting of an alternation of homogeneous isotropic concentric layers is further proposed based on the effective medium theory. This cloak is characterised by a low reflection and good efficiency over a large bandwidth for both near and far fields, which approximates the ideal cloak with a inhomogeneous and anisotropic distribution of material parameters. The latter suffers from singular material parameters on its inner surface. This singularity depends upon the sharpness of corners, if the cloak has an irregular boundary, e.g. a polyhedron cloak becomes more and more singular when the number of vertices increases if it is star shaped. We thus analyse the acoustic response of a non-singular spherical cloak designed by blowing up a small ball instead of a point, as proposed in [Kohn, Shen, Vogelius, Weinstein, Inverse Problems 24, 015016, 2008]. The multilayered approximation of this cloak requires less extreme densities (especially for the lowest bound). Finally, we investigate another type of non-singular cloaks, known as invisibility carpets [Li and Pendry, Phys. Rev. Lett. 101, 203901, 2008], which mimic the reflection by a flat ground.Comment: Latex, 21 pages, 7 Figures, last version submitted to Wave Motion. OCIS Codes: (000.3860) Mathematical methods in physics; (260.2110) Electromagnetic theory; (160.3918) Metamaterials; (160.1190) Anisotropic optical materials; (350.7420) Waves; (230.1040) Acousto-optical devices; (160.1050) Acousto-optical materials; (290.5839) Scattering,invisibility; (230.3205) Invisibility cloak

Focusing: coming to the point in metamaterials

The point of the paper is to show some limitations of geometrical optics in the analysis of subwavelength focusing. We analyze the resolution of the image of a line source radiating in the Maxwell fisheye and the Veselago-Pendry slab lens. The former optical medium is deduced from the stereographic projection of a virtual sphere and displays a heterogeneous refractive index n(r) which is proportional to the inverse of 1+r^2. The latter is described by a homogeneous, but negative, refractive index. It has been suggested that the fisheye makes a perfect lens without negative refraction [Leonhardt, Philbin arxiv:0805.4778v2]. However, we point out that the definition of super-resolution in such a heterogeneous medium should be computed with respect to the wavelength in a homogenized medium, and it is perhaps more adequate to talk about a conjugate image rather than a perfect image (the former does not necessarily contains the evanescent components of the source). We numerically find that both the Maxwell fisheye and a thick silver slab lens lead to a resolution close to lambda/3 in transverse magnetic polarization (electric field pointing orthogonal to the plane). We note a shift of the image plane in the latter lens. We also observe that two sources lead to multiple secondary images in the former lens, as confirmed from light rays travelling along geodesics of the virtual sphere. We further observe resolutions ranging from lambda/2 to nearly lambda/4 for magnetic dipoles of varying orientations of dipole moments within the fisheye in transverse electric polarization (magnetic field pointing orthogonal to the plane). Finally, we analyse the Eaton lens for which the source and its image are either located within a unit disc of air, or within a corona 1<r<2 with refractive index $n(r)=\sqrt{2/r-1}$. In both cases, the image resolution is about lambda/2.Comment: Version 2: 22 pages, 11 figures. More figures added, additional cases discussed. Misprints corrected. Keywords: Maxwell fisheye, Eaton lens; Non-Euclidean geometry; Stereographic projection; Transformation optics; Metamaterials; Perfect lens. The last version appears at J. Modern Opt. 57 (2010), no. 7, 511-52

Level sets of functions and symmetry sets of smooth surface sections

We prove that the level sets of a real C^s function of two variables near a non-degenerate critical point are of class C^[s/2] and apply this to the study of planar sections of surfaces close to the singular section by the tangent plane at hyperbolic points or elliptic points, and in particular at umbilic points. We also analyse the cases coming from degenerate critical points, corresponding to elliptic cusps of Gauss on a surface, where the differentiability is now reduced to C^[s/4]. However in all our applications to symmetry sets of families of plane curves, we assume the C^infty smoothness.Comment: 15 pages, Latex, 6 grouped figures. The final version will appear in Mathematics of Surfaces. Lecture Notes in Computer Science (2005

Solvable Lie algebras are not that hypo

We study a type of left-invariant structure on Lie groups, or equivalently on Lie algebras. We introduce obstructions to the existence of a hypo structure, namely the 5-dimensional geometry of hypersurfaces in manifolds with holonomy SU(3). The choice of a splitting g^*=V_1 + V_2, and the vanishing of certain associated cohomology groups, determine a first obstruction. We also construct necessary conditions for the existence of a hypo structure with a fixed almost-contact form. For non-unimodular Lie algebras, we derive an obstruction to the existence of a hypo structure, with no choice involved. We apply these methods to classify solvable Lie algebras that admit a hypo structure.Comment: 21 pages; v2: presentation improved, typos corrected, notational conflicts eliminated. To appear in Transformation Group

A Unified Algebraic Approach to Classical Yang-Baxter Equation

In this paper, the different operator forms of classical Yang-Baxter equation are given in the tensor expression through a unified algebraic method. It is closely related to left-symmetric algebras which play an important role in many fields in mathematics and mathematical physics. By studying the relations between left-symmetric algebras and classical Yang-Baxter equation, we can construct left-symmetric algebras from certain classical r-matrices and conversely, there is a natural classical r-matrix constructed from a left-symmetric algebra which corresponds to a parak\"ahler structure in geometry. Moreover, the former in a special case gives an algebraic interpretation of the ``left-symmetry'' as a Lie bracket ``left-twisted'' by a classical r-matrix.Comment: To appear in Journal of Physics A: Mathematical and Theoretica