Condition Numbers for the Cube. I: Univariate Polynomials and Hypersurfaces

The condition-based complexity analysis framework is one of the gems of modern numerical algebraic geometry and theoretical computer science. One of the challenges that it poses is to expand the currently limited range of random polynomials that we can handle. Despite important recent progress, the available tools cannot handle random sparse polynomials and Gaussian polynomials, that is polynomials whose coefficients are i.i.d. Gaussian random variables. We initiate a condition-based complexity framework based on the norm of the cube that is a step in this direction. We present this framework for real hypersurfaces and univariate polynomials. We demonstrate its capabilities in two problems, under very mild probabilistic assumptions. On the one hand, we show that the average run-time of the Plantinga-Vegter algorithm is polynomial in the degree for random sparse (alas a restricted sparseness structure) polynomials and random Gaussian polynomials. On the other hand, we study the size of the subdivision tree for Descartes' solver and run-time of the solver by Jindal and Sagraloff (arXiv:1704.06979). In both cases, we provide a bound that is polynomial in the size of the input (size of the support plus logarithm of the degree) for not only on the average, but all higher moments.Comment: 34 pages. Version 1, conference version; from version 2, journal versio

The fundamental group and torsion group of Beauville surfaces

We give a survey on the fundamental group of surfaces isogenous to a higher product. If the surfaces are regular, e.g. if they are Beauville surfaces, the first homology group is a finite group. We present a MAGMA script which calculates the first homology groups of regular surfaces isogenous to a product.Comment: 14 pages; MAGMA script included; v2: minor corrections, final version to appear in the Proceedings of the Conference "Beauville Surfaces and Groups", Newcastle University (UK), 7-9th June 201

Livestock and Local Development: Going to a New Humananimal Relationship

Along the past ten years, the French National Agency for Research (ANR) has financed projects regarding livestock. Results of five projects were gathered to understand the long-term livestock trends. At the end of the 19th century, animal breeding was oriented towards the production of goods to meet the local, regional, national and global demand, according to the zone. The market gradually became the key-factor to norm both production and consumption. It is now integrating environmental norms and is starting to invest in the social domain. However, this economical vision of animal production does not take into account the other functions of livestock, from “farm fork” to “table fork”. So, in parallel to the multi-functionality of livestock at the farm level, which is mentioned by several authors, livestock has a significant role at the local scale. Furthermore, in the past four decades, animal production sector has known several serious scandals with severe consequences in human health. At the same time, the FAO scoop in 2006 about the significant environmental impact of animal breeding has chocked a large part of the human society. Hence, in parallel to the discredit of animal production towards the consumers, these successive crises have led a part of the local and global society to question the human-animal relationship. In this way, a large part of the urban population with no contact with the rural world, would easily believe in animal welfare, and break the supply chain leading to the slaughterhouse. And to confirm this trend, research institutes are already seeking alternatives to meat and animal proteins. Consequently, maybe it is time now to think imagine other farming systems based on other human-animal relationships and other environment-society interactions; and perhaps to establish an adequate set of policies to strengthen this perspective

Condition Numbers for the Cube. I: Univariate Polynomials and Hypersurfaces

International audienceThe condition-based complexity analysis framework is one of the gems of modern numerical algebraic geometry and theoretical computer science. One of the challenges that it poses is to expand the currently limited range of random polynomials that we can handle. Despite important recent progress, the available tools cannot handle random sparse polynomials and Gaussian polynomials, that is polynomials whose coefficients are i.i.d. Gaussian random variables. We initiate a condition-based complexity framework based on the norm of the cube, that is a step in this direction. We present this framework for real hypersurfaces. We demonstrate its capabilities by providing a new probabilistic complexity analysis for the Plantinga-Vegter algorithm, which covers both random sparse (alas a restricted sparseness structure) polynomials and random Gaussian polynomials. We present explicit results with structured random polynomials for problems with two or more dimensions. Additionally, we provide some estimates of the separation bound of a univariate polynomial in our current framework

Artificial reefs: from ecological processes to fishing enhancement tools

info:eu-repo/semantics/publishedVersio

An update on the Hirsch conjecture

The Hirsch conjecture was posed in 1957 in a letter from Warren M. Hirsch to George Dantzig. It states that the graph of a d-dimensional polytope with n facets cannot have diameter greater than n - d. Despite being one of the most fundamental, basic and old problems in polytope theory, what we know is quite scarce. Most notably, no polynomial upper bound is known for the diameters that are conjectured to be linear. In contrast, very few polytopes are known where the bound $n-d$ is attained. This paper collects known results and remarks both on the positive and on the negative side of the conjecture. Some proofs are included, but only those that we hope are accessible to a general mathematical audience without introducing too many technicalities.Comment: 28 pages, 6 figures. Many proofs have been taken out from version 2 and put into the appendix arXiv:0912.423

The Double Star Plasma Electron and Current Experiment

The Double Star Project is a collaboration between Chinese and European space agencies, in which two Chinese magnetospheric research spacecraft, carrying Chinese and European instruments, have been launched into equatorial (on 29 December 2003) and polar (on 25 July 2004) orbits designed to enable complementary studies with the Cluster spacecraft. The two Double Star spacecraft TC-1 and TC-2 each carry a Double Star Plasma Electron and Current Experiment (PEACE) instrument. These two instruments were based on Cluster Flight Spare equipment, but differ from Cluster instruments in two important respects. Firstly, a Double Star PEACE instrument has only a single sensor, which must be operated in a manner not originally envisaged in the Cluster context in order to sample the full range of energies. Secondly, the DPU hardware was modified and major changes of onboard software were implemented, most notably a completely different approach to data compression has been adopted for Double Star, which allows high resolution 3-dimensional distributions to be transmitted almost every spin, a significant improvement over Cluster. This paper describes these instruments, and includes examples of data collected in various magnetospheric regions encountered by the spacecraft which have been chosen to illustrate the power of combined Double Star and Cluster measurements