Fiedler Vectors and Elongation of Graphs: A Threshold Phenomenon on a Particular Class of Trees

Let $G$ be a graph. Its laplacian matrix $L(G)$ is positive and we consider eigenvectors of its first non-null eigenvalue that are called Fiedler vector. They have been intensively used in spectral partitioning problems due to their good empirical properties. More recently Fiedler vectors have been also popularized in the computer graphics community to describe elongation of shapes. In more technical terms, authors have conjectured that extrema of Fiedler vectors can yield the diameter of a graph. In this work we present (FED) property for a graph $G$, i.e. the fact that diameter of a graph can be obtain by Fiedler vectors. We study in detail a parametric family of trees that gives indeed a counter example for the previous conjecture but reveals a threshold phenomenon for (FED) property. We end by an exhaustive enumeration of trees with at most 20 vertices for which (FED) is true and some perspectives.Comment: 19 page

The Generic Multiple-Precision Floating-Point Addition With Exact Rounding (as in the MPFR Library)

We study the multiple-precision addition of two positive floating-point numbers in base 2, with exact rounding, as specified in the MPFR library, i.e. where each number has its own precision. We show how the best possible complexity (up to a constant factor that depends on the implementation) can be obtain.Comment: Conference website at http://cca-net.de/rnc6

The new Sunspot Number: assembling all corrections

The Sunspot Number, created by R.Wolf in 1849, provides a direct long-term record of solar activity from 1700 to the present. In spite of its central role in multiple studies of the solar dynamo and of the past Sun-Earth relations, it was never submitted to a global critical revision. However, various discrepancies with other solar indices recently motivated a full re-calibration of this series. Based on various diagnostics and corrections established in the framework of several Sunspot Number Workshops and described in Clette et al. 2014, we assembled all corrections in order to produce a new standard version of this reference time series. In this paper, we explain the three main corrections and the criteria used to choose a final optimal version of each correction factor or function, given the available information and published analyses. We then discuss the good agreement obtained with the Group sunspot Number derived from a recent reconstruction. Among the implications emerging from this re-calibrated series, we also discuss the absence of a rising secular trend in the newly-determined solar cycle amplitudes, also in relation with contradictory indications derived from cosmogenic radionuclides. As conclusion, we introduce the new version management scheme now implemented at the World Data Center - SILSO, which reflects a major conceptual transition: beyond the re-scaled numbers, this first revision of the Sunspot Number also transforms the former locked data archive into a living observational series open to future improvements

Discrete-time port-Hamiltonian systems: A definition based on symplectic integration

We introduce a new definition of discrete-time port-Hamiltonian systems (PHS), which results from structure-preserving discretization of explicit PHS in time. We discretize the underlying continuous-time Dirac structure with the collocation method and add discrete-time dynamics by the use of symplectic numerical integration schemes. The conservation of a discrete-time energy balance - expressed in terms of the discrete-time Dirac structure - extends the notion of symplecticity of geometric integration schemes to open systems. We discuss the energy approximation errors in the context of the presented definition and show that their order is consistent with the order of the numerical integration scheme. Implicit Gauss-Legendre methods and Lobatto IIIA/IIIB pairs for partitioned systems are examples for integration schemes that are covered by our definition. The statements on the numerical energy errors are illustrated by elementary numerical experiments.Comment: 12 pages. Preprint submitted to Systems & Control Letter

Weak Form of Stokes-Dirac Structures and Geometric Discretization of Port-Hamiltonian Systems

We present the mixed Galerkin discretization of distributed parameter port-Hamiltonian systems. On the prototypical example of hyperbolic systems of two conservation laws in arbitrary spatial dimension, we derive the main contributions: (i) A weak formulation of the underlying geometric (Stokes-Dirac) structure with a segmented boundary according to the causality of the boundary ports. (ii) The geometric approximation of the Stokes-Dirac structure by a finite-dimensional Dirac structure is realized using a mixed Galerkin approach and power-preserving linear maps, which define minimal discrete power variables. (iii) With a consistent approximation of the Hamiltonian, we obtain finite-dimensional port-Hamiltonian state space models. By the degrees of freedom in the power-preserving maps, the resulting family of structure-preserving schemes allows for trade-offs between centered approximations and upwinding. We illustrate the method on the example of Whitney finite elements on a 2D simplicial triangulation and compare the eigenvalue approximation in 1D with a related approach.Comment: Copyright 2018. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0

On Finite-Time Ruin Probabilities for Classical Risk Models

This paper is concerned with the problem of ruin in the classical compound binomial and compound Poisson risk models. Our primary purpose is to extend to those models an exact formula derived by Picard and Lefèvre (1997) for the probability of (non-)ruin within finite time. First, a standard method based on the ballot theorem and an argument of Seal-type provides an initial (known) formula for that probability. Then, a concept of pseudo-distributions for the cumulated claim amounts, combined with some simple implications of the ballot theorem, leads to the desired formula. Two expressions for the (non-)ruin probability over an infinite horizon are also deduced as corollaries. Finally, an illustration within the framework of Solvency II is briefly presented.ruin probability; finite and infinite horizon; compound binomial model; compound Poisson model; ballot theorem; pseudo-distributions; Solvency II; Value-at-Risk.

First Approach for the Modelling of the Electric Field Surrounding a Piezoelectric Transformer in View of Plasma Generation

This paper is about an open multi-physics modelling problem resulting from recent investigations into plasma generation by piezoelectric transformers. In this first approach, the electric field distribution surrounding the transformer is studied according to a weak coupling formulation. Electric potential distribution views obtained numerically are compared to real views of plasma generation observed experimentally