Lipschitz geometry of complex surfaces: analytic invariants and equisingularity

We prove that the outer Lipschitz geometry of a germ $(X,0)$ of a normal complex surface singularity determines a large amount of its analytic structure. In particular, it follows that any analytic family of normal surface singularities with constant Lipschitz geometry is Zariski equisingular. We also prove a strong converse for families of normal complex hypersurface singularities in $\mathbb C^3$: Zariski equisingularity implies Lipschitz triviality. So for such a family Lipschitz triviality, constant Lipschitz geometry and Zariski equisingularity are equivalent to each other.Comment: Added a new section 10 to correct a minor gap and simplify some argument

Lipschitz geometry does not determine embedded topological type

We investigate the relationships between the Lipschitz outer geometry and the embedded topological type of a hypersurface germ in $(\mathbb C^n,0)$. It is well known that the Lipschitz outer geometry of a complex plane curve germ determines and is determined by its embedded topological type. We prove that this does not remain true in higher dimensions. Namely, we give two normal hypersurface germs $(X_1,0)$ and $(X_2,0)$ in $(\mathbb C^3,0)$ having the same outer Lipschitz geometry and different embedded topological types. Our pair consist of two superisolated singularities whose tangent cones form an Alexander-Zariski pair having only cusp-singularities. Our result is based on a description of the Lipschitz outer geometry of a superisolated singularity. We also prove that the Lipschitz inner geometry of a superisolated singularity is completely determined by its (non embedded) topological type, or equivalently by the combinatorial type of its tangent cone.Comment: A missing argument was added in the proof of Proposition 2.3 (final 4 paragraphs are new

Innovation and research in organic farming: A multi‐level approach to facilitate cooperation among stakeholders

A wider range of stakeholders is expected to be involved in organic research. A decision‐support tool is needed to define priorities and to allocate tasks among institutions. Based on research and management experience in organic research, the authors have developed a framework for experimental and research projects. The framework is based on a multi‐level approach. Each level is defined according to the directness of the innovation impact on the organic systems. The projects carried out for each level were assessed over a ten-year period. Two applications are presented: analysis of crop protection strategies in horticulture and plant breeding programmes. When combined with four development models of organic farming, this multi‐level analysis appears to be promising for defining research agendas

The rich cluster of galaxies ABCG~85. IV. Emission line galaxies, luminosity function and dynamical properties

This paper is the fourth of a series dealing with the cluster of galaxies ABCG 85. Using our two extensive photometric and spectroscopic catalogues (with 4232 and 551 galaxies respectively), we discuss here three topics derived from optical data. First, we present the properties of emission line versus non-emission line galaxies, showing that their spatial distributions somewhat differ; emission line galaxies tend to be more concentrated in the south region where groups appear to be falling onto the main cluster, in agreement with the hypothesis (presented in our previous paper) that this infall may create a shock which can heat the X-ray emitting gas and also enhance star formation in galaxies. Then, we analyze the luminosity function in the R band, which shows the presence of a dip similar to that observed in other clusters at comparable absolute magnitudes; this result is interpreted as due to comparable distributions of spirals, ellipticals and dwarfs in these various clusters. Finally, we present the dynamical analysis of the cluster using parametric and non-parametric methods and compare the dynamical mass profiles obtained from the X-ray and optical data.Comment: accepted for publication in A&

Initial Conditions for Large Cosmological Simulations

This technical paper describes a software package that was designed to produce initial conditions for large cosmological simulations in the context of the Horizon collaboration. These tools generalize E. Bertschinger's Grafic1 software to distributed parallel architectures and offer a flexible alternative to the Grafic2 software for ``zoom'' initial conditions, at the price of large cumulated cpu and memory usage. The codes have been validated up to resolutions of 4096^3 and were used to generate the initial conditions of large hydrodynamical and dark matter simulations. They also provide means to generate constrained realisations for the purpose of generating initial conditions compatible with, e.g. the local group, or the SDSS catalog.Comment: 12 pages, 11 figures, submitted to ApJ

Minimal surface singularities are Lipschitz normally embedded

Any germ of a complex analytic space is equipped with two natural metrics: the {\it outer metric} induced by the hermitian metric of the ambient space and the {\it inner metric}, which is the associated riemannian metric on the germ. We show that minimal surface singularities are Lipschitz normally embedded (LNE), i.e., the identity map is a bilipschitz homeomorphism between outer and inner metrics, and that they are the only rational surface singularities with this property.Comment: This paper is a major revision of the 2015 version. It now builds on the paper arXiv:1806.11240 by the same authors which gives a general characterization of Lipschitz normally embedded surface singularitie

The three dimensional skeleton: tracing the filamentary structure of the universe

The skeleton formalism aims at extracting and quantifying the filamentary structure of the universe is generalized to 3D density fields; a numerical method for computating a local approximation of the skeleton is presented and validated here on Gaussian random fields. This method manages to trace well the filamentary structure in 3D fields such as given by numerical simulations of the dark matter distribution on large scales and is insensitive to monotonic biasing. Two of its characteristics, namely its length and differential length, are analyzed for Gaussian random fields. Its differential length per unit normalized density contrast scales like the PDF of the underlying density contrast times the total length times a quadratic Edgeworth correction involving the square of the spectral parameter. The total length scales like the inverse square smoothing length, with a scaling factor given by 0.21 (5.28+ n) where n is the power index of the underlying field. This dependency implies that the total length can be used to constrain the shape of the underlying power spectrum, hence the cosmology. Possible applications of the skeleton to galaxy formation and cosmology are discussed. As an illustration, the orientation of the spin of dark halos and the orientation of the flow near the skeleton is computed for dark matter simulations. The flow is laminar along the filaments, while spins of dark halos within 500 kpc of the skeleton are preferentially orthogonal to the direction of the flow at a level of 25%.Comment: 17 pages, 11 figures, submitted to MNRA

Reduced Gutzwiller formula with symmetry: case of a finite group

We consider a classical Hamiltonian $H$ on $\mathbb{R}^{2d}$, invariant by a finite group of symmetry $G$, whose Weyl quantization $\hat{H}$ is a selfadjoint operator on $L^2(\mathbb{R}^d)$. If $\chi$ is an irreducible character of $G$, we investigate the spectrum of its restriction $\hat{H}\_\chi$ to the symmetry subspace $L^2\_\chi(\mathbb{R}^d)$ of $L^2(\mathbb{R}^d)$ coming from the decomposition of Peter-Weyl. We give reduced semi-classical asymptotics of a regularised spectral density describing the spectrum of $\hat{H}\_\chi$ near a non critical energy $E\in\mathbb{R}$. If $\Sigma\_E:=\{H=E \}$ is compact, assuming that periodic orbits are non-degenerate in $\Sigma\_E/G$, we get a reduced Gutzwiller trace formula which makes periodic orbits of the reduced space $\Sigma\_E/G$ appear. The method is based upon the use of coherent states, whose propagation was given in the work of M. Combescure and D. Robert.Comment: 20 page

Relaxation times of unstable states in systems with long range interactions

We consider several models with long-range interactions evolving via Hamiltonian dynamics. The microcanonical dynamics of the basic Hamiltonian Mean Field (HMF) model and perturbed HMF models with either global anisotropy or an on-site potential are studied both analytically and numerically. We find that in the magnetic phase, the initial zero magnetization state remains stable above a critical energy and is unstable below it. In the dynamically stable state, these models exhibit relaxation time scales that increase algebraically with the number $N$ of particles, indicating the robustness of the quasistationary state seen in previous studies. In the unstable state, the corresponding time scale increases logarithmically in $N$.Comment: Minor change