Rémi Langevin

Habitat geometry of benthic substrata: effects on arrival and settlement of mobile epifauna

Abstract The effect of substratum complexity on the early stages of colonization by mobile epifauna was assessed through a comparative study based on the architecture of artificial substrata. We conducted field observations over 4 years, on six types of small plastic substrata placed in the low intertidal zone of an exposed rocky shore, for varied immersion periods (1, 2, 4 and 12 wk). The use of artificial substrata allowed us to manipulate independently structural and spatial features of the habitat, such as total area, amount of folds, intercepting area, total volume, and interstitial volume. The invertebrate fauna colonizing over 300 sample units was recorded, and their densities compared as a function of substrata type and immersion time. Microcrustaceans predominated during the initial stages in all substrata. In this category, harpacticoid copepods and amphipods were the most abundant taxa. The effect of the original substratum complexity seemed to be restricted to the early stages of colonization, since after 12 wk of immersion the original geometry was greatly modified by fouling organisms, particularly ascidians and epiphytic algae. The geometric characteristic that most influenced epifaunal composition and density was the substratum folding, a one-dimensional measure that evaluates the amount of filaments and folds in the substratums surface. Folding was correlated with high faunal densities and high initial colonization rates, and proved to be a better density predictor than total substratum area, or volume. This correlation was especially well defined for amphipods.

On curvature integrals and knots

THE TOTAL curvature of a curve is the integral of the absolute curvature of the curve. I. Fary and J. Milnor proved the total curvature of a knot in R’ is at least 4~ [3,8]; the total curvature of a standard embedding of S’ is 2n. Many generalisations of this theorem now exist [2.4.5,9], a good survey is [2]. We pursue this study here. Let C be a knot in R’ which bounds an orientable surface M in R3. Consider functions on M of the form p(z): x -+ z -x (the scalar product of z with X, where z is a point of S’). For almost all z of S* (in the sense of Lebesgue measure), p(z) is a Morse function on M. Denote by

Feuilletages deCP(n) : de l’holonomie hyperbolique pour les minimaux exceptionnels

Let ℱ be a holomorphic foliation ofCP(n). If ℱ has a leaf L, the closure L of which is disjoint from the singular set of the foliation, we prove that there exists a loop in a leaf contained in L with contracting hyperbolic holonomy.

Chthamalus bisinuatus (Cirripedia) and Brachidontes solisianus (Bivalvia) spatial interactions: A stochastic model

Abstract Chthamalus bisinuatus (barnacles) and Brachidontes solisianus (mussels) are species with a sessile mode of life which coexist in high densities in the mid-intertidal zone. Our aim was to study spatial interactions between these two species on the rocky shore. Weconstructed a stochastic model to describe the temporal evolution of both populations, taking into account recruitment and death patterns of the species. This model is a karkovian system with an infinite number of components interacting locally. To check the model, simulations were compared to field data through a family of functions which discriminate between different spatial chaotic configurations.

Complete minimal surfaces with long line boundaries

On etudie des surfaces minimales completes bornees par des lignes dans R 3 . On demontre des theoremes de Bernstein pour de telles surfaces

Courbure totale des feuilletages des surfaces

The sum of the total curvatures of two orientable orthogonal foliations on the unit sphereS2⊂R3 is at least 4Π. The total curvature of a foliation with saddle singularities on a closed hyperbolic surfaceM is at least (12 Log 2–6 Log 3) ... |χ(M)|.

Technical note: Iterative construction of Dupin cyclide characteristic circles using non-stationary Iterated Function Systems (IFS)

A Dupin cyclide can be defined, in two different ways, as the envelope of an one-parameter family of oriented spheres. Each family of spheres can be seen as a conic in the space of spheres. In this paper, we propose an algorithm to compute a characteristic circle of a Dupin cyclide from a point and the tangent at this point in the space of spheres. Then, we propose iterative algorithms (in the space of spheres) to compute (in 3D space) some characteristic circles of a Dupin cyclide which blends two particular canal surfaces. As a singular point of a Dupin cyclide is a point at infinity in the space of spheres, we use the massic points defined by Fiorot. As we subdivide conic arcs, these algorithms are better than the previous algorithms developed by Garnier and Gentil.

Blending canal surfaces along given circles using Dupin cyclides

We study blends between canal surfaces using Dupin cyclides via the space of spheres. We have already studied the particular case where it is possible to blend two canal surfaces using one piece of Dupin cyclide bounded by two characteristic circles, but this is not possible in the general case. That is why we solve this problem using two pieces of different cyclides, which is always possible. To get this conclusion and give the algorithms allowing to obtain such a result, we study, at first, the blend between two circles by a piece of cyclide. We impose to the cyclide to be tangent to a given sphere containing one of the circles. We give the existence condition on the previous circles to have a cyclide making the blend. Then, we show how to obtain a G1-blend between two canal surfaces using two Dupin cyclides by imposing conditions on the blending circles between the two cyclides.

Blending Planes and Canal Surfaces Using Dupin Cyclides

We develop two different new algorithms of G1-blending between planes and canal surfaces using Dupin cyclides. It is a generalization of existing algorithms that blend revolution surfaces and planes using a plane called construction plane. Spatial constraints were necessary to do that. Our work consist in building three spheres to determine the Dupin cyclide of the blending. The first algorithm is based on one of the definitions of Dupin cyclides taking into account three spheres of the same family enveloping the cyclide. The second one uses only geometric properties of Dupin cyclide. The blending is fixed by a circle of curvature onto the canal surface. Thanks to this one, we can determine a cyclide symmetry plane used for the blend. Each algorithm uses one of the two symmetry planes of a cyclide constructed with the centres of three spheres or orthogonal to the first plane and to the one containing the circle of curvature of the canal surface.

Holomorphic Maps and Pencils of Circles

The function / is called holomorphic if fr(z) exists at each point z of U. This condition has very strong consequences: the real and imaginary part of / are of class C?? (i.e., they possess everywhere in U continuous partial derivatives of all orders) and, moreover, / is analytic in U (i.e., for each point zo of U, there is an open disc i/o centered at zo in which / can be represented by a power series with complex coefficients: